The Teaching of Geometry

by David Eugene Smith

CHAPTER V

EUCLID

It is fitting that a chapter in a book upon the teaching of this subject should be devoted to the life and labors of the greatest of all textbook writers, Euclid,—a man whose name has been, for more than two thousand years, a synonym for elementary plane geometry wherever the subject has been studied. And yet when an effort is made to pick up the scattered fragments of his biography, we are surprised to find how little is known of one whose fame is so universal. Although more editions of his work have been printed than of any other book save the Bible,[23] we do not know when he was born, or in what city, or even in what country, nor do we know his race, his parentage, or the time of his death. We should not feel that we knew much of the life of a man who lived when the Magna Charta was wrested from King John, if our first and only source of information was a paragraph in the works of some historian of to-day; and yet this is about the situation in respect to Euclid. Proclus of Alexandria, philosopher, teacher, and mathematician, lived from 410 to 485 A.D., and wrote a commentary on the works of Euclid. In his writings, which seem to set forth in amplified form his lectures to the students in the Neoplatonist School

of Alexandria, Proclus makes this statement, and of Euclid's life we have little else:

Not much younger than these[24] is Euclid, who put together the "Elements," collecting many of the theorems of Eudoxus, perfecting many of those of Theætetus, and also demonstrating with perfect certainty what his predecessors had but insufficiently proved. He flourished in the time of the first Ptolemy, for Archimedes, who closely followed this ruler,[25] speaks of Euclid. Furthermore it is related that Ptolemy one time demanded of him if there was in geometry no shorter way than that of the "Elements," to whom he replied that there was no royal road to geometry.[26] He was therefore younger than the pupils of Plato, but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.[27]

Thus we have in a few lines, from one who lived perhaps seven or eight hundred years after Euclid, nearly all that is known of the most famous teacher of geometry that ever lived. Nevertheless, even this little tells us about when he flourished, for Hermotimus and Philippus were pupils of Plato, who died in 347 B.C., whereas Archimedes was born about 287 B.C. and was writing about 250 B.C. Furthermore, since Ptolemy I reigned from 306 to 283 B.C., Euclid must have been teaching about 300 B.C., and this is the date that is generally assigned to him.

Euclid probably studied at Athens, for until he himself assisted in transferring the center of mathematical

culture to Alexandria, it had long been in the Grecian capital, indeed since the time of Pythagoras. Moreover, numerous attempts had been made at Athens to do exactly what Euclid succeeded in doing,—to construct a logical sequence of propositions; in other words, to write a textbook on plane geometry. It was at Athens, therefore, that he could best have received the inspiration to compose his "Elements."[28] After finishing his education at Athens it is quite probable that he, like other savants of the period, was called to Alexandria by Ptolemy Soter, the king, to assist in establishing the great school which made that city the center of the world's learning for several centuries. In this school he taught, and here he wrote the "Elements" and numerous other works, perhaps ten in all.