#### Eucleides

（**Εὐκλείδης**) of ALEXANDRIA. 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.* 2.4), 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 foinnation 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 literature. 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 sect, 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 answer to Ptolemy, who asked if geometry could not be made easier, that there was no royal road (μὴ εἰναι βασιλικὴν ἄτραπον πρὸς γεωμετρίαν). ^{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 August) 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 (8.12) 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 crigin 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.* 11.9) that it was the custom to paint Euclid with the fingers extended (*laxatis,*) 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 geometry in order to recover the landmarks which the Nile destroyed year by year, and that the Phoenicians were equally obliged to invent arithmetic 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 (καθολικώτερον) and some in a perceptive or sensible manner (αἰσθητικώτερον). Proclus clearly refers to *physical* discovery in geometry, 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 (παιδείας ἐλευθέρον), took higher views of the subject, and investigated his theorems immaterially and intellectually (ἀν̈́λως καὶ νοερῶς): he also wrote on incommensurable quantities (ὰλόγων), and on the mundane figures (the five regular solids).

Barocius, whose Latin edition of Proclus has been generally followed, singularly enough translates ἄλογα βψ (*quae non exlpicari possunt*, 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 elements 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 (ἐπιστημονικωτέραν σύστασιν) to various theorems; Neocleides and his disciple Leon came after the preceding, the latter of whom increased both the extent and utility of the science, in particular by finding a test (διορισμόν) 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], increased 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 himself, 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. Cyzicinus of Athens was his contemporary; they took different sides on many common inquiries. Hermotimus of Colophon added to what had been done by Eudoxus and Theaetetus, discovered elementary propositions, and wrote something on loci. Philip (ὁ Μεταῖος , others read Μεδμαῖος, Barocius reads Mendaeus), the follower of Plato, made many mathematical inquiries connected with his master's philosophy.
Those who write on the history of geometry bring the completion of this science thus far. Here Proclus expressly refers to written history, 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 demonstration had been given, that something had been written on proportion, something on incommensurables, something on loci, something on solids; that analysis had been applied, that the conic sections 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 geometry must have been adopted; for on keeping within then, the difficulty of this problem depends.

#### Works

#### The Elements

It will be convenient to speak separately of the __Elements__of Euclid, as to their contents; and afterwards to mention them bibliographically, among the other writings. The book which passes under this name, as given by Robert Simson, unexceptionable 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

*aumending*the original. If the books of Livy were to be rewritten upon the basis of Niebuhr, and the result declared to be the real text, then Livy would no more than share the fate of Euclid; the only difference 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 beginniing of the fifth book, and accordingly he there inserts, not merely a definition, but, he assures us, the very one which Fuclid gave. Not a single manuscript supports him : how, then, did he know ? He saw that there

*ought*to have been such a detinition, 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 protest against the assumption that he could not have erred, whether by omission or commission.

Some of the characteristics of the Elements are brietly 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 incorrectness of many of the definitions.

Secondly. There is no distinction between propositions which require demonstration, and those which a logician would see to be nothing but different modes of starting 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 propositions 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 lie 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 parting 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 rifht angles, he takes no notice of this necessity, and no one cantellwhether it was an overshigt, 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 *unity* is considered as a *number,* and Simson has thought it necessary to supply the omission (see his book v. prop. A), and has shewn himself more Euclid tian 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 giving 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 : assumptions are made which are not formally set down among the postulates. Things which really ought to have been proved are sometimes 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 existence. 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 alteration 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 limitations 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 interpolators, 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 characteristics 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 which, according to Proclus, were the objects on which the Elements were really meant to be written.

At the commencement of the first book, under the name of definitions (ὅροι, are contained the assumption of such notions as the point, line, &c.. and a number of verbal explanations. Then follow, under the name of postulates or demands (αἰτήματα), 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 of *common notions* (κοιναὶ ἔννοιαι) are given, either as conmmin to all men or to all 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 *axiom* (which was not used till after Euclid) to them all.
The intention of Euclid seems to have been, to disinguish between that which his reader must grant, or seek another system, whatever may be his opinion as to the propriety of the assumption, and that which there is no question every one will grant.
The modern editor merely distinguishes the assumed *problem* (or construction) from the assumed *theorem.* Now there is no such distinction in Euclid as that of problem and theorem ; the common term πρότασις, translated *proposition,* includes both, and is the only one used.
An immense preponderance of manuscripts, the testimony 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 mintutely 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 any radius; 4-8 are the basis of the theory of equal triangles; 9-12 increase the power of construction; 13-15 are solely on relatiolis of angles; 16-21 examine the relations of parts of one triangle; 22-23 are additional constructios ; 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 multiplication and division, that a student or a teacher who is not fully alive to the existence and difficulty of incommensurables is apt to think that common arithmetic would be as rigorous as geometry. 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 alluded 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. 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.

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, according 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 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 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 arithmetical.
It is one of the most curious of the Greek speculations : the reader will find a synoptical account of it in the __Penny Cyclopaedia,__ article, " Irrational Quantities." Euclid has evidently in his mind the intention of classifying inco'nmmensurable 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 incommensurables 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 cant be represented by ✓（✓*a* ± ✓*b*), *a* and *b* representing 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 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 completeiness 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 geosnetry, 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 collected 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 κύριος στοιχειωτής.

#### >The Data

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 undoubtedly 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.

#### Assessment

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 lie 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.

#### Influence of Euclid

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 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 historical 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 has played a conspicuous and singular part in the history 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 ἔκδοσις, 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 proposition of the sixth book is his own addition: and it is found in all the manuscripts following the ὅπερ ἔδει δεῖξαι 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 ἐκ τῶν Θέωνος συνουσιῶν. 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 inter 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 of others, probably contributed to the notions about Then 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 Euclid had almost disappeared from the West. But nearly one hundred and fifty years followed the capture of Egypt by the Mohammeddans 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 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 progress which the Mohammedans nade 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 concentrating the geometry of Alexandria, the arithmetic and algebra of India, and the astronomy of both, to formn 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 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 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 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.

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 Boethius, 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 introdultore 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 boyhoud (A. D. . 1635) contains the statement that he had got "as far as the 32nd proposition of the first book" before he wits 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 date 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, 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 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. Boethius 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 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 τὰ μαλακώτερον δεικνύμενα τοῖς ἔμπροσθεν εἰς ἀνελέγκτους 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 demonstration 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 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 refntation of " the scholiast " may be safely abandoned.

#### Works attributed to Euclid

All the works that have been attributed to Euclid are as follows:
1. Στοιχεῖα, the __Elements,__ in 13 books, with a 14th and 15th added by HYPSICLES.

2. Δεδομένα, the __Data,__ which has a preface by Marines of Naples.

3. Εἰσαγωγὴ, Ἁρμονική, a *Treatise on Music ;* and 4. Κατατομὴ Κανόνος, *the Division of the Scale :* one of these works, most likely the former, must be rejected. Proclus says that Euclid wrote κατὰ μουσικὴν στοιχειώσεισς.

5. Φαινόμενα, the __Appearances__ (of the heavens). Pappus mentions them.

6. Ὀπτικά, *on Optics ;* and 7. Κατοητρικά, *on Catoptrics.* Proclus mentions both.

The preceding works are in existence; the following 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. Κωνικῶν Βιβλία δ́, *Four books on Conic Sections.* 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. Πορισμάτων Βιβλία γ́, *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 Ἐπιφάνειαν Βιβλία Β´, mentioned by Pappus. What these Τόποι πρὸς Ἐπιφάνειαν, or *Loci ad Superfuiem,* were, neither Pappus nor Eutocius inform us; the latter says they derive their name from their own ἰδιότης, 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 (διορατικῆς φρονήσεως) 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 Ψευδαρίων, 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 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.

#### Editions

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.

#### Latin Editions (1)

*The first, or editio princeps, of the*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 quā 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 the press, but that he had so completely overcome it, by great pains, that "qua facilitate litterarum elementa imprimuntur, ea etiam 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 (

__Elements__is that printed by Erhard Ratdolt at Venice in 1482, black letter, folio.**15**books) is 485, of which 18 are wanting here, and 30 appear which are not in Euclid; so that there are 497 propositions. The

*preface*to the 14th 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

*helmuariphe,*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 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 *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 philosophici Platonici, mathematicarum disciplinarū janitoris : habent in hoc votumine quicūque ad mathematicā substantiā aspirāt : elemētorum libros, (&c. &c. Zamberto Veneto Interprete. At the end is Impressum Venetis, &c. in edibus Joannis Tacuini, &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 show 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, 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 aculissimi mathematicorumque omnium sine controversia principis opera,* &c.
At the end, *Venetüs impressum per ... Payaninum de Payaninis ... anno...*MDVIIII... Paciolus adopts the Latin of Adelard, and occasionally quotes the comment of Campanus, introducing 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 laudatory correspondence.
This edition is less loaded with comment 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 Henry Stephens at Paris in 1516*, has the title __Contenta__ followed by beads 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. 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 Latin editions which preceded the publication of the Greek text. Thus we see five folio editions of the Elements produced in thirty-four years.

#### Greek Editions

*The first Greek text was published by Simon Gryne, or Grynoeus, Basle, 1533, folio:*(

^{6}containing, 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 reverend^{7}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.

With regard to Greek texts, the student must be on his guard against bibliographers. For instance, Harless ^{8} gives, from good catalogues, Εὐκλείδου Στοιχείων Βιβλία ιέ, Rome, 1545, 8vo., printed by Antonius Bladus Asulanus, containing enunciations only, without demonstrations or diagrams, edited by Angelus 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 element di Euclide, *di Greco tradotti* in lingua Thoscana." Here is another instance in which the editor believed he had given the whole of Euclid in giving the enunciations. From this edition another Greek text, Florence, 1545, was invented by another mistake. All the Greek and Latin editions which Fabricius, Murhard, &c., attribute to Dasypodius (Conrad Rauchfuss), only give the enunciations in Greek.
The same may be said of *Scheubel's edition of the first six books (Basle, folio, 1550)*, which nevertheless professes in the title-page to give *Euclid,* Gr. Lat. *There is an anonymous complete Greek and Latin text, London, printed by William Jones, 1620*, which has *thirteen* books in the title-page, but contains only six in all copies that we have seen : it is attributed to the celebrated mathematician Briggs.

*The Oxford edition, folio, 1703, published by David Gregory, with the title Εὐκλείδου τὰ σωζόμενα*, took its rise in the collection of manuscripts bequeathed by Sir Henry Savile to the University, and was a part of Dr. Edward Bernard's plan (see his life in the

__Penny Cyclopaedia__) for a large republication of the Greek geometers. His intention was, that the first four volumes should contain Euclid, Apollonius, 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 additional supply of manuscripts was consulted, but various readings are not given. It contains all the reputed works of Euclid, the Latin work of Mohammed 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 have 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 intention to publish the texts of Euclid, Apollonius, 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 manuscript.
This document was part of the plunder sent from Rome to Paris by Napoleon, and had belonged to the Vatican Library. When restitution was enforced by the allied armies in 1815, a special permission was given to Peyrard to retain this manuscript till he had finished the edition on which he was then engaged, and of which one volume had already appeared. Peyrard was a worshipper of this manuscript, No. 190, and had a contempt 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 very 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, 1826, is the work of E. F. August*, and contains the thirteen books of the Elements, with various readings from Peyrard, 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 disseritatios, 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 edition, being Euclid word for word, with the additional 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, Berlin, 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.

#### Latin Editions (2)

Of Latin editions, which must have a slight notice, we have the six books by*Orontius Finoeus, Paris, 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 (Flussas 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);

*the fifteen books of Christopher Clavius, with conmmentary, and Candalla's sixteenth book annexed, Rome, 1574, fol.*(Fabr., Murhard);

*thirteen books, by Ambrosius 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 editor'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, Dechâles, Schott, &c., &c., which generally gave a tolerably complete edition of the Elements. Commandine and Clavius are the progenitors of a large school of editors, among whom Robert Simson stands conspicuous.

#### English Translations

We now proceed to English translations.
We find in Tanner (*Bibl. Brit. Hib.* p. 149) the following short statement : " Candish, Richardus, patria Suffolciensis, in linguam patriam transtulit Euclidis geometriam, lib. xv. Claruit ^{9} A. D. MDLVI. Bal. par. post. p. 111." Richard Candish is mentioned elsewhere as a translator, but we are confident that his translation was never published.

*Before 1570, all that had been published in English was Robert Recorde's Pathway 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 Billingsley's translation of the fifteen books, with Candalla's sixteenth, London, folio.*This book has a long preface by John Dee, the magician, whose picture is at the beginning : so that it has often been taken for Dee's translation ; 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 drudgery at least. On looking into Anthony Wood's account of Billingsley (

*Ath. Oxon. in verb.*) we find it stated (and also how the information was obtained) that he studied three years at Oxford before he was apprenticed to a haberdasher, and there made acquaintance with all " eminent mathematician" 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 him. " When Whytehead died, he gave his scholar all his mathematical observations that lie 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 Arabico-Latin versions, and is, for the time, a very good one.

*It was reprinted, London, folio, 1661.*Billingsley died in 1606, at a great age.

*Edmund Scarburgh (Oxford, folio, 1705) translated 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 progress of opinion about the Elementts.*

*Dr. Robert Simson published the first six, and eleventh and twelfth books, in two separate quarto editions. (Latin, Glasgow, 1756. English, London, 1756.)* *The translation of the Data was added to the first octavo edition (called 2nd edition), Glasgow, 1762*: other matters unconnected with Euclid have been added to the numerous succeeding editions.
With the exception of the editorial fancy about the perfect restoration of Euclid, there is little to object to in this celebrated edition.
It might indeed have been expected that sone 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 fashioned upon it, particularly to those which have introduced algebraical symbols into the demonstrations in such a manner as to confuse geometrical 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 solid 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 abandonment of Euclid's notation, though his definition and method are retained.

* Euclid's 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 Euclid.

*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.

#### Translations into other European Languages

*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, Kästner).

*In French we have Errard, nine books, Paris, 1598, 8vo. (Fabr.)*; *fifteen books by Henrion, Paris, 1615 ((Fabr.), 1623 (Murh.), about 1627 (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 Dou, Leyden, 1606 (Fabr.), 1608 (Murh.).* *Dou was translated into German, Amsterdam, 1634, 8vo.* *Also an anonymous translation of Clavius, 1663 (Murh.).*

*In Italian, Tartaglia'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 Strömer, 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.

#### Editions of the other works

#### The Data

*The first Greek edition of the*

__Data__is Εὐκλείδου δεδομένα, &c., by Clandius Hardy, Paris, 1625, 4to., Gr. Lat., with the preface of Marinus prefixed. Murhard speaks of a second edition, Paris. 1695, 4to.*Dasypodius had previously published them in Latin, Strasburg, 1570.*(Fabr.) We have already spoken of Zamberti's Latin, and of the Greek of Gregory and Peyrard.

*There is also*

*Euclidis Dalorum Liber*by Horsley, Oxford, 1803, 8vo.

#### The Phaenomena

The __Phaenomena__is an astronomical work, containing 25 geometrical propositions on the doctrine of the sphere. Pappus (lib. vi.

*praef.*) refers to the second proposition of this work of Euclid, and the second proposition of the book which has come down to us contains the matter of the reference. We have referred to the Latin of Zamberti and the Greek of Gregory.

*Dasypodius gave an edition (Gr. Lat., so said; but we suppose with only the enunciations Greek), Strasburg, 1 1571, 4to.*(?) (Weidler), and

*another appeared (Lat.) by Joseph Auria, with the comment of Maurolycus, Rome, 1591, 4to.*(Lalande and Weidler)

*The book is also in Mersenne's Synopsis, Paris, 1644, 4to.*(Weidler.) Lalande names it (

*Bibl. Astron.*p. 188) as part of

*a very ill-described astronomical collection, in 3 vols. Paris, 1626, 16mo.*

#### The Works on Music

Of the two works on music, the__Harmonics__and the

__Division of the Canon__(or scale), it is unlikely that Euclid should have been the author of both. The former is a very dry description of the interminable musical nomenclature of the Greeks, and of their modes. It is called Aristoxenean [ARISTOXENUS] : it does not contain any discussion of the proper ultimate authority in musical matters, though it does, in its wearisome enumeration, adopt some of those intervals which Aristoxenus retained, and the Pythagoreans rejected. The style and matter of this treatise, we strongly susspect, belong to a later period than that of Euclid. The second treatise is an arithmetical description and demonstration of the mode of dividing the scale. Gregory is inclined to think this treatise cannot be Euclid's, and one of his reasons is that Ptolemy does not mention it; another, that the theory followed in it is such as is rarely, if ever, mentioned before the time of Ptolemy. If Euclid did write either of these treatises, we are satisfied it must have been the second. Both are contained in Gregory (Gr. Lat.) as already noted;

*in the collection of Greek musical authors by Meibomius (Gr. Lat.), Amsterdam, 1652, 4to.*;

*and in a separate edition (also Gr. Lat.) by J. Pena, Paris, 1537, 4to. (Fabr.), 1557 (Schweiger).*

*Possevinus has also a corrected Latin edition of the first in his*

__Bibl. Set.__Colon. 1657.
*Forcadel translated one treatise into French, Paris, 1566, 8vo. (Schweiger.)*

#### Optics

The book on __Optics__treats, in 61 propositions, on the simplest geometrical characteristics of vision and perspective: the

__Catoptrics__have 31 propositions on the law of reflexion as exemplified in plane and spherical mirrors. We have referred to the Gr. Lat. of Gregory and the Latin of Zamberti ;

*there is also the edition of J. Pena (Gr. Lat.), Paris, 1557, 4to. (Fabr.)*;

*that of Dasypodius (Latin only, we suppose, with Greek enunciations), Strasburg, 1557, 4to.*(Fabr.);

*a reprint of the Latin of Pena, Leyden, 1599, 4to. (Fabr.)*;

*and some other reprint, Leipsic, 1607. (Fabr.)*

*There is a French translation by Rol. Freart Mans, 1663, 4to.*; and *an Italian one by Egnatio Danti, Florence. 1573, 4to.* (Schweiger.)

#### Further Information

Proclus Pappus; August*ed cit.;*Fabric.

*Bibl. Graec.*vol. iv. p. 44, &c.; Gregory,

*Pracf. edit. cit.*; Murhard,

*Bibl. Math. ;*Zamberti,

*ed. cit.;*Savile,

*Prelect. in Eucl.;*Heilbronner,

*Hist. Mathes. Unit. ;*Schweiger,

*Handb. der Classisch. Bibl. ;*Peyrard,

*ed. cit.,*&c. &c.: tall editions to which a reference is not added having been actually consulted.

[A. DE M.]