PYTHAGOKEANISM. 584 PYTHIAS. proof of the proposition known by his nnme (Euclid i. 47: see IIvi-otenvse) . The scliool was foiicerned chiefly with the questions 'liow mnny' and 'how great,' i.e. with number and maj;nitude, and the Pythagorean geometry is mostly con- cerned with those relations of areas, volumes, and lines which admit of arithmetical expres- sion. Geometry was to them a means for inves- tigation in the' theory of numbers. This is seen in the remarks concerning gnomon-ntmibers. Among the Pythagoreans a square out of which a corner was cut in the shape of a square was called a gnomon. The gnomon-number of the Pythagoreans is in + 1, since the square on n can be made equal to the square on n + 1 by adding the scpiare 1 -1 and the two rectangles 1 •», we then have )r -f 2» + 1 = (n -f 1) =. Ex- pressions like plane and solid numbers used for the contents of sjiatial magnitudes of two and three dimensions also serve to indicate the con- stant tendency to objectify mathematical thought by means of geometry. The knowledge of the Pythagoreans in the field of elementary series was quite comprehensive (see Series), and the three proi)ortions. arithmetical, geometrical, and harnionical, were known to them. The so-called most perfect or mtisical proportions, e.g. (5:8^ 9 : 12, was invented by the Babylonians and is said to have been first brought to Greece by Pytjiagoras. By im))rovement in definition, by systematization. and by the use of deduction, the study of geometry at the hands of the Pytha- goreans was made a factor of liberal education. PYTHAGOREAN PROPOSITION. See Pytiiagoh.vs ; Hypotexuse. PYTH'EAS (Lat., from Gk. nuS^as). A Greek navigator, born at ilassilia (Marseilles) in the fourth century B.C. He is said to have sailed around the west coast of Europe, and through the English Channel to Thule, the most northern land known to the ancients, probably ^Mainland, the largest of the Shetland Islands. Nothing is accurately known about his life. He gave an account of his first voyage in his Description of the Ocean, in which he stated that he traveled through Britain, and that its eirciunference was over 40,000 stadia. In regard to the island of Thule, he said there was neither air, nor land, nor sea, but a composition of all of them, in which the whole universe was suspended. This substance, which could not be penetrated by land or sea, he had himself seen, and was told that it was "a connecting link of the universe." He puts Thule six days' sail from Britain. He .says that the sun never sets during the summer sol- stice in Thule. On a second voyage he skirted the shore of Europe, from Cadiz to a river which he called Tanais. This may have been the Elbe, which he may have confused with the Don, the classical name of which was Tanais. He was also a mathematician and astronomer, and was the first to determine the meridian altitude of the sun at the summer solstice by the use of a gnomon. The fragments of his writings were collected and published by Arvedson at Upsala in 1824. Polybius and Strabo speak of him con- temptuously, but modern geographers are more favorable in their judgment. It is supposed that he was sent out by the ]Massilians for the purpose of increasing their commercial connections. PYTHaA (Lat.. from Gk. Tlveia). The priest- ess of Apollo at Delphi. See Delphi. PYTHIAN GAMES (Lat. J'ythia, from Gk. UvSla). The second of the four great national festivals of the Greeks, held in the Crissa-an Plain, near Delphi. Their origin was attributed to Apollo, in celebration of his destruction of the dragon Python. At first they were celebrated under the superintendence of the priests of Del- phi every ninth year, i.e. once in each cycle, and consisted solely of a musical contest between singers to the cithara. After the first Sacred War, the character of the festival was changed, and the Amphictyons assumed charge. The first of the new series was held in n.c. 580, but it was not till the second celebration in B.C. 582 that the laurel wreath was given as a prize, and from this date the Pythian series was reckoned. They were from this time held in the summer of the third year of each Olympiad, probably in August, and seem usually to have occupied four days. The first day was occui)ied with the musical contests which always held the chief place. Among them the most important was the Pythian Nomos, a solo on the double ilute. which represented the victory of Apollo over the dragon. On the second day came athletic games, much like those at Olympia (q.v.), and on the third the horse-racing. The latter contests were held in the Crisssean plain. On the fourth day seem to have come the festival procession and sacrifices. The mtisical contests were increased in later times ; even poets and historians competed. Con- sult: Krause, Die Pythicn, yemecn mid Isth- mien- (Leipzig, 1841); Mommsen, Dciphiku (ib., 1878) ; Schijmann-Lipsius, (Iricchische Alter- turner, vol. ii. (Berlin, 11102); Stengel, "Grie- chische Kultus-Altertumer," in iliiller's Band- hiich der klas.sischen Altertumswissenschaft, vol. V. (Munich, 1808). PYTHIAN ODES. See Pindar. PYTH'IAS. See Damon and Phintias. PYTHIAS, Knights of. A fraternal and beneficial order founded in Washington, D. C, in 1804. by Justus H. Rathbone and five associ- ates. Its objects are the practice of friendship, benevolence, and charity toward the members. It is asserted that the ritual, which is mainly Rathbone's work, is purely American. Its most binding obligation is complete and absolute secrecy. The first lodge instituted was Washing- ton Lodge. A Grand Lodge for the District of Co- lumbia was created on April 12th, and the work of creating subordinate lodges began with the institution of Franklin Lodge No. 2. In 1868 a new constitution was adopted at Washington under which was organized, as the central gov- erning body, the Su])rcme Lodge Knights of Pythias of the World. The growth of the order was not rapid in the earlier years of its exist- ence. In 186() it had a membership of only about 380. In 1867 there were only 074 on the roll, but when the twentieth year was reached the mem- bership had increased to more than 100,000, while in 189.5 the membership was over 450,000, and at the end of 1902. 587..506. The endowment or insurance branch was estab- lished in 1877. This branch is under the man- agement of a president and board of officers, hut is subject to the control of the Su])reme Lodge. The members of this branch enjoy no greater privileges than other members outside of the insurance which they pay for. It had at the close of 1902 a total membership of nearly
