Proper name: EUCLEIDES (Εὐκλείδης) of Alexandrria. 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. ii. 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 ​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 ​generalized many particular propositions. Cyzici- uus of Athens was his contemporary ; they took different sides on many common inquiries. Hermo- timus of Colophon added to what had been done by Eudoxus and Theaetetus, discovered elementary propositions, and wrote something on loci. Philip (o Meraios, others read Med/mTos, Barocius reads Mendaeus), the follower of Plato, made many ma- thematical inquiries connected with his master's philosophy. Those who write on the history of geo.uetry bring the completion of this science thus far. Here Proclus expressly refers to written his- tory, and in another place he particularly mentions the history of Eudemus the Peripatetic. 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 demon- stration had been given, that something had been written on proportion, something on incoramensu- rables, something on loci, something on solids ; that analysis had been applied, that the conic sec- tions 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 geo- metry must have been adopted ; for on keeping within them, the difficulty of this problem depends. It will be convenient to speak separately of the Eletnents of Euc/id, as to their contents; and after- wards to mention them bibliographically, among the other writings. The book which passes under this name, as given by Robert Simson, unexcep- tionable as Elements of Geometry^ is not calculated to give the scholar a proper idea of the elements of Euclid ; but it is admirably adapted to confuse, in the mind of the young student, all those notions of sound criticism which his other instructors are endeavouring to instil. The idea that Euclid must be perfect had got possession of the geometrical world ; accordingly each editor, when he made what he took to be an alteration for the better, assumed that he was restoring, not amending, the original. If the books of Livy were to be re- written upon the basis of Niebuhr, and the result declared to be the real text, then Livy Avould no more than share the fate of Euclid ; the only dif- ference being, that the former would undergo a larger quantity of alteration than editors have seen fit to inflict upon the latter. This is no caricature ; e.g., Euclid, says Robert Simson, gave, without doubt, a definition of compound ratio at the be- ginning of the fifth book, and accordingly he there inserts, not merely a definition, but, he assures us, the very one which Euclid gave. Not a single manu- script supports him : how, then, did he know ? He saw that there ought to have been such a defi- nition, and he concluded that, therefore, there had been one. Now we by no means uphold Euclid as an all-sufficient guide to geometry, though we feel that it is to himself that we owe the power of amending his writings ; and we hope we may pro- test against the assumption that he could not have erred, whether by omission or commission. Some of the characteristics of the Elements are briefly as follows:— First. There is a total absence of distinction between the various ways in which we know the meaning of terms : certainty, and nothing more, is the thing sought. The definition of straightness, an idea which it is impossible to put into simpler words, and which is therefore described by a more difficult circumlocution, comes under the same heading as the explanation of the word ' parallel.' Hence disputes about the correctness or incorrect- ness of many of the definitions. Secondly. There is no distinction between pro- positions which require demonstration, and those which a logician would see to be nothing but different modes of stating a preceding proposition. When Euclid has proved that everything which is not A is not B, he does not hold himself entitled to infer that every B is A, though the two propo- sitions are identically the same. Thus, having shewn that every point of a circle which is not the centre is not one from which three equal straight lines can be drawn, he cannot infer that any point from which three equal straight lines are drawn is the centre, but has need of a new demonstration. Thus, long before he wants to use book i. prop. 6, he has proved it again, and independently. Thirdly. He has not the smallest notion of admitting any generalized use of a word, or of part- ing with any ordinary notion attached to it. Setting out with the conception of an angle rather as the sharp corner made by the meeting of two lines than as the magnitude which he afterwards shews how to measure, he never gets rid of that corner, never admits two right angles to make one angle, and still less is able to arrive at the idea of an angle greater than two right angles. And when, in the last proposition of the sixth book, his definition of proportion absolutely requires that he should reason on angles of even more than four right angles, he takes no notice of this neces- sity, and no one can tell whether it was an over- sight, whether Euclid thought the extension one which the student could make for himself, or whether (which has sometimes struck us as not unlikely) the elements were his last work, and he did not live to revise them. In one solitary case, Euclid seems to have made an omission implying that he recognized that natural extension of language by which uriity is considered as a numl)er, and Simson has thought it necessary to supply the omission (see his book v. prop. A), and has shewn himself more Euclid than Euclid upon the point of all others in which Euclid's philosophy is defective. Fourthly. There is none of that attention to the forms of accuracy with which translators have endeavoured to invest the Elements, thereby giv- ing them that appearance which has made many teachers think it meritorious to insist upon their pupils remembering the very words of Simson. Theorems are found among the definitions : assump​assumptions are made which are not formally set down among the postulates. Things which really ought to have been pioved are souietimes passed over, and whether this is by mistake, or by intention of supposing them self-evident, cannot now be known : for Euclid never refers to previous propositions by name or number, but only by simple re-assertion without reference ; except that occasionally, and chiefly when a negative proposition is referred to, such words as 'it has been demonstrated' are employed, without further specification. Fifthly. Euclid never condescends to hint at the reason why he finds himself obliged to adopt any particular course. Be the difficulty ever so great, he removes it without mention of its exist- ence. Accordingly, in many places, the unassisted student can only see that much trouble is taken, without being able to guess why. What, then, it may be asked, is the peculiar merit of the Elements which has caused them to retain their ground to this day.^ The answer is, that the preceding objections refer to matters which can be easily mended, without any alter- ation of the main parts of the work, and that no one has ever given so easy and natural a chain of geometrical consequences. There is a never erring truth in the results ; and, though there may be here and there a self-evident assumption used in demonstration, but not formally noted, there is never any the smallest departure from the limit- ations of construction which geometers had, from the time of Plato, imposed upon themselves. The strong inclination of editors, already mentioned, to consider Euclid as perfect, and all negligences as the work of unskilful commentators or interpo- lators, is in itself a proof of the approximate truth of the character they give the work ; to which it may be added that editors in general prefer Euclid as he stands to the alterations of other editors. The Elements consist of thirteen books written by Euclid, and two of which it is supposed that Hypsicles is the author. The first four and the sixth are on plane geometry ; the fifth is on the theory of proportion, and applies to magnitude in general ; the seventh, eighth, and ninth, are on arithmetic ; the tenth is on the arithmetical cha- racteristics of the divisions of a straight line ; the eleventh and twelfth are on the elements of solid geometry; the thirteenth (and also the fourteenth and fifteenth) are on the regular solids, which were so much studied among the Platonists as to bear the name of Platonic, and Avhich, according to Proclus, were the objects on which the Elements were really meant to be written. At the commencement of the first book, undor the name of definitions (Spot), are contained the assumption of such notions as the point, line, &c.. and a number of verbal explanations. Then fol- low, under the name of postulates or demands (aiT'/inaTa), all that it is thought necessary to state as assumed in geometry. There are six postulates, three of which restrict the amount of construction granted to the joining two points by a straight line, the indefinite lengthening of a terminated straight line, and the drawing of a circle with a given centre, and a given distance measured from that centre as a radius ; the other three assume the equality of all right angles, the much disputed property of two lines, which meet a third at angles less than two right angles (we mean, of course, much disputed as to its propriety as an assumption, not as to its truth), and that two straight lines cannot inclose a space. Lastly, under the name oi common notions {koipuI ei-uoiai) are given, either as com.mon to all men or to u'il sciences, such assertions as that — things equal to the same are equal to one another — the whole is greater than its part — &c. Modern editors have put the last three postulates at the end of the common notions, and applied the term aaiom (which was not used till after Euclid) to them all. The in-' tention of Euclid seems to have been, to dip^Jn- guish between that which his reader must grant, or seek another system, whatever may be his opi- nion as to the propriety of the assumption, and that which there is no question ever}-- one will grant. The modern editor merely distinguishes the assumed problem (or construction) from the assumed theorem. Now there is no such distinct tion in Euclid as that of problem and theorem ; the common term irpSraa-is, translated proposition^ includes both, and is the only one used. An im- mense preponderance of manuscripts, the testi- mony of Proclus, the Arabic translations, the summary of Boethius, place the assumptions about right angles and parallels (and most of them, that about two straight lines) among the postulates ; and this seems most reasonable, for it is certain that the first two assumptions can have no claim to rank among common notions or to be placed in the same list with ' the whole is greater than its part.' Without describing minutely the contents of the first book of the Elements, we may observe that there is an arrangement of the propositions, which will enable any teacher to divide it into sections. Thus propp. 1 — 3 extend the power of construction to the drawing of a circle with any centre and ani/ radius ; 4 — 8 are the basis of the theory of equal triangles ; 9 — 12 increase the power of construction ; 13 — 15 are solely on rela- tions of angles; 16 — 21 examine the relations of parts of one triangle ; 22 — 23 are additional con- structions ; 23 — 26 augment the doctrine of equal triangles ; 27 — 31 contain the theory of parallels;[3] 32 stands alone, and gives the relation between the angles of a triangle ; 33 — 34 give the first properties of a parallelogram ; 35 — 41 consider parallelograms and triangles of equal areas, but different forms; 42 — 46 apply what precedes to augmenting power of construction ; 47 — 48 give the celebrated property of a right angled triangle and its converse. The other books are all capable of a similar species of subdivision. The second book shews those properties of the rectangles contained by the parts of divided straight lines, which are so closely connected with the common arithmetical operations of multipli- cation and division, that a student or a teacher who is not fully alive to the existence and diffi- culty of incommensurables is apt to think that common arithmetic would be as rigorous as geo- metry. Euclid knew better. The third book is devoted to the consideration of the properties of the circle, and is much cramped in several places by the imperfect idea already al- luded to, which Euclid took of an angle. There are some places in which he clearly drew upon experimental knowledge of the form of a circle, ​and made tacit assumptions of a kind which are rarely met with in his writings. The fourth book treats of regular figures. Eu- clid'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. 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 exist- ence of incommensurable quantities obliges him to introduce a definition of proportion which seems at first not only difficult, but uncouth and inele- gant ; 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- cording to his custom ; few students therefore un- derstand 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 propor- tion 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 propor- tion, and adds to the first four books the proposi- tions 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 construct 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 of numbers, on which the rules of arithmetic must be founded. But Euclid goes further than is ne- cessary merely to constnict a system of computa- tion, 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 mo- dern systems began to prevail, these books of Eu- clid were abandoned to the antiquary : our elemen- tary 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- count of it in the Feiiny Cyclopaedia, article, ' Ir- rational 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+^^h), a and b repre- senting two commensurable lines. He divides lines 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 anj' species can belong to that species in two different ways, or for two different sets of values of a and I. He shews how to form other classes of incommen- surables, in number how many soever, no one of which can contain an individual line which is com- mensurable with an individual of any other class ; and he demonstrates the incommensurability of a square and its diagonal. This book has a com- pleteness 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 dis- covery, producing results which are now usually referred to the differential calculus : while in those of Euclid it was only the mode of proving proposi- tions which must have been seen and beHeved be- fore they were proved. The method of these books is clear and elegant, with some striking imperfec- tions, 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 regu- lar 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 com- plete his intended structure. Proclus was an en- thusiastic 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 specula- tion : 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 col- lected 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 was called Kvpios (TToix€iooti^s. The Data of Euclid should be mentioned in con- nection with the Elements. This is a book contain- ing 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 undoubtedly advantageous to have a rapid perception of the steps which will reach the result, if they can be succes- sively 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 ma}^ be that the first person, had he perceived it, would have immediately found D. The formation of data consequential, as our ancestors would per- haps 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 proposi- tions shews that the angle it makes with the pa- rallels IS given in magnitude. There is not much more in this book of Data than an intelligent stu- dent 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 Ele- ments themselves, notions upon consequential data which the moderns acquire from the application of arithmetic and algebra : perhaps it was the percep- tion 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 re- mind 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 merelj' shewing his contem- poraries that a connected system of demonstration might be made without taking more than a certain number of postulates out of a collection, the neces- sity 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 posi- tion which they occupy, and we can find no fault with his tacit admission of many others, the neces- sity 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 ap- proach. The literary history of the writings of Euclid would contain that of the rise and progress of geo- metry in every Christian and Mohammedan na- tion : our notice, therefore, must be but slight, and various points of it will be confirmed by the biblio- graphical 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 Aeneaa of Hierapolis, who made an epitome of the whole. Theon the younger (of Alexandria) lived a little before 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 lumi- nary in ages more dark than his own. But Theon was a light of another sort, and his name has 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 altera- tions : he tells us so himself, and uses the word e»c5o(r{y, as applied to his own edition, in his com- mentary 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 oVep iSet diilai 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 manu- scripts. Again, in several manuscripts the whole work is headed as e/c tuv Siuucs awovcricjuv. 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 re- siding 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 altera- tions 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 of others, probably contributed to the notions about 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. 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 Eu- clid had almost disappeared from the West. But 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 geome- ters were sedulously translated, and a great im- pulse was given by them. Commentaries, and 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- 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 al- lowance 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 ​mentioned. Honein ben Ishak (died A. D. 873) published an edition which was afterwards cor- rected by Thabet ben Corrah, a well-known astro- nomer. After him, according to D'Herbelot, Othraan of Damascus (of uncertain date, but before the thirteenth century) saw at Rome a Greek ma- nuscript containing many more propositions than he had been accustomed to find: he had been used to 1 90 diagrams, and the manuscript contained 40 more. If these numbers be correct, Honein could 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 an- other edition, which is now accessible, having been printed in Arabic at Rome in 1594. It is tolera- bly complete, but yet it is not the edition from which the earliest European translation was made, as Peyrard found by comparing the same proposi- tion in the two. The first European who found Euclid in Arabic, and translated the Elements into Latin, was Athe- lard or Adelard, of Bath, who was certainly alive in 11 30. (See 'Adelard,' in the Biofir. bid. 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 ver}' 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 hand- writing 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 manu- scripts 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 Kvpios (rroix^iwri^s down to the end of the seventeenth century, and that in so ab- solute 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 deiie scietitie mathematice: and algebra was not at that time con- sidered as entitled to the name of a science by those who had been formed on the Greek model; ' arfe maggiore ' was its designation. The siory about Pascal's discovery of geometry in his boy- hood (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 exaggeratora (for much exaggerated this very circumstance sliewa the truth must have been) not having the slightest idea that a new invented system could proceed iu any other order than that of Euclid. The vernacular translations of the Elements date from the middle of the sixteenth century,from which. time the history of mathematical science divides itself into that of the several countries v/here it flourished. By slow steps, the continent of Europe has almost entirely abandoned the ancient Ele- ments, 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 cur- rency, 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 ma- thematical reasoning has declined wherever Euclid has been abandoned. We are not so much of the old opinion as to say that this nmst necessarily have happened; but, feeling quite sure that all the al- terations 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 be- ing 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, been much inclined to do something of the same sort, less openly. But here, at least, writers have left it to teachers to shirk[4] 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 difticulty by any false assumption. At the time of the invention of printing, two errors were current with respect to Euclid person- ally. 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. Boethius per- petuated it. The second was that Theon was the demonstrator of all the propositions, and that Euclid only left the definitions, postulates, &.C., with the ​enunciations in their present order. So completely was this notion received, that editions of Euclid, 60 called, contained only enunciations ; all that contained demonstrations were said to be Euclid with the commentary of Theon, Campanus, Zam- bertus, or some other. Also, when the enunciations were given in Greek and Latin, and the demon- strations 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: tut Proclus, who not only describes Euclid as rci fiaKaKwrepou SeiKvifieua to7s %jj.ttpo<tQ^u eis dvc- yKTovs dTToSei'l^ts dvayayuv, 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[5] 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 negatl the third, that of Buteo of Dauphiny, a geometer of merit, who attributes the whole to Euclid, quae opinio aut vera est, aut veritati eerie jiroxima. 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 exa- mination ; and the endorsement of an obscure scho- liast, which was perhaps the mere whim of a tran- scriber, was allowed to rank with the clearest as- sertions 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. Srotxf'a, the Elements, in 13 books, with a 14th and 15th added by Hypsicles. 2. AeSojueca, the Data, which has a preface by Marinas of Naples. 3. Eia-aywyiii 'ApfjLoviic^, a Treatise on Music; and 4. KaraTOfiil Kavovos, the Division oftlie Scale : one of these works, most likely the former, must be rejected. Proclus says that Euclid wrote Kara. fiovaiKrv (TToix^iwcreis. 5. ^aipSfiem, the Appearances (of the heavens). Pappus mentions them. 6. 'OiTTiKd, on Optics ; and 7. KaroTrrpiKd, on Catoptrics. Proclus mentions both. The preceding works are in existence ; the fol- lowing are either lost, or do not remain in the original Greek. 8. liepX Aiaipeffewv fiiSKlov, On Divisions. Pro- clus (I. 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. KauiKuv Pi§la S', Four hooks on Conic Sec- tions. Pappus (lib. vii. praef.) affirms that Euclid wrote four books on conies, which Apollonius en- larged, adding four others. Archimedes refers to ilie 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. Tlopi(Tp.dTu>v ^L^Kiay' , Three boohs of Porisms. These are mentioned by Proclus and by Pappus (/. c), the latter of whom gives a description which is so corrupt as to be unintelligible. 11. ToTvtav 'ETTtTreSwi/ fii§ia j8', Two hooks on Plane Loci. Pappus mentions these, but not Eu- tocius, as Fabricius affirms. {Comment, in Apoll. lib. i. lemm.) 12. ToTTwi/ irpos 'ETTKpdvetav Pt€la 0, men- tioned by Pappus. What these Toitoi irpos 'Ettl- (pdveiav, or Loci ad Siiperfidem, were, neither Pappus nor Eutocius inform us ; the latter says they derive their name from their own tSiOTTjs, 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. riept ^^vBapiccv, On Fallacies. On this work Proclus says, ' He gave methods of clear judgment (Sioport/cTjs (ppovfin^ws) the possession of which enables us to exercise those who are begin- ning geometry in the detection of false reasonings, and to keep them free from delusion. And the book which gives us this preparation is called ^GvZapiiav, 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.' It thus appears that Euclid did not intend his Ele- ments to be studied without any preparation, but tliat 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 explana- tions of the course adopted in tlie Elements which it cannot but have contained. 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 Elamenis is that printed by Erhard Ratdolt at Venice in 1482, Ijlack letter, folio. It is the Latin of the fifteen books of the Elements, from Adelard, with the commentary of Campanus following the demon- strations. It has no title, but, after a short intro- duction by the printer, opens thus : 'Preclarissimus liber elementorum Euclidis perspicacissimi : in artem geometric incipit qua foelicissime : Punctus est cujus ps nn est,' &c. Ratdolt states in the introduction that the difficulty of printing diagrams ​had prevented books of geometry from going throiigli the press, but that he had so completely overcome it, by great pains, that ' qua facilitate litterarum elenienta imprimuntur, ea etiara geometrice figure conficerentur/' These diagrams are printed on the margin, and though at first sight they seem to be woodcuts, yet a closer inspection makes it probable that they are produced from metal lines. The number of propositions in Euclid (15 books) is 485, of which 18 are wanting here, and 30 appear which are not in Euclid ; so that there are 497 proposi- tions. The preface to the l4th book, by which it is made almost certain that Euclid did not write it (for Euclid's books have no prefaces) is omitted. Its Arabic origin is visible in the words helmuaym and hehnuariplw^ which are used for a rhombus and 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 magis- trum 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 snarcer. Both these editions call Euclid Megarensis. 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 philo- sophici Ftatonici, matliemaiicarum disciplinaru janitoris : habent in hoc volumine quicuque ad nia- thematica substantia aspirdt : elemeioriun iibros, ^c. <^c. Zamberto Veneto Interprete. At the end is Impressum Veneiis, <S[c. in edibus Joannis Ta- cuini, ^c, M. D. V. VIII. Klendas Novtbris — 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 trans- lated 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 manu- scripts. He states that the propositions have the trposition of Theon or Hypsicles, by which he pro- bably means that Theon or Hypsicles gave the demonstrations. The preceding editors, whatever their opinions may have been, do not expressl}' 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. Fabri- cius 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, 1509), 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 matluiinaticorumque omnium sine controversia principis opera. Sec. At the end, Venetiis impressum per . .. Payaninum de Paganinis . . . a7mo...JADvnii . ., Paciolus adopts the Latin of Adelard, and occa- sionally quotes the comment of Campanus, intro- ducing his own additional comments with the head ' Castigator.' He opens the fifth book with the account of a lecture which he gave on that book in a church at Venice, August 11, 1508, giving the names of those present, and some subsequent lau- datory correspondence. This edition is less loaded Avith connnent than either of those which precede. It is extremely scarce, and is beautifully printed : the letter is a curious intermediate step between the old thick black letter and that of the Roman type, and makes the derivation of the latter from the former very clear. The fifth edition (Elements, Latin, Roman letter, folio), edited by Jacobus Faber, and printed by Ileary 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 E?mnciations (see the preceding re- marks 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 Zam- bertus ; 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. From it we date the time when a list of enunciations merely was universally called .the complete work of Euclid. With these editions the ancient series, as we may call it, terminates, meaning the complete La- tin 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 Gryne, or Grynoeus, Basle, 1533, folio:[6] contain- ing, 6/f Twi/ @iwvos cvvovaiciv (the title-page has this statement), the fifteen books of the Elements, and the connnentary of Proclus added at the end, so far as it remains ; all Greek, without Latin. On Grynoeus and his reverend[7] care of manuscripts, see Anthony Wood. {Athen. Oocon. 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. With regard to Greek texts, the student must be on his guard against bibliographers. For in- stance, Harless[8] gives, from good catalogues, Eu​K(i5ov troix^iuv fit§ia te', Rome, 1545, 8vo., printed by Antonius Bladus Asulanus, containing enimciations only, without demonstrations or dia- grams, edited by Angelas Cujanus, and dedicated to Antonius Altovitus. We happen to possess a little volume agreeing in every particular with this description, except only that it is in Italian, being ' I quindici libri degli elementi di Euclide, di Greco tradotti in lingua Thoscana.' Here is another in- stance in which the editor believed he had given the whole of Euclid in giving the enunciations. From this edition another Greek text, Florence, 1 545, was invented by another mistake. All the Greek and Latin editions which Fabricius, Mur- hard, &c., attribute to Dasj^podius (Conrad Rauch- fuss), only give the enunciations in Greek. The same may be said of Scheubel's edition of tha first six books (Basle, folio, 1550), which nevertheless professes in the title-page to give Eudid, Gr. Lat. There is an anonymous complete Greek and Latin text, London, printed by William Jones, lb'20, Avhich has ihhieen books in the title-page, but contains only six in all copies that we have seen : it is attributed to the celebrated mathematician Rriggs. The Oxford edition, folio, 1703, published by David Gregory, with the title Eu«:AeiSou to aw^o- /jLeva, took its rise in the collection of manuscripts bequeathed by Sir Henry Savile to the University, and was a part of Dr. Edward Beniard's plan (see his life in the Penny Cyclopaedia) for a large republication of the Greek geometers. His inten- tion was, that the first four volumes should contain Euclid, ApoUonius, Archimedes, Pappus, and Heron ; and, by an undesigned coincidence, the University has actually published the first three volumes in the order intended : we hope Pappus and Heron will be edited in time. In this Oxford text a large addi- tional supply of manuscripts was consulted, but various readings are not given. It contains all the leputed works of Euclid, the Latin work of Mo- hammed of Bagdad, above mentioned as attributed by some to Euclid, and a Latin fragment De Levi et Ponderoso, which is wholly unworthy of notice, but which some had given to Euclid. The Latin of this edition is mostly from Commandine, with the help of Henry Savile 's papers, which seem to liave nearly amounted to a complete version. As an edition of the whole of Euclid's works, this stands alone, there being no other in Greek. Peyrard, who examined it with every desire to find errors of the press, produced only at the rate of ten for each book of the Elements. The Paris edition was produced under singular circumstances. It is Greek, Latin, and French, in 3 vols. 4to. Paris, 1814-16-18, and it contains fifteen books of the Elements and the Data ; for, though professing to give a complete edition of Euclid, Peyrard would not admit anything else to be genuine. F. Peyrard had published a translation of some books of Euclid in 1804, and a complete translation of Archimedes. It was his in- tention to publish the texts of Euclid, ApoUonius, and Archimedes ; and beginning to examine the manuscripts of Euclid in the Royal Library at Paris, 23 in number, he found one, marked No. 190, which had the appearance of being written in the ninth century, and which seemed more complete and trustworthy than any single known manu- script. This document was part of the plunder sent from Rome to Paris by Napoleon, and had belonged to the Vatican Library. When restitu- tion was enforced by the allied armies in 1815, a special pennission was given to Peyrard to retain this manuscript till he had finished the edition on which he was then engaged, and of which one vo- lume had already appeared. Peyrard was a wor- shipper of this manuscript. No. 190, and had a con- tempt for all previous editions of Euclid. He gives at the end of each volume a comparison of the Paris edition with the Oxford, specifying what has been derived from the Vatican manuscript, and making a selection from the various readings of the other 22 manuscripts which were before him. This edition is therefore very valuable ; but it is veiy incorrectly printed: and the editor's strictures upon his predecessors seem to us to require the support of better scholarship than he could bring to bear upon the subject. (See the Dublin Review^ No. 22, Nov. 1841, p. 341, &c.) The Berlin edition, Greek only, one volume in two parts, octavo, Berlin, 1 826, is the work of E. F. August, and contains the thirteen books of the Elements, with various readings from Peymrd, and from three additional manuscripts at Munich (making altogether about 35 manuscripts consulted by the four editors). To the scholar who wants one edition of the Elements, we should decidedly recommend this, as bringing together all that has been done for the text of Euclid's greatest work. We mention here, out of its place, The Elements of Euclid with dissertations, by James Williamson, B.D. 2 vols. 4to., Oxford, 1781, and London, 1788. This is an English translation of thirteen books, made in the closest manner from the Oxford edi- tion, being Euclid word for word, with the addi- tional words required by the English idiom given in Italics. This edition is valuable, and not very scarce : the dissertations may be read with profit by a modern algebraist, if it be true that equal and opposite errors destroy one another. Camerer and Hauber published the first six books in Greek and Latin, with good notes, Ber- lin, 8vo. 1824. We believe we have mentioned all the Greek texts of the Elements ; the liberal supply with which the bibliographers have furnished the world, and which Fabricius and others have perpetuated, is, as we have no doubt, a series of mistakes arising for the most part out of the belief about Euclid the enunciator and Theon the demonstrator, which we have described. Of Latin editions, which must have a slight notice, we have the six books by Orontius Finoeus, Paiis, 1536, folio (Fabr., Murhard) ; the same by Joachim Camerarius, Leipsic, 1549, 8vo (Fabr., Murhard); the fifteen books by Steph. Gracilis, Paris, 1557, 4to. (Fabr., who calls it Gr. Lat., Murhard); the fifteen books of Franc, de Foix de Candale(Flussa8 Candalla), who adds a sixteenth, Paris, 1566, folio, and promises a seventeenth and eighteenth, which he gave in a subsequent edition, Paris, 1578, folio (Fabr., Murhard) ; Frederic ​Commandine's first edition of the fifteen books, with commentaries, Pisauri, 1572, fol. (Fabr., Murhard); tiie fifteen books of Christopher Clavius, with com- mentary, and Candalla's sixteenth book annexed, Korae, 1574, fol. (Fabr., Murhard); thirteen books, by Ambrosiiw Rhodius, Witteberg, 1609, 8vo. ( Fabr., Murh.) ; thirteen books by the Jesuit Claude Richard, Antwerp, 1645, folio (Murh.) ; twelve books by Horsley, Oxford, 1802. We have not thought it necessary to swell this article with the various reprints of these and the old Latin editions, nor with editions which, though called Elements of Euclid, have the demonstrations given in the edi- tor's own manner, as those of Maurolycus, Barrow, Cotes, &c., &c., nor with the editions contained in ancient courses of mathematics, such as those of Herigonius, Dechales, Schott, &c., &c., which ge- nerally gave a tolerably complete edition of the Elements. Commandine and Clavius are the pro- genitors of a large school of editors, among whom Robert Simson stands conspicuous. We now proceed to English translations. We find in Tanner {Bibl. Brit. I lib. p. 149) the fol- lowing short statement : ' Candish, Richardus, patria SufFolciensis, in linguam patriam transtulit Euclidis geometriara, lib. xv. Claruit[9] a.d. mdlvi. Bal. par. post. p. 111.' Richard Candish is men- tioned elsewhere as a translator, but we are confi- dent that his translation was never published. Before 1570, all that had been published in Eng- lish was Robert Recorders PcUhway to Knowledge, 1551, containing enunciations only of the first four books, not in Euclid's order. Recorde considers demonstration to be the work of Theon. In 1570 appeared Henry Biilingsley's translation of the fif- teen books, with Candalla's sixteenth, London, folio. This book has a long preface by John Dee, the magician, whose picture is at the beginning : 60 that it has often been taken for Dee's transla- tion ; but he himself, in a list of his own works, ascribes it to Billingsley. The latter was a rich citizen, and was mayor (with knighthood) in 1591. We always had doubts whether he was the real translator, imagining that Dee had done the drud- gery at least. On looking into Anthony Wood's account of Billingsley {^Aih. Occon. in verb.) we find it stated (and also how the information was ob- tained) that he studied three years at Oxford be- fore he was apprenticed to a haberdasher, and there made acquaintance with an 'eminent mathema- tician' called Whytehead, an Augustine friar. When the friar was 'put to his shifts' by the dissolution of the monasteries, Billingsley received and maintained him, and learnt mathematics from hira. 'When Whytehead died, he gave his scho- lar all his mathematical observations that he had made and collected, together with his notes on Euclid's Elements.' This was the foundation of the translation, on which we have only to say that it was certainly made from the Greek, and not from any of the Arab ico- Latin versions, and is, for the time, a very good one. It was reprinted, Lon- don, folio, 1661. Billingsley died in 1606, at a great age. Edmund Scarburgh (Oxford, folio, 1705) trans- lated six books, with copious annotations. We omit detailed mention of Whiston's translation of Tacquet, of Keill, Cunn, Stone, and other editors, whose editions have not much to do with the pro gress of opinion about the Elements. Dr. Robert Simson published the first six, and eleventh and twelfth books, in two separate quarto editions. (Latin, Glasgow, 1 756. English, London, 1756.) The translation of the Data was added to the first octavo edition (called 2nd edition), Glas- gow, 1762 : other matters unconnected with Euclid have been added to the numerous succeeding edi- tions. With the exception of the editorial fancy about the perfect restoration of Euclid, there is lit- tle to object to in this celebrated edition. It might indeed have been expected that some notice would have been taken of various points on which Euclid has evidently fallen short of that formality of rigour which is tacitly claimed for him. We prefer this edition very much to many which have been fasliioned upon it, particularly to those which have introduced algebraical symbols into the de- monstrations in such a manner as to confuse geo- metrical demonstration with algebraical operation. Simson was first translated into German by J. A. Matthias, Magdeburgh, '1799, 8vo. Professor John Playfair's Elements of Geometry contains the first six books of Euclid ; but the so- lid geometry is supplied from other sources. The first edition is of Edinburgh, 1795, octavo. This is a valuable edition, and the treatment of the fifth book, in particular, is much simplified by the aban- donment of Euclid's notation, though his definition and method are retained. EucliiCs Elements of Plane Geometry, by John Walker, London, 1827, is a collection containing very excellent materials and valuable thoughts, but it is hardly an edition of EucHd. We ought perhaps to mention W. Halifax, whose English Euclid Schweiger puts down as printed eight times in London, between 1685 and 1752. But we never met with it, and cannot find it in any sale[10] catalogue, nor in any English enumeration of editors. The Diagrams of Euclid's Elements by the Rev. W. Taylor, York, 1828, 8vo. size (part i. containing the first book; we do not know of any more), is a collection of lettered diagrams stamped in relief, for the use of the blind. The earliest German print of Euclid is an edition by Scheubel or Scheybl, who published the seventh, eighth, and ninth books, Augsburgh, 1555, 4to. (Fabr. from his own copy) ; the first six books by W. Holtzmann, better known as Xylander, were published at Basle, 1562, folio (Fabr., Murhard, Kastner). In French we have Errard, nine books, Paris, 1598, 8vo. (Fabr.) ; fifteen books by Hen- rion, Paris, 1615 ((Fabr.), 1623 (Murh.), about 1 627 (necessary inference from the preface of the fifth edition, of 1649, in our possession). It is a close translation, with a comment. In Dutch, six books by J. Petersz Don, Leyden, 1 606 (Fabr.), 1608 (Murh.). Dou was translated into German, Amsterdam, 1634, 8vo. Also an anonymous trans- ladon of Clavius, 1663 (Murh.). In Italian, Tar- taglia's edition, Venice, 1543 and 1565. (Murh., Fabr.) In Spanish, by Joseph Saragoza, Valentia 1673, 4to. (Murh.) In Swedish, the first six books, by Martin Stromer, Upsal, 1753. (Murh.) The remaining writings of Euclid are of small interest compared with the Elements, and a shorter account of them will be sufficient. (Wikisource | public domain)