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 (circle-and-square) 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.
Etymology The etymology comes from the Venetian language. In his 1436 atlas, Venetian captain and cartographer Andrea Bianco introduced a table of numbers which he called the toleta de marteloio ("table of marteloio"), and the method of using it as the raxon de marteloio ("reason of marteloio").
The meaning of marteloio itself is uncertain. The most widely accepted hypothesis, first forwarded by A.E. Nordenskiöld, is that marteloio relates to "hammer" ("martelo" in Venetian), referring to the small hammer that was used to hit the on-board ship's bell to mark the passage of time.Nordenskiöld (1897: p.51ff); Ruge (1900: p.177). It has been suggested that the -oio suffix implies that marteloio meant not quite the hammer itself nor the hammerer, but rather "the hammering", intending to indicate "the hammering, the din, the racket" from the change of the watch every four hours. As there were many hands on deck during a change of the watch, it would be an opportune moment for the ship's pilot to order a change in bearing (if necessary).Kelley (1995: p. 2)
Alternative hypotheses (not nearly as accepted) are that "marteloio" is a corruption of mari logio (meaning "rule of the sea"),This was forwarded by Giuseppe Toaldo (1782: p. 44) or from mare tela (meaning "sea network"),This was proposed by Desimoni (1888: p. 15) or that it derives from the Greek homartologium (όμαρτόλογίον, meaning "companion piece"),Morelli (1810: p.42). Morelli's interpretation is cited already by Formaleoni (1783: p.28). or from the Greek imeralogium (ήμερόλογίον, meaning "daily calculation")This was proposed by Fincati, as reported by Albertis (1893) or that it might be from the northern French matelot, which in turn comes from Breton martolod (meaning "sailors").This is proposed by Breusing (1881: p. 130)
Purpose
The "rule of marteloio" was used in European navigation in the Middle Ages, most notably in the Mediterranean Sea between the 14th and 16th centuries, although it may have older roots. It was an integral part of navigation by "compass and chart", before the advent of geographical coordinates and the development of celestial navigation in Europe.Taylor (1956); Parry (1974)
Medieval navigation relied on two parameters, direction and distance. On board ship, direction was determined by the mariner's compass (which emerged around 1300).Aczel (2001: p.76) Distance was measured by dead reckoning, (i.e., distance = speed × time), where time was measured by a half-hour-glass, and speed readings were taken with by some form of a chip log (the archaic method, used in the 14th and 15th centuries, involved heaving a piece of wood or flotsam overboard; the crew engaged in a rhythmic chant to mark the time it took for the chip to float past the length of the ship).Kelley (1995: p.12). This is slightly different from the later chip log, where the wood was tied to a rope with regularly spaced knots; the rope was allowed to unspool freely until a sandglass ran out, and the "knots" counted up in the aftermath. The chip log was first mentioned by William Bourne in 1574.
Plotting a course required knowing the compass direction and distance between point A and point B. Knowledge of where ports lay relative to each other was acquired by navigators by long experience at sea. This information was sometimes collected and written down in a pilot's handbook, known as a portolano ("port book", in Italian, equivalent to the Greek periplus, the Portuguese roteiro and the English rutter). These handbooks were used to construct a class of nautical maps known as portolan charts. Portolan charts began being produced in Genoa in the late 13th century, and soon spread to Venice and Majorca. Portolan charts were not gridded by longitude and latitude lines, but rather by a web of compass rhumb lines, giving mariners an idea of only the distance and direction between places.
By a handbook or a portolan chart, a navigator could see immediately that, for example, Pisa lay 85 miles southeast ("Scirocco" in the traditional compass rose nomenclature) of Genoa, and so a ship that set out from Genoa to Pisa would simply maintain that bearing for that distance. However, most sailing courses were not nearly that neat. A mariner wishing to sail from Majorca to Naples could tell the latter was due east ("Levante") by some 600 miles – but the island of Sardinia lies in the way, therefore the ship's bearing must be changed along the route. This is easier said than done, as geographical coordinates did not exist during this era. The only way to determine the exact position of the ship at sea would be to calculate via past bearing and distance travelled.Taylor (1956: p.123, 159, 167); Parry (1974: p.37)
Islands were a predictable obstacle – circumventing Sardinia would be simply a matter of sailing southeast for a set distance then changing the bearing to northeast ("Greco") for the remainder. More problematic is if the ship were blown off its intended route by fitful winds, or had to engage in tacking, changing bearing repeatedly. In order to return to its intended course, the rule of marteloio would then be utilized.
The traverse problem
The rule of marteloio addressed the problem of changing bearing at sea. More specifically, it helped a navigator plot the traverse from one navigational course to another.Taylor (1956: p.116ff.) Taylor (1960: p.10) For example, suppose a ship was to sail from Corsica to Genoa, a course bearing straight north ("Tramontana") for some 130 miles. But the winds are not cooperative, and the ship was forced to sail northwest ("Maestro") for some 70 miles. How does it return to its original route? Re-setting its bearing to northeast ("Greco") seems sensible enough, but how long should it sail on that bearing? How would a navigator know when the ship had reached its old route and should turn north again? How to avoid overshooting or undershooting the old course?
This is a mathematical problem of solving a triangle. If a navigator knows how long the ship has sailed on the erroneous course, he can calculate its current distance from its intended course, and estimate how long it must sail back on a new bearing until it recovers its old course. In the Corsica-to-Genoa example, there is an implied triangle ACD, with one side given (AC = 70 miles on actual NW course), a 45° angle at A (angle of difference between actual course NW and intended course N) and another angle of 90° at C (angle of difference between actual course NW and return course NE). The challenge to the navigator is to find how long one must sail on the NE return course (the length of side CD, what is called the ritorno) and how far one has advanced on the intended course by the time one straightens out (the length of the hypotenuse AD, or what is called the total avanzo).
This is elementary trigonometry, solving for two sides given one side (70) and two angles (45° and 90°). This is quickly done by applying the law of sines:
yielding up the solutions ritorno = 70 miles and total avanzo = 98.99 miles. This means that if the ship bears NE from its current position (C), it will reach its original intended course after 70 miles of sailing on the NE bearing. By the time it reaches its junction point (D), it will have covered 98.99 miles of its original 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.Van Brunnelen (2010, p.67) As a result, Medieval navigators needed a 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.This interpretation is originally due to Taylor (1956: pp. 117–19). See also Cortesão (1969: v. 1, pp. 206–7), Cotter (1978: pp. 6–7) and Campbell (1987: pp. 441–42).
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).Ramom Llull, 1517 ed., Part 10, "De Navigatione", fo. 93, p. 213. See also Cotter (1978: p. 7) 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 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."To a mathematician familiar with contemporary mathematical knowledge and practice, it is quite clear that Lull did not fully understand what he was writing about. He used his shipboard observations during his travels to support transcendental ideas about the circle, triangle and square. His value is as a witness to late-thirteenth century mariners doing vector navigations with chart or plotting board." (Kelley, 1995: p. 3) This is perhaps unsurprising – although trigonometry was only in its infancy in Christian Europe, sine and cosine tables were already known in Arab mathematics.Taylor (1960: p. 10) 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.Taylor (1956: p. 114); Kelley (1995: p. 3); Vernet (2008) 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.