373
EUCLID
In the meantime we may observe that, where a proposition
is worked out by analysis followed by synthesis, the analysis
comes between the definition and the construction of the
proposition; and it should not be forgotten that reductio ad
absurdum (called in Greek 77 ei’y to dSwarov duaycoyri,
‘reduction to the impossible’, or 77 Sid too dSvvdrov
or dnoSeL^Ls, ‘proof per impossible’), a method of proof
common in Euclid as elsewhere, is a variety of analysis.
For analysis begins with reduction (dTraycoyy) of the original
proposition, which we hypothetically assume to be true, to
something simpler which we can recognize as being either
true or false; the case where it leads to a conclusion known
to be false is the reductio ad absurdum..
(5) Case, objection, porism, lemma.
Other terms connected with propositions are the following.
A proposition may have several cases according to the different
arrangements of points, lines, &c., in the figure that may
result from variations in the positions of the elements given;
the word for case is 7ttcoctls. The practice of the great
geometers was, as a rule, to give only one case, leaving the
others for commentators or pupils to supply for themselves.
But they were fully alive to the existence of such other
cases; sometimes, if we may believe Proclus, they would even
give a proposition solely with a view to its use for the purpose
of proving a case of a later proposition which is actually
omitted. Thus, according to Proclus, 1 the second part of I. 5
(about the angles, beyond the base) was intended to enable
the reader to meet an objection (eWrao-iy) that might be
raised to I. 7 as given by Euclid on the ground that it was
incomplete, since it took no account of what was given by
Proclus himself, and is now generally given in our text-books,
as the second case.
What we call a corollary was for the Greeks a porism
(ivopiapa), i. e. something provided or ready-made, by which
was meant some result incidentally revealed in the course
of the demonstration of the main proposition under discussion,
a sort of incidental gain ’ arising out of the demonstration,
1 Proclus on Eucl. I, pp. 248. 8-11 ; 268. 4-8.
