Disappointingly very little is known about the life and the personality of Euclid except that he was a professor of mathematics at the University of Alexandria and apparently the founder of the illustrious and long-lived Alexandrian School of Mathematics. Even the date and location of his birth are not known, but it seems probable that he received his mathematical training in the Platonic school at Athens.
Many years later, when comparing Euclid with Apollonius, to the latter’s discredit, Pappus praised Euclid for his modesty and consideration of others. Proclus augmented his Eudemian Summary with the frequently told story of Euclid’s reply to Ptolemy’s request for a short cut to geometric knowledge that “there is no royal road in geometry.” But the same story has been told of Menaechmus when serving as instructor to Alexander the Great. Stobaeus told another story of a student studying geometry under Euclid who questioned what he would get from learning the subject, whereupon Euclid ordered a slave to give the fellow a penny, “ since he must make gain from what he learns.”
Although Euclid was the author of at least ten works, and fairly complete texts of five of these have come down to us, his reputation rests mainly on his Elements. It appears that this remarkable work immediately and completely superseded all previous Elements; in fact, no trace remains of the earlier efforts. As soon as the work appeared it was accorded the highest respect, and from Euclid’s successors on up to modern times, the mere citation of Euclid’s book and proposition numbers was regarded as sufficient to identify particular theorem or construction. No work, except the Bible, has been more widely used, edited, or studied, and probably no work has exercised a greater influenced on scientific thinking. Over a thousand editions of Euclid’s Elements have appeared since the first one was printed in 1482, and for more than two millennia this work has dominated all teaching of geometry.
The Elements contains thirteen books, or chapters, which describe and prove a good part of all that the human race knows, even now, about lines, points, circles and the elementary solid shapes. All these information Euclid deduced, by the mind-sharpening logic, from just ten simple premises-five postulates and five axioms. These are detailed below. Out of these premises Euclid constructed not only the geometry normally taught today in high school but also a great deal of other mathematics. His chapters on line lengths and areas give geometric methods for solving many problems that are now taken up as algebra. His handling of the Zeno-plague concept of infinity and of the technique for summing up areas under circular arcs involves ideas now studied in calculus. His discussion of prime numbers-numbers which cannot be evenly divided except by themselves or by 1-is now a classic of college-level “number theory.”
Things equal to the same thing are equal.
If equals are added to equals, the sums are equal.
If equals are subtracted from equals, the remainders are equal.
Things which coincide with one another are equal to one another.
The whole is greater than the part.
A straight line can be drawn from any point to any other point.
A finite straight line can be drawn continuously in a straight line.
A circle can be described with any point at the center, and with a radius equal to any finite straight drawn form the center.
All right angles are equal to each other.
Given a straight line and any point not on this line, there is, through that point, one and only one line that is parallel to the given line.