the poarms 
4 35 
with other classes, e. g. (V) that such and such a line is given 
in position, (VI) that such and such a line verges to a given point, 
(XXVII) that there exists a given point such that straight 
lines drawn from it to such and such (circles) will contain 
a triangle given in species, we may conclude that a usual form 
of a porism was ‘ to prove that it is possible to find a point 
with such and sucli a property ’ or ‘ a straight line on which 
lie all the points satisfying given conditions and so on. 
The above exhausts all the positive information which we 
have about the nature of a porism and the contents of Euclid's 
Porism s. It is obscure and leaves great scope for speculation 
and controversy ; naturally, therefore, the problem of restoring 
the Porisms has had a great fascination for distinguished 
mathematicians ever since the revival of learning. But it lias 
proved beyond them all. Some contributions to a solution have, 
it is true, been made, mainly by Simson and Chasles. The first 
claim to have restored the Porisms seems to be that of Albert 
Girard (about 1590—1633), who spoke (1626) of an early pub 
lication of his results, which, however, never saw the light. 
The great Fermat (1601-65) gave his idea of a ‘porism’, 
illustrating it by five examples which are very interesting in 
themselves 1 ; but he did not succeed in connecting them with 
the description of Euclid’s Porisms by Pappus, and, though he 
expressed a hope of being able to produce a complete restoration 
of the latter, his hope was not realized. It was left for Robert 
Simson (1687-1768) to make the first decisive step towards the 
solution of the problem. 1 2 He succeeded in explaining the mean 
ing of the actual porisms enunciated in such general terms by 
Pappus. In his tract on Porisms he proves the first porism 
given by Pappus in its ten different cases, which, according to 
Pappus, Euclid distinguished (these propositions are of the 
class connected with loci)', after this he gives a number of 
other propositions from Pappus, some auxiliary proposi 
tions, and some 29 ‘ porisms ’, some of which are meant to 
illustrate the classes I, VI, XV, XXVII-XXIX distin 
guished by Pappus. Simson was able to evolve a definition 
of a porism which is perhaps more easily understood in 
Chasles’s translation : ‘ Le porisme est une proposition dans 
1 Œuvres de Fermat, eel. Tannery and Henry, I, p. 76-84. 
2 Robert! Simson Opera quaedam reliqua, 1776, pp. 315-594. 
F f 2