We must next speak of the origin of geometry in the present world cycle. For, as the remarkable Aristotle tells us, the same ideas have repeatedly come to men at various periods of the universe. It is not, he goes on to say, in our time or in the time of those known to us that the sciences have first arisen, but they have appeared and again disappeared, and will continue to appear and to disappear, in various cycles, of which the number both past and future is countless. But since we must speak of the origin of the arts and sciences with reference to the present world cycle, it was, we say, among the Egyptians that geometry is generally held to have been discovered. It owed its discovery to the practice of land measurement. For the Egyptians had to perform such measurements because the overflow of the Nile would cause the boundary of each person's land to disappear. Furthermore, it should occasion no surprise that the discovery both of this science and of the other sciences proceeded from utility, since everything that is in the process of becoming advances from the imperfect to the perfect. The progress, then, from sense perception to reason and from reason to understanding is a natural one. And so, just as the accurate knowledge of numbers originated with the Phoenicians through their commerce and their business transactions, so geometry was discovered by the Egyptians for the reason we have indicated.
It was Thales, who, after a visit to Egypt, first brought this study to Greece. Not only did he make numerous discoveries himself, but laid the foundations for many other discoveries on the part of his successors, attacking some problems with greater generality and others more empirically. After him Mamercus, the brother of the poet Stesichorus, is said to have embraced the study of geometry, and in fact Hippias of Elis writes that he achieved fame in that study.
After these Pythagoras changed the study of geometry, giving it the form of a liberal discipline, seeking its first principles in ultimate ideas, and investigating its theorems abstractly and in a purely intellectual way. It was he who discovered the subject of proportions and the construction of the cosmic figures. After him Anaxagoras of Clazomenae devoted himself to many of the problems of geometry, as did Oenopides of Chios, who was a little younger than Anaxagoras. Plato in The Rivals mentions them both as having achieved repute in mathematics. After them Hippocrates of Chios the discoverer of the quadrature of the lune, and Theodorus of Cyrene gained fame in geometry. For Hippocrates is the first man on record who also composed Elements.
Plato, who lived after Hippocrates and Theodorus, stimulated to a very high degree the study of mathematics and of geometry in particular because of his zealous interest in these subjects. For he filled his works with mathematical discussions, as is well known, and everywhere sought to awaken admiration for mathematics in students of philosophy.
At this time lived also Leodamas of Thasos, Archytas of Tarentum, and Theaetetus of Athens, all of whom increased the number of theorems and made progress towards a more scientific ordering thereof. Neoclides and his pupil, Leon, who were younger than Leodamas, made many additions to the work of their predecessors. As a result the Elements composed by Leon are more carefully worked out both in the number and in the usefulness of the propositions proved. Leon also first discussed the diorismi (distinctions), that is the determination of the conditions under which the problem posed is capable of solution, and the conditions under which it is not.
A little younger than Leon was Eudoxus of Cnidus, an associate of Plato's school, who first increased the number of the general theorems, as they are called, and added three new proportions to the three then known. Using the method of analysis he greatly extended the theory of the section, a subject which had originated with Plato. Amyclas of Heraclea, a friend of Plato, Menaechmus, a pupil of Eudoxus who had also been associated with Plato, and Dinostratus, a brother of Menaechmus, brought the whole of geometry to an even higher degree of perfection. Theudius of Magnesia achieved a reputation for excellence both in mathematics and in the other parts of philosophy, for his Elements were excellently arranged and many of the (theretofore) limited propositions were put in more general form. Furthermore, Athenaeus of Cyzicus, a contemporary, distinguished himself in mathematics generally and in geometry in particular. These men spent their time together in the Academy and collaborated in their investigations. Hermotimus of Colophon extended the work that had been done by Eudoxus and Theaetetus, made many discoveries in the Elements, and put together some material on the subject of Loci. Philippus of Mende who, as a pupil of Plato, was inspired by him to the study of mathematics, conducted investigations along lines suggested by Plato, and in particular set before himself researches which he thought would contribute to the philosophy of Plato. Now those who have written histories trace the development of the science of geometry down to Philippus.
Euclid, who was not much younger than Hermotimus and Philippus, composed Elements, putting in order many of the theorems of Eudoxus, perfecting many that had been worked on by Theaetetus, and furnishing with rigorous proofs propositions that had been demonstrated less rigorously by his predecessors. Euclid lived in the time of the first Ptolemy, for Archimedes, whose life overlapped the reign of this Ptolemy too, mentions Euclid. Furthermore, there is a story that Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by the study of the Elements. Whereupon Euclid answered that there was no royal road to geometry. He is, then, younger than Plato's pupils and older than Eratosthenes and Archimedes, who, as Eratosthenes somewhere remarks, were contemporaries.
By choice Euclid was a follower of Plato and connected with this school of philosophy. In fact he set up as the goal of the Elements as a whole the construction of the so-called Platonic figures.
There are, in addition, many other mathematical works by Euclid, written with remarkable accuracy and scientific insight, such as the Optics, the Catoptrics, works on the Elements of Music, and the book On Divisions. But he is most to be admired for his Elements of Geometry because of the choice and arrangement of the theorems and problems made with regard to the elements. For he did not include all that he might have included, but only those theorems and problems which could fulfil the functions of elements .... If you seek to add or subtract anything, are you not unwittingly cast adrift from science and carried away toward falsehood and ignorance?
Now there are many things that seem to be grounded in truth and to follow from scientific principles but actually are at variance with these principles and deceive the more superficial. It was for this reason that Euclid set forth methods for intelligent discrimination in such matters, too. With these methods not only shall we be able to train beginners in this study to detect fallacies, but we shall be able to escape deception ourselves. The work in which he gives us this preparation he entitled Pseudaria. In it he recounts in order various types of fallacies, training us to understand each type by all sorts of theorems, setting the true alongside the false, and linking the refutation of the fallacy with empirical examples. This book, therefore, is for the purification and training of the understanding, while the Elements contain the complete and irrefutable guide to the scientific study of the subject of geometry.