There are one or two methods that might be of much use in marking-out patterns for circular equal-tapering articles, but which, unfortunately, are little known in practice. They depend upon a few important properties of the circle, which we will mention before proceeding to show their application in the setting-out of patterns. Fig. 99.

Referring to Fig. 99, in the segment A D C B it will be found that if points such as B and C be joined to the extremities of the chord A B the angles A B D and A C D will be equal; that is, "angles in the same segment of a circle are equal," and conversely if pairs of lines be drawn all containing the same angle, as in Fig. 103, then the points of intersection of the lines will lie on the arc of a circle. The way this can be used in practice is explained by Fig. 100. If two strips of timber or two lengths of hoop-iron be screwed or riveted together to form a bevel, and one arm of the bevel be allowed to slide along the nail at A and the other along the nail at B, then the scriber or pencil which is held against the joint will mark out an arc of a circle A C B as shown.

Referring again to Fig. 99. If a chord, E K, be drawn, and a tangent, E F, then the line E H, which is drawn to the middle point of the arc E H K, will bisect the angle between the chord E K and the tangent E F. It is also useful to remember that the line G F is divided by the line E H into two parts in the ratio of E F to E G. Thus, suppose E F =10 and E G = 8. Then -

FH= GF x EF / EF + EG= 6 X 10 / 18=3 1/3 And - GH = GF X EG / EF + EG = 6 X 8 / 18 = 2 2/3

From this it will be seen that if we have the lines K E and G F or K E and E F, a third point, H, can always be determined which lies on the arc of a circle. This particular property we shall find of much. use in setting out patterns for circular articles of long taper, and we shall be able to obtain the pattern without the use of the pattern circle centre. Fig. 100.

Fig. 101 indicates a pattern so set out. A Centre line is drawn as shown, and the girth of the large end set along the line A B, half being measured along each side of the centre line. Then with centres A and B and radius equal to the slant height of the pipe or vessel, arcs of circles are drawn. Two lines are now set down parallel to the centre line, and at a distance apart equal to the circumference of the small end of the tapered pipe or article. Where these lines cut the arcs, as previously drawn, will give the points E and D. The line B K is drawn square to B D, and the line B C drawn to bisect the angle A B K. Thus we now have three points, A, C, and B, which'will come on the curve of the pattern. Set the bevel to the angle A C B, and fix pins or nails at A and B, and slide along, thus marking the pattern curve as previously explained by reference to Fig. 100. The curve at the small end can be described in the same manner, the bevel being kept set at the same angle or rake as used at the large end. There is really no need to determine the point F, as shown by the construction lines, the bevel giving the correct height of curve. It must not be forgotten that the length of the bevel arm should not be less than the line A B. When the article has little taper, the point C will come as near as possible at the middle point of L K; there will, therefore, be no need to bisect the angle A B K. The pattern will come slightly wider than it ought to be on account of the curve being a little longer than the girth lines. But when the difference between the end diameters is small, and the article long, it will be quite near enough for ordinary work in practice. Fig. 101.

The exact size and shape of pattern, however, can be marked out by this method, and the reader who can follow a few simple calculations will readily understand the construction involved. The essential thing to determine is the exact length of the chord A B (Fig. 102) which is required to give the correct length of the arc, A C B. To obtain this, let us take, for the sake of clearness, an actual example. Suppose we require to set out a plate for a pipe 4 ft. diameter at one end, 3 ft. diameter at the other, and the slant length 5 ft. The slant height of the complete cone, of which this is a frustum, can be obtained from the following rule:- "Multiply the diameter of the large end by the slant height, and divide by the difference of the diameters." Thus, the slant height of cone equals large diameter x slant height /difference of diameters 4 x 5 / 4 - 3 = 20 ft.

The angle made by the two outside lines of the pattern with each other can be found by the following rule: - "Multiply the diameter of large end of the pipe by 360, and divide by twice the slant height of cone."Thus, the angle between end lines of pattern equals large diameter x 360 / twice slant height of cone = 4 x 360 / 2X 20 = 36°

The angle between the chord A B (Fig. 102) and the end line, B D, of pattern can be calculated from this rule: -

"Deduct half the angle between end lines of pattern from 90°." Thus, angle between chord and end line equals -

90- 36 / 2 = 72°

Having obtained the above particulars, let us set out the pattern, first going over a construction which will give us the correct length of A B. Draw a line across, and mark a point B anywhere upon it. Now set off a line B D at 72°, and make it 5 ft. in length. Drop a perpendicular on A B from the point D which will cut off a length H B. The half-length of the chord A B will be equal to -

LB = H B x slant height of cone / slant height of pipe

=H B x 20 / 5 = 4 X H B, which means to obtain the point L three additional lengths of H B must be marked along from H. The line L F is drawn square to A B, and points A and E determined. To scribe out the curves, we shall first have to obtain the angle at which to set the bevel. The rule for this is: "Deduct half the angle between the end lines of pattern from 180°."

Bevel angle = 180 - 36 / 2 = 162°.

Set the bevel at this angle; fix pins at A and B, and slide along as in Fig. 100. To mark the curve at the bottom of the pattern, fix pins at E and D and slide bevel along, keeping it at the same angle as used for describing the curve at the top.

Those readers who can understand the use of mathematical tables will be able to calculate the lengths of A B and E D in a simple manner, and thus set out the pattern quite easily. Referring to a table of chords, the chord of 36° is given as 0.618. The length of A B will therefore be -

0.618 x 20 = 12.36 ft., and - E D = 0.618 x 15 = 9.27 ft.

For the sake of comparison we will calculate the length of the arc ACB, which will, of course be -

4 x 3.1416 = 12.57 ft.

Thus the difference in the length of the chord A B, and the arc AC B, will be as near as possible

2.5 in. Hence, where accurate work of the above description is required, it will be necessary to follow the last-named method.

Fig. 102 (a) shows the various angles set out; but in practice the only lines required are those on the pattern, as indicated in (b).

If it is required to build an article up in several pieces, as in large plate work, the pattern can be subdivided when set out, or a pattern for the required segment can be marked out by either of the methods explained.

When the angle in the segment of the circle is determined, it is sometimes convenient to obtain a few points that would lie on the curve, and then join them up by bending a lath of timber along the points and scribing along. Points that would lie on the curve can be obtained Fig. 102 in the way illustrated by Fig. 103. Thus, suppose the angle in the segment is 100°, then, as the three angles in a triangle are together equal to 180°, the sum of the base angles must be -

180 - 100 = 80°.

Thus, if a line making 10° with A C is set at one end, and one at 70° with C A at the other, then the point of intersection of the two lines will give a point on the curve. In the same way, further points can be obtained by marking angles of 20° and 60°, 30° and 50°, and so on, as explained by the diagram. Fig. 103.

Possibly this chapter is a little more difficult to follow than the preceding ones, on account of the calculations introduced; but the reader who is interested should make an effort to understand all that has been stated. It is exceedingly important to the sheet and plate metal worker, especially to the latter.