**PORISM.** 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 (εἰς
πορισμὸν αὐτοῦ τοῦ προτεινομένου). 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 τὸ λεῖπον ὑποθέσει τοπικοῦ θεωρήματος, 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 12*n*(*n*−1) points, in (*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 12*n* (*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, 12 (*n*−1) (*n*−2) in number, describes 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 λ to the first segment AM. The rest of the enunciation's 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 (1) 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; (2) the proof of the harmonic properties of a complete quadrilateral; (3) the theorem that, if the six vertices of a hexagon lie three and three on two straight lines, the three points of concourse of opposite sides lie on a straight line. During the last three centuries this subject seems to have had
great fascination for mathematicians, and many geometers have
attempted to restore the lost porisms. Thus Albert Girard says in
his *Traité de trigonométrie* (1626) that he hopes to publish a restoration.
About the same time P. de Fermat wrote a short work under
the title *Porismatum euclidaeorum renovata doctrina et sub forma*
*isagoges recentioribus geometris exhibita* (see *Oeuvres de Fermat*, i.,
Paris, 1891); but two at least of the five examples of porisms which
he gives do not fall within the classes indicated by Pappus. 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*
*tractatus; quo doctrinam porismatum satis explicatam, et in posterum*
*ab oblivione 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 datas esse, 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 affection em quandam communem in propositione
descriptam. Porisma etiam in forma problematis enuntiari potest,
si nimirum ex quibus data demonstranda sunt, 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 has
the support of the great authority of Michael Chasles. However,
in *Liouville's Journal de mathémaliques 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 F. 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 ease 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. 45-47 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.

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: Aug. Richter, *Porismen nach Simson bearbeitet*
(Elbin, 1837); M. Cantor, “Ueber die Porismen des Euklid und
deren Divinatoren,” in *Schlömilch'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. Buchbinder,
*Euclids Porismen und Data* (Programm der kgl. Landesschule
Pforta, 1866). (T. L. H.)