# Extensive Definition

The subject of porisms is perplexed by the
multitude of different views which have been held by geometers as to what a porism
really was and is.

The treatise which has given rise to the
controversies on this subject is the Porisms of Euclid, the author
of the Elements.
For as much as we know of this lost treatise we are indebted to the
Collection of Pappus
of Alexandria, who mentions it along with other geometrical
treatises, and gives a number of lemmas
necessary for understanding it. Pappus states that the porisms of
Euclid are neither theorems nor problems, but are in some sort
intermediate, so that they may be presented either as theorems or
as problems; and they were regarded accordingly by many geometers,
who looked merely at the form of the enunciation, as being actually
theorems or problems, though the definitions given by the older
writers showed that they better understood the distinction between
the three classes of propositions.

The older geometers regarded a theorem as
directed to proving what is proposed, a problem as directed to
constructing what is proposed, and finally a porism as directed to
finding what is proposed (). Pappus goes on to say that this last
definition was changed by certain later geometers, who defined a
porism on the ground of an accidental characteristic as , that
which falls short of a locus-theorem by a (or in its) hypothesis.
Proclus points out that the word was used in two senses. One sense
is that of "corollary," as a result unsought, as it were, but seen
to follow from a theorem. On the "porism" in the other sense he
adds nothing to the definition of "the older geometers" except to
say (what does not really help) that the finding of the center of a
circle and the finding of the greatest common measure are porisms
(Proclus,
ed. Friedlein, p.301).

Pappus gives a complete enunciation of a porism
derived from Euclid, and an extension of it to a more general case.
This porism, expressed in modern language, asserts that —
given four straight lines of which three turn about the points in
which they meet the fourth, if two of the points of intersection of
these lines lie each on a fixed straight line, the remaining point
of intersection will also lie on another straight line. The general
enunciation applies to any number of straight lines, say (n+1), of
which n can turn about as many points fixed on the (n+1)th. These n
straight lines cut, two and two, in 1/2n(n-1) points, 1/2n(n-1)
being a triangular number whose side is (n-1). If, then, they are
made to turn about the n fixed points so that any (n-1) of their
1/2n(n-1) points of intersection, chosen subject to a certain
limitation, lie on (n-1) given fixed straight lines, then each of
the remaining points of intersection, 1/2n(n-1)(n-2) in number,
describes a straight line. Pappus gives also a complete enunciation
of one porism of the first book of Euclid's treatise.

This may be expressed thus: If about two fixed
points P, Q we make turn two straight lines meeting on a given
straight line L, and if one of them cut off a segment AM from a
fixed straight line AX, given in position, we can determine another
fixed straight line BY, and a point B fixed on it, such that the
segment BM' made by the second moving line on this second fixed
line measured from B has a given ratio X to the first segment AM.
The rest of the enunciations given by Pappus are incomplete, and he
merely says that he gives thirty-eight lemmas for the three books
of porisms; and these include 171 theorems. The lemmas which Pappus
gives in connexion with the porisms are interesting historically,
because he gives:

- the fundamental theorem that the cross or an harmonic ratio of a pencil of four straight lines meeting in a point is constant for all transversals
- the proof of the harmonic properties of a complete quadrilateral
- the theorem that, if the six vertices of a hexagon lie three and three on two straight lines, the three points of concourse 9f opposite sides lie on a straight line

Robert
Simson was the first to throw real light upon the subject. He
first succeeded in explaining the only three propositions which
Pappus indicates with any completeness. This explanation was
published in the Philosophical Transactions in 1723. Later he
investigated the subject of porisms generally in a work entitled De
porismatibus traclatus; quo doctrinam porisrnatum satis explicatam,
et in posterum ab oblivion tutam fore sperat auctor, and published
after his death in a volume, Roberti Simson opera quaedam reliqua
(Glasgow, 1776).

Simson's treatise, De porismatibus, begins with
definitions of theorem, problem, datum, porism and locus.
Respecting the porism Simson says that Pappus's definition is too
general, and therefore he will substitute for it the
following:

"Porisma est propositio in qua proponitur
demonstrate rem aliquam vel plures Batas ease, cui vel quibus, ut
et cuilibet ex rebus innumeris non quidem datis, sed quae ad ea
quae data sunt eandem habent relationem, convenire ostendendum est
affectionem quandam communem in propositione descriptam. Porisma
etiam in forma problematis enuntiari potest, si nimirum ex quibus
data demonstranda aunt, invenienda proponantur."

A locus (says Simson) is a species of porism.
Then follows a Latin translation of Pappus's note on the porisms,
and the propositions which form the bulk of the treatise. These are
Pappus's thirty-eight lemmas relating to the porisms, ten cases of
the proposition concerning four straight lines, twenty-nine
porisms, two problems in illustration and some preliminary
lemmas.

John
Playfair's memoir (Trans. Roy. Soc. Edin., 1794, vol. iii.), a
sort of sequel to Simson's treatise, had for its special object the
inquiry into the probable origin of porisms, that is, into the
steps which led the ancient geometers to the discovery of them.
Playfair remarked that the careful investigation of all possible
particular cases of a proposition would show that (1) under certain
conditions a problem becomes impossible; (2) under certain other
conditions, indeterminate or capable of an infinite number of
solutions. These cases could be enunciated separately, were in a
manner intermediate between theorems and problems, and were called
"porisms." Playfair accordingly defined a porism thus: "A
proposition affirming the possibility of finding such conditions as
will render a certain problem indeterminate or capable of
innumerable solutions."

Though this definition of a porism appears to be
most favoured in England, Simson's view has been most generally
accepted abroad, and had the support of Michel
Chasles. However, in Liouville's
Journal de mathematiques pures et appliquées (vol. xx., July,
1855), P.
Breton published Recherches nouvelles sur les porismes
d'Euclide, in which he gave a new translation of the text of
Pappus, and sought to base thereon a view of the nature of a porism
more closely conforming to the definitions in Pappus. This was
followed in the same journal and in La Science by a controversy
between Breton and A. J. H.
Vincent, who disputed the interpretation given by the former of
the text of Pappus, and declared himself in favour of the idea of
Schooten, put forward in his Mathematicae exercitationes (1657), in
which he gives the name of "porism" to one section. According to
Frans
van Schooten, if the various relations between straight lines
in a figure are written down in the form of equations or
proportions, then the combination of these equations in all
possible ways, and of new equations thus derived from them leads to
the discovery of innumerable new properties of the figure, and here
we have "porisms."

The discussions, however, between Breton and
Vincent, in which C. Housel also
joined, did not carry forward the work of restoring Euclid's
Porisms, which was left for Chasles. His work (Les Trois livres de
porismes d'Euclide, Paris, 1860) makes full use of all the material
found in Pappus. But we may doubt its being a successful
reproduction of Euclid's actual work. Thus, in view of the
ancillary relation in which Pappus's lemmas generally stand to the
works to which they refer, it seems incredible that the first seven
out of thirty-eight lemmas should be really equivalent (as Chasles
makes them) to Euclid's first seven Porisms. Again, Chasles seems
to have been wrong in making the ten cases of the four-line Porism
begin the book, instead of the intercept-Porism fully enunciated by
Pappus, to which the "lemma to the first Porism" relates
intelligibly, being a particular case of it.

An interesting hypothesis as to the Porisms was
put forward by H.
G. Zeuthen (Die Lehre von den Kegelschnitten im Altertum, 1886,
ch. viii.). Observing, e.g., that the intercept-Porism is still
true if the two fixed points are points on a conic, and the
straight lines drawn through them intersect on the conic instead of
on a fixed straight line, Zeuthen conjectures that the Porisms were
a by-product of a fully developed projective geometry of conics. It
is a fact that Lemma 31 (though it makes no mention of a conic)
corresponds exactly to Apollonius's
method of determining the foci of a central conic (Conics, iii.
4547 with 42). The three porisms stated by Diophantus in
his Arithmetica are propositions in the theory of numbers which can
all be enunciated in the form "we can find numbers satisfying such
and such conditions"; they are sufficiently analogous therefore to
the geometrical porism as defined in Pappus and Proclus.

## See also

### References

A valuable chapter on porisms (from a philological standpoint) is
included in
J. L. Heiberg's Litterargeschichtliche Studien über Euklid
(Leipzig, 1882) ; and the following books or tracts may also be
mentioned:

- August Richter, Porismen nach Simson bearbeitet (Elbing, 1837)
- M. Cantor, "Über die Porismen des Euklid and deren Divinatoren," in Schlomilch's Zeitsch. f. Math. u. Phy. (1857), and Literaturzeitung (1861), p. 3 seq.
- Th. Leidenfrost, Die Porismen des Euklid (Programm der Realschule zu Weimar, 1863)
- Fr. Buch-binder, Euclids Porismen und Data (Programm der kgl. Landesschule Pforta, 1866).