Scaling Triangles

Before discussing how such offsets were incorporated into the ship's design, it is necessary to address the practical matter of scaling an offset sequence to subdivide any specified length or dimension. While with the mezzaluna method the length of the largest offset, AB, is the starting point of the procedure, with the arithmetic series the length of the final offset is dependent on the size of the starting base unit. Therefore, for the final distance to correspond to a desired length, AB in this case, it has to ultimately be equal to the number of subunits corresponding to the specific arithmetic series being used (21 in the case of Figure 32a and 36 for Figure 33a). For this reason, Crescentio's first step in creating such a series of offsets is to subdivide a staff equal in length to the desired final offset into the necessary number of subunits. This staff is then used to mark off the offset staff. Unfortunately, in his description of the "infinite stick" method, Crescentio does not state how this is to be done.

This task could be accomplished by trial and error, stepping off various distances with a pair of dividers until getting the correct number of subdivisions. However, by the time of La Belle's construction in the last quarter of the seventeenth century, various methods and devices had already been developed with which to scale any series of measurements. Figure 35a–c depicts two basic types of scaling triangles that are based on the geometric principle that similar triangles maintain a proportional relationship between the lengths of their corresponding sides. For these examples I used the scale from Figure 33a. It is marked with an offset sequence of square numbers and has the same size base unit as the triangular number series in Figure 32a. However, any size base unit such as a cm, inch, or an arbitrarily chosen distance could have been used to create this starting scale.

In Figure 35a two such scales are used to create a triangle whose base is made equal to the length of the largest offset, AB in this case. The lengths between corresponding divisions on the two arms provide proportionally scaled down offsets for dividing the distance AB by a sequence of square numbers. The scaling triangle in Figure 35b operates essentially on the same principle and clearly shows how the resulting offsets subdivide AB. The advantage of the first variation is that there is no need to project any lines. In fact, from the last years of the sixteenth century through the mid nineteenth century, a variety of scaling tools known as sectors were invented based on the concept of having two scaled rulers hinged at point H (Figure 35a) (Hambly 1988:135). For example, Galileo Galilei describes the sector he developed (1597–1599) in his 1606 pamphlet Le Operazioni del Compasso Geometrico et Militare (Boyer 1985:351–352; Hambly 1988:135; Sarsfield 1984:88). NEXT