Therefore, for the study of La Belle's design, frame shapes were reconstructed at both the adjoining and open faces of the floor timbers and second futtocks, but in the final drawings for this essay, only the sections along the labeled and surmarked faces are depicted.

Diagonal Planes and Curve Offsets

As is shown in Figure 20e, at this stage in the hull's design frame spacing is already established, but this only provides locations for evenly spaced design planes with no additional data as to the hull's curvature. It is the diagonal lines fb' and fa in the body plan (Figures 27f, 28b) that conceptually define longitudinally oriented inclined planes on which curves can be drawn from the midship station to the ends of the vessel (Figure 20f). Although sharing a common point on the midship frame curve, the forward and after components of La Belle's diagonals and the planes they define have different inclinations from the horizontal. These differences in inclination reflect the designer beginning to differentiate the curvature between the forward and after parts of the vessel. The established frame spacing can be used to subdivide these inclined planes into evenly spaced increments as in Figure 20g. It is possible at this point to conceive the plotting of offsets for longitudinal curves on these subdividing lines. Of course, La Belle's design was not developed in an isometric drawing, therefore the real "secret" to the design process was how the diagonals were subdivided in the two-dimensional body plan.

Shipbuilding documents and treatises from the fifteenth to the eighteenth century reveal that various geometric and arithmetic methods of generating offsets for longitudinal curves played a central role in the quantification of hull curvature. Figure 31 illustrates the mezzaluna, or half-moon, one of the oldest and best known of these techniques. It first appears in notes on Italian shipbuilding from the fifteenth century (Trombetta ca. 1445:fol. 45r), and archaeologist Éric Rieth has deduced its use in the design of the Culip VI vessel, whose remains dating to the late thirteenth or early fourteenth century were discovered off the Catalan coast (Rieth 1996:149–164). To create a mezzaluna, a half circle is drawn with a radius, AB, equal in length to the final and largest offset (Figure 31). The perimeter of each quarter circle is then evenly divided into as many parts as the number of offsets desired. The corresponding points on each quarter circle are then joined with straight lines that are parallel to the baseline CD and intersect line AB. The distances from the apex, A, to each of the points of intersection gives a progression of offsets for a curve. Note that the mezzaluna method is purely a geometric construction with no arithmetic involved in generating the offset distances. NEXT