432 •
EUCLID
respect of its hypothesis. Of this kind of porisms loci are
a species, and they abound in the Treasury of Analysis; but
this species has been collected, named, and handed down
separately from the porisms, because it is more widely diffused
than the other species] ... But it has further become charac
teristic of porisms that, owing to their complication, the enun
ciations are put in a contracted form, much being by usage
left to be understood; so that many geometers understand
them only in a partial way and are ignorant of the more
essential features of their content.
c [Now to comprehend a number of propositions in one
enunciation is by no means easy in these porisms, because
Euclid himself has not in fact given many of each species, but
chosen, for examples, one or a few out of a great multitude.
But at the beginning of the first book he has given some pro
positions, to the number of ten, of one species, namely that
more fruitful species consisting of loci.] Consequently, finding
that these admitted of being comprehended in our enunciation,
we have set it out thus :
If, in a system of four straight lines which cut one
another two and two, three points on one straight line
be given, while the rest except one lie on different straight
lines given in position, the remaining point also will lie
on a straight line given in position.
‘ This has only been enunciated of four straight lines, of
which not more than two pass through the same point, but it
is not known (to most people) that it is true of any assigned
number of straight lines if enunciated thus:
If any number of straight lines cut one another, not
more than two (passing) through the same point, and all
the points (of intersection situated) on one of them be
given, and if each of those which are on another (of
them) lie on a straight line given in position—
or still more generally thus :
if any number of straight lines cut one another, not more
than two (passing) through the same point, and all the
points (of intersection situated) on one of them be given,
while of the other points of intersection in multitude
equal to a triangular number a number corresponding
to the side of this triangular number lie respectively on
straight lines given in position, provided that of these
latter points no three are at the angular points of a
triangle (sc. having for sides three of the given straight