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of these writings differ from each other, as manuscripts will do; and when the best has been made of them which criticism will allow, the errors of humanity may be seen peeping through the manifold merits which they contain. The other Euclid was a native of Utopia, and though probably as ancient as his namesake of Alexandria, was hardly known till after the invention of printing. He wrote works on geometry which were absolutely perfect; a fact so certain, that no one editor of this Euclid ever scrupled at rejecting, adding, or altering, wherever there appeared occasion for either process. And what could be more proper ? Euclid was perfection; this sentence is not perfection, therefore this sentence is not Euclid. And though editors did sometimes differ about the true mode of turning imperfection into perfection, this proved, of course, not the fallibility of Euclid, but their own. Each of them could see it in the rest, and so it happens that many others can see it in all. After the battle of Salamis, each commander thought Themistocles only second to himself; for which they were laughed at, and Themistocles placed first: every editor of Euclid of Utopia thinks Euclid of Alexandria better than the first Euclid in the hands of any but himself; the inference is as clear. The perfect Euclid is better known in our country than the human one, according to the perfection of Robert Simson, a profoundly learned geometer of the last century. This excellent man (we have as much of right to make him complete as he had to do the same to Euclid)

dreamed three times that Theon, a contemporary of the Emperor Theodosius, had translated “ Molly put the kettle on” into Greek, and distributed the fragments through the books of the perfect Euclid, altering the context so as to make no violent appearance of transition. He awoke only to set about an edition, in which, by supernatural assistance (for human he had none), he not only threw out the vile kitchen song, but " restored to him those things which Theon, or others, had suppressed, and which had then many ages been buried in oblivion.” If any reader doubt our story, and require us to produce authority for it, we will do so as soon as he shall produce any one single manuscript, or set of manuscripts, which collectively bear out Robert Simson's restorations, but not till then.

This preface may serve as well as another, to express that we intend to treat of Euclid of Alexandria,—who is either the Homer of geometry, or else Homer is the Euclid of poetry. It has been the good fortune of both never to be surpassed ; and to complete the parallel, one Pope served Homer just as Simson served Euclid-set him forth as he ought to have written instead of as he did write. It cannot be denied that an Englishman with a head full of Pope and Simson, has very good notions, both of poetry and geometry ; but, for all that, he who would write on Homer must discard the first, while one who would describe Euclid must make light of the second, or at least of his omissions and restorations.

The little we know of the rise of geometry in Greece comes from Proclus, in his commentary on Euclid; a writer who lived, it is true, five centuries after the Christian era, but who appears to have had access to sources of historical information which are now lost. Passing over his story of the floods of the Nile obliging the Egyptians to invent geometry, we come, among several minor names, to the mention of Pythagoras, Eudoxus, and Euclid. The first, it is said, changed geometry into the form of a liberal science; and looked at its principles, and considered its theorems, immaterially and intellectually (aúlwç kai vospūç): we suppose Proclus means to say that Pythagoras was the inventor of demonstration, and that his predecessors were experimental geometers. He also wrote on incommensurables, * and on the regular solids. Eudoxus generalized many propositions, and added three proportions to the three already known, mean what it may: he also employed analysis in augmenting the properties of Plato's sections (the conic sections). Then comes Euclid, who collected the elements (ó croixeia ouvayaywv), put many propositions of Eudoxus into order, and perfected others; strengthening many previously weak demonstrations. He lived in the time of the first Ptolemy, for (Proclus has no other reason) Archimedes mentions him in his first and other books. And they report that when Ptolemy asked him, if there were no easier mode of learning geometry, he answered that there was no royal road. There is nothing else of any importance either in Proclus or elsewhere; and we must confess that the account of that writer is so pithy and cautious,

'Alóywv is the Greek word, which always meant incommensurables. But Barocius, whose Latin is highly esteemed, translated it qua explicari non possunt, and the late Thomas Tavlor, the Platonist, who translated Proclus with the love of a disciple, follows Barocius, and cites Fabricius, who thought the word should be ávalóywv, proportionals. But surely " incommensurables” makes perfect sense, and we know that some rather acute ideas of incommensurables must have preceded Euclid's theory of proportion. The words of Proclus are, την των αλόγων πραγματείαν και την των κοσμικών σχημάτων σύστασιν ανεύρε.

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that we are inclined to give its details more credit than has been usually accorded to them. If Proclus had been given to collect hearsay, he would hardly have written so briefly on the author whom he was annotating: he would, for example, at least have copied the eulogium of Pappus (A.D. 370, or thereabouts) on the suavity of Euclid's man

We conclude, then, that about the year 300 B. C. Euclid collected the scattered elements of geometry, which had been prepared by his predecessors, and organized them into the system which bears his name.

The first editor of Euclid was Theon, who lived A.D. 380, or thereabouts, and who, as he himself informs us in his commentary on the Syntaxis, had given an edition (éndoors) of Euclid; and, among other things, had added to the last proposition of the sixth book. The addition has evidently been made, and follows the “ which was to be proved,” with which Euclid always ends. This Theon had nearly run off with all the merit; for many of the manuscripts of the Elements head them as if they had been collected by him ; and one (mentioned by Savile) has in the margin a distinct statement that Theon was the

person

who arranged them. There is answer enough to this, first in the silence of the best authorities upon this point, secondly in a quotation of Alexander Aphrodiseus, a commentator on Aristotle prior to Theon, who quotes both Euclid and a particular proposition. He certainly makes the number of this proposition one earlier than it is in our present edition, which seems to indicate (if he have not quoted wrongly) that some one later than himself has made an insertion. But Euclid has been signally avenged ; for since the time of Savile, and more particularly since that of Simson, Theon has been made to bear the blame of everything which appeared to any editor short of perfection. Every schoolboy in England, who has looked into the notes to his Simson, has been taught to connect “ Theon” and “ some unskilful editor.” Every editor, from Grynæus downwards, has felt himself able to please his fancy, with an assurance to his readers that he was only undoing Theon.

It is difficult to say when or how Euclid disappeared, any more than other Greek writings: but it is certain that by the seventh century no trace of him was left in Europe. Boethius is said to have translated the first book; but in all probability this pretended translation only refers to the mere description of the four first books which that writer gave, and which continued for a long time to be the only text book on the subject. The Saracens, who are reported to have destroyed the library of Alexandria (though their subsequent acquaintance with Greek literature would make one suspect they took the books out first), found the treasures of geometry, which the northern barbarians had extirpated throughout the West, and began the task of translation, though not until they had been in possession of Alexandria nearly a century and a half. Translations of Euclid were made under the auspices of the caliphs Haroun al Raschid and al Mamon (we follow D'Herbelot in the spelling); and there was a considerable number of commentaries and abridgments. There were also, a little later, two celebrated translations, which became known in Europe. The first by Honein Ben Ishak (who died A.D. 873), which was corrected by Thabet Ben Corrah, an astronomer of unlucky fame (A.D. 950), who revived a notion of some of the Greeks, which gave a large motion of trepidation (as it has been called) to the ecliptic. The second was by Nassireddin (died A.D. 1276) an astronomer of note, and for a long time the sole authority for Asiatic longitudes and latitudes among the Westerns. The Mahometans returned Euclid into Christian hands again, in the following manner. Athelard, or Adelard, a Benedictine of Bath, who travelled all over Europe and the East for his improvement, brought back with him Euclid, and probably other translations from the Greek. His epoch is well settled, since Bale describes him, as stating himself (in one of his treatises) to have been living in the year 1130. He is mentioned as a man of very extensive knowledge, and a devoted follower of Aristotle (a writer only then beginning to be generally read). He translated Euclid into Latin ; and his version, instead of having lain manuscript to this day, as was once supposed, has been sufficiently shown to have been that which was first printed, and which kept its ground until the introduction of the Greek text. The first printed edition appeared in 1482 ; it was printed by Ratdolt of Venice, who informs us that the difficulty of printing diagrams was then overcome for the first time: and it bears the name of Campanus, but in an equivocal manner: at the end it is stated that the work of Euclid of Megara,* and the comment of Campanus, are finished. This Campanus is known to be the author of an almanac for the year 1200, though some have placed him later, and some earlier.

* A very common mistake of the time.

It was at one time supposed that the translation of Euclid was first made from that of Nassireddin, and, probably on such a supposition, that work was printed in Arabic at Rome in 1594. But a comparison of dates will show this to be impossible, be it either Campanus or Adelard who made it. Nassireddin was certainly in the prime of life when he accompanied the Tartar chief Hulaku, the grandson of Jenghis Khan, in the invasion of Persia, his native country (some said the renegade was the adviser of the expedition). This was about A.D. 1260, and his translation was most probably subsequent to his settlement as the chief astronomer of the conqueror.

It

may be, then, that the translation of Honein, or Thabet, by whichever name it is to be called, is the one which was used: there is, it is stated, a manuscript of this translation in the Bodleian Library, from which the question might be settled. M. Peyrard procured a proposition out of the printed Nassireddin to be translated, and found no very close agreement between it and the corresponding proposition of Adelard : besides, the Arabic work is a translation with a commentary, the Latin one a translation with a different commentary. There is, however, yet something to be said. According to D’Herbelot, Othman of Damascus, a writer whom he places between Thabet Ben Corrah and Nassireddin, without giving any more precise date, saw a Greek manuscript of Euclid at Rome, and found it to contain much more (forty diagrams more) than the Arabic versions to which he had been accustomed, which only contained one hundred and ninety diagrams.* He accordingly made a new translation, and as D'Herbelot does not mention Nassireddin at all as a translator, but only as a commentator, we are left to infer that in all probability Adelard obtained either the translation of Othman or some one based upon it, and that Nassireddin was but an arranger and commentator of the same.

The translation and commentary of Adelard (called that of Campanus) was printed in 1482, 1491, and again by the celebrated Lucas Paciolus, with additional comments, in 1509. As yet there was no news of any Greek text, until 1505, when Bartholomew Zamberti, of Venice, published a new

*
So
says

D'Herbelot, but there must be some numerical confusion ; for 190 diagrams would be the first six books, or thereabouts, and forty diagrams more would not serve for all the other books. The Easterns furnished Adelard with 497 propositions, being the thirteen books of Euclid, and the two additional books of Hypsicles. The Greek of all this contains only 485 propositions; and there are 18 wanting, and 30 redundant, in the Arabic.

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