French mathematician Blaise Pascal describes the properties of such a number triangle in his treatise *Traité du triangle arithmétique* (1653), but the triangle itself had already been known for at least 600 years and appeared in print for the first time in the West in 1527 (Boyer 1985:327–328, 397–398). Thus by the time of *La Belle*'s construction, knowledge of such number series would have been fairly widespread in mathematical circles, and the evidence from *La Belle* suggests that it may have been known in shipbuilding circles as well.

Since the offsets for such arithmetic curves can be generated by addition, I subjected *La Belle*'s archaeological offset distances to a simple subtractive analysis. This entails geometrically subtracting each offset distance from the previous one and then superimposing these differences on a new line (Figure 37). After two rounds of subtraction (*a*–*b* and *b*–*c* in Figure 37) a terminal sequence is reached in which the first increment is two units (2*x*) and each subsequent increment is one unit (1*x*) (Figure 37c, d). Reversing the process by switching back to addition generates the following unit values for *La Belle*'s archaeologically documented offsets: * = 0*x*, *IIID* = 2*x*, *VID* = 7*x*, *VIIIID* = 16*x* and *XIID* = 30*x*. This process also predicts the offset values of 50*x* for *XVD* and 77*x* for *XVIIID* (Figure 37e, 38a, b). The equilateral scaling triangle in Figure 39 was constructed with the resulting mother sequence of 0*x*, 2*x*, 7*x*, 16*x*, 30*x*, 50*x*, 77*x*.

In Figure 40a this equilateral triangle is superimposed on the after floor diagonal in the cross-sectional drawing of *La Belle*'s frames. Maintaining the base of the triangle parallel to the diagonal the triangle was shifted until each of the frame sections aligned with the corresponding offset ray. In this orientation the final ray predicted for *XVIIID* corresponds exactly with the upper endpoint of this diagonal (Figures 20h, 40a). Since the upper endpoint was established prior to the "discovery" of the mother offset sequence, this correspondence supports the validity of the sequence itself.

The mathematical logic of generating this sequence is an expansion on the ideas of figurate numbers—specifically the idea that one sequence (polygonal numbers) can be used to generate another (polyhedral numbers). However, the starting increment in the terminal series above is double the next value, thus deviating it from regular figurate geometry (Figure 38). Unfortunately, I have not discovered any discussion of such more complex arithmetic sequences in any shipbuilding treatise. The lack of corroborating documentary evidence does not mean such sequences were not used, but it did motivate me to continue searching for known historic methods that might yield a similar set of offsets.

In *Traité du Navire*, published in 1746, Pierre Bouguer presents another geometric method of generating offsets for diagonals known as the method of convexity of arcs (Boudriot 1994:42–43; Bouguer 1746:44–46, Pl. 2, Figures 10–12; Duhamel 1758:xxix, 260). NEXT

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