While Mungo Murray in his Treatise on Ship-building and Navigation (1765) specifically describes the use of the sector (the tool) in ship design (Murray 1765:106–109, 129–130, 146–153, 176–178), the incremental triangle illustrated in the fifteenth-century Trombetta manuscript is such a graphic scaling triangle (Trombetta ca.1445:fol. 45r).

Figure 35c illustrates a different variation of a scaling triangle that appears in François Coulomb's draft from 1684 (Figure 7). As with the other scaling triangles, the two depicted in this draft do not themselves generate the starting offset series. To draw such a triangle, a baseline is divided by offsets for a curve that were already determined in some other way. I refer to this set of offsets as the "mother sequence." In Figure 35c the same starting scale is used as in the previous examples. An equilateral triangle is then drawn with the starting scale as the baseline. Rays are then drawn from the apex of the triangle to each of the subdivision points on the baseline. Any line drawn parallel to the baseline, such as AB from the previous examples, is subdivided by the rays proportionally to the original mother sequence. Figure 36a–c with a mezzaluna mother sequence shows that such triangles can be used to scale any series of increments whether they are originally arithmetically or geometrically generated.

Subdividing the Diagonals

As is shown in the Toulon flute draft (Figure 7), using such a scaling triangle makes the procedure for subdividing the diagonals in the body plan with a series of offsets amazingly simple. The length of a diagonal is plotted as a horizontal line on the equilateral triangle with its two endpoints on the two outside rays. The increments into which the other rays divide the line are then transferred back onto the diagonal in the body plan. Each of the resulting points on the now subdivided diagonal indicates the point of intersection with a frame. When the process is repeated for all the diagonals, the designer almost "magically" has a series of guide points for drawing each of the predesigned frames. In the Toulon flute draft the lengths of diagonals AA, BB, CC and BB, AA, LL are shown superimposed on two equilateral triangles (Figure 7).

In the Toulon flute draft two separate triangles are used to subdivide the diagonals before and abaft amidships (Figure 7). The two triangles are based on different sequences of increments. By using the same type of sequence for all the diagonals to either side of the midship frame, the designer assures that the curves defined by them will have common characteristics and thus define a fair hull shape. The isometric drawings in Figure 20h & i–k illustrate how the process of subdividing a diagonal in the body plan both defines the offsets for a longitudinal curve and also provides a guide point on each of the transverse frame planes through which the curved shape of the frame will be drawn—the circled point in Figure 20k. NEXT