P O R P O H 519 between which and the toes boughs and other objects can be firmly grasped as with a hand. The last genus is Chsetomys, distinguished by the shape of its skull and the greater complexity of its teeth. It contains only one species, C. subspinosus, a native of the hottest parts of Brazil. PORDENONE, IL (1483-1539), whose correct name was GIOVANNI ANTONIO LICINIO, or LICINO, was an eminent painter of the Venetian school. He was commonly named 11 Pordenone from having been born in 1483 at Corticelli, a village near Pordenone, a city of Italy, in the province of Udine (Friuli). He himself ultimately dropped the name of Licinio, having quarrelled with his brothers, one of whom had wounded him in the hand ; he then called him self Regillo, or De Regillo. His signature runs " Antonius Portunaensis," or " De Portunaonis." He was created a cavaliere by Charles V. As a painter Licinio was a scholar of Pellegrino da S. Daniele, but the leading influence which governed his style was that of Giorgione ; the popular story that he was a fellow-pupil with Titian under Giovanni Bellini is incorrect. The district about Pordenone had been some what 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 affect 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 else where in Friuli, and in Cremona and Venice as well ; 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 con nexion 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 with out suspicion of poison. His latest works are compara tively 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 II Sacchiense, his son-in- law Pomponio Amalteo, and Giovanni Maria Calderari. _ The following may be named among Pordenone s works : the Sicture of S. Luigi Giustiniani and other Saints, originally in S. laria dell Orto, Venice ; a Madonna and Saints, 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. PORIFERA. See SPONGES. PORISM. The subject of porisms is perplexed by the multitude of different views which have been held by famous geometers as to what a porism really was and is. This article must therefore be limited to a short historical account (1) of the principal works of the Greek mathe maticians which we know to have been called Porisms, and ^ (2) of some of the principal contributions to the elucidation of these works, and conjectures as to the true signification of the term. 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 enuncia tion, 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, namely, defined a theorem as TO TrpoTetvo/zevov ei? a.7r68eiiv avrov TOV , a problem as TO Trpo/3aXX6/j.evov ei s KO.TO.- avrov TOV TrpoTeivo/xevou, and finally a porism as TO TrpoTecvo/j.evov ets iropurfjiuv OLVTOV TOV TrpoTetvo/xevou. 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 TO AeiTrov VTroOecrei TOTTIKOV Oewp i i/JLaTos. Proclus gives a definition of a porism which agrees very well with the fact that Euclid used the same word Trop 107/0. in his Elements for what is now called by the Latin name "corollary." Proclus s definition is To 8e 7ro ptcrp;a AeyeTou jj.lv CTTi Trpoflrj/j.dT(DV TLVWV, ofov TO. Eu/cAei<5e6 TTopicr/zaTa. AeyeTai Se t oYws, orai/ e/< TWV aTro ciAAo Ti fo-wa^avT} [crwaTro^aj/^ (?)] Oewprjfj.a, fj.r) irpoOe- /xeFwv rj/jitiiv, o KGU SLOL TOVTO Tro picr/za KKAr;Kacrt wcrTrep TI KepSos ov TT}S eTTio-TT^oyiKT/s aTroSei^fcos Trapepyov (Prod., Comment. End., p. 58 ; cf. p. 80). 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 ivhich three turn about the points in which they meet the fourth, if two of the points of inter section of these lines lie each on a fixed straight line, the remaining point of intersection will also lie on another straight line ; or, If the sides of a triangle are made to turn each about one of three fixed points in a straight line, and if two of the vertices are made to move on two fixed straight lines, taken arbitrarily, the third vertex describes a third 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 (re+l)th. These n n(n - 1 ) n(n - 1 ) straight lines cut, two and two, in ^ points, ^^ -J *J being a triangular number whose side is (n - 1 ). If, then, they are made to turn about the n fixed points so that j7{% 1 ) any (n - 1 ) of their --^ = - points of intersection lie on (n - 1 ) given fixed straight lines, then each of the remaining (n - }(n - 2) points of intersection, - in number, describes a straight line. Pappus gives also a complete enuncia tion 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 A 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 (1)