Early Geometry

Proclus Diadochus, AD 410-485.

(From his book: Commentary on Euclid's Elements I)

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 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 foundation 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.

[He then mentions several who developed this abstract approach further: Anaxagoras, Hippocrates, Theodorus, etc.]

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.

[He then lists several mathematicians, including Eudoxus and Theatetus, who discovered many new geometric theorems, and began to arrange them in logical sequences-this process culminated in the work of Euclid, called his Elements (of geometry) about 300 BC. ]

Euclid composed Elements, putting in order many of the theorems of Eudoxus, perfecting many that had been worked out by Theatetus, and furnishing with rigorous proofs propositions that had been demonstrated less rigorously by his predecessors … the Elements contain the complete and irrefutable guide to the scientific study of the subject of geometry.