foimation of a just opinion of Euclid's writings ; he was, we see, a younger contemporary of Aristotle (b. c. 384 — 322) if we suppose him to have been of mature age when Ptolemy began to patronise litera- ture ; and on this supposition it is not likely that Aristotle's writings, and his logic in particular, should have been read by Euclid in his youth, if at all. To us it seems almost certain, from the structure of Euclid's writings, that he had not read Aristotle : on this supposition, we pass over, as perfectly natural, things which, on the contrary one, would have seemed to shew great want of judgment.
Euclid, says Proclus, was younger than Plato, and older than Eratosthenes and Archimedes, the latter of whom mentions him. He was of the Platonic eect, 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 an- swer to Ptolemy, who asked if geometry could not be made easier, that there was no royal road (/UT) elvai fiaaiXiK-fju arpairov irpbs yecoixerpiav).[1] 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 Au- gust) 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 (viii. 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 origin 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 xi. 9) that it was the custom to paint Euclid with the fingers extended (laaiatis), 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 geo- metry in order to recover the landmarks which the Nile destroyed year by year, and that the Phoenicians were equally obliged to invent arith- metic 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 (KadoXiKciripov) and some in a perceptive or sensible manner (atVflrjTi/cwTepoi/). Proclus clearly refers to physical discovery in geo- metry, 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 (TratSetas eAeuflepov), took higher views of the subject, and investigated his theorems immaterially and intellectually {aoKois Koi voepoos) : he also wrote on incommensurable quantities (dXoyoov), and on the mundane figures (the five regular solids).
Barocius, whose Latin edition of Proclus has been generally followed, singularly enough trans- lates dhoya by quae non eaplicari possimt, 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 ele- ments 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 (ewKrrrjiJ.ovLKQyrepav avaraaiv) to va- rious theorems ; Neocleides and his disciple Leon came after the preceding, the latter of whom increas- ed both the extent and utility of the science, in par- ticular by finding a test (SLopiafxcv) of whether the thing proposed be possible[2] or impossible. Eudoxus of Cnidus, a little younger than Leon, and the companion of those about Plato [Eudoxus], in- creased 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 him- self, 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 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.
- ↑ We cannot well understand whether by δυνατόν Proclus means geometrically soluble, or possible in the common sense of the word.
