PYTHAGORAS.

gathered in later ages round his name, that it is the assertion that they had to maintain silence for very difficult to arrive at anything like certainty two or even five years is an exaggeration of later regarding his history and character. That he was times. Among the members of the society we are a native of the island of Samos, the son of Mnesar- told there were several gradations, and there was also chus, a merchant, or, according to other accounts, a more general division of his disciples under the a signet-engraver, we know on good authority. names Esoteric and Exoteric--the former being apThe date of his birth is very uncertain, but is plied to all who were admitted to the more abstruse usually placed about the year 570 B. C.; and all doctrines and sublimer teaching of their master, the authorities agree that he flourished in the times of latter to those who received only the instruction Polycrates and Tarquinius Superbus (540-510 B.C.). open to all. The mode of life seems to have been He is said to have been a disciple of Pherecydes of regulated by P. in its minutest details. It is well Syros of Thales, and Anaximander, and, like other known that he is said to have forbidden all animal illustrious Greeks, to have undertaken extensive food-a consequence, perhaps, of the doctrine of travels for the purpose of adding to his knowledge; Metempsychosis-and also particularly beans (but in the course of which-lasting, we are told, for these statements cannot be relied on), and there is nearly 30 years-he visited Egypt (bringing with no doubt that temperance of all kinds was strictly him according to the usual story, letters of intro- enjoined. In the course of instruction, great attenduction from Polycrates to Amasis the king) and tion was paid to mathematics, music, and astrothe more important countries of Asia, including nomy; and gymnastics formed an important part of even India. We have every reason to believe that the training. Religious teaching was inculcated in he did, at all events, visit Egypt, and there availed the so-called Pythagorean Orgies or Mysteries; and himself of all such mysterious lore as the priests while he outwardly conformed to the usual mode could be induced to impart; from whom possibly of worship, there is reason to believe that in secret he learned the doctrine of Metempsychosis, or the he taught a purer faith. The result of the whole transmigration of souls (which was, as is well system seems to have been an unbounded reverence known, one of the most famous tenets of the Pytha- on the part of the disciples for their master (of gorean school), and whose influence may perhaps be which the well-known ipse dixit is a sufficient traced in the mystic rites, asceticism, and peculi- attestation); in the members of the order an elearities of diet and clothing which formed some of vated tone of character, exhibited in serenity of its chief characteristics-though we may consider mind and self-possession, extreme attachment to it as nearly certain that his philosophic and each other, and also supreme contempt for all the religious system was much less indebted to the outer world. But it was natural that political influence of other countries than the ancients gener- power uniformly exercised in one direction by an ally believed. During his travels, we may believe, aristocratic and exclusive society such as this should P. matured the plans which he afterwards carried in the end excite a wide-spread feeling of jealousy into action; but finding, on his return to his native and hatred, which at length, when opportunity was island, that the tyranny established there by Poly-given, caused the overthrow of the fraternity. A crates unfitted it for his abode, he quitted Samos, war between the cities of Croton and Sybaris, in and eventually settled in the city of Croton, in which the Pythagoreans took a prominent part, Southern Italy. Here he is said to have acquired ended in the total destruction of the latter city in a short time unbounded influence over the (510 B. C.); and on this success they seem to have inhabitants, as well as over those of the neigh-presumed so greatly, that they proceeded to more bouring states; and here he established the famous active measures against the popular party than they Pythagorean fraternity or order, which has often had yet attempted. A violent outbreak was the been compared with the still more celebrated order consequence; the house in which the leading founded by Ignatius Loyola in modern times. The Pythagoreans were assembled was set on fire, and adherents of P. were chiefly found among the noble many perished in the flames. Similar commoand the wealthy; these, to the number of 300, he tions ensued in other cities of Southern Italy in formed into a select society, bound by a sort of vow which Pythagorean clubs had been formed, and the to himself and to each other, for the purpose of study-result was that, as a political organisation, the ing the philosophical system of their master, and cultivating the ascetic observances and religious rites enjoined by him. They thus formed at once a philosophical school and a religious brotherhood, which gradually assumed the character and exercised the power of a political association also. This political influence, which undoubtedly became very great, was constantly exerted on the side of aristocracy and to carry out the principles of this form of government, understood in the best sense of the word, seems to have been the ultimate aim of Pythagoras. He is said also to have increased his influence by a practice unknown to the other sages of the ancient world-the admission of women, not probably into his society, but to attendance on his lectures and teaching. Of the internal arrangement and discipline of this fraternity we really know but little. All accounts agree that what was done and taught among the members was kept a profound secret from the outer world. In the admission of members, P. is said to have exercised the greatest care, and to have relied much on his skill in physiognomy. They then had, it is said, to pass through a long period of probation, intended apparently to test especially their powers of endurance and self-restraint-though probably

Pythagorean order was everywhere suppressed; though, as a philosophical sect, it continued to exist for many years after. Of the fate of P. himself different accounts are given; but he is generally supposed to have escaped to Metapontum, and died there (504 B. C.), where his tomb was shewn in the time of Cicero.

P. is said to have been the first to assume the title of Philosopher ('Lover of wisdom') in place of the name Sophos ('Wise '), by which the sages had before been known. Various discoveries in music, astronomy, and mathematics are attributed to him; among others, the proposition now known as the 47th of Euclid, Book I. We have good ground for believing that he was a man of much learning and great intellectual powers, which were specially exerted in the way of mathematical research, as is evinced by the general tendency of the speculations of his school. There is no doubt that he maintained the doctrine of the transmigration of souls into the bodies of men and other animals-which seems to have been regarded in the Pythagorean system as a process of purification and he is said to have asserted that he had a distinct recollection of having himself previously passed through other stages of existence. We are told that on seeing a

PYTHIAN GAMES-PYTHON.

dog beaten, and hearing him howl, he bade the striker desist, saying, 'It is the soul of a friend of mine, whom I recognise by his voice.'

Brandis, and Tennemann on the History of Phi losophy; in Lowes's Biographical History of Phi losophy; and a complete summary of the whole in Smith's Dictionary of Greek and Roman Biography.

Respecting the system of philosophy actually taught by P., we have but little trustworthy testimony. P. himself, it is all but certain, wrote national festivals of the Greeks, held in the Crissæan PYTHIAN GAMES, one of the four great nothing, and the same seems to have been the case with his immediate successors; we are therefore, valent mythological legend) to have been instituted plain, near Delphi, are said (according to the prein endeavouring to form an idea of the Pythagorean by Apollo after vanquishing the snaky monster, philosophy, obliged to rely almost entirely on the Python, and were certainly in the earliest times compilations of later writers (mainly Diogenes celebrated in his honour every ninth year. They Laërtius, and the Neo-Platonists, Porphyrius and Iamblichus, all of them long subsequent to the were at first under the management of the Delphians, Christian era), who often but imperfectly under-intrusted with the conduct of them, and arranged but about 590-586 B. C. the Amphictyons were stood the details they gave. The tendency of that they should be held every fifth year. Some

the school was towards the consideration of
abstractions as the only true materials of science'
(Lewes's Biographical History of Philosophy), and to
Number was allotted the most prominent place in
their system. They taught that in Number only is
absolute certainty to be found; that Number is
the Essence of all things; that things are only a
copy of Numbers; nay, that in some mysterious
way, Numbers are things themselves. This Number
theory was probably worked out from the funda-
mental conception, that, after destroying or
disarranging every other attribute of matter, there
still remains the attribute Number; we still can
predicate that the thing is one. With this doctrine
of Number was intimately connected that of the
Finite and the Infinite, corresponding respectively
with the Odd and the Even in Number; and from a
combination of this Finite and Infinite it was taught
that all things in the Universe result. The abstract
principle of all perfection was One and the Finite;
of imperfection, the Many and the Infinite. Essen-
tially based also on the same doctrine, was the
Theory of Music; the System of the Universe,
which was conceived as a Kosmos, or one harmonious
whole, consisting of ten heavenly bodies revolving
round a Central Fire, the Hearth or Altar of the
Universe; and the celebrated doctrine of the
Harmony of the Spheres the music produced, it
was supposed, by the movement of these heavenly
bodies, which were arranged at intervals according
with the laws of harmony-forming thus a sublime
Musical Scale. The Soul of Man was believed to
partake of the nature of the Central Fire, possessing
three elements, Reason, Intelligence, and Passion
the first distinctive of Man, the two last common to
Man and Brutes.

The Ethical teaching of the Pythagoreans was of the purest and most spiritual kind; Virtue was regarded as a harmony of the soul, a conformity with or approximation to the Deity; Self-restraint, Sincerity, and Purity of Heart were especially commended; and Conscientiousness and Uprightness in the affairs of life would seem to have been their distinguishing characteristics.

The Pythagorean system was carried on by a succession of disciples down to about 300 B. C., when it seems to have gradually died out, being superseded by other systems of philosophy; it was revived about two centuries later, and lasted for a considerable time after the Christian era-disfigured by the admixture of other doctrines, and an exaggeration of the mysticism and ascetic practices, without the scientific culture of the earlier school.

writers state that it was only after this date that
they were called Pythian. Originally, the contests
of cithern-playing, but the Amphictyons added the
were restricted to singing, with the accompaniment
flute, athletic contests, and horse-racing. By and
by, contests in tragedy, and other kinds of poetry,
in historical recitations, and in works of art, were
introduced, and long continued a distinguishing
feature of these games, which are believed to have
lasted down to nearly the end of the 4th c. A. D.
The prize was a laurel wreath and the symbolic
relate to victors in the Pythian Games.
palm-branch. Several of Pindar's extant odes

PYTHON, a genus of serpents of the family Boida (see BOA), differing from the true boas in having the plates on the under surface of the tail double. The tip of the muzzle is plated; the lips are grooved. The species are all natives of the Old World. They are all large; some of them very large, and rivalled in size by no serpents except the boas of America. The name Boa is often popularly given to the pythons, and in its ancient use belongs to them. Some of the pythons are known in the East Indies by the name of ROCK SNAKE, as P. molurus, a species very extensively diffused. This name is given to some species which belong to the genus or subgenus Hortulia, one of which, the NATAL ROCK SNAKE (H. Natalensis), is said to attain so large a size that its body is as thick as

that of a man. Although a native of Natal, it is already unknown in the settled parts of the colony. Python reticulatus is probably the largest snake of India and Ceylon. It is found also in more eastern regions. What size it attains is not well known. Specimens of 15 or 20 feet long are common, but it certainly attains a much larger size. It seems to In addition to the writers above mentioned, be this snake which is sometimes called ANACONDA. scattered and scanty notices - affording, however, It is rather brilliantly coloured; its body being really the most trustworthy information that we covered with gold and black, finely intermixed. possess as to the life and doctrines of P.-occur in The forehead is marked by a longitudinal brown Herodotus, Plato, Aristotle (the latter especially), stripe. Although sluggish for some time after & and a few other authors. Fuller details on the repast, it is at other times very active, and easily subject will be found in the Histories of Greece scales the highest garden walls. It feeds on deer by Thirlwall and Grote, in the works of Ritter, and smaller animals; but the largest pythons are

PYX.

said to seize buffaloes, tigers, and even elephants, and to crush them in their coils. In this there is perhaps some exaggeration; but there are wellauthenticated stories of snakes in the East Indies quite capable of killing at least the buffalo and the tiger (see My Indian Journal, by Colonel Walter Campbell; Edin. 1864, pp. 126, 127).

PYX (Gr. pyxis, a box, properly of boxwood), the sacred vessel used in the Catholic Church to contain

Corpo

兒

with gold. Like all the other sacred utensils connected with the administration of the eucharist, it must be blessed by a bishop, or a priest delegated | by a bishop.

PYX, TRIAL OF THE, the final trial by weight and assay of the gold and silver coins of the United Kingdom, prior to their issue from the Mint. It is so called from the Pyx, i. e., box or chest, in which are deposited specimen coins. When the coins are the consecrated eucha-weighed into bags at the Mint, two pieces are taken ristic elements, which out of each bag, one for assay within the Mint, the are preserved after con- other for the pyx. The latter are sealed up by secration, whether for three officers and deposited in the chest or pyx. The the communion of the trial takes place about once in three years by a jury sick or for the adora- of goldsmiths, summoned by the Lord Chancellor. tion of the faithful in The jury are charged by the Lord Chancellor, at the churches. Its form the Exchequer Office, Whitehall, in presence of has varied very much several privy councillors, and of the officers of the at different times. Mint. Being furnished with a piece of gold and Anciently it was some- silver from the trial plates deposited in the times of the form of a Exchequer, they are required to declare to what dove, which was hung degree the coin under examination deviates from suspended over the them. The jury then proceed to Goldsmiths' Hall, altar. More commonly, where assaying apparatus is in readiness, and the however, it was, as its sealed packets of coin being delivered to them by name implies, a simple the officers of the Mint, are first tried by weight, box. generally of the after which a certain number of pieces taken from precious metals, or, at the whole are melted into a bar, from which the least, of metal plated with gold or silver. At assay trials are taken. A favourable verdict present, the pyx is commonly cup-shaped, with a relieves the officers of the Mint from responsibility, close-itting cover of the same material. The and constitutes a public attestation of the standard interior is ordered to be of gold, or at least plated purity of the coin.

Pyx, Ashmolean Museum, Oxford. (Copiea from Parker's Glossary.)

40

Q

THE 17th letter of the Latin,
English, and other western Alpha-
bets, is identical in power with
the letter K (q. v.). It is always
followed by u.

half that of the rectangle with the same base and height; that of any parabolic segment is two-thirds of the corresponding triangle, whose sides are the chord and the tangents at its extremities; that of the cycloid three times that of its generating circle, QUADRAGE'SIMA (Lat. &c. 'fortieth day'), the name of the Lenten season, or more properly of the first Sunday of the Lent. It is so called by analogy with the three Sundays which precede Lent, and which are called respectively Septuagesima, 70th; Sexagesima, 60th; and Quinquagesima, 50th.

QUADRA'NGLE, an open square, or courtyard having four sides. Large public buildings-such as Somerset House and the colleges of Oxford and Cambridge-are usually planned in this form.

QUADRANT (Lat. quadrans, a fourth part), literally the fourth part of a circle, or 90°; but signifying, in Astronomy, an instrument used for the determination of angular measurements. The quadrant consisted of a limb or arc of a circle equal to the fourth part of the whole circumference, graduated into degrees and parts of degrees. The quadrant employed by Ptolemy was of stone, with one smooth and polished side, on which the graduations were made; the quadrant was firmly placed in a meridian plane, with one radius vertical, and the other horizontal.

The term is also applied in a special sense in cases in which an area or other quantity is expressed by an integral, whose value cannot be determined exactly; and it then means the process of approximation by which the value of the integral can be gradually arrived at.

All the practical rules for approximating to the areas of curvilinear figures, and the volumes of various solids-such as occur in land-measuring, gauging, engineering, &c.-are, in this sense, cases of quadrature, except in those very special cases in which an area or a volume can be assigned exactly as a finite function of its dimensions. See

MENSURATION.

but more

The physical question involved in the Perpetual Motion (q. v.) is treated of under that head; and we shall now take the opportunity of noticing the mathematical questions involved in the other problems above mentioned; especially that of the quadrature of the circle, in which the difficulty is of a different nature from that involved in the other two geometrical ones. A few words about them, however, will help as an introduction to the subject.

QUADRATURE OF THE CIRCLE. This is one of the grand problems of antiquity, which, unsolved and probably unsolvable, continue to occupy even in the present day the minds of many curious speculators. The trisection of an angle, the duplication of the cube, and the perpetual motion have found, in every age of the world since geometry and physics were thought of, their hosts of patient devotees. Tycho Brahe, who has a right to be considered as the first great practical astronomer of modern times, fixed his quadrant on a wall, and employed it for the determination of meridian altitudes; he also adjusted others on vertical axes for the measurement of azimuths. Picart was the first who applied telescopic sights to this instrument. About this time the large mural quadrant (of 6 to 8 feet radius) began to be introduced into observatories. These quadrants were adjusted in the same way as the mural circle (see CIRCLE, MURAL). Various innate defects of the quadrant as an instrument-such as the impossibility of securing exactness of the whole arc, concentricity of the centre of motion with the centre of division, and perfect stability of the centrework-led to its being superseded by the repeating circle, otherwise called the Mural Circle (q. v.).

Hadley's Quadrant is more properly an octant, as its limb is only the eighth part of a circle, though it measures an arc of 90°. Its principle is that of the SEXTANT (q. v.).

QUADRATIC EQUATIONS. See EQUATIONS. QUA'DRATURE. This term is employed in Mathematics to signify the process of determining the area of a surface. Its derivation sufficiently indicates its nature-i. e., it consists in determining a square (the simplest measure of surface) whose area is equal to that of the assigned surface. In many cases, of which the Triangle (q. v.), the Parabola (q. v.), and the Cycloid (q. v.) are perhaps the simplest, the area is easily assigned in terms of some simple unit. Thus, the area of a triangle is

According to the postulates of ordinary geometry, all constructions must be made by the help of the circle and straight line. Straight lines intersect each other in but one point; and a straight line and circle, or two circles, intersect in two points only. From the analytical point of view we mination of the intersection of two straight lines may express these facts by saying that the deterinvolves an equation of the first degree only; while that of the intersection of a straight line and a circle, or of two circles, is reducible to an equation of the second degree. But the trisection of an angle, or the duplication of the cube, requires for its accomplishment the solution of an equation of the third degree; or, geometrically, requires the intersections of a straight line and a curve of the third degree, or of two conics, &c., all of which are excluded by the postulates of the science. If it were allowed that a parabola or ellipse could be described with a given focus and directrix, as it is allowed that a circle can be described with a given radius about a given centre, the trisection of an angle and the duplication of the cube would be at once brought under the category of questions resolvable by pure

QUADRATURE OF THE CIRCLE

Vieta shortly afterwards gave the ratio in a form true to the tenth decimal place, and was the first that to give, though of course in infinite terms, an exact formula. Designating, as is usual in mathematical works, the ratio of the circumference to the diameter by, Vieta's formula is—

} √ễ × √ễ + žv§ × No1⁄2 + &v§ + &v‡ × &c. Shortly afterwards, Adrianus Romanus, by calcu