LIBRA.

INTRODUCTION.

CHAPTER I.

EUCLID AND THE TRADITIONS ABOUT HIM.

As in the case of the other great mathematicians of Greece, so in Euclid's case, we have only the most meagre particulars of the life and personality of the man.

Most of what we have is contained in the passage of Proclus' summary relating to him, which is as follows1:

"Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived' in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. He is then younger than the pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says."

This passage shows that even Proclus had no direct knowledge of Euclid's birthplace or of the date of his birth or death. He proceeds by inference. Since Archimedes lived just after the first 1 Proclus, ed. Friedlein, p. 68, 6—20.

66

The word yeyove must apparently mean flourished," as Heiberg understands it (Litterargeschichtliche Studien über Euklid, 1882, p. 26), not "was born," as Hankel took it: otherwise part of Proclus' argument would lose its cogency.

• So Heiberg understands επιβαλών τῷ πρώτῳ (sc. Πτολεμαίῳ). Friedlein's text has καὶ between ἐπιβαλών and τῷ πρώτῳ; and it is right to remark that another reading is καὶ ἐν τῷ πρώτῳ (without ἐπιβαλών) which has been translated in his first book," by which is understood On the Sphere and Cylinder 1., where (1) in Prop. 2 are the words "let BC be made equal to D by the second (proposition) of the first of Euclid's (books)," and (2) in Prop. 6 the words "For these things are handed down in the Elements" (without the name of Euclid). Heiberg thinks the former passage is referred to, and that Proclus must therefore have had before him the words "by the second of the first of Euclid": a fair proof that they are genuine, though in themselves they would be somewhat suspicious.

The same story is told in Stobaeus, Ecl. (11. p. 228, 30, ed. Wachsmuth) about Alexander and Menaechmus. Alexander is represented as having asked Menaechmus to teach him geometry concisely, but he replied: "O king, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all."

H. E.

Ptolemy, and Archimedes mentions Euclid, while there is an anecdote about some Ptolemy and Euclid, therefore Euclid lived in the time of the first Ptolemy.

We may infer then from Proclus that Euclid was intermediate between the first pupils of Plato and Archimedes. Now Plato died in *347, Archimedes lived 287-212, Eratosthenes 276-194 B.C. Thus Euclid must have flourished c. 300 B.C., which date agrees well with the fact that Ptolemy reigned from 306 to 283 B.C.

It is most probable that Euclid received his mathematical training in Athens from the pupils of Plato; for most of the geometers who could have taught him were of that school, and it was in Athens that the older writers of elements, and the other mathematicians on whose works Euclid's Elements depend, had lived and taught. He may himself have been a Platonist, but this does not follow from the statements of Proclus on the subject. Proclus says namely that he was of the school of Plato and in close touch with that philosophy'. But this was only an attempt of a New Platonist to connect Euclid with his philosophy, as is clear from the next words in the same sentence, "for which reason also he set before himself, as the end of the whole Elements, the construction of the so-called Platonic figures." It is evident that it was only an idea of Proclus' own to infer that Euclid was a Platonist because his Elements end with the investigation of the five regular solids, since a later passage shows him hard put to it to reconcile the view that the construction of the five regular solids was the end and aim of the Elements with the obvious fact that they were intended to supply a foundation for the study of geometry in general, "to make perfect the understanding of the learner in regard to the whole of geometry?." To get out of the difficulty he says that, if one should ask him what was the aim (akoπós) of the treatise, he would reply by making a distinction between Euclid's intentions (1) as regards the subjects with which his investigations are concerned, (2) as regards the learner, and would say as regards (1) that "the whole of the geometer's argument is concerned with the cosmic figures." This latter statement is obviously incorrect. It is true that Euclid's Elements end with the construction of the five regular solids; but the planimetrical portion has no direct relation to them, and the arithmetical no relation at all; the propositions about them are merely the conclusion of the stereometrical division of the work.

One thing is however certain, namely that Euclid taught, and founded a school, at Alexandria. This is clear from the remark of Pappus about Apollonius": "he spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought."

It is in the same passage that Pappus makes a remark which might, to an unwary reader, seem to throw some light on the

1 Proclus, p. 68, 20, καὶ τῇ προαιρέσει δὲ Πλατωνικός ἐστι καὶ τῇ φιλοσοφίᾳ ταύτῃ οἰκεῖος. 2 ibid. p. 71, 8. 8 ibid. p. 70, 19 $99.

4

Pappus, vii. p. 678, 10-12, συσχολάσας τοῖς ὑπὸ Εὐκλείδου μαθηταῖς ἐν ̓Αλεξανδρείᾳ πλεῖστον χρόνον, ὅθεν ἔσχε καὶ τὴν τοιαύτην ἕξιν οὐκ ἀμαθῆ.

4

personality of Euclid. He is speaking about Apollonius' preface to the first book of his Conics, where he says that Euclid had not completely worked out the synthesis of the "three- and four-line locus," which in fact was not possible without some theorems first discovered by himself. Pappus says on this1: "Now Euclidregarding Aristaeus as deserving credit for the discoveries he had already made in conics, and without anticipating him or wishing to construct anew the same system (such was his scrupulous fairness and his exemplary kindliness towards all who could advance mathematical science to however small an extent), being moreover in no wise contentious and, though exact, yet no braggart like the other [Apollonius] -wrote so much about the locus as was possible by means of the conics of Aristaeus, without claiming completeness for his demonstrations." It is however evident, when the passage is examined in its context, that Pappus is not following any tradition in giving this account of Euclid: he was offended by the terms of Apollonius' reference to Euclid, which seemed to him unjust, and he drew a fancy picture of Euclid in order to show Apollonius in a relatively unfavourable light.

Another story is told of Euclid which one would like to believe true. According to Stobaeus, "some one who had begun to read geometry with Euclid when he had learnt the first theorem, asked Euclid, 'But what shall I get by learning these things?' Euclid called his slave and said 'Give him threepence, since he must make gain out of what he learns.'"

In the middle ages most translators and editors spoke of Euclid as Euclid of Megara. This description arose out of a confusion between our Euclid and the philosopher Euclid of Megara who lived about 400 B.C. The first trace of this confusion appears in Valerius Maximus (in the time of Tiberius) who says that Plato, on being appealed to for a solution of the problem of doubling the cubical altar, sent the inquirers to "Euclid the geometer." There is no doubt about the reading, although an early commentator on Valerius Maximus wanted to correct "Eucliden" into "Eudoxum," and this correction is clearly right. But, if Valerius Maximus took Euclid the geometer for a contemporary of Plato, it could only be through confusing him with Euclid of Megara. The first specific reference to Euclid as Euclid of Megara belongs to the 14th century, occurring in the vπоμvnμatioμol of Theodorus Metochita (d. 1332) who speaks of "Euclid of Megara, the Socratic philosopher, contemporary of Plato," as the author of treatises on plane and solid geometry, data, optics etc. and a Paris MS. of the 14th century has "Euclidis philosophi Socratici liber elementorum." The misunderstanding was general in the period from Campanus' translation (Venice 1482) to those of Tartaglia (Venice 1565) and Candalla (Paris 1566). But one Constantinus Lascaris (d. about 1493) had already made the proper

1 Pappus, VII. pp. 676, 25—678, 6. Hultsch, it is true, brackets the whole passage pp. 676, 25-678, 15, but apparently on the ground of the diction only.

2 Stobaeus, .c.

3

VIII. 12, ext. 1.

distinction by saying of our Euclid that "he was different from him of Megara of whom Laertius wrote, and who wrote dialogues"; and to Commandinus belongs the credit of being the first translator' to put the matter beyond doubt: "Let us then free a number of people from the error by which they have been induced to believe that our Euclid is the same as the philosopher of Megara" etc.

Another idea, that Euclid was born at Gela in Sicily, is due to the same confusion, being based on Diogenes Laertius' description of the philosopher Euclid as being "of Megara, or, according to some, of Gela, as Alexander says in the Aiadoxai."

In view of the poverty of Greek tradition on the subject even as early as the time of Proclus (410-485 A.D.), we must necessarily take cum grano the apparently circumstantial accounts of Euclid given by Arabian authors; and indeed the origin of their stories can be explained as the result (1) of the Arabian tendency to romance, and (2) of misunderstandings.

We read that "Euclid, son of Naucrates, grandson of Zenarchus, called the author of geometry, a philosopher of somewhat ancient date, a Greek by nationality domiciled at Damascus, born at Tyre, most learned in the science of geometry, published a most excellent and most useful work entitled the foundation or elements of geometry, a subject in which no more general treatise existed before among the Greeks: nay, there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine. Hence also Greek, Roman and Arabian geometers not a few, who undertook the task of illustrating this work, published commentaries, scholia, and notes upon it, and made an abridgment of the work itself. For this reason the Greek philosophers used to post up on the doors of their schools the well-known notice: 'Let no one come to our school, who has not first learned the elements of Euclid.'" The details at the beginning of this extract cannot be derived from Greek sources, for even Proclus did not know anything about Euclid's father, while it was not the Greek habit to record the names of grandfathers, as the Arabians commonly did. Damascus and Tyre were no doubt brought in to gratify a desire which the Arabians always showed to connect famous Greeks in some way or other with the East. Thus Naṣiraddin, the translator of the Elements, who was of Ţūs in Khurāsān, actually makes Euclid out to have been "Thusinus" also. The readiness of the Arabians to run away with an idea is illustrated by the last words

1 Letter to Fernandus Acuna, printed in Maurolycus, Historia Siciliae, fol. 21 r. (see Heiberg, Euklid-Studien, pp. 22-3, 25).

2 Preface to translation (Pisauri, 1572).

Diog. L. II. 106, p. 58 ed. Cobet.

• Casiri, Bibliotheca Arabico-Hispana Escurialensis, I. p. 339- Casiri's source is alQifti (d. 1248), the author of the Ta'rikh al-Ḥukamā, a collection of biographies of philosophers, mathematicians, astronomers etc.

The Fihrist says "son of Naucrates, the son of Berenice (?)" (see Suter's translation in Abhandlungen zur Gesch. d. Math. vi. Heft, 1892, p. 16).

The same predilection made the Arabs describe Pythagoras as a pupil of the wise Salomo, Hipparchus as the exponent of Chaldaean philosophy or as the Chaldaean, Archimedes as an Egyptian etc. (Ḥāji Khalfa, Lexicon Bibliographicum, and Casiri).

of the extract. Everyone knows the story of Plato's inscription over the porch of the Academy: "let no one unversed in geometry enter my doors"; the Arab turned geometry into Euclid's geometry, and told the story of Greek philosophers in general and "their Academies."

Equally remarkable are the Arabian accounts of the relation of Euclid and Apollonius'. According to them the Elements were originally written, not by Euclid, but by a man whose name was Apollonius, a carpenter, who wrote the work in 15 books or sections". In the course of time some of the work was lost and the rest became disarranged, so that one of the kings at Alexandria who desired to study geometry and to master this treatise in particular first questioned about it certain learned men who visited him and then sent for Euclid who was at that time famous as a geometer, and asked him to revise and complete the work and reduce it to order. Euclid then re-wrote it in 13 books which were thereafter known by his name. (According to another version Euclid composed the 13 books out of commentaries which he had published on two books of Apollonius on conics and out of introductory matter added to the doctrine of the five regular solids.) To the thirteen books were added two more books, the work of others (though some attribute these also to Euclid) which contain several things not mentioned by Apollonius. According to another version Hypsicles, a pupil of Euclid at Alexandria, offered to the king and published Books XIV. and XV., it being also stated that Hypsicles had "discovered" the books, by which it appears to be suggested that Hypsicles had edited them from materials left by Euclid.

We observe here the correct statement that Books XIV. and xv. were not written by Euclid, but along with it the incorrect information that Hypsicles, the author of Book XIV., wrote Book XV. also.

The whole of the fable about Apollonius having preceded Euclid and having written the Elements appears to have been evolved out of the preface to Book XIV. by Hypsicles, and in this way; the Book must in early times have been attributed to Euclid, and the inference based upon this assumption was left uncorrected afterwards when it was recognised that Hypsicles was the author. The preface is worth quoting :

"Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of their common interest in mathematics. And once, when

1 The authorities for these statements quoted by Casiri and Ḥāji Khalfa are al-Kindi's tract de instituto libri Euclidis (al-Kindi died about 873) and a commentary by Qāḍīzāde ar-Rumi (d. about 1440) on a book called Ashkal at-ta' sis (fundamental propositions) by Ashraf Shamsaddin as-Samarqandi (c. 1276) consisting of elucidations of 35 propositions selected from the first books of Euclid. Naşiraddin likewise says that Euclid cut out two of 15 books of elements then existing and published the rest under his own name. According to Qādizade the king heard that there was a celebrated geometer named Euclid at Tyre: Naşiraddin says that he sent for Euclid of Ṭūs.

So says the Fihrist. Suter (op. cit. p. 49) thinks that the author of the Fihrist did not suppose Apollonius of Perga to be the writer of the Elements, as later Arabian authorities did, but that he distinguished another Apollonius whom he calls "a carpenter." Suter's argument is based on the fact that the Fihrist's article on Apollonius (of Perga) says nothing of the Elements, and that it gives the three great mathematicians, Euclid, Archimedes and Apollonius, in the correct chronological order.

ぐ