THE FOMISMS 
43 7 
restoration of Euclid’s work, Chasles’s Porisrns cannot be re 
garded as satisfactory. One consideration alone is, to my 
mind, conclusive on this point. Chasles made ‘ porisrns 3 out 
of Pappus’s various lemmas to Euclid’s porisrns and com 
paratively easy deductions from those lemmas. Now we 
have experience of Pappus’s lemmas to books which still 
survive, e.g. the Conics of Apollonius; and, to judge by these 
instances, his lemmas stood in a most ancillary relation to 
the propositions to which they relate, and do not in the 
least compare with them in difficulty and importance. Hence 
it is all but impossible to believe that the lemmas to the 
porisrns were themselves porisrns such as were Euclid’s own 
porisrns ; on the contrary, the analogy of Pappus’s other sets 
of lemmas makes it all but necessary to regard the lemmas in 
question as merely supplying proofs of simple propositions 
assumed by Euclid without proof in the course of the demon 
stration of the actual porisrns. This being so, it appears that 
the problem of the complete restoration of Euclid’s three 
Books still awaits a solution, or rather that it will never be 
solved unless in the event of discovery of fresh documents. 
At Hie same time the lemmas of Pappus to the Porisrns 
are by no means insignificant propositions in themselves, 
and, if the usual relation of lemmas to substantive proposi 
tions holds, it follows that the Porisrns was a distinctly 
advanced work, perhaps the most important that Euclid ever 
wrote ; its loss is therefore much to be deplored. Zeuthen 
has an interesting remark à propos of the proposition which 
Pappus quotes as the first proposition of Book I, ‘ If from two 
given points straight lines be drawn meeting on a straight 
line given in position, and one of them cut off from a straight 
line given in position (a segment measured) towards a given 
point on it, the other will also cut off from another (straight 
line a segment) bearing to the first a given ratio.’ This pro 
position is also true if there be substituted for the first given 
straight line a conic regarded as the ‘locus with respect to 
four lines ’, and the proposition so extended can be used for 
completing Apollonius’s exposition of that locus. Zeuthen 
suggests, on this ground, that the Porisrns were in part by 
products of the theory of conics and in part auxiliary means 
for the study of conics, and that Euclid called them by the