about Pordenone had been somewhat fertile in capable painters; but Licinio excelled them all in invention and design, and more especially in the powers of a vigorous chiaroscurist and flesh painter. Indeed, so far as mere flesh-painting is concerned he was barely inferior to Titian in breadth, pulpiness and tone; and he was for a while the rival of that great painter in public regard. The two were open enemies, and Licinio would sometimes afiect to wear arms while he was painting. He excelled Giorgione in light and shade and in the effect of relief, and was distinguished in perspective and in portraits; he was equally at home in fresco and in oil-colour. He executed many works in Pordenone and elsewhere in Friuli, and in Cremona and Venice; at one time he settled in Piacenza, where is one of his most celebrated church pictures, “ St Catherine disputing with the Doctors in Alexandria ”; the figure of St Paul in connexion with this picture is his own portrait. He was formally invited by Duke Hercules II. of Ferrara to that court; here soon afterwards, in 1539, he died, not without suspicion of poison. His latest works are comparatively careless and superficial; and generally he is better in male figures than in female—the latter being somewhat too sturdy—and the composition of his subject pictures is scarcely on a level with their other merits. Pordenone appears to have been a vehement self-asserting man, to which his style as a painter corresponds, and his morals were not unexceptionable. Three of his principal scholars were Bernardino Licinio, named Il Sacchiense, his son-in-law Pomponio Amalteo, and Giovanni Maria Calderari.

The following may be named among Pordenone's works: the picture of “ S Luigi Giustiniani and other Saints,” originally in S Maria dell' Orto, Venice; a “ Madonna and Saints ” (both of these in the Venice academy); the “ Woman taken in Adultery,” in the Berlin museum; the “ Annunciation,” at Udine, regarded by Vasari as the artist's masterpiece, now damaged by restoration. In Hampton Court is a duplicate work, the “ Painter and his Family ”; and in Burghley House are two fine pictures now assigned to Pordenone—the “ Finding of Moses ” and the “ Adoration of the Kings.” These used to be attributed to Titian and to Bassano respectively.

**PORDENONE**, a town of the province-of Udine, Venetia, Italy, 30 m. W. by S. of Udine on the railway to Treviso. Pop. (1901), 8425 (town); 12,409 (commune). It was the birthplace of the painter generally known as Il Pordenone (*q.v.*). Paintings from his brush adorn the cathedral (which has a fine brick
campanile), and others are preserved in the gallery of the town
hall. Cotton industries are active, and silk and pottery are
manufactured.

**PORE**, a small opening or orifice, particularly used of the openings
of the ducts of the sweat-glands in the skin or of the stomata
in the epidermis of plants or those through which the pollen
or seed are discharged from anthers or seed capsules. The
word is an adaptation through the French from Lat. *porus*,
Gr. πόρος, passage. In the sense of to look closely at, to read
with persistent or close attention, “ pore ” is of obscure origin.
It would seem to be connected with “ peer, ” to look closely
into, and would point to an O. Eng. *purian* or *pyrian*. There
is no similar word in Old French.

**PORFIRIUS, PUBLILIUS OPTATIANUS**, Latin poet, possibly a native of Africa, flourished during the 4th century A.D. He
has been identified with Publilius Optatianus, who was *praefectus*
*urbi* (329 and 333), and is by some authorities included
amongst the Christian poets. For some reason he had been
banished, but having addressed a panegyric to the Emperor
Constantine the Great, he was allowed to return. Twenty-eight
poems are extant under his name, of which twenty were
included in the panegyric. They have no value except as
curiosities and specimens of perverted ingenuity. Some of
them are squares (the number of letters in each line being equal),
certain letters being lubricated so as to form a pattern or figure,
and at the same time special verses or maxims; others represent
various objects (a syrinx, an organ, an altar); others have
special peculiarities in each line (number of words or letters);
while the 28th poem (the *versus anacyclici*) may be read backwards
without any effect upon sense or metre. A complimentary
letter from the emperor and letter of thanks from the author are
also extant. The best edition of the poem is by L. Müller (1877).

See also O. Seeck, “ Das Leben des Dichters Porphyrius " in
*Rheinisches Museum* (1908), lxiii. 267.

**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 (els
ITU/JLO', U.6V ai/roi) ro0 1rpo'reL1/otiéi/ou). 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 -ro λe'Evro1/ 1§ 1ro0é<rel 'l'0(7|'LKOl'l 0ewp1j/.aa-ras, 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 ½*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 ½*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, ½ (*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.