Therefore, as I have said, from Lisbon, whence we started, the distance from the equinoctial line is 39°, and we navigated beyond the equinoctial line to 50°, which together make 90°, which is one quarter of a great circle, according to the true measurement handed down to us by the ancients, so that it is manifest that we must have navigated over a fourth part of the earth. By this reasoning, we who inhabit Lisbon, at a distance of 39° from the equinoctial line in north latitude, are to those who live under 50° beyond the same line, in meridional length, angularly 5° on a transverse line. I will explain this more clearly: a perpendicular line, while we stand upright, if suspended from a point of the heavens exactly vertical, hangs over our heads; but it hangs over them sideways. Thus, while we are on a right line, they are on a transverse line. An orthogonal triangle is thus formed, of which we have the right line, but the base and hypotenuse to them seems the vertical line, as in this figure it will appear. This will suffice as regards cosmography.
Vertex[127] of our heads. _.-'| _.-' | Hypotenuse. _.-' | Us. _.-' | Vertex[127] of _.-' | their heads. _.-'____________________| Them.
These are the most notable things that I have seen in this my last navigation, or, as I call it, the third voyage. For the other two voyages were made by order of the Most Serene King of Spain to the west, in which I noted many wonderful works of God, our Creator; and if I should have time, I intend to collect all these singular and wonderful things into a geographical or cosmographical book, that my record may live with future generations; and the immense work of the omnipotent God will be known, in parts still unknown, but known to us. I also pray that the most merciful God will prolong my life that, with His good grace, I may be able to make the best disposition of this my wish. I keep the other two journeys in my sanctuary, and the Most Serene King restoring to me the third journey, I intend to return to peace and my country. There, in consultation with learned persons, and comforted and aided by friends, I shall be able to complete my work.
I ask your pardon for not having sooner been able to send you this my last navigation, as I had promised in my former letters. I believe that you will understand the cause, which was that I could not get the books from this Most Serene King. I think of undertaking a fourth voyage in the same direction, and promise is already made of two ships with their armaments, in which I may seek new regions of the East on a course called Africus. In which journey I hope much to do God honour, to be of service to this kingdom, to secure repute for my old age, and I expect no other result with the permission of this Most Serene King. May God permit what is for the best, and you shall be informed of what happens.