THE POT? I SMS 
483 
lines)—each of the remaining points will lie on a straight 
line given in position. 1 
‘ It is probable that the writer of the Elements was not 
unaware of this, but that he only set out the principle; and 
he seems, in the ease of all the porisms, to have laid down the 
principles and the seed only [of many important things], 
the kinds of which should be distinguished according to the 
differences, not of their hypotheses, but of the results and 
the things sought. [All the hypotheses are different from one 
another because they are entirely special, but each of the 
results and things sought, being one and the same, follow from 
many different hypotheses.] 
‘ We must then in the first book distinguish the following 
kinds of things sought: 
‘ At the beginning of the book is this proposition: 
I. If from two given 'points straight lines he drawn 
meeting on a straight line given in position, and one cut 
off from a straight line given in position {a segment 
measured) to a given point on it, the other will also cut 
off from another {straight line a segment) having to the 
first a; given ratio. 
‘ Following on this (we have to prove) 
II. that such and such a point lies on a straight line 
given in position; 
III. that the ratio of such and such a pair of straight 
lines is given ’; 
Ac. Ac. (up to XXIX). 
‘The three books of the porisms contain 38 lemmas; of the 
theorems themselves there are 171/ 
Pappus further gives lemmas to the Porisms. 2 
With Pappus’s account of Porisms must be compared! the 
passages of Proclus on the same subject. Proclus distinguishes 
1 Loria (Le scienze esatte nell'antica Grecia, pp. 256-7) gives the mean 
ing of this as follows, pointing out that Simson first discovered it: ‘If 
a complete «-lateral be deformed so that its sides respectively turn about 
n points on a straight line, and (« — 1) of its \n{n — 1) vertices move on 
as many straight lines, the other -|(n —1) (n — 2) of its vertices likewise 
move on as many straight lines: Tut it is necessary that it should be 
impossible to form with the (« — 1) vertices any triangle having for sides 
the sides of the polygon.’ 
2 Pappus, vii, pp. 866-918; Euclid, ed. Heiberg-Menge, vol. viii, 
pp. 243-74. \ 
F f