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
