exercises were performed. The common meal consisted principally of bread, honey and water. The remainder of the day was devoted to public and domestic affairs, conversation, bathing, and religious performances. After the destruction of the association, and the flight of his scholars from Lower Italy, Lysis and Archippus deemed it necessary to collect the doctrines of their master in a systematic treatise, and preserve them from oblivion; but the greatest secrecy was nevertheless recommended. Thus Plato purchased from Philolaus a writing on the philosophy of Pythagoras, and received from Archytas his commentaries on the verses and tenets of his master. The accounts that we possess of the doctrines of Pythagoras are very scanty, and, with the exception of what we learn from Aristotle, and from some fragments of the Pythagoreans, very uncertain. Neither can we accurately discriminate between his doctrines and those of his scholars. Later writes represent him as making it the object of all philosophy to exalt the mind to the contemplation of immutable truth, to the knowledge of divine and spiritual objects. This can only be effected by degrees, on account of familiarity with sensual things. The first step to wisdom is the study of mathematics, the foundation of which appeared to him to be the doctrine of numbers. Numbers are, in his view, the first and most essential of things. They are, as it were, the model, according to which the world is formed in all parts. The odd numbers are limited and perfect; the even unlimited and imperfect. The monad, or unity, is the source of all numbers. The dyad is according to the later Pythagorean doctrines, imperfect and passive, and the cause of increase and division. The triad, compounded of the monad and dyad, partakes of the nature of both. The tetras, or number four, is in the highest degree perfect. The decad, which contains the sum of the four prime numbers, and is therefore called tetractys, comprehends all musical and arithmetical proportions, and denotes the system of the world. The real meaning of the Pythagorean doctrine of numbers is not well understood; numhers were probably, in this system, the symbolical or allegorical representations of the first principles or forms of nature. As Pythagoras could not express abstract ideas in simple language, he seems to have made use of numbers, as geometers do of a diagram, to assist the comprehension of his scholars. He perceived some analogies between numbers and the attributes of

the divine understanding, and made the former the symbols of the latter. As the numbers proceed from the monad, or unity, undergo various combinations, and in their progress assume new properties, so he regarded the pure and simple essence of the Deity as the common source of all the forms of nature, which, according to their various modifications, possess different properties. Pythagoras is also said to have invented the multiplication table (abacus), thence called the Pythagorean table. Next to numbers, music belongs to the preparatory exercises of the Pythagorean school, by which the mind was elevated above the dominion of passion, and fitted for contemplation. Pythagoras considered music not only as an art to be judged of by the ear, but as a science to be reduced to mathematical maxims and relations, and allied to astronomy. Tradition makes him the inventor of a musical (Pythagorean lyre, octochordum Pythagora), which, after his death, was engraved in brass, and preserved in the temple of Juno at Samos. The invention of the harmonic canon, or monochord-an instrument of a single string-which served for the measurement of musical intervals, has also been ascribed to him by ancient and modern writers. He believed that the heavenly spheres, in which the planets move, dividing the ether in their course, produced tones, and that the tones must be different according to their size, velocity and distance. That these relations were in concord, that these tones produced the most perfect harmony (music of the spheres), he necessarily believed, in consequence of his notions of the supreme perfection of the universe. The real meaning of this doctrine was, that he regarded the world as a harmonically arranged whole (Kocpos), in which the relations of numbers were realized. His followers took occasion from this doctrine to say of their master, that he was the only mortal whom the gods had permitted to hear the harmony of the spheres. Geometry, which he had learned in Egypt, he reduced, more than any of his predecessors and contemporaries, to the form of a regular science. According to his notion, the geometrical point was simple, the line double, the area threefold, and solids quadruple; and in this way, also, he applied the doctrine of numbers. Of the geometrical theorems which are ascribed to him, the following are the most important: The three angles of a triangle are together equal to two right angles; and in a right-angled triangle, the square of the

hypothenuse is equal to the sum of the squares of the two sides. This last is still called the Pythagorean theorem (also magister matheseos), although it is doubtful whether Pythagoras invented it. In astronomy he taught the following: The word heaven denotes either the spheres of the fixed stars, or the whole space between the fixed stars and the moon, or the whole world, including both the heavenly spheres and the earth. Agreeably to the arithmetical hypothesis, there are ten heavenly spheres, of which nine are visible to us, viz. the sphere of the fixed stars, the seven spheres of the seven planets (including the sun and moon), and the sphere of the earth. The tenth earth, called by him Anticthon (anti-earth), is invisible, but necessary to the perfection of the harmony of nature, since the decad is the perfection of the numerical harmony. By this anti-earth he explains the eclipses of the moon. In the middle of the universe is the central fire, principle of warmth and life. The earth is one of the planets, moving around the sphere of fire. The atmosphere of the earth is a gross, immovable mass, but the ether is pure, clear, always in motion, and the region of all divine and immortal natures. The distances of the various heavenly spheres from the earth correspond to the proportions of the musical scale. His moon and stars are gods, or inhabited by gods. Pythagoras, therefore, rendered important services to the mathematical sciences, and first established a mathematical philosophy. His disciples Philolaus, Archytas, Ecphantus, Ocellus, Timæus, carried it farther. Philolaus, in particular, whose fragments are the most valuable relics of the Pythagorean school, distinguished himself by his astronomical system. With mathematics were also connected the natural sciences. With respect to philosophy, Pythagoras taught, that true knowledge embraced those subjects which are in their nature immutable, eternal and indestructible, and of which alone it can be properly predicated, that they exist. He who devotes himself to this study is a philosopher. The object of philosophy is, by contemplation, to render the human mind similar to the divine, and make it fit to enter the assembly of the gods. For this purpose it is necessary to invoke, in prayer, the assistance of the Divinity and good demons. Contemplative wisdom cannot be fully attained without entire abstraction from common things, without entire tranquillity, and freedom of mind. Hence the necessity of founding a society

separate from the world, for contemplatice and study. The theoretical philosophy Pythagoras, which treats of nature and s origin, was enveloped in the most pr found obscurity, and we know nothing it, but what may be conjectured fro single intimations of the ancients. Int opinion of Pythagoras, God is the u versal spirit, diffused in all directions fre the centre, the source of all animal life, the actual and inward cause of all motion, substance similar to light, the first pr ple of the universe, incapable of suffe invisible, indestructible, and to be con hended by the mind alone. To the Div ity there were subordinate, according the notions of the Pythagoreans, tar kinds of intelligences, gods, demons heroes, emanations of the supreme Go varying in dignity and perfection, in pr portion as they were more or less remov from their source. The heroes he be lieved to be clothed with a body of subt matter. Besides these three kinds, t was a fourth-the human mind; likew→ an emanation of the Divinity. As Go one, and the origin of all variety, he w represented as a monad, and the sub nate spirits as numbers derived from a... contained in unity. Thus the num of Pythagoras resembled the ideas of P to, excepting that they are contained in things themselves. The regions of air the Pythagoreans thought filled w spirits, demons and heroes, who were cause of health or sickness to men a animals, and, by means of dreams a other kinds of divinations, imparted: knowledge of future events. The s according to him, was likewise a nun and by numbers it first has perception= Philolaus says, of the world; it is an anation of the central fire, and, cose quently, always in motion, and indestruct ble. Of man, the Pythagoreans believed at least the later, that, since he consiste of an elementary nature, of a divine rational principle, he was a microcr that his soul was a self-moving pri ple, and consisted of two parts, the rac al, which was a portion of the unive soul, an emanation of the central f and had its seat in the brain, and irrational, which comprised the passions and lived in the heart; that in both, m had something in common with brutes, who, on account of their body structure and the want of language, incapable of acting reasonably; tha sensitive soul (pos) perishes, but tha rational mind (peres, vous) is immortai, be cause it has its origin in au imara

source; that the latter, when freed from the fetters of the body, assumes an ethereal vehicle, and passes to the habitations of the dead, where it remains till it returns to the world, to dwell in some other human or animal body, and that at last, when sufficiently purified, it returns to the source from which it proceeded. This doctrine of the transmigration of souls (metempsychosis), which was originally Egyptian, and connected with the idea of the reward and punishment of human actions, was the chief cause why the Pythagoreans killed no animals. His morality Pythagoras taught in symbolic maxims and ascetic precepts, in connexion with his contemplative views. The powers of the mind are reason and passion; where the latter is obedient to the former, virtue reigns. The mind possesses unity, harmony, and a resemblance to God. Right consists in retribution. The following maxims are also ascribed to him. "Youth should be habituated to obedience, for it will then find it easy to obey the authority of reason. It should be trained in the best course of life; habit will soon make it the most pleasant." "Silence is better than unmeaning words." The wise man should be prepared for every thing that does not lie within his control." "Do what you consider right: whatever the people think of you, despise its censure and its praise." "It is cowardly to quit the post assigned us by God, before he permits us." "Strength of mind rests on sobriety, for this keeps the reason unclouded by passion." "No one is to be deemed free, who has not perfect self-command." "Intoxication is a temporary madness." "The desire for the superfluous is folly, for it has no bounds," &c. The Pythagoreans recommended, especially, the virtue of friendship. In it, Pythagoras requires the absence of all dissension, perfect confidence, aid under all circumstances, and a mutual endeavor to make each other perfect. To true friends every thing is common. True friendship is imperishable. In performing the usages of religion, he required piety of soul. The gods are to be worshipped by symbols corresponding to

their nature, by simple purifications and offerings, and with purity of heart. An oath should never be violated. The dead must not be burned. Next to the gods and dæmons, the highest respect belongs to parents and lawgivers. The laws and customs of our country are to be sacredly observed. The Pythagorean philosophy had a great influence on the Platonic. In later times, it was revived and intermingled with New Platonism.-See Geschichte der Pythagoräischen Philosophie, by Ritter (Hamburg, 1826), and Bökh's Disputat. de Platonico Systemate Cœlest. Glob., &c. (Heidelb., 1810, 4to.).

(See Pythagoras.)

PYTHAGOREAN LYRE. PYTHAGOREAN TABLE. PYTHAGOREAN THEOREM. PYTHIA, and PYTHONISSA. (See Delphi.) PYTHIAN GAMES; one of the four great Grecian games, instituted in early times, in honor of Apollo, the conqueror of the Python. They were celebrated in the Crissean fields near Delphi (formerly called Pytha), at first every nine years, but afterwards, by the command of the Amphictyons, every five years. Poems in honor of Apollo were sung to the flute or the lyre, and poets contended for the prize, which was a crown of laurel or oak. The Amphictyons were the judges in the contests. Ŏther musical and gymnastic contests were afterwards added. In later times, these games were celebrated in other Grecian cities, and were kept up at Delphi as late as the third century A. D.

PYTHIAS. (See Damon.)

PYTHON; a dreadful dragon, which sprang from the mud left by the flood of Deucalion, and dwelt near Crissa, on Parnassus, watching the future oracle of Delphi. Acquainted with the future, he foresaw that the son of Latona would kill him, and he persecuted her with the greatest violence. Apollo slew him with an arrow, the first day after his birth, threw his bones into a deep chasm, possessed himself of the oracle, and received from this circumstance the surname of Pythian," the slayer of the Python." This fable was probably meant to indicate the power of the sun over the noxious vapors, remaining after a great flood.

tude of a heavenly body. Its name indicates that it consists of an arc of ninety degrees; the degrees are subdivided into smaller divisions. The quadrant is provided with glasses attached to a straight rod, through which the heavenly body is to be seen, and the position of which on the graduated are, determines the altitude of the body. In modern times, this instrument has been improved by the superior accuracy of the graduation, and by the use of a telescope, instead of simple dioptric glasses, for sights. Instead of the quadrant, it is now more common to use an entire circle. Quadrants are movable or fixed. The former are for common use, set in a vertical plane, and are of two sorts; in the one, the glasses are attached to a side of the quadrant, and a plumb line, suspended from the vertex, plays along the graduated arc; in the other, the quadrant itself remains stationary, and the rod to which the glasses are attached, moves upon the arc. The fixed quadrants are larger, and are set in a wall of an observatory in the plane of the meridian. The observations made by them are more accurate. (See Godfrey, Thomas.)

Quadrant, Gunter's. (See Gunter's Quadrant.)

QUADRAT, in printing; a piece of metal cast like the letters, to fill up the void spaces between words, &c. There are quadrats of various sizes, called m quadrats, n quadrats, &c.

QUADRATIC EQUATIONS. (See Equations.)

QUADRATRIX, in the higher geometry; a transcendental curve, which Dinostrates, and in modern times, Tschirnhausen, made use of to find the quadrature of the circle by approximation.

QUADRATURE, in astronomy; that aspect of the moon when she is ninety degrees distant from the sun; or when she is in the middle point of her orbit, between the points of conjunction and opposition, namely, in the first and third quarters.

QUADRIVIUM. (See Schools.) QUADRUPEDS, in zoology; a class of land animals, with hairy bodies, and four limbs or legs proceeding from the trunk of their bodies; the females are viviparous, or bring forth their young alive, and nourish them with milk from their teats. They constitute with man (bimana), the monkeys (quadrumana), and the cetaceous animals, the division mammalia. (See Animals.)

QUADRUPLE and QUINTUPLE ALLIANCE. The natural, but undue influence, which European states have mutually ex

ercised upon each other, has at times produced alliances more complicated than any which history elsewhere records, and which could be produced only by a combination of various interests. Alliances of this nature indicate the existence of powerful interests and counter interests, to trace which to their origin is one of the chief purposes of history. The first quadruple alliance, so called from the number of the contracting parties, was the alliance which was concluded October 28, 1666, between the states-general (Holland), Denmark, the duke of BrunswickLüneburg, and the elector of Brandenburg. The second was concluded at London, August 2, 1718, between Great Britain, France, and Austria, and was called quadruple because acceded to by Holland, February 16, 1719. The object of this league was to force Spain to consent to the peace of Utrecht. It continued to be so called even after the duke of Savoy and Spain had joined the alliance. The quadruple alliance of Austria, Russia, Great Britain, and Prussia, at Chaumont, March 1, 1814, originated from their coalition, which had effected the dissolution of the French empire. (See Coalition, and Chaumont.) It was less an alliance, in the diplomatic sense of the word, than an armed union for the restoration of the independence of its members. After effecting its object, it became the basis of the European political system which prevailed with little effectual opposition until 1830, having been confirmed by the congress of Vienna, the Holy Alliance (q.v.), and the congress of Aix-la-Chapelle, in October and November, 1818, when the alliance became, in a certain respect, quintuple, as France joined the union professedly for the maintenance of peace in Europe: England joined the three other powers for the overthrow of Napoleon; but when the alliance became obviously directed against the national independence which had been originally its professed object, and religious sophistry was blended with political, to deceive the people, and the right of armed interference was boldly pronounced, and in several instances carried into effect, England naturally separated more and more from the other powers in consequence of its constitutional system, until at length Canning proclaimed the principle of non-intervention. (See Intervention.) History will yet speak of quadruple and quintuple alliances in the great struggle between the friends of liberty and the friends of despotism. (See Alliance.)