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