THE POT? I SMS
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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. \
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