It remains for me to lay before your Majesty a Cosmographical exposition of our voyage.  Taking our departure, an I before observed, from the above mentioned desert rocks, which lie on the extreme verge of the west, as known to the ancients, in the meridian of the Fortunate Islands, and in the latitude of 32 degrees north from the equator, and steering a westward course, we had run, when we first made land, a distance of 1200 leagues or 4800 miles, reckoning, according to nautical usage, four miles to a league.  This distance calculated geometrically, upon the usual ratio of the diameter to the circumference of the circle, gives 92 degrees; for if we take 114 degrees as the chord of an arc of a great circle, we have by the same ratio 95 deg., as the chord of an arc on the parallel of 34 degrees, being that on which we first made land, and 300 degrees as the circumference of the whole circle passing through this plane.  Allowing then, as actual observations show, that 62 1/2 terrestrial miles correspond to a celestial degree, we find the whole circumference of 300 deg., as just given, to be 18,759 miles, which divided by 360, makes the length of a degree of longitude in the parallel of 34 degrees to be 52 miles, and that is the true measure.  Upon this basis, 1200 leagues, or 4800 miles meridional distance, on the parallel of 34, give 92 degrees, and so many therefore have we sailed farther to the west than was known to the ancients.  During our voyage we had no lunar eclipses or like celestial phenomenas, we therefore determined our progress from the difference of longitude, which we ascertained by various instruments, by taking the sun’s altitude from day to day, and by calculating geometrically the distance run by the ship from one horizon to another; all these observations, as also the ebb and flow of the sea in all places, were noted in a little book, which may prove serviceable to navigators; they are communicated to your Majesty in the hope of promoting science.

My intention in this voyage was to reach Cathay, on the extreme coast of Asia, expecting however, to find in the newly discovered land some such an obstacle, as they have proved to be, yet I did not doubt that I should penetrate by some passage to the eastern ocean.  It was the opinion of the ancients, that our oriental Indian ocean is one and without interposing land.  Aristotle supports it by arguments founded on various probabilities; but it is contrary to that of the moderns and shown to be erroneous by experience; the country which has been discovered, and which was unknown to the ancients, is another world compared with that before known, being manifestly larger than our Europe, together with Africa and perhaps Asia, if we might rightly estimate its extent, as shall now be briefly explained to your Majesty.  The Spaniards have sailed south beyond the equator on a meridian 20 degrees west of the Fortunate Islands to the latitude of 54, and there still found land; turning about they steered