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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 Ésoteric 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 illustrimus 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 atten. duction 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 Pythagorean order was everywhere suppressed ; cultivating the ascetic observances and religious rites though, as a philosophical sect, it continued to exist enjoined by him. They thus formed at once a philo- for many years after. Of the fate of P. himself sophical school and a religious brotherhood, which different accounts are given; but he is generally gradually assumed the character and exercised the supposed to have escaped to Metapontum, and died power of a political association also. This political there (504 B.C.), where his tomb was shewn in the influence, which undoubtedly became very great, time of Cicero. was constantly exerted on the side of aristocracy; P. is said to have been the first to assume the and to carry out the principles of this form of title of Philosopher ('Lover of wisdom ') in place of government, understood in the best sense of the the name Sophos ("Wise '), by which the sages had word, seems to have been the ultimate aim of before been known. Various discoveries in music, Pythagoras. He is said also to have increased his astronomy, and mathematics are attributed to him; influence by a practice unknown to the other sages among others, the proposition now known as the of the ancient world—the admission of women, 47th of Euclid, Book I. We have good ground for not probably into his society, but to attendance on believing that he was a man of much learning and his lectures and teaching. Of the internal arrange- great intellectual powers, which were specially exment and discipline of this fraternity we really erted in the way of mathematical research, as is know but little. All accounts agree that what was evinced by the general tendency of the speculations done and taught among the members was kept a of his school. There is no doubt that he maintained profound secret from the outer world. In the the doctrine of the transmigration of souls into the admission of members, P. is said to have exercised | bodies of men and other animals -- which seems to the greatest care, and to have relied much on have been regarded in the Pythagorean system as his skill in physiognomy. They then had, it is a process of purification -- and he is said to have said, to pass through a long period of probation, asserted that he had a distinct recollection of intended apparently to test especially their powers having himself previously passed through other of endurance and self-restraint-though probably stages of existence. We are told that on seeing a PYTHIAN GAMES-PYTHON.

dog beaten, and hearing him howl, he bade the Brandis, and Tennemann on the History of Phi striker desist, saying, 'It is the soul of a friend of losophy; in Lewes's Biographical History of Phi. mine, whom I recognise by his voice.'

losophy; and a complete summary of the whole in Respecting the system of philosophy actually Smith's Dictionary of Greek and Roman Biography. taught by P., we have but little trustworthy testi

PYTHIAN GAMES, one of the four great mony. P. himself, it is all but certain, wrote

national festivals of the Greeks, held in the Crissæan nothing, and the same seems to have been the case

plain, near Delphi, are said (according to the prewith his immediate successors; we are therefore, in endeavouring to form an idea of the Pythagorean

valent mythological legend) to have been instituted philosophy, obliged to rely almost entirely on the

by Apollo after vanquishing the snaky monster, compilations of later writers (mainly Diogenes

Python, and were certainly in the earliest times Laërtius, and the Neo-Platonists, Porphyrius and

celebrated in his honour every ninth year. They Iamblichus, all of them long subsequent to the

were at first under the management of the Delphians,

but about 590—586 B. C. the Amphictyons were Christian era), who often but imperfectly under

intrusted with the conduct of them, and arranged 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' th

writers state that it was only afier this date that

(they were called Pythian. Originally, the contests (Lewes's Biographical History of Philosophy), and to

were restricted to singing, with the accompaniment Number was allotted the most prominent place in their system. They taught that in Number only is

of cithern-playing, but the Amphictyons added the absolute certainty to be found ; that Number is

flute, athletic contests, and horse-racing. By and the Essence of all things ; that things are only a liñ'historical recitations, and in works of art. were

by, contests in tragedy, and other kinds of poetry, copy of Numbers ; nay, that in some mysterious in

introduced, and long continued a distinguishing way, Numbers are things themselves. This Number

feature of these games, which are believed to have theory was probably worked out from the funda

lasted down to nearly the end of the 4th c. A. D. mental conception, that after destroying or disarranging every other attribute of matter, there

The prize was a laurel wreath and the symbolic still remains the attribute Number; we still can

palm-branch. Several of Pindar's extant odes predicate that the thing is one. With this doctrine

relate to victors in the Pythian Games. of Number was intimately connected that of the PY'THON, à genus of serpents of the family Finite and the Infinite, corresponding respectively Boido (see Boa), differing from the true boas in with the Odd and the Even in Number; and from a having the plates on the under surface of the tail combination of this Finite and Infinite it was taught double. The tip of the r’uzzle is plated; the lips that all things in the Universe result. The abstract are grooved. The species are all natives of the Old principle of all perfection was One and the Finite; World. They are all large; some of them very of imperfection, the Many and the Infinite. Essen large, and rivalled in size by no serpents except the tially based also on the same doctrine, was the boas of America. The name Boa is often popularly Theory of Music; the System of the Universe, given to the pythons, and in its ancient use belongs which was conceived as a Kosmos, or one harmonious to them. Some of the pythons are known in the whole, consisting of ten heavenly bodies revolving East Indies by the name of RocK SNAKE, as P. round a Central Fire, the Hearth or Altar of the molurus, a species very extensively diffused. This Universe ; and the celebrated doctrine of the name is given to some species which belong to the Harmony of the Spheres--the music produced, it genus or subgenus Hortulia, one of which, the was supposed, by the movement of these heavenly NATAL ROCK SNAKE (H. Natalensis), is said to bodies, which were arranged at intervals according attain so large a size that its body is as thick as 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.

T'he Pythagorean system was carried on by a succes Python, or Rock Snake (Hortulia Natalensis). sion of disciples down to about 300 B. C., when it seems to have gradually died out, being superseded by other that of a man. Although a native of Natal, it is systems of philosophy; it was revived about two cen- already unknown in the settled parts of the colony. turies later, and lasted for a considerable time after Python reticulatus is probably the largest snake of the Christian era --- disfigured by the admixture of India and Ceylon. It is found also in more eastern other doctrines, and an exaggeration of the mysticism regions. What size it attains is not well known. and ascetic practices, without the scientific culture of Specimens of 15 or 20 feet long are common, but it the earlier school.

certainly attains a much larger size. It seems to In addition to the writers above mentioned, I be this snake which is soinetimes 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 a 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 garder, walls. It feeds on deer by Thirlwall and Grote, in the works of Ritter, I and smaller animals; but the largest pythons are

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said to seize buffaloes, tigers, and even elephants, with gold. Like all the other sacred utensils conand to crush them in their coils. In this there is nected with the administration of the eucharist, it perhaps some exaggeration ; but there are well. must be blessed by a bishop, or a priest delegated authenticated stories of snakes in the East Indies by a bishop. quite capable of killing at least the buffalo and the PYX; TRIAL OF THE, the final trial by weight and tiger (see My Indian Journal, by Colonel Walter assay of the gold and silver coins of the United Campbell; Edin. 1864, pp. 126, 127).

Kingdom, prior to their issue from the Mint. It is PYX (Gr. pyxis, a box, properly of boxwood), the so called from the Pyx, i. e., box or chest, in which sacred vessel used in the Catholic Church to contain 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 Pyx, Ashmolean Museum, however, it was, as its sealed packets of coin being delivered to them by Oxford.

name implies, a simple the officers of the Mint, are first tried by weight, (Copieu from Parker's Glossary.) 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 ccin.

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THE 17th letter of the Latin, 1 half that of the rectangle with the same baso and English, and other western Alpha- height; that of any parabolic segment is two-thirds bets, is identical in power with of the corresponding triangle, whose sides are the the letter K (q. v.). It is always chord and the tangents at its extremities ; that of followed by u.

| the cycloid three times that of its generating circle, QUADRAGE'SIMA (Lat. &c.

''fortieth day'), the name of the. The term is also applied in a special sense in cases CS Lenten season, or more properly of the in which an area or other quantity is expressed by

first Sunday of the Lent. It is so called an integral, whose value cannot be determined el by analogy with the three Sundays exactly, and it then means the process of approxi. Gu which precede Lent, and which are called mation by which the value of the integral can be

respectively Septuagesima, 70th; Sexagesina, gradually arrived at. 60th; and Quinquagesima, 50th.

All the practical rules for approximating to the QUADRA'NGLE, an open square, or courtyard

areas of curvilinear figures, and the volumes of having four sides. Large public buildings—such as

various solidls—such as occur in land-measuring, Somerset House and the colleges of Oxford and gauging, engineering, &c.-are, in this sense, cases Cambridge-are usually planned in this form.

of quadrature, except in those very special cases QUADRANT (Lat. quadrans, a fourth part), 1 ;

in which an area or a volume can be assigned literally the fourth part of a circle, or 90°; but (Mrver

parol, exactly as a finite function of its dimensions. See signifying, in Astronomy, an instrument used for

MENSURATION. the determination of angular measurements. The

QUADRATURE OF THE CIRCLE. This is quadrant consisted of a limb or arc of a circle equal

one of the grand problems of antiquity, which, to the fourth part of the whole circumference, unsolved and probably unsolvable, continue to graduated into de rees and parts of degrees. The occupy even in the present day the minds of many quadrant employed by Ptolemy was of stone, with curiou

with curious speculators. The trisection of an angle, the one smooth and polished side, on which the duplig

duplication of the cube, and the perpetual motion graduations were made ; the quadrant was firmly |

have found, in every age of the world since geometry placed in a meridian plane, with one radius vertical, and physics were thought of, their hosts of patient and the other horizontal. Tycho Brahe, who devotees. The physical question involved in the has a right to be considered as the first great Perpetual Motion (q. v.) is treated of under that practical astronomer of modern times, fixed his head ; and we shall now take the opportunity of avadrant on a wall, and employed it for the deter. noticing the mathematical questions involved in mination of meridian altitudes; he also adjusted

the other problems above mentioned; but more others on vertical axes for the measurement of

| especially that of the quadrature of the circle, in azimuths. Picart was the first who applied tele.

which the difficulty is of a different nature from scopic sights to this instrument About this time that involved in the other two geometrical ones. the large mural quadrant (of 6 to 8 feet radius)

| A few words about them, however, will help as an began to be introduced into observatories. These

introduction to the subject. quadrants were adjusted in the same way as the

According to the postulates of ordinary geometry, mural circle (see CIRCLE, MURAL). Various innate

all constructions must be made by the help of the defects of the quadrant as an instrument-such as

circle and straight line. Straight lines intersect the impossibility of securing exactness of the whole

each other in but one point; and a straight line arc, concentricity of the centre of motion with the

and circle, or two circles, intersect in two points centre of division, and perfect stability of the centre

only. From the analytical point of view we work – led to its being superseded by the repeating

may express these facts by saying that the detercircle, otherwise called the Mural Circle (q. v.).

mination of the intersection of two straight lines Hadley's Quadrant is more properly an octant, as

involves an equation of the first degree only; while its. limb is only the eighth part of a circle, though

that of the intersection of a straight line and a it measures an arc of 90°. Its principle is that of circle, or of two circles, is reducible to an equation the SEXTANT (q. V.).

of the second degree. But the trisection of an QUADRATIC EQUATIONS. See EQUATIONS. its accomplishment the solution of an equation of

angle, or the duplication of the cube, requires for QUA'DRATURE. This term is employed in the third degree; or, geometrically, requires the Mathematics to signify the process of determining intersections of a straight line and a curve of the the area of a surface. Its deriyation sufficiently third degree, or of two conics, &c., all of which are indicates its nature--i. e., it consists in determining excluded by the postulates of the science. If it were a square (the simplest measure of surface) whose allowed that a parabola or ellipse could be described area is equal to that of the assigned surface. In with a given focus and directrix, as it is allowed many cases, of which the Triangle (q. v.), the Para- that a circle can be described with a given radius bola (q. v.), and the Cycloid (q. v.) are perhaps the about a given centre, the trisection of an angle and simplest, the area is easily assigned in terms of the duplication of the cube would be at once brought some simple unit. Thus, the area of a triangle is I under the category of questions resolvable by pure QUADRATURE OF THE CIRCLE

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geometry; so that the difficulty in these cases is one lating the length of the side of an equilateral of mere restriction of the postulates of what is to be inscribed polygon of 1073741824 sides, determined called geometry.

the value of 5 to 16 significant figures; and It is very different in the case of the quadrature Ludolph von Ceulen, his contemporary, by calcuof the circle, which the reader of the preceding lating that of the polygon of 36893488147419103232 article will see at once) means the determination sides, arrived (correctly) at 36 significant figures. of the area of a circle of given radius-literally, the It is scarcely possible to give, in the present day, assigning of the side of a square whose area shall an idea of the enormous labour which this mode of be equal to that of the given circle.

procedure entails even when only 8 or 10 figures The common herd of squarers of the circle,' are sought; and when we consider that Ludolph which grows more numerous every day, and which was ignorant of logarithms, we wonder that a life includes many men of undoubted sanity, and time sufficed for the attainment of such a result by even of the very highest business talents, rarely the method he employed. have any idea of the nature of the problem they

%. The value of x was thus determined to insi course to shew first of all what has been done towards

of its amount, a fraction of which, after Montucla, the solution of the problem ; we shall then venture

we shall attempt to give an idea, thus : Suppose a a few remarks as to what may yet be done, and in circle whose radius is the distance of the nearest wliat direction philosophic squarers of the circle' fixed star (250,000 times the earth's distance from must look for real advance.

the sun), the error in calculating its circumference In the first place, then, we observe that mechanical by Ludolph's result would be so excessively processes are utterly inadmissible. A fair approxi

small a fraction of the diameter of a human hair mation may, no doubt, be got by measuring the as to be utterly invisible, not merely under the diameter of a circular disc of uniform material, and

most powerful microscope yet made; but under comparing the weight of the disc with that of a any which future generations may be able to square portion of the same material of given side.

Oi given side. | construct. But it is almost impossible to execute any measure These results were, as we have pointed out, all ment to more than six places of significant figures ; derived by common arithmetical operations, based hence, as will soon be shewn, this process is at best on the obvious truth that the circumference of a but a rude approximation. The same is to be circle is greater than that of any inscribed, and less said of such obvious processes as wrapping a string than that of any circumscribed polygon. They round a cylindrical post of known diameter, and involve none of those more subtle ideas connected comparing its length with the diameter of the with Limits, Infinitesimals, or Differentials, which cylinder: only a rude approximation to the ratio of seem to render more recent results suspected by the circumference of a circle to its diameter can modern squarers.' If one of that unhappy body thus be obtained.

would only consider this simple fact, he could hardly Before entering on the history of the problem, it have the presumption to publish his 3.125, or what. must be remarked that the Greek geometers knew ever it may be, as the accurate value of a quantity that the area of a circle is half the rectangle under which by common arithmetical processes, founded on its radius and circumference (see CIRCLE), so that an obvious geometrical truth, was seversal centuries the determination of the length of the circumfer- ago shewn to be greater than ence of a circle of given radius is precisely the same 3.14159265359979323846264338327950288, problem as that of the quadrature of the circle. Confining ourselves strictly to the best ascer

and less than

and de tained steps in the history of the question, we 3.14159265358979323846264338327950289. remark that Archimedes proved that the ratio of We now know, by far simpler processes, its exact the diameter to the circumference is greater than value to more than 600 places of decimals ; but 1 to 314, and less than 1 48 to 3. The difference the above result of Von Ceulen is much more between these two extreme limits is less than tho than sufficient for any possible practical applicaTobo of the whole ratio. Archimedes's process tion even in the most delicate calculations in depends upon the obvious truth, that the circum- astronomy. ference of an inscribed polygon is less, while that of Snellius, Huyghens, Gregory de Saint Vincent, a circumscribed polygon is greater, than that of the and others, suggested simplifications of the polygon circle. His calculations were extended to regular process, which are in reality some of the approximate polygons of 96 sides.

expressions derived from modern trigonometry. Little more seems to have been done by mathematicians till the end of the 16th c., when P. Métius demonstration of the impossibility of effecting exgave the expression for the ratio of the circumfer- actly the quadrature of the circle, which, although ence to the diameter as the fraction 3:08, which, in objected to by Huyghens, is now received as quite decimals, is true to the seventh significant figure satisfactory. inclusive. Curiously enough, it happens that this We may merely advert to the speculations of is one of the convergent fractions which express in Fermat, Roberval, Cavalleri, Wallis, Newton, and the lowest possible terms the best approximations others as to quadrature in general. Their most to the required number. Métius seems to have valuable result was the invention of the Differential employed, with the aid of far superior arithmetical and Integral Calculus by Newton, under the name notation, a process similar to that of Archimedes. lof Fluxions and Fluents. Wallis, however, by

Vieta shortly afterwards gave the ratio in a form an ingenious process of interpolation, sliewed true to the tenth decimal place, and was the first that to give, though of course in infinite terms, an exact

* _ 2.4.4.6.6.8.8.10.10. &o. formula. Designating, as is usual in mathematical

$ 3.3.5.5.7.7.9.9.11. &c. works, the ratio of the circumference to the diameter by 7, Vieta's formula is—

which is interesting, as being the first reco ded

example of the determination, in a finite form, of = { vāxv# + f vtx \} + V + $V$ * &c. the value of the ratio of two intinite products.

| Lord Brouncker, being consulted by Wallis as Shortly afterwards, Adrianus Romanus, by calcu. I to the value of the above expression, put it

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