The “tondo e quadro” (“circle and square”) is a geometric diagram that allowed sailors to calculate distances with a set square and a divider, avoiding the tedious arithmetic required by the “tablet of marteloio”.
The diagram is formed by a thirty-two point wind rose and a square grid divided into eight sections per side. A double scale of miles assigns each of the 8 sections a value either of 20 miles or 100 miles, depending on the sailor’s need.
If the ship traveled three quarter-winds off course for a distance of 100 miles, the helmsman could open the divider to a width on the diagram equivalent to 100 miles and mark a point crossing the third quarter-wind on the diagram. Using a set square, he would then trace a perpendicular line to the side of the square, measure that length with the divider, and apply that length to the side of the square, where he would find the allargo was something over 40 miles. By measuring with the divider the excess segment, and checking against the scale of miles, the helmsman was able to measure with precision the distance the ship had traveled off course, as well as verifying the miles of progress the ship had made towards its destination.
As seen in a drawing by Andrea Bianco, a navigator and cartographer contemporary to Fra Mauro, the values of the allargo and the avanzo could also be calculated from the wind rose by attributing a value of 100 miles to the diameter of the rose. For example, tracing from point A the line corresponding to the third quarter-wind—which touches the circumference at point C, forming a right angle with the line joining C to B—the helmsman could directly ascertain the allargo measuring the line BC, and the avanzo measuring the line AC.