page_title: Rule of marteloio
section_title: Circle and square
hierarchical_section_title: Rule of marteloio / Rules / Circle and square
caption_attribution_description: English: Illustration of the use of the tondo e quadro (circle and square) tool introduced in the 1436 atlas of Venetian captain Andrea Bianco. Ship intended to go on an East course, but has been instead been sailing 120 miles on an East-southeast (ESE) course - that is, two quarter-winds below the intended course. Using a pair of dividers and the map scale, navigator constructs a right-angled triangle ABC, by which he can calculate the alargar (current distance from intended course) and avanzar (how much of the intended course he has already made good).
context_page_description: The rule of marteloio is a medieval technique of navigational computation that uses compass direction, distance and a simple trigonometric table known as the toleta de marteloio. The rule told mariners how to plot the traverse between two different navigation courses by means of resolving triangles with the help of the Toleta and basic arithmetic.
  Those uncomfortable with manipulating numbers could resort to the visual tondo e quadro and achieve their answer with dividers. The rule of marteloio was commonly used by Mediterranean navigators during the 14th and 15th centuries, before the development of astronomical navigation.
context_section_description: 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 rays – that is, Bianco includes half-quarter winds, or eighth-winds (otava), so that the emanating rays are at intervals of 5.625 degrees. Other constructions of the circle-and-square, e.g. the Cornaro Atlas, use only eight rays emanating at quarter-wind distances (11.25 degrees). Visually, these rays replicate the bottom right quarter of a 32-wind compass rose: East (0q), E by S (1q), ESE (2q), SE by E (3q), SE (4q), SE by S (5q), SSE (6q), S by E (7q) and South (8q).
  Above the grid is a distance bar scale, notched with sub-units. There are two sets of numbers on the scale, one for measuring each grid square by 20 miles, another for measuring each grid square by 100 miles (see diagram). The top bar is the 20m-per-square scale, with every black dot denoting one mile. The bottom bar is the 100m-per-square scale, where the length of a unit square is divided into two equal 50m sub-squares, and a set of dots and red lines break it down further into lengths of 10 miles. So depending on which scale one chooses, the length of the side of the entire grid (eight squares) could be measured up to 160 miles (using the 20 m-per-square scale) or up to 800 miles (using the 100m-per-square scale).
  The cherub with the dividers suggests how a navigator is supposed to use the grid to calculate alargar and avanzar by visual measurement rather than manipulating numbers.
  Example: suppose the ship has travelled 120 miles at two quarter-winds below the intended course (e.g. traveled at ESE, when the intended course is East).   Using the dividers and the 20m scale, the navigator can measure out 120 miles with his dividers. Then setting one end at the top left corner (A), he lays out the dividers along the ESE ray (= two quarter-winds below the East ray, or horizontal top of the grid) and marks the spot (point B on the diagram). Then using a straightedge ruler draws a line up to the East ray, and marks the corresponding spot C.
  It is easy to see immediately that a right-angled triangle ABC has been created. The length BC is the alargar (distance from intended course), which can be measured as 46 miles (this can be visually seen as two grid squares plus a bit, that is 20m + 20m and a little bit which can be assessed as 6m by 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.
caption: Solving a triangle with the tondo e quadro
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