At the end of the fifteenth century, celestial navigation was just being developed in Europe, primarily by the Portuguese. Prior to the development of celestial navigation, sailors navigated by "deduced" (or "dead") reckoning, hereafter called DR. This was the method used by Columbus and most other sailors of his era. In DR, the navigator finds his position by measuring the course and distance he has sailed from some known point. Starting from a known point, such as a port, the navigator measures out his course and distance from that point on ....