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.

[The distances of these itineraries are shown in evidence of the preceding.  Maps of India made in Portugal “at the time when there was no suspicion that so great a number of leagues was to be subtracted as is proved now to have been the case,” are cited and distances taken therefrom in proof of the assertions made by the Castilian deputies.  As a result of these distances it is shown that the distance between the Moluccas and the island of Sant Antonio would be one hundred and eighty-four degrees to the eastward, to which number “must be added the degrees contained in the said three hundred and seventy leagues from the island of Sant Antonio to the line of demarcation.”  The following deductions are made:]

It is quite evident from the above that the distance of the navigation eastward assigned by the Portuguese in the proceedings is short by more than fifty degrees, being proved by the said old Portuguese relations and maps, which are not to be doubted.  And it is evident that our calculation is true, both eastward and westward, and that from the said divisional line commencing from the island of Sant Antonio, the distance westward to the Malucos is not more than the said one hundred and fifty degrees.