out on an East-southeast (ESE) bearing. That is six quarter-winds above South (remember: unlike the marteloio, the Regiment of the Leagues always measures quarter-winds away from the north–south meridian). Looking at any regiment of the leagues table (e.g. Martín Cortés de Albacar, 1551), for q = 6, the table gives the relevar as 45 leagues and the afastar as 42 leagues. This means that a ship sailing on the ESE bearing will have to sail 45.73 leagues to cover one degree of latitude (17.5 leagues of alargar from the east bearing, to use the marteloio language), and the corresponding afastar (avanzar in marteloio terms) will be 42.25 leagues. If, instead, the ship had set out on a SE bearing, that is four quarter-winds above South, the corresponding values of the Regiment of the Leagues table at q = 4 are relevar = 24 and the afastar = 17 . Notice that the SE bearing reaches the 1 degree alargar faster (i.e. smaller relevar) than that the ESE bearing, and will have less afastar (closer to the N–S meridian). Mathematically, relevar = 17.5/cos θ afastar = 17.5 × tan θ where θ = 11.25 × number of quarters-winds away from the north-south axis. Despite the difference in terminology, notably the use of latitude degrees, the rule of marteloio and the Regiment of the Leagues are very similar – they are both about solving triangles on a plane chart. The advantage of the regiment over the marteloio is the introduction of latitude parallels in the table, so that the position can be checked by astronomical observation (via quadrant, astrolabe, etc.), and not have to rely wholly on sailor estimations of distance and direction. With the regiment, geographical coordinates can also be used to guide navigation. For instance, the search for the Todesillas line (meridian 370 leagues west of Cape Verde) is much simplified by reference to a precise latitude. For instance, suppose two ships depart from Cape Verde (17° N), one on a West by North bearing (WbN, that is one quarter above West, or q = 7 from North axis), the other by a west-northwest bearing (WNW, two quarters above west, or q = 6 from the North axis). Using the Regiment of the Leagues, it is possible to calculate the precise latitudes when they will cross the Tordesillas meridian – simply divide 370 leagues west by the implied afastar at