solving ratios. For example, if the ship had sailed 65 miles on southeast-by-east, then calculating the alargar from the intended Eastward course is simply a matter of solving the following for : where 55 is the alargar for 100 miles (as given in the second column of the table at q = 3). This is easily done by the simple "Rule of Three", a method of cross-multiplication, using three numbers to solve for the fourth by successive multiplication and division: So, sailing for 65 miles on SE by E implies alargar = = 35.75 miles. The avanzar, etc. can be figured out analogously. While the "rule of three" was already known in the 14th century, skill in executing multiplication and division could be elusive for Medieval sailors drawn from what was a largely illiterate society. Nonetheless, it was not inaccessible. As Andrea Bianco urged, navigators should "know how to multiply well and divide well" ("saver ben moltiplichar e ben partir") It is here where we see the important interface of commerce and navigation. The mathematics of commerce – Arabic numerals, multiplication, division, fractions, the tools needed to calculate purchases and sales of goods and other commercial transactions – was essentially the same as the mathematics of navigation. And this kind of mathematics was taught at the abacus schools which were set up in the 13th century in the commercial centers of northern Italy to train the sons of merchants, the very same class where Italian navigators were drawn from. As historian E.G.R. Taylor notes, "sailors were the first professional group to use mathematics in their everyday work" Circle and square For those troubled by the high art of manipulating numbers, there was an alternative. This was the visual device known as the "circle and square" (tondo e quadro), also supplied by Andrea Bianco in his 1436 atlas. The circle was a 32-wind compass rose (or gathering of rhumb-lines). The circle was inscribed with an 8 × 8 square grid. The compass rose in the center can be overlooked – indeed, the circle itself can be ignored, as it seems to have no other purpose than the construction of the rays that run across the grid. The rose of interest is in the upper left corner of the square grid. From that corner, emanate a series of compass rhumb lines. In his original 1436 tondo e quadro, Bianco has sixteen emanating