every 100 miles it sails on the current course. Notice that in his passages, Ramon Lull is not recommending the rule, but reporting it, insinuating that this rule was already known and used by contemporary sailors in practice. This is perhaps unsurprising – although trigonometry was only in its infancy in Christian Europe, sine and cosine tables were already known in Arab mathematics. The Kingdom of Majorca, under Muslim rule until the 1230s, remained a multicultural center in Lull's time, with flourishing Jewish communities, many of whom dabbled in mathematics and astronomy, and whose seafarers had extensive contact across the Mediterranean Sea. That Majorcan navigators had some sort of trigonometric table at hand is not improbable. Nonetheless, the exact content and layout of this table implied by Ramon Llull in 1295 is uncertain. Andrea Bianco's "toleta" We get our first glimpse of a mariner's trigonometric table more than a century after Llull. In the first folio of his 1436 portolan atlas, the Venetian captain Andrea Bianco explains the raxon de marteloio, how to calculate the traverse and recover the course. He lays out a simple trigonometric table he calls the toleta de marteloio and recommends that mariners commit the table to memory. The toleta de marteloio is set out as follows: The numbers in the Toleta can be approximated by the modern formulas: Alargar = Avanzar = Ritorno = Avanzo di ritorno = where q = number of quarter winds (angle of difference expressed in number of quarter winds). (notice that the numbers work with quarter-winds set at 11.15° intervals, rather than 11.25°, the usual definition of a quarter wind). The Toleta is a simple table with several columns of numbers. In the first column is the angle of difference between the actual and intended courses, expressed by number of quarter-winds. Once that difference is determined, the second column gives the Alargar (the "Widening", the current distance the ship is from the intended course) while the third column tells the Avanzar (the "Advance", how much of the distance on the intended course has already been covered by sailing on the current bearing – this is equivalent of Ramon Llull's miliaria di mari). The Alargar and Avanzar numbers are shown on the Bianco's table for 100 miles of sailing on the current course. Example: suppose a ship intended to sail bearing east ("Levante") from point A to point B. But suppose