using the dividers and the 20m bar scale). The length AC is the avanzar (distance made good), which is 111 miles – visually, five grid squares and a bit, or (20 × 5) + 11, measured by dividers and scale again. This is how the "circle and square" dispenses manipulating numbers by multiplication and division or the rule of three. The navigator can assess the avanzar and alargar visually, by measurement alone. This method can be used for any intended bearing and deviation, as the only purpose is to solve the triangle by dividers and scale. e.g. using our first Corsica-to-Genoa example, where intended bearing was North but the ship actually sailed Northwest, the navigator would set the dividers at length 70 miles and lay it along the fourth quarter wind (= SE ray in the tondo e quadro, as NW is four quarter winds away from North). He would calculate the alargar and avanzar in exactly the same way – draw a line to the horizontal top of the grid, measure the squares, etc. The tondo e quadro device is very similar to the Arab sine quadrant (Rubul mujayyab), with the corner rays replicating the role of the adjustable plumb line. Other applications While the toleta de marteloio (and its visual counterpart, the tondo e quadro) are designed for the explicit task of recovering an intended course, they can be used in more ways, for many classes of navigational problems, e.g. plotting out a course with multiple-bearing changes, etc. Triangulation One of the interesting applications of the rule of marteloio is for triangulation, e.g. determining the distance of the ship from shore landmark. (This was the final exercise attempted in the notebook of the Venetian navigator Michael of Rhodes, which we replicate here.) Example: Suppose a ship sailing NW ("Maestro") spots a landmark due West ("Ponente") one evening, but distance unknown. Suppose the ship continues sailing on the NW route overnight, and the next morning, 40 miles later, it notices that landmark is now west-southwest (WSW, "Ponente-Libeccio") of its current position. Finding the distance of the landmark from the ship is just an application of the rule of marteloio. To solve the problem, start from the evening position (A on the map) and treat the distance between the ship and the landmark (length AB) as the intended course, and the actual route of the ship (NW) as a deviation.