Figures 32 and 33 illustrate two types of similar arithmetic offset sequences, and Figure 32 depicts a simple mechanical procedure for generating them. Bartolomeo Crescentio presents such a mechanical method of marking offsets on one staff with the aid of another, the Neapolitan "infinite stick" technique, in his 1602 book on shipbuilding, *Nautica Mediterranea* (Bloesch 1983; Crescentio 1602:21–22; Sarsfield 1984).

In Figure 32a each subsequent offset is equal to the value of the previous offset plus an amount that increases by one unit from one offset to the next. In other words, the differences between the offsets are a simple arithmetic progression of 1, 2, 3, 4, and 5 consecutive integer units. The use of such an offset sequence is already documented in the fifteenth-century Trombetta manuscript; it forms the basis of an incremental triangle that is illustrated along with the *mezzaluna* (Anderson 1925:153–154; Sarsfield 1984:87; Trombetta ca.1445:fol. 45r). This is also the arithmetic series presented by Crescentio with his "infinite stick" method (Anderson 1925:153–154; Crescentio 1602:21–22; Sarsfield 1984:87).

In Figure 33a each subsequent offset is equal to the value of the previous offset plus an amount that increases by two units from one offset to the next. In this case, the differences between the offsets is the arithmetic progression of 1, 3, 5, 7 and 9 units. The offsets in this second series are the square numbers 1, 4, 9, 16, 25 and 36, but they were derived by addition and not multiplication. This additive approach is described by Duhamel du Monceau (Duhamel 1758:226); it is both mathematically simple and the underlying logic opens up the possibility for the use of other series of increments whose offset differences increase by 3, 4 or more units. As with the two series in Figures 32a and 33a , each series would yield a slightly different curve (Figure 34).

The study of such sequences of numbers (i.e., the numeric values of the individual offsets) dates back at least to the time of Pythagoras (Boyer 1985:59–60). Figures 32b and 33b illustrate how these numbers, known as figurate numbers, are associated with polygonal geometry. The series in Figure 32a is actually a sequence of triangular numbers (Figure 32b), and the series in Figure 33a is a sequence of square numbers (Figure 33b). The amount added to transform one figurate number to the next in the series (the gray circle arrangements) is known as a gnomon.

It is critical to the discussion of ship design to emphasize that these sequences of numbers or increments only become offsets for curves when they are plotted on parallel lines, known as ordinates, at equally spaced intervals (dx in Figures 31, 32a, 33a) along a common baseline or axis. The equally spaced subdividing lines on the diagonal plane in Figure 20g provide such ordinates for plotting curve offsets for the vessel's design. During actual construction, the equidistant positioning of the design frames themselves achieves the same result in real-world three-dimensional space. NEXT

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