persons, were represented as Bartholomeus Venetus, and Zambertus Candalla. Such things, however, scem exceptions.

Thus on the whole it appears that the present text of the Elements of Euclid depends upon about thirty-five manuscripts, few of them however containing the whole; the results of which are presented in the four editions of Basle, Oxford, Paris, and Berlin.

The particular point which most strikes a reader of Peyrard, is his preference for the Vatican manuscript, and his contempt for the editions of Basle and Oxford. We do not wish to be considered as thinking lightly of the French editor, to whom, as admirers of Euclid, we feel under singular obligations. Every scholar will admit that, by the description given of the Vatican manuscript, it was most desirable that an edition should be founded upon it, and that there ought to be a decided partisan of the said manuscript to do it. All the various readings are given in such a manner that the reader has before him the Vatican manuscript, the Oxford edition, or a compost of the twenty-two manuscripts of the Royal Library, whichever he pleases. But, while acknowledging freely the real and substantial addition which Peyrard has made to our knowledge of Euclid, we are compelled to say, that he gives no testimony of that scholarship which would make his individual opinion valuable, nor of that care which would give him a right to speak as he has done of his predecessors. We are afraid, moreover, that the animosity which his countrymen naturally felt towards England in 1812-1818, has coloured his views materially. In an ephemeral production, we should not have thought it worth while to notice such a misère: but, having before us the very careful edition of Gregory in 1703, and finding by subtraction that from 1703 to 1816, it is one hundred and thirteen years, we look forward to A.D. 1929, and picture to ourselves the smile with which any critic of that day, French or English, will, after wondering what could make Peyrard undervalue an edition so much more correct than his own, suddenly recollect that the battle of Waterloo was fought in 1815.

The French, for the last half century, have not been conspicuous cultivators of Greek; and it was notorious that of all the savans of the Bonapartean era, no one but Delambre was tolerably well versed in that language. There was hardly such a thing as a school of classical criticism in the country: and this being taken into account, the merit of Peyrard is much enhanced by the very circumstances which prevented

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his book from being what it would have been, if he had been a German. As soon as the first volume of the translation was finished and printed, it was referred by the Minister of the Interior to the two classes of the Institute, that of litera ́ture, and that of mathematics. The latter class appointed a commission, consisting of Delambre and Prony, that is of Delambre, for Prony was not, we believe, a scholar. But if Peyrard himself had dictated the report (and we shall cite something curious on this point presently) he could not have had his ideas more completely adopted. The Oxford edition is the mere copy of that of Basle, though it passes for the best of all-M. Peyrard is a judicious editor,-the misprints, inevitable in a work of this nature, are much fewer than those of the Oxford edition of Archimedes-the work fulfils all the conditions that could be exacted-and the edition is evidently superior to all the rest. On the first point, namely, that the Oxford edition is a servile copy of that of Basle, Peyrard had forgotten to give his counsel proper instructions. Had he read the preface of Gregory, he would have known better. But the information that errors are fewer than in the Oxford Archimedes, is a curious little bit of information, and contains some generalship. Why did they not say fewer than the Oxford Euclid, which would have been more to the purpose; especially since Peyrard had signalized this as the incorrect Basle edition with new faults of its own? Why, simply because the reporters themselves had detected in the seven first books-about the third part of the whole-more than two-thirds as many misprints as Peyrard's research had detected in all the fifteen books of Gregory. It was much safer, therefore, to bring in the Archimedes, which they took on Peyrard's word to be full of faults (fourmille de fautes); though they did not see what a very modified compliment they thus paid. Peyrard's faults are worse than the menda crassissima of the Oxford edition; Gregory's eye, though it sometimes passed one Greek word for another, never let slip one that was not Greek: Peyrard let go σκέσις for σχέσις; τριῶσι for ποιῶσι ; μιγέθες for μεγέθες ; πρῶτως for πρῶτος; ἐφαπάπτηται for ἐφάπτηται. And yet the sheets were first read by himself, then by M. Jannet, then by M. Patris, and then by

He read one part, at least, very incorrectly. He tells us that Gregory admits that all the writings, except the elements and data, are very evidently not Euclid's. Gregory admits no such thing; of some he properly doubts; of some he expresses no doubt.

himself again; and no one was sent to press until every error had been corrected, or, as the printers say, a perfectly clean revise was always sent back. Besides this, M. Nicolopoulo, of Smyrna, read a large number of the proofs. All this reading rather surprised us; and it also puzzled us to understand how Delambre and Prony came to examine so minutely as to detect a misplaced accent, or a wrong aspirate. Did Peyrard furnish them with a list of his own, to make their report look more minute? We should not breathe such a suspicion, if it were not for a curious circumstance which we will now explain.

Peyrard sometimes forgets that he is editing Euclid of Alexandria, and shows some disposition to restore Euclid of Utopia. In all manuscripts, the seventh of the first book has only one case, that in which the vertex of one triangle falls inside the other not being considered. Of course all commentators have supplied the deficiency; Grynous and Gregory let Euclid stand. The case is plain enough; aliquando bonus dormitat geometria Homerus, and Euclid took the case of the vertex of one triangle falling within the other as obviously impossible. Peyrard thought that he could make one demonstration do for both cases, by drawing the second figure, and adding a few words: this he informs us in his preface he has done, and Delambre and Prony assure us in their report that he has drawn the new figure, and added a line, which they tell us is entre deux crochets. Looking to the various readings at the end, in which Peyrard puts his own text in one column, and that of the Oxford and the manuscripts in two others, we find that, at the reference 3, the words καὶ αἱ ΒΓ, ΒΔ ἐκβεβλήσθωσαν ἐπ ̓ εὐθείας ἐπ' [sic] τὰ ΕΖ are part of the text; on which Peyrard remarks, Desunt in omnibus codicibus et in omnibus editionibus. Well then, we turn to the text; we find no such words in the whole proposition, we find no second figure added, and, to three words or so, everything as in the Oxford! Grant for a moment that the reporters looked at the various readings instead of the text, as would have been their best plan in the first instance, where did they find the crochets? They were evidently examining a printed work, for they detected misprints; where were the crochets? Perhaps such things would not remain in the text, but flew off, by the laws of attraction, into the heads of the examiners, carrying with them the intercepted words. And if they got their information from the various readings, how came they to overlook in'ra for éni rà, they

who made eighteen corrections, by their own account, in these very various readings. Or was this the state of the case; did Peyrard furnish them with the materials of the report, and a list of errata to look business-like, telling them what he meant to do with the seventh proposition, and did do in the list of various readings, but forgot to do in the text? We regret very much being forced upon this supposition, but we ask any candid reader how it is to be avoided?

The class of literature evaded the question of the minister, in a short letter from their secretary, in which they administer what may be called a rap on the knuckles to the worthy, but too self-sufficient, editor. After referring to the report of the other class, with which the subject had most to do, they observe that the text seems (lui a paru) more correct in the new edition--but that the Basle edition (no mention of the Oxford one, February 26, 1814) though containing some misprints, not so many as is commonly thought, and easily corrected, will always be precious to the lovers of Greek literature that the new edition was carefully done, but that some errors had crept in, particularly towards the end of the volume.

The Berlin editor, E. F. August, has insinuated his opinion in the following manner. After describing the Oxford preface, he adds, "Atque revera tanta cura hæc editio instituta est, ut digna esset, qua per totum seculum matheseos studiosi nec Græci sermonis inperiti uterentur." After a similar description of Peyrard's preface, with a preliminary compliment to his labour and industry, he says-not one word. In the fifth proposition of the sixth book (the only one which he thus treats) August has pointed out five misprints, no one of which is in the Oxford edition. We ourselves sat down with the determination to read till we came to an erratum not noticed in the list: we took the first proposition of the fifth book, and at the eighteenth word of the demonstration we found our mark; πολλαπλάσιον for πολλαπλάσια, the Latin is multiplices. We feel then, from all these things, that Peyrard's Euclid is by far the most incorrectly printed edition which exists. For ordinary mathematical students, we should decidedly recommend the Berlin edition, which is more easily obtained than the Oxford, of which it possesses the merit, without the inaccuracies of the Paris edition. It also gives the principal points of the Vatican manuscript. At the same time, the critical scholar will feel that he is not in possession of Euclid unless he have by him the edition of Peyrard, for the sake of the manuscript just mentioned, the

twenty-two others, and their comparison with the Oxford edition. And though Peyrard was not what he imagined himself to be, yet from that to absolute insignificance is a longum intervallum, of which a little indulgence, no more than due to his intentions and industry, may put him at the point of bisection.

From the Latin and the Greek we may pass to the English. The first English dream of geometry was the Pathway to Knowledge, by Robert Recorde, published in 1551, containing no system of demonstration, but "one book of conclusions geometricall," and "one book of theorems geometricall." The first contains the problems of the first four books of Euclid constructed; the second the theorems in the same books described without demonstration. This is done after the example of Rheticus, and "that wittie clarke" Boethius. Euclid is mentioned once, in a manner which shows that Recorde considers all demonstration to be the work of "Theon and others that write on Euclide:" the old story again. This work of Recorde is as much an edition of four books of Euclid as some others that went by that name in his day. But nothing that can properly be called by the name of Euclid was published until 1570, in which year Sir Henry Billingsley (who Dee tells us was the translator) published an edition containing the whole of the fifteen books, with all manner of commentaries, and an additional book on solids by Flussas; together with a long preface and notes by John Dee. Had it not been for Dee himself, in the catalogue which he subsequently published (in his epistle to the archbishop of Canterbury), it would never have been known that the worshipful Sir Henry Billingsley was the translator: and considering that the plan, preface, and notes are Dee's, and that the worshipful knight is altogether unknown, it must be presumed that he worked under Dee's advice and direction. The name of Billingsley does not occur either in the first edition or the second and last (1671); and we have always had a firm persuasion, that the knight was either Dee's pupil, working under his directions in the mechanical translation, or his patron, who had bought the credit of the edition. shall not speak here of Scarburgh, Barrow, Cotes, Simson, Horsley, &c., except in general comments where occasion arises we shall merely add, on this branch of the subject, that the Clarendon press, besides the best Greek version, has also produced the most English Euclid, in the most Euclidean English; we speak of the translation of the thirteen books,

We