To develop the pattern for a circular article which has very little taper, by the ordinary method, is somewhat inconvenient in practice on account of the long radius required. A way that it can be done, and a plan that is often adopted, is by the

## Method Of Triangulation

This is nothing more or less than the method used by surveyors in measuring-up the exact shape and area of land. The sheet and plate metal worker should be most familiar with the application of this system to the scores of cases that crop up in his own particular line. The use of this method of triangulation is a plan that can be followed in obtaining the shapes of patterns for any and every kind of job where it is possible to obtain the development of a pattern. Also, in some cases, where the pattern is not strictly developable, it will give us considerable aid in obtaining an approximation. It is not by any means in all cases the shortest way of getting out a pattern; but this defect is more than compensated by its universal application. Essentially, the method consists in dividing up any surface, for which a pattern is required, into a series of triangles and then obtaining the true lengths of the sides of each triangle and plotting or setting-out their true shape in the flat. If the three sides of a triangle be given, then one shape of triangle only can be marked out from these. Thus, suppose three links, A B, B C, and

C D (Fig. 95) are hinged together to form a triangle, and one link, say A C, is held fast, then we shall find it impossible to alter the shape of the triangle by pushing it either one way or the other. Thus, it will be impossible to move the two sides so as to cause B to come into the position B1: hence the triangle will remain of constant shape. To illustrate the above, suppose we wish to reproduce the area (a) in Fig. 96, either full size or to scale. Divide the figure up into triangles as in (b). Now carefully measure the sides of the triangle A, and reconstruct it as in (c), then obtain the lengths of the two remaining sides of the triangle B, and thus construct this triangle on A. Continue the process by adding triangles C, D, E, etc. These triangles will give a series of points, all of which will lie on the curved outline. Join the points together with an even curve, and the figure (a) will be reproduced in (c).

In applying the above kind of work to the development of patterns, the operation is not quite so simple, as before the pattern can be laid out, some construction work is necessary to obtain the true lengths of the sides of the triangles.

Fig 95.

Fig. 96.

A practical application of the method will now be shown in connection with the setting out of a pattern for a conical vessel of long taper. A sketch of the article is given (Fig. 97) showing lines drawn on the surface to represent its division into triangles. In this case the method is quite easily applied, as only two different shaped triangles are required - viz., a o 1 and a o b. In practice the plan followed for setting out the pattern is shown in

Fig. 98. A half-elevation and a quarter-plan are drawn as indicated, the latter consisting of two quarter-circles respectively representing a quarter of the top circle and a quarter of the bottom. These quartercircles are divided into three equal parts, and numbered and lettered as shown. Join a to 1, and using a as centre swing this length on to the base line. The length A 1 will then give the true length of the diagonal line required in setting out the pattern. To mark out the pattern, draw a centre line, a o, and make it equal in length to the line A o in the elevation. Set the compasses to a radius equal in length to the curve o 1 in plan, and from centre o on pattern mark arcs as shown, and with radius equal to curve a b do the same from centre a on pattern. Now stretch out compasses to length A 1 in elevation, and from centres o and a cut the arcs in 1 1 and b b. In the same manner determine points 2, 3, and c d. Join the points up with an even curve and the net pattern is complete. Allowances for jointing, etc., can be added as required. It should be remembered that in practice there is no need to draw any lines on the pattern, those shown being put there to illustrate the method. It is most important that lengths of lines should be found with some degree of accuracy, or else the resulting pattern will not be of much use. This system of "triangulation" will be further explained in connection with more difficult patterns in subsequent chapters.

Fig. 97.

Fig. 98.