Figure 223 is a drawing of a locomotive frame, which the student may as well draw three or four times as large as the engraving, which brings us to the subject of enlarging or reducing scales.

REDUCING SCALES.

[Ill.u.s.tration: Fig. 223.]

[Ill.u.s.tration: Fig. 224.]

[Ill.u.s.tration: Fig. 225.]

It is sometimes necessary to reduce a drawing to a smaller scale, or to find a minute fraction of a given dimension, such fraction not being marked on the lineal measuring rules at hand. Figure 224 represents a scale for finding minute fractions. Draw seven lines parallel to each other, and equidistant draw vertical lines dividing the scale into half-inches, as at _a_, _b_, _c_, etc. Divide the first s.p.a.ce _e d_ into equal halves, draw diagonal lines, and number them as in the figure. The distance of point 1, which is at the intersection of diagonal with the second horizontal line, will be 1/24 inch from vertical line _e_. Point 2 will be 2/24 inch from line _e_, and so on. For tenths of inches there would require to be but six horizontal lines, the diagonals being drawn as before. A similar scale is shown in Figure 225. Draw the lines A B, B D, D C, C A, enclosing a square inch. Divide each of these lines into ten equal divisions, and number and letter them as shown. Draw also the diagonal lines A 1, _a_ 2, B 3, and so on; then the distances from the line A C to the points of intersection of the diagonals with the horizontal lines represent hundredths of an inch.

Suppose, for example, we trace one diagonal line in its path across the figure, taking that which starts from A and ends at 1 on the top horizontal line; then where the diagonal intersects _horizontal_ line 1, is 99/100 from the line B D, and 1/100 from the line A C, while where it intersects _horizontal_ line 2, is 98/100 from line B D, and 2/100 from line A C, and so on. If we require to set the compa.s.ses to 67/100 inch, we set them to the radius of _n_, and the figure 3 on line B D, because from that 3 to the vertical line _d_ 4 is 6/10 or 60/100 inch, and from that vertical line to the diagonal at _n_ is seven divisions from the line C D of the figure.

In making a drawing to scale, however, it is an excellent plan to draw a line and divide it off to suit the required scale. Suppose, for example, that the given scale is one-quarter size, or three inches per foot; then a line three inches long may be divided into twelve equal divisions, representing twelve inches, and these may be subdivided into half or quarter inches and so on. It is recommended to the beginner, however, to spend all his time making simple drawings, without making them to scale, in order to become so familiar with the use of the instruments as to feel at home with them, avoiding the complication of early studies that would accompany drawing to scale.

CHAPTER X.

_PROJECTIONS._

In projecting, the lines in one view are used to mark those in other views, and to find their shapes or curvature as they will appear in other views. Thus, in Figure 225_a_ we have a spiral, wound around a cylinder whose end is cut off at an angle. The pitch of the spiral is the distance A B, and we may delineate the curve of the spiral looking at the cylinder from two positions (one at a right-angle to the other, as is shown in the figure), by means of a circle having a circ.u.mference equal to that of the cylinder.

The circ.u.mference of this circle we divide into any number of equidistant divisions, as from 1 to 24. The pitch A B of the spiral or thread is then divided off also into 24 equidistant divisions, as marked on the left hand of the figure; vertical lines are then drawn from the points of division on the circle to the points correspondingly numbered on the lines dividing the pitch; and where line 1 on the circle intersects line 1 on the pitch is one point in the curve. Similarly, where point 2 on the circle intersects line 2 on the pitch is another point in the curve, and so on for the whole 24 divisions on the circle and on the pitch. In this view, however, the path of the spiral from line 7 to line 19 lies on the other side of the cylinder, and is marked in dotted lines, because it is hidden by the cylinder. In the right-hand view, however, a different portion of the spiral or thread is hidden, namely from lines 1 to 13 inclusive, being an equal proportion to that hidden in the left-hand view.

[Ill.u.s.tration: Fig. 225 _a_.]

The top of the cylinder is shown in the left-hand view to be cut off at an angle to the axis, and will therefore appear elliptical; in the right-hand view, to delineate this oval, the same vertical lines from the circle may be carried up as shown on the right hand, and horizontal lines may be drawn from the inclined face in one view across the end of the other view, as at P; the divisions on the circle may be carried up on the right-hand view by means of straight lines, as Q, and arcs of circle, as at R, and vertical lines drawn from these arcs, as line S, and where these vertical lines S intersect the horizontal lines as P, are points in the ellipse.

Let it be required to draw a cylindrical body joining another at a right-angle; as for example, a Tee, such as in Figure 226, and the outline can all be shown in one view, but it is required to find the line of junction of one piece, A, with the other, B; that is, find or mark the lines of junction C. Now when the diameters of A and B are equal, the line of junction C is a straight line, but it becomes a curved one when the diameter of A is less than that of B, or _vice versa_; hence it may be as well to project it in both cases. For this purpose the three views are necessary. One-quarter of the circle of B, in the end view, is divided off into any number of equal divisions; thus we have chosen the divisions marked _a_, _b_, _c_, _d_, _e_, etc.; a quarter of the top view is similarly divided off, as at _f_, _g_, _h_, _i_, _j_; from these points of division lines are projected on to the side view, as shown by the dotted lines _k_, _l_, _m_, _n_, _o_, _p_, etc., and where these lines meet, as denoted by the dots, is in each case a point in the line of junction of the two cylinders A, B.

[Ill.u.s.tration: Fig. 226.]

[Ill.u.s.tration: Fig. 227.]

Figure 227 represents a Tee, in which B is less in diameter than A; hence the two join in a curve, which is found in a similar manner, as is shown in Figure 227. Suppose that the end and top views are drawn, and that the side view is drawn in outline, but that the curve of junction or intersection is to be found. Now it is evident that since the centre line 1 pa.s.ses through the side and end views, that the face _a_, in the end view, will be even with the face _a'_ in the side view, both being the same face, and as the full length of the side of B in the end view is marked by line _b_, therefore line _c_ projected down from _b_ will at its junction with line _b'_, which corresponds to line _b_, give the extreme depth to which _b'_ extends into the body A, and therefore, the apex of the curve of intersection of B with A. To obtain other points, we divide one-quarter of the circ.u.mference of the circle B in the top view into four equal divisions, as by lines _d_, _e_, _f_, and from the points of division we draw lines _j_, _i_, _g_, to the centre line marked 2, these lines being thickened in the cut for clearness of ill.u.s.tration. The compa.s.ses are then set to the length of thickened line _g_, and from point _h_, in the end view, as a centre, the arc _g'_ is marked. With the compa.s.ses set to the length of thickened line _i_, and from _h_ as a centre, arc _i'_ is marked, and with the length of thickened line _j_ as a radius and from _h_ as a centre arc _j'_ is marked; from these arcs lines _k_, _l_, _m_ are drawn, and from the intersection of _k_, _l_, _m_, with the circle of A, lines _n_, _o_, _p_ are let fall. From the lines of division, _d_, _e_, _f_, the lines _q_, _r_, _s_ are drawn, and where lines _n_, _o_, _p_ join lines _q_, _r_, _s_, are points in the curve, as shown by the dots, and by drawing a line to intersect these dots the curve is obtained on one-half of B.

Since the axis of B is in the same plane as that of A, the lower half of the curve is of the same curvature as the upper, as is shown by the dotted curve.

[Ill.u.s.tration: Fig. 228.]

In Figure 228 the axis of piece B is not in the same plane as that of D, but to one side of it to the distance between the centre lines C, D, which is most clearly seen in the top view. In this case the process is the same except in the following points: In the side view the line _w_, corresponding to the line _w_ in the end view, pa.s.ses within the line _x_ before the curve of intersection begins, and in transferring the lengths of the full lines _b_, _c_, _d_, _e_, _f_ to the end view, and marking the arcs _b'_, _c'_, _d'_, _e'_, _f'_, they are marked from the point _w_ (the point where the centre line of B intersects the outline of A), instead of from the point _x_. In all other respects the construction is the same as that in Figure 227.

[Ill.u.s.tration: Fig. 229.]

In these examples the axis of B stands at a right-angle to that of A.

But in Figure 229 is shown the construction where the axis of B is not at a right-angle to A. In this case there is projected from B, in the side view, an end view of B as at B', and across this end at a right-angle to the centre line of B is marked a centre line C C of B', which is divided as before by lines _d_, _e_, _f_, _g_, _h_, their respective lengths being transferred from W as a centre, and marked by the arcs _d'_, _e'_, _f'_, which are marked on a vertical line and carried by horizontal lines, to the arc of A as at _i_, _j_, _k_. From these points, _i_, _j_, _k_, the perpendicular lines _l_, _m_, _n_, _o_, are dropped, and where these lines meet lines _p_, _q_, _r_, _s_, _t_, are points in the curve of intersection of B with A. It will be observed that each of the lines _m_, _n_, _o_, serves for two of the points in the curve; thus, _m_ meets _q_ and _s_, while _n_ meets _p_ and _t_, and _o_ meets the outline on each side of B, in the side view, and as _i_, _j_, _k_ are obtained from _d_ and _e_, the lines _g_ and _h_ might have been omitted, being inserted merely for the sake of ill.u.s.tration.

In Figure 230 is an example in which a cylinder intersects a cone, the axes being parallel. To obtain the curve of intersection in this case, the side view is divided by any convenient number of lines, as _a_, _b_, _c_, etc., drawn at a right-angle to its axis A A, and from one end of these lines are let fall the perpendiculars _f_, _g_, _h_, _i_, _j_; from the ends of these (where they meet the centre line of A in the top view), half-circles _k_, _l_, _m_, _n_, _o_, are drawn to meet the circle of B in the top view, and from their points of intersection with B, lines _p_, _q_, _r_, _s_, _t_, are drawn, and where these meet lines _a_, _b_, _c_, _d_ and _e_, which is at _u_, _v_, _w_, _x_, _y_, are points in the curve.

[Ill.u.s.tration: Fig. 230.]

[Ill.u.s.tration: Fig. 231.]

It will be observed, on referring again to Figure 229, that the branch or cylinder B appears to be of elliptical section on its end face, which occurs because it is seen at an angle to its end surface; now the method of finding the ellipse for any given degree of angle is as in Figure 231, in which B represents a cylindrical body whose top face would, if viewed from point I, appear as a straight line, while if viewed from point J it would appear in outline a circle. Now if viewed from point E its apparent dimension in one direction will obviously be defined by the lines S, Z. So that if on a line G G at a right angle to the line of vision E, we mark points touching lines S, Z, we get points 1 and 2, representing the apparent dimension in that direction which is the width of the ellipse. The length of the ellipse will obviously be the full diameter of the cylinder B; hence from E as a centre we mark points 3 and 4, and of the remaining points we will speak presently.

Suppose now the angle the top face of B is viewed from is denoted by the line L, and lines S', Z, parallel to L, will be the width for the ellipse whose length is marked by dots, equidistant on each side of centre line G' G', which equal in their widths one from the other the full diameter of B. In this construction the ellipse will be drawn away from the cylinder B, and the ellipse, after being found, would have to be transferred to the end of B. But since centre line G G is obviously at the same angle to A A that A A is to G G, we may start from the centre line of the body whose elliptical appearance is to be drawn, and draw a centre line A A at the same angle to G G as the end of B is supposed to be viewed from. This is done in Figure 231 _a_, in which the end face of B is to be drawn viewed from a point on the line G G, but at an angle of 45 degrees; hence line A A is drawn at an angle of 45 degrees to centre line G G, and centre line E is drawn from the centre of the end of B at a right angle to G G, and from where it cuts A A, as at F, a side view of B is drawn, or a single line of a length equal to the diameter of B may be drawn at a right angle to A A and equidistant on each side of F. A line, D D, at a right angle to A A, and at any convenient distance above F, is then drawn, and from its intersection with A A as a centre, a circle C equal to the diameter of B is drawn; one-half of the circ.u.mference of C is divided off into any number of equal divisions as by arcs _a_, _b_, _c_, _d_, _e_, _f_. From these points of division, lines _g_, _h_, _i_, _j_, _k_, _l_ are drawn, and also lines _m_, _n_, _o_, _p_, _q_, _r_. From the intersection of these last lines with the face in the side view, lines _s_, _t_, _u_, _t_, _w_, _x_, _y_, _z_ are drawn, and from point F line E is drawn. Now it is clear that the width of the end face of the cylinder will appear the same from any point of view it may be looked at, hence the sides H H are made to equal the diameter of the cylinder B and marked up to centre line E.

[Ill.u.s.tration: Fig. 231 _a_.]

[Ill.u.s.tration: Fig. 232.]

It is obvious also that the lines _s_, _z_, drawn from the extremes of the face to be projected will define the width of the ellipse, hence we have four of the points (marked respectively 1, 2, 3, 4) in the ellipse.

To obtain the remaining points, lines _t_, _u_, _v_, _w_, _x_, _y_ (which start from the point on the face F where the lines _m_, _n_, _o_, _p_, _q_, _r_, respectively meet it) are drawn across the face of B as shown. The compa.s.ses are then set to the radius _g_; that is, from centre line D to division _a_ on the circle, and this radius is transferred to the face to be projected the compa.s.s-point being rested at the intersection of centre line G and line _t_, and two arcs as 5 and 6 drawn, giving two more points in the curve of the ellipse. The compa.s.ses are then set to the length of line _h_ (that is, from centre line D to point of division _b_), and this distance is transferred, setting the compa.s.ses on centre line G where it is intersected by line _u_, and arcs 7, 8 are marked, giving two more points in the ellipse. In like manner points 9 and 10 are obtained from the length of line _i_, 11 and 12 from that of _j_; points 13 and 14 from the length of _k_, and 15 and 16 from _l_, and the ellipse may be drawn in from these points.

It may be pointed out, however, that since points 5 and 6 are the same distance from G that points 15 and 16 are, and since points 7 and 8 are the same distance from G that points 13 and 14 are, while points 9 and 10 are the same distance from G that 11 and 12 are, the lines, _j_, _k_, _l_ are unnecessary, since _l_ and _g_ are of equal length, as are also _h_ and _k_ and _i_ and _j_. In Figure 232 the cylinders are line shaded to make them show plainer to the eye, and but three lines (_a_, _b_, _c_) are used to get the radius wherefrom to mark the arcs where the points in the ellipse shall fall; thus, radius _a_ gives points 1, 2, 3 and 4; radius _b_ gives points 5, 6, 7 and 8, and radius _c_ gives 9, 10, 11 and 12, the extreme diameter being obtained from lines S, Z, and H, H.

CHAPTER XI.

_DRAWING GEAR WHEELS._

The names given to the various lines of a tooth on a gear-wheel are as follows:

In Figure 233, A is the face and B the flank of a tooth, while C is the point, and D the root of the tooth; E is the height or depth, and F the breadth. P P is the pitch circle, and the s.p.a.ce between the two teeth, as H, is termed a s.p.a.ce.

[Ill.u.s.tration: Fig. 233.]

[Ill.u.s.tration: Fig. 234.]

It is obvious that the points of the teeth and the bottoms of the s.p.a.ces, as well as the pitch circle, are concentric to the axis of the wheel bore. And to pencil in the teeth these circles must be fully drawn, as in Figure 234, in which P P is the pitch circle. This circle is divided into as many equal divisions as the wheel is to have teeth, these divisions being denoted by the radial lines, A, B, C, etc. Where these divisions intersect the pitch circle are the centres from which all the teeth curves may be drawn. The compa.s.ses are set to a radius equal to the pitch, less one-half the thickness of the tooth, and from a centre, as R, two face curves, as F G, may be marked; from the next centre, as at S, the curves D E may be marked, and so on for all the faces; that is, the tooth curves lying between the outer circle X and the pitch circle P. For the flank curves, that is, the curve from P to Y, the compa.s.ses are set to a radius equal to the pitch; and from the sides of the teeth the flank curves are drawn. Thus from J, as a centre flank, K is drawn; from V, as a centre flank, H is drawn, and so on.

The proportions of the teeth for cast gears generally accepted in this country are those given by Professor Willis, as average practice, and are as follows:

Depth to pitch line, 3/10 of the pitch.

Working depth, 6/10 " "

Whole depth, 7/10 " "

Thickness of tooth, 5/11 " "

Breadth of s.p.a.ce, 6/11 " "

Instead, however, of calculating the dimensions these proportions give for any particular pitch, a diagram or scale may be made from which they may be taken for any pitch by a direct application of the compa.s.ses. A scale of this kind is given in Figure 235, in which the line A B is divided into inches and parts to represent the pitches; its total length representing the coa.r.s.est pitch within the capacity of the scale; and, the line B C (at a right-angle to A B) the whole depth of the tooth for the coa.r.s.est pitch, being 7/10 of the length of A B.

[Ill.u.s.tration: Fig. 235.]