intended course. There it can straighten its bearing N and sail the remaining 30 miles or so to Genoa. Unfortunately, Medieval sailors with the rudimentary educational levels of the 14th and 15th centuries, were not likely to know the Law of Sines or manipulate it with ease. As a result, Medieval navigators needed simpler and more accessible method of calculation. Rules Ramon Llull's "miliaria" The scholar-cleric Ramon Llull of Majorca, was the first writer to refer to a rule to solve the traverse problem of navigation. In his Arbor Scientiae (1295), in the section of questions on geometry, Llul writes: What Llull seems to be trying to explain is that a ship actually sailing E, but intending to sail SE, it can figure out how much of its intended southeastward distance it has already made good – what Italians called the "avanzar", but Lull seems to call the "miliaria in mari". Llull does not explain exactly how, but refers only to an "instrument", presumably some sort of trigonometric table. Lull is implying that mariners can calculate the miliaria on the intended course by multiplying the distance actually sailed on the erroneous course by the cosine of the angle between the two routes. Miliaria in mari = distance sailed × cos(θ) where θ is the angle of difference between the two routes. Using Lull's example, a ship that intended to sail southeast ("Exaloch" is Catalan for "Scirocco") but was instead forced to sail east ("Levant"), then the angle of difference is θ = 45°. After 100 miles on the erroneous route, the miliaria on the intended route is 100 × cos 45° = 70.71. Doubling the sailing on the erroneous route to 200 miles will double the miliaria on the intended route to 141.42 miles (= 200 cos 45°). (Diagramatically, Lull's miliaria in mari is measured by constructing a right-angled triangle by running a cord from the distance sailed on the actual course to the intended course, meeting the latter at a 90° angle). Llull is a little more explicit in his Ars magna generalis et ultima (written c. 1305). Reversing his example, with a ship actually sailing Southeast but intending to sail East, Llull notes that for every four miles on the southeast bearing, it "gains three miles" (2.83 actually) on the intended eastward route. Thus, Lull notes, the ship "loses 25 miles" (29 actually) of its intended course for