C and made tacit assumptions of a kind which are rarely met with in his writings. The fourth book treats of regular figures. Euclid's original postulates of construction give him, by this time, the power of drawing them of 3, 4, 5, and 15 sides, or of double, quadruple, &c., any of these numbers, as 6, 12, 24, &c., 8, 16, &c. &c. count of it in the Penny Cyclopaedia article, "Irrational Quantities." Euclid has evidently in his mind the intention of classifying incommensurable quantities: perhaps the circumference of the circle, which we know had been an object of inquiry, was suspected of being incommensurable with its diameter; and hopes were perhaps entertained that a searching attempt to arrange the incommen surables which ordinary geometry presents might enable the geometer to say finally to which of them, if any, the circle belongs. However this may be, Euclid investigates, by isolated methods, and in a manner which, unless he had a concealed algebra, is more astonishing to us than anything in the Elements, every possible variety of lines which can be represented by √(√a±√√b), a and b repre The fifth book is on the theory of proportion. It refers to all kinds of magnitude, and is wholly independent of those which precede. The existence of incommensurable quantities obliges him to introduce a definition of proportion which seems at first not only difficult, but uncouth and inelegant; those who have examined other definitions know that all which are not defective are but various readings of that of Euclid. The reasons for this difficult definition are not alluded to, ac-senting two commensurable lines. He divides lines cording to his custom; few students therefore understand the fifth book at first, and many teachers decidedly object to make it a part of the course. A distinction should be drawn between Euclid's definition and his manner of applying it. Every one who understands it must see that it is an application of arithmetic, and that the defective and unwieldy forms of arithmetical expression which never were banished from Greek science, need not be the necessary accompaniments of the modern use of the fifth book. For ourselves, we are satisfied that the only rigorous road to proportion is either through the fifth book, or else through something much more difficult than the fifth book need be. The sixth book applies the theory of proportion, and adds to the first four books the propositions which, for want of it, they could not contain. It discusses the theory of figures of the same form, technically called similar. To give an idea of the advance which it makes, we may state that the first book has for its highest point of constructive power the formation of a rectangle upon a given base, equal to a given rectilinear figure; that the second book enables us to turn this rectangle into a square; but the sixth book empowers us to make a figure of any given rectilinear shape equal to a rectilinear figure of given size, or briefly, to econstruct a figure of the form of one given figure, and of the size of another. It also supplies the geometrical form of the solution of a quadratic equation. The seventh, eighth, and ninth books cannot have their subjects usefully separated. They treat of arithmetic, that is, of the fundamental properties a of numbers, on which the rules of arithmetic must be founded. But Euclid goes further than is necessary merely to construct a system of computation, about which the Greeks had little anxiety. He is able to succeed in shewing that numbers which are prime to one another are the least in their ratio, to prove that the number of primes is infinite, and to point out the rule for constructing what are called perfect numbers. When the modern systems began to prevail, these books of Euclid were abandoned to the antiquary: our elementary books of arithmetic, which till lately were all, and now are mostly, systems of mechanical rules, tell us what would have become of geometry if the earlier books had shared the same fate. The tenth book is the development of all the power of the preceding ones, geometrical and arith metical. It is one of the most curious of the Greek speculations: the reader will find a synoptical ac which can be represented by this formula into 25 species, and he succeeds in detecting every possible species. He shews that every individual of every species is incommensurable with all the individuals of every other species; and also that no line of any species can belong to that species in two different ways, or for two different sets of values of a and l. He shews how to form other classes of incommensurables, in number how many soever, no one of which can contain an individual line which is commensurable with an individual of any other class; and he demonstrates the incommensurability of a square and its diagonal. This book has a completeness which none of the others (not even the fifth) can boast of: and we could almost suspect that Euclid, having arranged his materials in his own mind, and having completely elaborated the tenth book, wrote the preceding books after it, and did not live to revise them thoroughly. The eleventh and twelfth books contain the elements of solid geometry, as to prisms, pyramids, &c. The duplicate ratio of the diameters is shewn to be that of two circles, the triplicate ratio that of two spheres. Instances occur of the method of exhaustions, as it has been called, which in the hands of Archimedes became an instrument of discovery, producing results which are now usually referred to the differential calculus: while in those of Euclid it was only the mode of proving propositions which must have been seen and believed before they were proved. The method of these books is clear and elegant, with some striking imperfections, which have caused many to abandon them, even among those who allow no substitute for the first six books. The thirteenth, fourteenth, and fifteenth books are on the five regular solids: and even had they all been written by Euclid (the last two are attributed to Hypsicles), they would but ill bear out the assertion of Proclus, that the regular solids were the objects with a view to which the Elements were written: unless indeed we are to suppose that Euclid died before he could complete his intended structure. Proclus was an enthusiastic Platonist: Euclid was of that school; and the former accordingly attributes to the latter a particular regard for what were sometimes called the Platonic bodies. But we think that the author himself of the Elements could hardly have considered them as a mere introduction to a favourite speculation: if he were so blind, we have every reason to suppose that his own contemporaries could have set him right. From various indications, it can be coilected that the fame of the Elements was almost coeval with their publication; and by the time of Marinus we learn from that writer that Euclid epitome of the whole. was called κύριος στοιχειωτής. The Data of Euclid should be mentioned in connection with the Elements. This is a book containing a hundred propositions of a peculiar and limited intent. Some writers have professed to see in it a key to the geometrical analysis of the ancients, in which they have greatly the advantage of us. When there is a problem to solve, it is undoubtediy advantageous to have a rapid perception of the steps which will reach the result, if they can be successively made. Given A, B, and C, to find D: one person may be completely at a loss how to proceed; another may see almost intuitively that when A, B, and C are given, E can be found; from which it may be that the first person, had he perceived it, would have immediately found D. The formation of data consequential, as our ancestors would perhaps have called them, things not absolutely given, but the gift of which is implied in, and necessarily follows from, that which is given, is the object of the hundred propositions above mentioned. Thus, when a straight line of given length is intercepted between two given parallels, one of these propositions shews that the angle it makes with the parallels is given in magnitude. There is not much more in this book of Data than an intelligent student picks up from the Elements themselves; on which account we cannot consider it as a great step in geometrical analysis. The operations of thought which it requires are indispensable, but they are contained elsewhere. At the same time we cannot deny that the Data might have fixed in the mind of a Greek, with greater strength than the Elements themselves, notions upon consequential data which the moderns acquire from the application of arithmetic and algebra: perhaps it was the perception of this which dictated the opinion about the value of the book of Data in analysis. While on this subject, it may be useful to remind the reader how difficult it is to judge of the character of Euclid's writings, as far as his own merits are concerned, ignorant as we are of the precise purpose with which any one was written. For instance was he merely shewing his contemporaries that a connected system of demonstration might be made without taking more than a certain number of postulates out of a collection, the necessity of each of which had been advocated by some and denied by others? We then understand why he placed his six postulates in the prominent position which they occupy, and we can find no fault with his tacit admission of many others, the necessity of which had perhaps never been questioned. But if we are to consider him as meaning to be what his commentators have taken him to be, a model of the most scrupulous formal rigour, we can then deny that he has altogether succeeded, though we may admit that he has made the nearest approach. The literary history of the writings of Euclid would contain that of the rise and progress of geometry in every Christian and Mohammedan nation our notice, therefore, must be but slight, and various points of it will be confirmed by the bibliographical account which will follow. In Greece, including Asia Minor, Alexandria, and the Italian colonies, the Elements soon became the universal study of geometers. Commentators were not wanting; Proclus mentions Heron and Pappus, and Aeneas of Hierapolis, who made an "heon the younger d Alexandria) lived a little wefore Proclus (who died about A. D. 485). The latter has made his feeble commentary on the first book valuable by its his torical information, and was something of a luminary in ages more dark than his own. But Theon was a light of another sort, and his name ha played a conspicuous and singular part in the his tory of Euclid's writings. He gave a new edition of Euclid, with some slight additions and alterations he tells us so himself, and uses the word ěkdoσis, as applied to his own edition, in his commentary on Ptolemy. He also informs us that the part which relates to the sectors in the last propo sition of the sixth book is his own addition: and it is found in all the manuscripts following the oneр del deîşa with which Euclid always ends. Alexander Aphrodisiensis (Comment. in priora Analyt. Aristot.) mentions as the fourth of the tenth book that which is the fifth in all manuscripts. Again, in several manuscripts the whole work is headed as ek Tŵv Oéwvos σuvovσiv. We shall presently see to what this led but now we must remark that Proclus does not mention Theon at all; from which, since both were Platonists residing at Alexandria, and Proclus had probably seen Theon in his younger days, we must either infer some quarrel between the two, or, which is perhaps more likely, presume that Theon's alterations were very slight. The two books of Geometry left by BOETHIUS contain nothing but enunciations and diagrams from the first four books of Euclid. The assertion of Boethius that Euclid only arranged, and that the discovery and demonstration were the work o. others, probably contributed to the notions abou Theon presently described. Until the restoration of the Elements by translation from the Arabic, this work of Boethius was the only European treatise on geometry, as far as is known. But to F t ཝོ The Arabic translations of Euclid began to be made under the caliphs Haroun al Raschid and Al Mamun; by their time, the very name of Euclid had almost disappeared from the West. nearly one hundred and fifty years followed the capture of Egypt by the Mohammedans before the latter began to profit by the knowledge of the Greeks. After this time, the works of the geometers were sedulously translated, and a great impulse was given by them. Commentaries, and E even original writings, followed; but so few of these are known among us, that it is only from the Saracen writings on astronomy (a science which always carries its own history along with it) that we can form a good idea of the very striking pro- N gress which the Mohammedans made under their Greek teachers. Some writers speak slightingly of this progress, the results of which they are too apt to compare with those of our own time they ought rather to place the Saracens by the side of their own Gothic ancestors, and, making some allowance for the more advantageous circumstances under which the first started, they should view the second systematically dispersing the remains of Greek civilization, while the first were concentrat ing the geometry of Alexandria, the arithmetic and algebra of India, and the astronomy of both, to form a nucleus for the present state of science. The Elements of Euclid were restored to Europe by translation from the Arabic. In connection with this restoration four Eastern editors may be h DA the E mentioned. Honein hen Ishak (died A. D. 873) | published an edition hich was afterwards corrected by Thabet ben Corrah, a well-known astronomer. After him, according to D'Herbelot, Othman of Damascus (of uncertain date, but before the thirteenth century) saw at Rome a Greek manuscript containing many more propositions than 2 he had been accustomed to find: he had been used to 190 diagrams, and the manuscript contained 40 more. If these numbers be correct, Honein could El only have had the first six books; and the new translation which Othman immediately made must have been afterwards augmented. A little after A. D. 1260, the astronomer Nasireddin gave another edition, which is now accessible, having been printed in Arabic at Rome in 1594. It is tolerably complete, but yet it is not the edition from which the earliest European translation was made, as Peyrard found by comparing the same proposition in the two. Σ The first European who found Euclid in Arabic, and translated the Elements into Latin, was Athelard or Adelard, of Bath, who was certainly alive in 1130. (See "Adelard," in the Biogr. Dict. of the Soc. D. U. K.) This writer probably obtained his original in Spain: and his translation is the one which became current in Europe, and is the first which was printed, though under the name of Campanus. Till very lately, Campanus was supposed to have been the translator. Tiraboschi takes it to have been Adelard, as a matter of course; Libri pronounces the same opinion after inquiry; and Scheibel states that in his copy of Campanus the authorship of Adelard was asserted in a handwriting as old as the work itself. (A. D. 1482.) Some of the manuscripts which bear the name of Adelard have that of Campanus attached to the commentary. There are several of these manuscripts in existence; and a comparison of any one of them with the printed book which was attributed to Campanus would settle the question. The seed thus brought by Adelard into Europe was sown with good effect. In the next century Roger Bacon quotes Euclid, and when he cites Boe thius, it is not for his geometry. Up to the time of printing, there was at least as much dispersion of the Elements as of any other book: after this period, Euclid was, as we shall see, an early and frequent product of the press. Where science flourished, Euclid was found; and wherever he was found, science flourished more or less according as more or less attention was paid to his Elements. As to writing another work on geometry, the middle ages would as soon have thought of composing another New Testament: not only did Euclid preserve his right to the title of κύριος στοιχειωτής down to the end of the seventeenth century, and that in so absolute a manner, that then, as sometimes now, the young beginner imagined the name of the man to be a synonyme for the science; but his order of demonstration was thought to be necessary, and founded in the nature of our minds. Tartaglia, whose bias we might suppose would have been shaken by his knowledge of Indian arithmetic and algebra, calls Euclid solo introduttore delle scientie mathematice: and algebra was not at that time considered as entitled to the name of a science by those who had been formed on the Greek model; arte maggiore" was its designation. The story About Pascal's discovery of geometry in his boyhod (A. D. 1635) contains the statement that he had got as far as the 32nd proposition of the first book" before he was detected, the exaggerators (for much exaggerated this very circumstance shews the truth must have been) not having the slightest idea that a new invented system could proceed in any other order than that of Euclid. The vernacular translations of the Elements dato from the middle of the sixteenth century, from which time the history of mathematical science divides itself into that of the several countries where it flourished. By slow steps, the continent of Europe has almost entirely abandoned the ancient Elements, and substituted systems of geometry more in accordance with the tastes which algebra has introduced: but in England, down to the present time, Euclid has held his ground. There is not in our country any system of geometry twenty years old, which has pretensions to anything like currency, but it is either Euclid, or something so fashioned upon Euclid that the resemblance is as close as that of some of his professed editors. We cannot here go into the reasons of our opinion; but we have no doubt that the love of accuracy in mathematical reasoning has declined wherever Euclid has been abandoned. We are not so much of the old opinion as to say that this must necessarily have happened; but, feeling quite sure that all the alterations have had their origin in the desire for more facility than could be obtained by rigorous deduction from postulates both true and evident, we see what has happened, and why, without being at all inclined to dispute that a disposition to depart from the letter, carrying off the spirit, would have been attended with very different results. Of the two best foreign books of geometry which we know, and which are not Euclidean, one demands a right to "imagine" a thing which the writer himself knew perfectly well was not true; and the other is content to shew that the theorems are so nearly true that their error, if any, is imperceptible to the senses. It must be admitted that both these absurdities are committed to avoid the fifth book, and that English teachers have, of late years, beer much inclined to do something of the same sort, less openly. But here, at least, writers have left it to teachers to shirk* truth, if they like, without being wilful accomplices before the fact. In an English translation of one of the preceding works, the means of correcting the error were given: and the original work of most note, not Euclidean, which has appeared of late years, does not attempt to get over the difficulty by any false assumption. At the time of the invention of printing, two errors were current with respect to Euclid personally. The first was that he was Euclid of Megara, a totally different person. This confusion has been said to take its rise from a passage in Plutarch, but we cannot find the reference. Boëthius perpetuated it. The second was that Theon was the demonstrator of all the propositions, and that Euclid only left the definitions, postulates, &c., with the * We must not be understood as objecting to the teacher's right to make his pupil assume anything he likes, provided only that the latter knows what he is about. Our contemptuous expression (for such we mean it to be) is directed against those who substitute assumption for demonstration, or the particular for the general, and leave the student in ignorance of what has been done. enunciations in their present order. So completely was this notion received, that editions of Euclid, so called, contained only enunciations; all that contained demonstrations were said to be Euclid with the commentary of Theon, Campanus, Zambertus, or some other. Also, when the enunciations were given in Greek and Latin, and the demonstrations in Latin only, this was said to constitute an edition of Euclid in the original Greek, which has occasioned a host of bibliographical errors. We have already seen that Theon did edit Euclid, and that manuscripts have described this editorship in a manner calculated to lead to the mistake: but Proclus, who not only describes Euclid as rd μαλακώτερον δεικνύμενα τοῖς ἔμπροσθεν εἰς ἀνε λέγκτους ἀποδείξεις ἀναγαγών, and comments on the very demonstrations which we now have, as on those of Euclid, is an unanswerable witness; the order of the propositions themselves, connected as it is with the mode of demonstration, is another; and finally, Theon himself, in stating, as before noted, that a particular part of a certain demonstra-| tion is his own, states as distinctly that the rest is not. Sir Henry Savile (the founder of the Savilian chairs at Oxford), in the lectures on Euclid with which he opened his own chair of geometry before he resigned it to Briggs (who is said to have taken up the course where his founder left off, at book i. prop. 9), notes that much discussion had taken place on the subject, and gives three opinions. The first, that of quidam stulti et perridiculi, above discussed: the second, that of Peter Ramus, who held the whole to be absolutely due to Theon, propositions as well as demonstrations, false, quis negat? the third, that of Buteo of Dauphiny, a geometer of merit, who attributes the whole to Euclid, quae opinio aut vera est, aut veritati certe proxima. It is not useless to remind the classical student of these things: the middle ages may be called the "ages of faith" in their views of criticism. Whatever was written was received without examination; and the endorsement of an obscure scholiast, which was perhaps the mere whim of a transcriber, was allowed to rank with the clearest assertions of the commentators and scholars who had before them more works, now lost, written by the contemporaries of the author in question, than there were letters in the stupid sentence which was allowed to overbalance their testimony. From such practices we are now, it may well be hoped, finally delivered but the time is not yet come when refutation of "the scholiast " may be safely abandoned. : All the works that have been attributed to Euclid are as follows: 1. ETOXEîα, the Elements, in 13 books, with a 14th and 15th added by HYPSICLES. 2. Aedouéva, the Data, which has a preface by Marinus of Naples. 3. Εἰσαγωγὴ 'Αρμονική, a Treatise on Music; and 4. Kaтaтoμr Kavóvos, the Division of the Scale: one of these works, most likely the former, must be rejected. Proclus says that Euclid wrote Karà μουσικὴν στοιχειώσεις. 5. Pavóueva, the Appearances (of the heavens). Pappus mentions them. 6. 'OTTIкá, on Optics; and 7. Kатожтρikά, on Catoptrics. Proclus mentions both. * Praelectiones tresdecin in principium elementorum Euclidis; Oxonii habitae M.DC.XX. Oxoniae, 1621. The preceding works are in existence; the for lowing are either lost, or do not remain in the original Greek. 8. Περὶ Διαιρέσεων βιβλίον, On Divisions. Proclus (l. c.) There is a translation from the Arabic, with the name of Mohammed of Bagdad attached, which has been suspected of being a translation of the book of Euclid: of this we shall see more. 9. Kwvikŵv Biểλla d, Four books on Conic Seo. tions. Pappus (lib. vii. praef.) affirms that Euclid wrote four books on conics, which Apollonius enlarged, adding four others. Archimedes refers to the elements of conic sections in a manner which shews that he could not be mentioning the new work of his contemporary Apollonius (which it is most likely he never saw). Euclid may possibly have written on conic sections; but it is impossible that the first four books of APOLLONIUS (see his life) can have been those of Euclid. 10. Порioμáтшv Bibλlay, Three books of Porisms. These are mentioned by Proclus and by Pappus (l. c.), the latter of whom gives a description which is so corrupt as to be unintelligible. 11. Τόπων Ἐπιπέδων βιβλία β', Two books on Plane Loci. Pappus mentions these, but not Eutocius, as Fabricius affirms. (Comment. in Apoll lib. i. lemm.) 12. Τόπων πρὸς Επιφάνειαν βιβλία β', mentioned by Pappus. What these TÓTоι πρÓS 'ETIpávelav, or Loci ad Superficiem, were, neither Pappus nor Eutocius inform us; the latter says they derive their name from their own idións, which there is no reason to doubt. We suspect that the books and the meaning of the title were as much lost in the time of Eutocius as now. 13. Περὶ Ψευδαρίων, On Fallacies. On this work Proclus says, "He gave methods of clear judgment (diopatiks pрovnσews) the possession of which enables us to exercise those who are beginning geometry in the detection of false reasonings, and to keep them free from delusion. And the book which gives us this preparation is called Yevdapiwv, in which he enumerates the species of fallacies, and exercises the mental faculty on each species by all manner of theorems. He places truth side by side with falsehood, and connects the confutation of falsehood with experience." thus appears that Euclid did not intend his Elements to be studied without any preparation, but that he had himself prepared a treatise on fallacious reasoning, to precede, or at least to accompany, the Elements. The loss of this book is much to be regretted, particularly on account of the explanations of the course adopted in the Elements which it cannot but have contained. It We now proceed to some bibliographical account of the writings of Euclid. In every case in which we do not mention the source of information, it is to be presumed that we take it from the edition itself. The first, or editio princeps, of the Elements is that printed by Erhard Ratdolt at Venice in 1482, black letter, folio. It is the Latin of the fifteen books of the Elements, from Adelard, with the commentary of Campanus following the demonstrations. It has no title, but, after a short introduction by the printer, opens thus: "Preclarissimus liber elementorum Euclidis perspicacissimi : in artem geometrie incipit qua foelicissime: Punctus est cujus ps nñ est," &c. Ratdolt states in the introduction that the difficulty of printing diagrams had prevented books of geometry from going through | principis opera, &c. At the end, Venetiis impressum the press, but that he had so completely overcome per... Paganinum de Paganinis... anno...MDVIIII... it, by great pains, that "qua facilitate litterarum Paciolus adopts the Latin of Adelard, and occaelementa imprimuntur, ea etiam geometrice figure sionally quotes the comment of Campanus, introconficerentur." These diagrams are printed on the ducing his own additional comments with the head margin, and though at first sight they seem to be Castigator." He opens the fifth book with the woodcuts, yet a closer inspection makes it probable account of a lecture which he gave on that book in that they are produced from metal lines. The a church at Venice, August 11, 1508, giving the number of propositions in Euclid (15 books) is 485, names of those present, and some subsequent lauof which 18 are wanting here, and 30 appear which datory correspondence. This edition is less loaded are not in Euclid; so that there are 497 proposi- with comment than either of those which precede. tions. The preface to the 14th book, by which it It is extremely scarce, and is beautifully printed: rs made almost certain that Euclid did not write it the letter is a curious intermediate step between (for Euclid's books have no prefaces) is omitted. the old thick black letter and that of the Roman Ìts Arabic origin is visible in the words helmuaym type, and makes the derivation of the latter from and helmuariphe, which are used for a rhombus and the former very clear. a trapezium. This edition is not very scarce in England; we have seen at least four copies for sale in the last ten years. The second edition bears "Vincentiae 1491," Roman letter, folio, and was printed "per magistrum Leonardum de Basilea et Gulielmum de Papia socios." It is entirely a reprint, with the introduction omitted (unless indeed it be torn out in the only copy we ever saw), and is but a poor specimen, both as to letter-press and diagrams, when compared with the first edition, than which it is very much scarcer. Both these editions call Euclid Meyarensis. The fifth edition (Elements, Latin, Roman letter, folio), edited by Jacobus Faber, and printed by Henry Stephens at Paris in 1516, has the title Contenta followed by heads of the contents. There are the fifteen books of Euclid, by which are meant the Enunciations (see the preceding remarks on this subject); the Comment of Campanus, meaning the demonstrations in Adelard's Latin; the Comment of Theon as given by Zambertus, meaning the demonstration in the Latin of Zambertus; and the Comment of Hypsicles as given by Zambertus upon the last two books, meaning the demonstrations of those two books. This edition is fairly printed, and is moderately scarce. it we date the time when a list of enunciations merely was universally called the complete work of Euclid. From With these editions the ancient series, as we may call it, terminates, meaning the complete Latin editions which preceded the publication of the Greek text. Thus we see five folio editions of the Elements produced in thirty-four years. The first Greek text was published by Simon ing, ἐκ τῶν Θέωνος συνουσιῶν (the title-page has this statement), the fifteen books of the Elements, and the commentary of Proclus added at the end, so far as it remains; all Greek, without Latin. On Grynoeus and his reverendt care of manuscripts, see Anthony Wood. (Athen. Oxon. in verb.) The Oxford editor is studiously silent about this Basle edition, which, though not obtained from many manuscripts, is even now of some value, and was for a century and three-quarters the only printed Greek text of all the books. The third edition (also Latin, Roman letter, folio,) containing the Elements, the Phaenomena, the two Optics (under the names of Specularia and Perspectiva), and the Data with the preface of Marinus, being the editio princeps of all but the Elements, has the title Euclidis Megarensis philosophici Platonici, mathematicarum disciplinaru janitoris: habent in hoc volumine quicūque ad mathematica substantiā aspirāt: elemētorum libros, &c. &c. Zamberto Veneto Interprete. At the end is Impressum Venetis, &c. in edibus Joannis Ta-Gryne, or Grynoeus, Basle, 1533, folio:* containcuini, &c., M. D.V.VIII. Klendas Novēbris that is, 1505, often read 1508 by an obvious mistake. Zambertus has given a long preface and a life of Euclid: he professes to have translated from a Greek text, and this a very little inspection will shew he must have done; but he does not give any information upon his manuscripts. He states that the propositions have the exposition of Theon or Hypsicles, by which he probably means that Theon or Hypsicles gave the demonstrations. The preceding editors, whatever their opinions may have been, do not expressly state Theon or any other to have been the author of the demonstrations: but by 1505 the Greek manuscripts which bear the name of Theon had probably come to light. For Zambertus Fabricius cites Goetz. mem. bibl. Dresd. ii. p. 213: his edition is beautifully printed, and is rare. He exposes the translations from the Arabic with unceasing severity. Fabricius mentions (from Scheibel) two small works, the four books of the Elements by Ambr. Jocher, 1506, and something called "Geometria Euclidis," which accompanies an edition of Sacrobosco, Paris, H. Stephens, 1507. Of these we know nothing. The fourth edition (Latin, black letter, folio, *509), containing the Elements only, is the work of the celebrated Lucas Paciolus (de Burgo Sancti Sepulchri), better known as Lucas di Borgo, the first who printed a work on algebra. The title is Euclidis Megarensis philosophi acutis simi mathematicorumque omnium sine controversia With regard to Greek texts, the student must be on his guard against bibliographers. For instance, Harless gives, from good catalogues, E * Fabricius sets down an edition of 1530, by the same editor: this is a misprint. +"Sure I am, that while he continued there (i. e. at Oxford), he visited and studied in most of the libraries, searched after rare books of the Greek tongue, particularly after some of the books of commentaries of Proclus Diadoch. Lycius, and having found several, and the owners to be careless of them, he took some away, and conveyed them with him beyond the seas, as in an epistle by him written to John the son of Thos. More, he confesseth." Wood. Schweiger, in his Handbuch (Leipsig, 1830). gives this same edition as a Greek one, and makes the same mistake with regard to those of Dasypodius. Scheubel, &c. We have no doubt that the |