The true shape of the longitudinal curve defined by the offsets on a diagonal can only be depicted if drawn from a viewpoint perpendicular to the diagonal plane. Such auxiliary views are used in modern ship drawings. However, in the Toulon flute draft, as in the other seventeenth-century drafts, the diagonal curves are shown as they would appear when viewed from above and the sides of the vessel (Figures 7, 18, 19) (for *La Belle* see Figure 8). In these views, any one diagonal is broken up into its narrowing (*y*) and rising (*z*) components (Figure 20l). The spacing of the frame design planes provides the equivalent of the *x* coordinate for both the narrowing and rising curves (Figure 20l). By first identifying the plane of each diagonal in the body plan, the French ship designers were able to define the *y* (narrowing) and *z* (rising) coordinates in each frame plane with one set of curve offsets. Formal Cartesian coordinates do not appear in these drafts, nor is ship design presented within the context of coordinate geometry in shipbuilding treatises. However, I think it must be appreciated in terms of the history of design that shipwrights were using elegant concepts of three-dimensional mathematical curve plotting in order to achieve their very utilitarian goals.

La Belle*'s Mother Offset Sequences*

*La Belle*'s archaeological surmark evidence for diagonals subdivided with curve offsets conforms to what is depicted in the Toulon flute draft. However, identifying *La Belle*'s mother sequence and reconstructing its equilateral triangles would greatly strengthen such an association.

The archaeology provides offset lengths from the midship frame to each surmarked frame position along two diagonals both forward and aft. These offset distances are evident in the reconstructed body plan of the superimposed remains of *La Belle*'s surmarked frames (Figure 5a, b). I compared these archaeological offset sequences with various different offset sequences known to me from shipbuilding treatises and from the work of other researchers to "crack the code" of *La Belle*'s mother sequence. Ultimately, the one additional offset value for the frames abaft amidships as well as insights gained from Duhamel du Monceau's treatise helped yield some promising results.

I observed that the curve defined by the offsets along *La Belle*'s after floor diagonal has the same general shape as the curves of polyhedral figurate numbers (Figure 34). Polyhedral figurate numbers result from adding successive polygonal numbers. For example, in Figure 32c a sequence of tetrahedral (triangular pyramidal) numbers was generated using the triangular numbers from Figure 32b. Knowledge of polyhedral numbers dates back at least to classical antiquity (Boyer 1985:60), and both triangular and tetrahedral numbers appear in Pascal's triangle. NEXT

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