In my opinion, this correspondence strongly supports the conclusion that La Belle was designed by the same method of geometric fairing with diagonals as the Profond and the other vessels in the French drafts in this essay. The Profond draft's documentary support for the reconstructed offset sequence also helps confirm that the heights listed in the devis for La Belle's floor diagonal were indeed offset from the centerline 4 inches.

The sequence used for La Belle's floor diagonal forward was much more difficult to discern, and thus the following discussion is more speculative. Frames VIA and VIIIID are La Belle's balancing frames. These timbers have the same width between their floor timber surmarks. Having subdivided the after floor diagonal, La Belle's designer could determine at what point the curve for frame VIA would have to intersect with the floor diagonal to have the same width as VIIIID. This would give the designer three points along the length of the floor diagonal forward—the two endpoints and the point for VIA. These points would then be marked on the edge of a piece of paper—for this discussion represented by line f'b' in Figure 40b. This line is then positioned on the equilateral triangle at such an angle that these three points fall on the appropriate rays, and the intersections with the intervening rays provide the guide points for frames IIIA and VIIIIA. The guide points thus obtained correspond exactly with the archaeologically documented intersections of frames IIIA and VIIIIA with the floor diagonal (Figure 40b). The same basic procedure for establishing the offsets for the forward floor diagonal is described by Duhamel du Monceau (Duhamel 1758:246–248). This procedure highlights another aspect of the utility of the equilateral scaling triangle. By adjusting the angle at which the diagonal is placed on the triangle, it is possible to alter the characteristics of the resulting curve while still remaining within the same family of curves defined by the mother sequence.

I experimented with the convexity of arcs method to see if a similar set of offsets could be generated. Using a radius three times the diagonal's length results in a similar sequence, although in this case with a fairly large deviation at IIIA from the arithmetically generated sequence (Figure 41b).

Figure 20m–o shows how the offsets established on the forward diagonal in the body plan define the forward part of the longitudinal floor curve, whose curvature abaft the midship station was already established with the after floor diagonal offsets. The dark short lines or tick marks projecting from the floor curve in Figure 20o correspond to surmarks. The archaeologically documented floor timber surmarks are not only found at these exact locations but are also carved at the same angles of inclination as the diagonal planes in this drawing. NEXT