FORMAL DIVISIONS OF A PROPOSITION 371 
In particular cases some of these formal divisions may be 
absent, but three are always found, the enunciation, proof 
and conclusion. Thus in many propositions no construction 
is needed, the given figure itself sufficing for the proof; 
again, in IV. 10 (to construct an isosceles triangle with each 
of the base angles double of the vertical angle) we may, in 
a sense, say with Proclus 1 that there is neither setting-out nor 
definition, for there is nothing given in the enunciation, and 
we set out, not a given straight line, but any straight line AB, 
while the proposition does not state (what might be said by 
way of definition) that the required triangle is to have AB for 
one of its equal sides. 
((3) The Siopiargos or statement of conditions of possibility. 
« 
Sometimes to the statement of a problem there has to be 
added a Siopurpos in the more important and familiar sense of 
a criterion of the conditions of possibility or, - in its most 
complete form, a criterion as to ‘ whether what is sought 
is impossible or possible and how far it is practicable and in 
how many ways ’. 2 Both kinds of SiopLcrpos begin with the 
words <5et Srj, which should be translated, in the case of the 
definition, ‘ thus it is required (to prove or do so and so) ’ and, 
in the case of the criterion of possibility, ‘ thus it is necessary 
that . . (not ‘ but it is necessary . . .’). Cf. I. 22, ‘ Out of 
three straight lines which are equal to three given straight 
lines to construct a triangle: thus it is necessary that two 
of the straight lines taken together in any manner should be 
greater than the remaining straight line 
(y) Analysis, synthesis, reduction, reductio ad absurdum. 
The Elements is a synthetic treatise in that it goes directly 
forward the whole way, always proceeding from the known 
to the unknown, from the simple and particular to the more 
complex and general; hence analysis, which reduces the 
unknown or the more complex to the known, has no place 
in the exposition, though it would play an important part in 
the discovery of the proofs. A full account of the Greek 
analysis and synthesis will come more conveniently elsewhere. 
1 Proclus on Eucl. I, p. 203. 28 sq. 
B b 2 
2 lb., p. 202. 3.