Preparatory to obtaining the varying width of the pattern of the side, a number of points must be fixed upon in the curves of both top and bottom from which to take the measurements. As one-quarter of the top is already divided into spaces, another quarter, O P, may be divided into the same number of spaces (also dividing O" P" into the same space as O" N"). If N" O" P" is the normal curve of the top it would very naturally be divided into equal spaces by the dividers, as is usual in such cases, while the spacing in N O P would be the result of the operation of raking. It is advisable to have the spaces in N"O"P" all equal to each other, as it is from this curve that the stretch-out of the top of the pattern is to be derived, the convenience of which will become apparent when the pattern is developed.
The quarter O P is used in connection with the quarter 0 N, because these two combined constitute a half of the top curve lying on one side of the line A C of the plan which divides the article into symmetrical halves, it being only necessary, when the shape of an article permits, to obtain the pattern of one-half and then to duplicate by any convenient means to obtain the other half.
The corresponding half of the plan of the base, therefore, ABC, must also be divided into the same number of equal spaces as were used at the top, all as shown in the drawing, and both sets of points should be numbered alike, beginning at the same side.
Having thus fixed the points from which measurements across the pattern of the side are to be taken, next draw lines across the plan connecting points of like number, as shown by the full lines in the plan. This divides the entire side of the article into a number of four-sided figures; but as it is necessary, as shown above, to have it divided into triangles, each four-sided figure may now be redivided by a line drawn through its opposite angles, thus cutting it into two triangles. In other words, each point in the base should be connected with a point of the next lower number (or higher, according to circumstances) in the curve of the top, and these lines should be dotted instead of full lines for the sake of distinction and to avoid confusion in subsequent parts of the work. Thus 1 of the base is connected by a dotted line with 0' of the top, 2 of the base with 1' of the top, etc.
In respect to which is the best way to run the dotted lines, common sense will be the best guide.
Thus, in the space bounded by the lines 4 4' and 5 5', it is plainly to be seen that there would be greater advantage and less liability of error in connecting 5 of the bottom curve with 4' of the top than in crossing the line from 4 of the bottom to 5' of the top, for the reason that in the former case the triangles produced would be less scalene or acute.
The next step is to devise a means of determining the true lengths which these lines representor, in other words, their real length as they could be measured if a full size model of the article were cut from a block of wood or clay upon which these lines had been marked, as shown upon the drawing.
The lines upon the plan, of course, only show the horizontal distances between the points which they connect. The vertical hight above the base of any of the points in the upper curve can easily be found by measuring from its position upon the line F G of the elevation perpendicularly to the base E H. Therefore, having both the vertical and the horizontal distance given between any two points, it is only necessary to construct with these dimensions a right angle triangle, and the hypothenuse will give their true distance apart Thus in Fig. 262 a b is equal to the line 4 4' of the plan, while a c is made equal to 4 4' of the elevation. Consequently c b represents the true distance between the points 4' of the top and 4 of the base. Therefore, to obtain all of these hypothenuses in the simplest possible manner, it will be necessary to construct one or two diagrams of triangles. To avoid confusion it is better to make two; one for obtaining the distances represented by the full or solid lines drawn across the plan and the other for those of the dotted lines. To do this extend the base line E H of the elevation, as shown at the left, at any convenient points, in which, as R and S, erect two perpendicular lines. Project lines horizontally from all the points in F G, cutting these two lines as shown, and number the points of intersection. (Some of the figures are omitted in the drawing for lack of space.) From R set off on the base line distances equal to the lengths of the solid lines of the plan 1 1', 2 2', 3 3', etc., numbering the points 1, 2, 3, etc., as shown, and connect points of similar number upon the base with those upon the perpendicular. From S set off on the base line distances equal to the lengths of the dotted lines of the plan 1 0', 2 1', 3 2', etc., and number them to correspond with figure upon the line of the base ABC. Thus make 1 S equal to 1 0' of the plan, 2 S equal to 2 1' of the plan, 3 S equal to 3 2', etc., and connect each point in the base with the point of next lower number upon the perpendicular by a dotted line, as 1 on the base with 0 on the perpendioular, 2 with 1, 3 with 2, etc. The entire surface of the piece for which a pattern is required has thus been cut up into two sets of triangles, one set having the spaces upon the base line A B C, which are all equal, for their bases, and the other set having the spaces in the curve N" O" P" of the top, also equal to each other, as their bases, and each separate triangle having one solid line and one dotted line as its sides.
Fig. 263. - Top, Back and Bottom for a Model of One-half the Article Shown in Fig. 261.
In all of this work the student's powers of mental conception are called into play. The shape of the surface, which is yet to be developed, has been spoken of as if it really existed - in fact, it must exist in the mind or imagination of the operator in order to make him intelligent as to what he is doing. If this fails him, he can resort to a model which can easily be con-structed (full size or to scale, according to convenience) as follows: Describe upon a piece of cardboard or metal the shape E F G H, Fig. 261, to which add on its lower side, E H, one-half of the plan of the bottom, ABC, with the curve NOP and the solid lines connecting it with the outside curve traced thereon. Also add on its upper side, F G, one-half the shape of top, N" O" P", marking the points 1, 2, 3, etc., upon its edge. Now cut out the entire shape in one piece, as shown in Fig. 263, and bend the same at right angles, on the lines F G and E H. Small triangles of the shape and size of each of the triangles shown in the diagram of solid lines, Fig. 261, as 0 0 R, 1 1 E, 2 2 R, etc., can now be cut out and placed upon the portion representing the bottom, each with its base upon the solid line which it represents, at the same bringing the apex of each to the corresponding number on the top. These can be fastened in place by bits of sealing wax, or if cut from metal the whole can be soldered together.