434
EUCLID
the two senses of the word rropLaya. The first is that of
a corollary, where something appears as an incidental result
of a proposition, obtained without trouble or special seeking,
a sort of bonus which the investigation has presented us
with. 1 The other sense is that of Euclid’s Porisms. In
this sense
‘ porism is the name given to things which are sought, but
need some finding and are neither pure bringing into existence
nor simple theoretic argument. For (to prove) that the angles
at the base of isosceles triangles are equal is matter of theoretic
argument, and it is with reference to things existing that sucli
knowledge is (obtained). But to bisect an angle, to construct
a triangle, to cut off, or to place—all these things demand the
making of something; and to find the centre of a given circle,
or to find the greatest common measure of two given commen
surable magnitudes, or the like, is in some sort intermediate
between theorems and problems. For in these cases there is
no bringing into existence of the things sought, but finding
of them; nor is the procedure purely theoretic. For it is
necessary to bring what is sought into view and exhibit it
to the eye. Such are the porisms which Euclid wrote and
arranged in three books of Porisms.’ 2
Proclus’s definition thus agrees well enough with the first,
the ‘ older ’, definition of Pappus. A porism occupies a place
between a theorem and a problem; it deals with something
already existing, as a theorem does, but has to find it (e.g. the
centre of a circle), and, as a certain operation is therefore
necessary, it partakes to that extent of the nature of a problem,
which requires us to construct or produce something not
previously existing. Thus, besides III. 1 and X, 3, 4 of the
Elements mentioned by Proclus, the following propositions are
real porisms: III. 25, VI. 11-13, YU. 33, 34, 36, 39, VIII. 2, 4,
X. 10, XIII. 18. Similarly, in Archimedes's On the sphere and
Cylinder, I. 2-6 might be called porisms.
The enunciation given by Pappus as comprehending ten of
Euclid’s propositions may not reproduce the form of Euclid’s
enunciations; but, comparing the result to be proved, that
certain points lie on straight lines given in position, with the
class indicated by II above, where the question is of such and
such a point lying on a straight line given in position, and
1 Proclus on Each I, pp. 212. 14; 301. 22. 2 lb., p. 301. 25 sq.