Therefore, as is apparent from the said information of modern navigators and cosmographers, both Portuguese and those of other nations, and from the relation of the said pilots and sailors, it has been proved completely that the said distances and routes, declared by the said deputies of Portugal, are neither just nor true, and that the deputies have reported them much shorter than, in sober truth, they are. From this it can be presumed, that inasmuch as they shorten the said route each day, the said mistake of fifty degrees proceeds doubtless from their eastern part and not from our western part. _Item:_ it is to be observed that, notwithstanding the said distances, expressed, as is shown by the said pilots who determined them, as they should, on a spherical body, the said Malucos fall many degrees within the limits of our lord, the Emperor, and that they lie a much greater number of degrees east of the island of La Sal, than they had declared, inasmuch as, according to geometrical reasoning, the lands situated along the said eastern voyage, placed on a plane surface, and the number of leagues being reckoned by equinoctial degrees, are not in their proper location as regards the number and quantity of their degrees, for it is well known in cosmography that a lesser number of leagues along parallels other than the equinoctial, occupy a greater quantity of degrees. Now then as almost all the lands from the Cabo Verde islands to the Malucos, are, for the most part quite distinct from the equinoctial, it will take a much greater number of degrees when they are transferred and drawn on the spherical body. Calculating by geometrical proportion, with the arc and chord, whereby we pass from a plane to a spherical surface, so that each parallel is just so much less as its distance from the equinoctial is increased, the number of degrees in the said maps is much greater than the said pilots confess, and consequently these lands fall by a greater number of degrees inside their Majesties' limits. In order to verify the above we must examine the itineraries and navigation routes, and the angles and intersections made by the routes with the meridians and parallels encountered, which are styled angles _positionis_ among cosmographers. This is the most certain method of determining lands on a spherical body, when calculating them from the plane surface, as the following will show.