(Εὐκλείδης) of ALEXANDRIA. The length of this article will not be blamed by any one who considers that, the sacred writers excepted, no Greek has been so much read or so variously translated as Euclid. To this it may be added, that there is hardly any book in our language in which the young scholar or the young mathematician can find all the information about this name which its celebrity would make him desire to have.
Euclid has almost given his own name to the science of geometry, in every country in which his writings are studied; and yet all we know of his private history amounts to very little. He lived, according to Proclus (Comm. in Eucl. 2.4), in the time of the first Ptolemy, B. C. 323-283. The forty years of Ptolemy's reign are probably those of Euclid's age, not of his youth; for had he been trained in the school of Alexandria formed by Ptolemy, who invited thither men of note, Proclus would probably have given us the name of his teacher: but tradition rather makes Euclid the founder of the Alexandrian mathematical school than its pupil. This point is very material to the
Euclid, says Proclus, was younger than Plato, and older than Eratosthenes and Archimedes, the latter of whom mentions him. He was of the Platonic sect, and well read in its doctrines. He collected the Elements, put into order much of what Eudoxus had done, completed many things of Theaetetus, and was the first who reduced to unobjectionable demonstration the imperfect attempts of his predecessors. It was his answer to Ptolemy, who asked if geometry could not be made easier, that there was no royal road (μὴ εἰναι βασιλικὴν ἄτραπον πρὸς γεωμετρίαν). [*](* This celebrated anecdote breaks off in the middle of the sentence in the Basle edition of Proclus. Barocius, who had better manuscripts, supplies the Latin of it; and Sir Henry Savile, who had manuscripts of all kinds in his own library, quotes it as above, with only ἐπὶ for πρὸς. August, in his edition of Euclid, has given this chapter of Proclus in Greek, but without saying from whence he has taken it.) This piece of wit has had many imitators; " Quel diable" said a French nobleman to Rohault, his teacher of geometry, " pourrait entendre cela ?" to which the answer was " Ce serait un diable qui aurait de la patience." A story similar to that of Euclid is related by Seneca (Ep. 91, cited by August) of Alexander.
Pappus (lib. vii. in praef.) states that Euclid was distinguished by the fairness and kindness of his disposition, particularly towards those who could do anything to advance the mathematical sciences: but as he is here evidently making a contrast to Apollonius, of whom he more than insinuates a directly contrary character, and as he lived more than four centuries after both, it is difficult to give credence to his means of knowing so much about either. At the same time we are to remember that he had access to many records which are now lost. On the same principle, perhaps, the account of Nasir-eddin and other Easterns is not to be entirely rejected, who state that Euclid was sprung of Greek parents, settled at Tyre; that he lived, at one time, at Damascus; that his father's name was Naucrates, and grandfather's Zenarchus. (August, who cites Gartz, De Interpr. Eucl. Arab.) It is against this account that Eutocius of Ascalon never hints at it.
At one time Euclid was universally confounded with Euclid of Megara, who lived near a century before him, and heard Socrates. Valerius Maximus has a story (8.12) that those who came to Plato about the construction of the celebrated Delian altar were referred by him to Euclid the geometer. This story, which must needs be false, since Euclid of Megara, the contemporary of Plato, was not a geometer, is probably the crigin of the confusion. Harless thinks that Eudoxus should be read for Euclid in the passage of Valerius.
In the frontispiece to Whiston's translation of Tacquet's Euclid there is a bust, which is said to be taken from a brass coin in the possession of Christina of Sweden; but no such coin appears in the published collection of those in the cabinet of the queen of Sweden. Sidonius Apollinaris says (Epist. 11.9) that it was the custom to paint Euclid with the fingers extended (laxatis,) as if in the act of measurement.
The history of geometry before the time of Euclid is given by Proclus, in a manner which shews that he is merely making a summary of well known or at least generally received facts. He begins with the absurd stories so often repeated, that the Aegyptians were obliged to invent geometry in order to recover the landmarks which the Nile destroyed year by year, and that the Phoenicians were equally obliged to invent arithmetic for the wants of their commerce. Thales, he goes on to say, brought this knowledge into Greece, and added many things, attempting some in a general manner (καθολικώτερον) and some in a perceptive or sensible manner (αἰσθητικώτερον). Proclus clearly refers to physical discovery in geometry, by measurement of instances. Next is mentioned Ameristus, the brother of Stesichorus the poet. Then Pythagoras changed it into the form of a liberal science (παιδείας ἐλευθέρον), took higher views of the subject, and investigated his theorems immaterially and intellectually (ἀν̈́λως καὶ νοερῶς): he also wrote on incommensurable quantities (ὰλόγων), and on the mundane figures (the five regular solids).
Barocius, whose Latin edition of Proclus has been generally followed, singularly enough translates ἄλογα βψ (quae non exlpicari possunt, and Taylor follows him with " such things as cannot be explained." It is strange that two really learned editors of Euclid's commentator should have been ignorant of one of Euclid's technical terms. Then come Anaxagoras of Clazomenae, and a little after him Oenopides of Chios; then Hippocrates of Chios, who squared the lunule, and then Theodorus of Cyrene. Hippocrates is the first writer of elements who is recorded. Plato then did much for geometry by the mathematical character of his writings; then Leodamos of Thasus, Archytas of Tarentum, and Theaetetus of Athens, gave a more scientific basis (ἐπιστημονικωτέραν σύστασιν) to various theorems; Neocleides and his disciple Leon came after the preceding, the latter of whom increased both the extent and utility of the science, in particular by finding a test (διορισμόν) of whether the thing proposed be possible [*](* We cannot well understand whether by δυνατόν προξλυς means geometrically soluble, or possible in the common sense of the word.) or impossible. Eudoxus of Cnidus, a little younger than Leon, and the companion of those about Plato [EUDOXUS], increased the number of general theorems, added three proportions to the three already existing, and in the things which concern the section (of the cone, no doubt) which was started by Plato himself, much increased their number, aud employed analyses upon them. Amyclas Heracleotes, the companion of Plato, Menaechmus, the disciple of Eudoxus and of Plato, and his brother Deinostratus, made geometry more perfect. Theudius of Magnesia
This history of Proclus has been much kept in the background, we should almost say discredited, by editors, who seem to wish it should be thought that a finished and unassailable system sprung at once from the brain of Euclid; an armed Minerva from the head of a Jupiter. But Proclus, as much a worshipper as any of them, must have had the same bias, and is therefore particularly worthy of confidence when he cites written history as to what was not done by Euclid. Make the most we can of his preliminaries, still the thirteen books of the Elements must have been a tremendous advance, probably even greater than that contained in the Principia of Newton. But still, to bring the state of our opinion of this progress down to something short of painful wonder, we are told that demonstration had been given, that something had been written on proportion, something on incommensurables, something on loci, something on solids; that analysis had been applied, that the conic sections had been thought of, that the Elements had been distinguished from the rest and written on. From what Hippocrates had done, we know that the important property of the right-angled triangle was known; we rely much more on the lunules than on the story about Pythagoras. The dispute about the famous Delian problem had arisen, and some conventional limit to the instruments of geometry must have been adopted; for on keeping within then, the difficulty of this problem depends.