LOST GEOMETRICAL WORKS 
431 
We now come to the lost works belonging to higher 
geometry. The most important was evidently 
(J3) The For isms. 
Our only source of information about the nature and con 
tents of the Porisms is Pappus. In his general preface about 
the books composing the Treasury of Analysis Pappus writes 
as follows 1 (I put in square brackets the words bracketed by 
Hultsch). 
‘ After the Tangencies (of Apollonius) come, in three Books, 
the Porisms of Euclid, a collection [in the view of many] most 
ingeniously devised for the analysis of the more weighty 
problems, [and] although nature presents an unlimited num 
ber of such porisms, [they have added nothing to what was 
originally written by Euclid, except that some before my time 
have shown their want of taste by adding to a few (of the 
propositions) second proofs, each (proposition) admitting of 
a definite number of demonstrations, as we have shown, and 
Euclid having given one for each, namely that which is the 
most lucid. These porisms embody a theory subtle, natural, 
necessary, and of considerable generality, which is fascinating 
to those who can see and produce results]. 
‘ Now all the varieties of porisms belong, neither to theorems 
nor problems, but to a species occupying a sort of intermediate 
position [so that their enunciations can be formed like those of 
either theorems or problems], the result being that, of the great 
number of geometers, some regarded them as of the class of 
theorems, and others of problems, looking only to the form of 
the proposition. But that the ancients knew better the differ 
ence between these three things is clear from the definitions. 
For they said that a theorem is that which is proposed with a 
view to the demonstration of the very thing proposed, a pro 
blem that which is thrown out with a view to the construction 
of the very thing proposed, and a porism that which is pro 
posed with a view to the producing of the very thing proposed. 
[But this definition of the porism was changed by the more 
recent writers who could not produce everything, but used 
these elements and proved only the fact that that which is 
sought really exists, but did not produce it, and were accord 
ingly confuted by the definition and the whole doctrine. They 
based their definition on an incidental characteristic, thus : 
A porism is that which falls short of a locus-theorem in 
1 Pappus, vii, pp. 648-60.