the different bearings. The WbN ship will reach the meridian when it achieves latitude 21° 21' N, while the WNW ship will reach it when it achieves latitude 29° N. So rather than counting leagues with hourglass and speed readings, the ships can just maintain bearing, and take periodic astronomical observations to assess their latitude. Relation to "traverse sailing" The toleta de marteloio is ancestral to the modern "traverse table" used in more modern navigation. In the modern nomenclature, the traverse is the "crooked path made by a ship when she sails in several successive directions" and resolving the traverse is "the method of finding a single course and distance which would bring a ship to the same place as two or more courses and distances". In marteloio language, when "resolving the traverse", the known information given is the "actual course" and the "ritorno", while the unknowns are the "intended bearing" and "total avanzo". Traverse tables use three values for each of the crooked course segments – the Distance (Dist.), Difference of Latitude (D.Lat., movement along N–S axis) and the Departure (Dep., movement along E–W axis), the latter two calculated by the formulas: Difference of latitude = distance × cos θ Departure = distance × sin θ where θ is the angular difference of the course from the N–S axis if the values of θ are less than 45°; if, however, the angle exceeds 45°, then θ is expressed as the angle of difference from the E–W axis, and the formulas are flipped, i.e. the Difference of Latitude formula becomes the Departure, and the Departure formula is the Difference of Latitude). Or, even more simply, calculate θ as the angle of difference from the nearest principal wind (N, S, E, W), run the formulas and then place the larger number in the appropriate column (D.Lat. or Dep.). For each course segment, the navigator inserts the relevant trio (Dist., D.Lat., Dep.) and can calculate the implied bearing from the beginning to the endpoint and the distance made good on that bearing. He then combines, by addition and subtraction, all the differences of latitude and departure, to get the overall difference of latitude and departure, and converts that back to overall bearing and distance made good. Manuscript sources Ramon Llull's suggestive 1295 remarks aside, the earliest known reference to marteloio is dated 1390, in the inventory of the estate of the mother