TECHNICAL TERMS 
373 
as Proclus says, 1 The name porism was also applied to a 
special kind of substantive proposition, as in Euclid’s separate 
work in three Books entitled Porisms (see below, pp. 431-8). 
The word lemma (Xrjjxpa) simply means something assumed. 
Archimedes uses it of what is now known as the Axiom of 
Archimedes, the principle assumed by Eudoxus and others in 
the method of exhaustion; but it is more commonly used 
of a subsidiary proposition requiring proof, which, however, 
it is convenient to assume in the place where it is wanted 
in order that the argument may not be interrupted or unduly 
lengthened. Such a lemma might be proved in advance, but 
the proof was often postponed till the end, the assumption 
being marked as something to be afterwards proved by some 
such words as ¿y e£fjs SeixOgaerai, ‘ as will be proved in due 
course ’. 
Analysis of the Elements. 
Book I of the Elements necessarily begins with the essential 
preliminary matter classified under the headings Definitions 
(6poL), Postulates (a/r^/iara) and Common Notions (kolvolI 
evvoLai). In calling the axioms Common Notions Euclid 
followed the lead of Aristotle, who uses as alternatives for 
‘ axioms ’ the terms ‘ common (things) ’, ‘ common opinions ’. 
Many of the Definitions are open to criticism on one ground 
or another. Two of them at least seem to be original, namely, 
the definitions of a straight line_(4) and of a plane surface (7); 
unsatisfactory as these are, they seem to be capable of a 
simple explanation. The definition of a straight line is 
apparently an attempt to express, without any appeal to 
sight, the sense of Plato’s definition ‘ that of which the middle 
covers the ends ’ (sc. to an eye placed at one end and looking 
along it); and the definition of a plane surface is an adaptation 
of the same definition. But most of the definitions were 
probably adopted from earlier text-books; some appear to be 
inserted merely out of respect for tradition, e. g. the defini 
tions of oblong, rhombus, rhomboid, which are never used 
in the Elements. The definitions of various figures assume 
the existence of the thing defined, e. g. the square, and the 
lb., p. 212. 16.