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-
