A Dictionary of Greek and Roman biography and mythology. By various writers. Ed. by William Smith. Illustrated by numerous engravings on wood.

EUCLEIDES. E UCLEIDES. 65
generalized many particular propositions. Cyzici- script supports him: how, then, did he know?
nus of Athens was his contemporary; they took He saw that there ouglht to have been such a defidifferent sides on many common inquiries. Hermo- nition, and he concluded that, therefore, there had
timus of Colophon added to what had been done been one. Now we by no means uphold Euclid
by Eudoxus and Theaetetus, discovered elementary as an all-sufficient guide to geometry, though we
-propositions, and wrote something on loci. Philip feel that it is to himself that we owe the power of
(d MeTra'os, others read Measors, Barocius reads amending his writings; and we hope we may proMendaeus), the follower of Plato, made many ma- test against the assumption that he could not have
thematical inquiries connected with his master's erred, whether by omission or commission.
philosophy. Those who write on the history of Some of the characteristics of the Elenments are
geo.netry bring the completion of this science thus briefly as follows:far. Here Proclus expressly refers to written his- First. There is a total absence of distinction
tory, and in another place he particularly mentions between the various ways in which we'know the
the history of Eudemus the Peripatetic. meaning of terms: certainty, and nothing more, is
This history of Proclus has been much kept in the thing sought. The definition of straightness,
the background, we should almost say discredited, an idea which it is impossible to put into simpler
by editors, who seem to wish it should be thought words, and which is therefore'described by a more
that a finished and unassailable system sprung at difficult'circumlocution, comes under the same
once from the brain of Euclid; an armed Minerva heading as the explanation of the word "parallel."
from the head of a Jupiter. But Proclus, as much Hence disputes about the correctness or incorrecta worshipper as any of them, must have had the ness of many of the definitions.
same bias, and is therefore particularly worthy of Secondly. There is no distinction between proconfidence when he cites written history as to positions which require demonstration, and those
what was -not done by Euclid. Make the most we which a logician would, see to be nothing but
can of his preliminaries, still the thirteen books of different modes of stating a preceding proposition.
the Elements must have been a tremendous advance, VWhen Euclid has proved that everything which'
probably even greater than that contained in the is not A is not B, he does'not hold himself entitled
Principia of Newton. But still, to bring the state to infer that every B is.A, though the two propoof our opinion of this progress down to something sitions are identically the same. Thus, having
short of painful wonder, we are told that demon- shewn that every point of a circle which is not the
stration'had been given, that something' had. been centre is not one from which three equal straight
written on proportion, something oni incommensu- lines canl be drawn, he cannot infer that any point
rables, something on loci, something on solids; from which three equal straight lines are drawn is
that analysis had been applied, that the conic see- the centre, but has need of a new demonstration.
tions had been thought of, that the Elements had Thus, long before he wants to use book i. prop. 69,
been distinguished from the rest and written on. he has proved it again, and independently.
From what Hippocrates had done, we know that Thirdly. He has not the smallest notion of
the- important property of the right-angled triangle admitting any generalized use of a word, or of partwas known; we rely. much more on the lunules ing with any ordinary notion attached to'it.
than on the story about Pythagoras. The dispute Setting out with the conception of an angle rather
about the famous Delian problem.had arisen, and as the sharp corner made by the meeting of two
some conventional limit to the instruments of geo- lines than as the magnitude which he afterwards
metry' must have been adopted; for on keeping shews how to measure, he never gets rid of that
within them, the difficulty of this problem depends. corner, never admits' two right angles to make
It will be convenient to speak separately of the one angle, and still less is able to arrive at the
Elemlents of Euclid, as to their contents; and after- idea of an angle greater than two right angles.
wards to mention them bibliographically, among And when, in the last proposition of-the sixth
the other writings. The book which passes under book, his definition of proportion absolutely requires
this name, as given by Robert Simson, unexcep- that he should reason on angles of even more than
tionable as Elements of Geometry, is not calculated four right angles, he takes no notice of this necesto give the scholar a proper idea of the'elements of sity, and no one can tell whetherit was an over-.
kEucdid; -but it is admirably: adapted to confuse, in sight, whether Euclid thought the extension one
the mind of the young student, all those notions of which the student could make for himself, or
sound criticism which his other instructors are whether (which has sometimes struck us as not
endeavouring to instil. The idea that Euclid must unlikely) the elements were his last work, and he
be perfect had got possession of the -geometrical did not live to revise them.
vorld; accordingly' each editor, when he made In one solitary case, Euclid seems to have made'
what he' took to be an alteration for the better, an omission implying that he recognized that.
assumed that he was restoring, not amending, the natural extension of language by which unity is.
original.'If the books of Livy were to be re- considered as a number, and Simson has'thought it
written upon the basis of Niebuhr, and the result necessary to'supply the omission (see his book v.
declared to be' the real text, then Livy would no prop. A), and has shewn himself more Euclid than
more than share the fate of Euclid; the only dif- Euclid upon the point of all others in which
ference being, that the former would undergo a Euclid's philosophy is defective.
larger quantity of alteration than editors have seen' Fourthly. There is none of that attention to
fit to inflict upon the latter. This is no caricature; the forms of accuracy with which translators have
e.g., Euclid, says Robert Simson, gave, without endeavoured to invest the Elements, thereby givdoubt, a definition of compound ratio at the be- ing them that appearance which has made manyi
ginning of the fifth book, and accordingly he there teachers think it meritorious to insist upon their
inserts, not merely a definition, but, he assures us,'pupils remembering the very words of Simson.
the very one which Euclid gave. N ot a single mann- Theorems are found among the definitions: assump'
VOL. II.' -'