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

66'EUCLEIDES.,EUCLEIDES.
tions' are made which are not formally set down as an assumption, not as to its truth), and that
among, the postulates. Things which really ought two straight lines cannot inclose a space. Lastly,
to have been proved are sometimes passed over, under the name of common notions (Koeval Gvooa)
and whether this is by mistake, or by intention of are given, either as common to all men or to all
supposing them self-evident, cannot now be known: sciences, such assertions as that-things equal to the
for Euclid never refers to previous propositions by same are equal to one another-the whole is greater
name or number, but only by simple re-assertion than its part-&c. Modem editors have put the
without reference; except that occasionally, and last three postulates at the end of the common
chiefly when a negative proposition is referred to, notions, and applied the term axiom' (which was
such words as "it has been demonstrated" are not used till after Euclid) to them all. The inemployed, without further specification. tention of Euclid seems to have been, to distinFifthly. Euclid never condescends to hint at guish between that which his reader must grant,
the reason why he finds himself obliged to adopt or seek another system, whatever may be his'opiany particular course. Be the difficulty ever' so nion as to the propriety of the assumption, and
great, he removes it without mention of its exist- that which there is no question every one will
ence. Accordingly, in many places, the unassisted grant. The modem editor merely distinguishes'
student can only see that much trouble is taken, the assumed problem (or construction) from the
without being able to guess why. assumed theorem. Now there is no such distincWhat, then, it may be asked, is the peculiar tion in Euclid as that of problem and theorem;
merit of the Elements which has caused them to the common term 7rpdacrs, translated proposition,
retain their ground to this day? The answer is, includes both, and is the only one used. An imthat the preceding objections refer to matters mense preponderance of manuscripts, the testiwhich can be easily mended, without any alter- mony of Proclus, the Arabic translations, the
ation of the main parts of the work, and that no summary of Boethius, place the assumptions about
one has ever given so easy and natural a chain of right angles and parallels (and most of them, that
geometrical consequences. There is a never erring about two straight lines) among the postulates;
truth- in the results; and, though there may be and this seems most reasonable, for it is certain
here and there a self-evident assumption used in that the first two assumptions can have no claim
demonstration, but not formally noted, there is to rank among common notions or to be placed in
never any the smallest departure from the:limit- the same list with " the whole is greater than its
ations of construction which geometers had, from part."
the time of Plato, imposed upon themselves. The Without describing minutely the contents of
strong inclination of editors, already mentioned, to the' first book of the Elements, we may observe
Consider Euclid as perfect, and all negligences as that there is an arrangement of the propositions,
the:work of unskilful'commentators or interpo- which will enable any teacher to divide it into
lators, is in itself a proof of the approximate truth sections. Thus propp. 1-3 extend the power of
of the character they give the work; to which it construction to the drawing of a circle with any
may be added that editors in general prefer Euclid centre and any radius; 4-8 are the basis of the
as he stands to the alterations of other editors. theory' of equal triangles; 9-12 increase the
The Elements consist of thirteen books written power of construction; 13-15 are solely on relaby Euclid, and two of which it is supposed that tions of angles; 16-21 examine'the relations of
Hypsicles is the author. The first four and the parts of one triangle; 22-23 are additional consixth are on plane geometry; the fifth is on the structions; 23-26 augment the doctrine of equal
theory of proportion, and applies to magnitude in triangles; 27-31 contain the theory of parallels;"
general; the seventh, eighth, and ninth, are on 32 stands alone, and gives the relation between
arithmetic; the tenth is on the arithmetical cha- the angles of a triangle; 33-34 give the first
racteristics of the divisions of a straight line; the properties of a parallelogram; 35-41 consider
eleventh and twelfth are on the elements of solid parallelograms and triangles of equal areas, but
geometry; the thirteenth (and also the fourteenth different forms; 42-46 apply what precedes to
and fifteenth) are on the regular solids, which augmenting power of construction; 47-48 give
Were so much studied among the Platonists as to the celebrated property of a right angled triangle
bear the name of Platonic, and which, according to and its converse. The other books are all capable
Proclus, were the objects on which the Elements of a'similar species of subdivision.
were really meant to be written. The second book shews those properties of the
At the commencement of the first book, under rectangles contained by the parts of divided
the name of definitions' (8pot), are contained the straight lines; which are so closely connected with
assumption of such notions as the point, line, &c.. the common arithmetical operations of multipliand'a'number of verbal explanations. Then fol- cation and division, that a student or a teacher
low, under the name of postulates or demands who is not fully alive to the existence and diffi(aizTa'ra), all that it is thought necessary to culty of incommensurables is apt to think that'
state as assumed in geometry. There are six common arithmetic would be as rigorous as geopostulates, three of which restrict the amount of metry. Euclid knew-better.
construction granted to the joining two points The third book is devoted to the consideration
by a straight line, the indefinite lengthening of a of the properties of the circle, and is much cramped
terminated straight line, and the drawing of a in several places by the imperfect idea already alcircle'with a given centre, and a given distance luded to, which Euclid took of an angle. There;
mieasuied from that'centre as a radius; the other are some places in which he clearly drew upon
three'assume the equality of all right angles, the, experimental knowledge of the form of a circle,
much disputed property of two lines, which meet
a third at angles less than two right angles (we * See Penny Cyclopaedia, art. "c Parallels," for
mean, of course, much disputed as to its propriety some account of this well-worn subject.