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.' Respecting the system of philosophy actually taught by P., we have but little trustworthy testimony. P. himself, it is all but certain, wrote nothing, and the same seems to have been the case with his immediate successors; we are therefore, in endeavouring to form an idea of the Pythagorean philosophy, obliged to rely almost entirely on the compilations of later writers (mainly Diogenes Laërtius, and the Neo-Platonists, Porphyrius and Iamblichus, all of them long subsequent to the Christian era), who often but imperfectly understood the details they gave. The tendency of 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 fundamental 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. Essentially 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.

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

national festivals of the Greeks, held in the Crissæan PYTHIAN GAMES, one of the four great valent mythological legend) to have been instituted plain, near Delphi, are said (according to the preby Apollo after vanquishing the snaky monster, Python, and were certainly in the earliest times celebrated in his honour every ninth year. They were at first under the management of the Delphians, but about 590-586 B. C. the Amphictyons were intrusted with the conduct of them, and arranged that they should be held every fifth year. Some 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 palm-branch. relate to victors in the Pythian Games. 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

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 the consecrated eucharistic elements, which are preserved after consecration, whether for the communion of the sick or for the adoration of the faithful in the churches. Its form has varied very much at different times.

aorpo

兒

Anciently it was sometimes of the form of a dove, which was hung suspended over the altar. More commonly, however, it was, as its name implies, a simple box, generally of the precious metals, or, at least, of metal plated with gold or silver. At present, the pyx is commonly cup-shaped, with a close-fitting cover of the same material. The interior is ordered to be of gold, or at least plated

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

40

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 weighed into bags at the Mint, two pieces are taken out of each bag, one for assay within the Mint, the other for the pyx. The latter are sealed up by three officers and deposited in the chest or pyx. The trial takes place about once in three years by a jury of goldsmiths, summoned by the Lord Chancellor. The jury are charged by the Lord Chancellor, at the Exchequer Office, Whitehall, in presence of several privy councillors, and of the officers of the Mint. Being furnished with a piece of gold and silver from the trial plates deposited in the Exchequer, they are required to declare to what degree the coin under examination deviates from them. The jury then proceed to Goldsmiths' Hall, where assaying apparatus is in readiness, and the sealed packets of coin being delivered to them by the officers of the Mint, are first tried by weight, after which a certain number of pieces taken from the whole are melted into a bar, from which the assay trials are taken. A favourable verdict relieves the officers of the Mint from responsibility, and constitutes a public attestation of the standard purity of the coin.

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

QUADRANGLE, 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.

QUA'DRANT (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. 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

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.

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.

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; but more 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.

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 may express these facts by saying that the determination of the intersection of two straight lines involves 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

geometry; so that the difficulty in these cases is one of mere restriction of the postulates of what is to be called geometry.

It is very different in the case of the quadrature of the circle, which (the reader of the preceding article will see at once) means the determination of the area of a circle of given radius-literally, the assigning of the side of a square whose area shal! be equal to that of the given circle.

The common herd of 'squarers of the circle,' which grows more numerous every day, and which includes many men of undoubted sanity, and even of the very highest business talents, rarely have any idea of the nature of the problem they attempt to solve. It will, therefore, be our best course to shew first of all what has been done towards the solution of the problem; we shall then venture a few remarks as to what may yet be done, and in what direction philosophic squarers of the circle' must look for real advance.

In the first place, then, we observe that mechanical processes are utterly inadmissible. A fair approximation may, no doubt, be got by measuring the diameter of a circular disc of uniform material, and comparing the weight of the disc with that of a square portion of the same material of given side. But it is almost impossible to execute any measurement to more than six places of significant figures; hence, as will soon be shewn, this process is at best but a rude approximation. The same is to be said of such obvious processes as wrapping a string round a cylindrical post of known diameter, and comparing its length with the diameter of the cylinder only a rude approximation to the ratio of the circumference of a circle to its diameter can

thus be obtained.

Before entering on the history of the problem, it must be remarked that the Greek geometers knew that the area of a circle is half the rectangle under its radius and circumference (see CIRCLE), so that the determination of the length of the circumference of a circle of given radius is precisely the same problem as that of the quadrature of the circle.

Confining ourselves strictly to the best ascertained steps in the history of the question, we remark that Archimedes proved that the ratio of the diameter to the circumference is greater than 1 to 39, and less than 143 to 3. The difference between these two extreme limits is less than the To of the whole ratio. Archimedes's process depends upon the obvious truth, that the circumference of an inscribed polygon is less, while that of a circumscribed polygon is greater, than that of the circle. His calculations were extended to regular polygons of 96 sides.

Little more seems to have been done by mathematicians till the end of the 16th c., when P. Métius gave the expression for the ratio of the circumference to the diameter as the fraction 355, which, in decimals, is true to the seventh significant figure inclusive. Curiously enough, it happens that this is one of the convergent fractions which express in the lowest possible terms the best approximations to the required number. Métius seems to have employed, with the aid of far superior arithmetical notation, a process similar to that of Archimedes.

lating the length of the side of an equilateral inscribed polygon of 1073741824 sides, determined the value of to 16 significant figures; and Ludolph von Ceulen, his contemporary, by calculating that of the polygon of 36893488147419103232 sides, arrived (correctly) at 36 significant figures. It is scarcely possible to give, in the present day, an idea of the enormous labour which this mode of procedure entails even when only 8 or 10 figures are sought; and when we consider that Ludolph was ignorant of logarithms, we wonder that a lifetime sufficed for the attainment of such a result by the method he employed.

1

The value of was thus determined to 3 × 1035 of its amount, a fraction of which, after Montucla, we shall attempt to give an idea, thus: Suppose a circle whose radius is the distance of the nearest fixed star (250,000 times the earth's distance from the sun), the error in calculating its circumference by Ludolph's result would be so excessively small a fraction of the diameter of a human hair as to be utterly invisible, not merely under the most powerful microscope yet made; but under any which future generations may be able to construct.

These results were, as we have pointed out, all derived by common arithmetical operations, based on the obvious truth that the circumference of a circle is greater than that of any inscribed, and less than that of any circumscribed polygon. They involve none of those more subtle ideas connected with Limits, Infinitesimals, or Differentials, which seem to render more recent results suspected by modern 'squarers.' If one of that unhappy body would only consider this simple fact, he could hardly have the presumption to publish his 3.125, or whatever it may be, as the accurate value of a quantity which by common arithmetical processes, founded on an obvious geometrical truth, was several centuries ago shewn to be greater than 3.14159265358979323846264338327950288,

and less than

3.14159265358979323846264338327950289.

We now know, by far simpler processes, its exact value to more than 600 places of decimals; but the above result of Von Ceulen is much more than sufficient for any possible practical application even in the most delicate calculations in astronomy.

Snellius, Huyghens, Gregory de Saint Vincent, and others, suggested simplifications of the polygon process, which are in reality some of the approximate expressions derived from modern trigonometry.

In 1668 the celebrated James Gregory gave a demonstration of the impossibility of effecting exactly the quadrature of the circle, which, although objected to by Huyghens, is now received as quite satisfactory.

We may merely advert to the speculations of Fermat, Roberval, Cavalleri, Wallis, Newton, and others as to quadrature in general. Their most valuable result was the invention of the Differential and Integral Calculus by Newton, under the name of Fluxions and Fluents. Wallis, however, by

Shortly afterwards, Adrianus Romanus, by calcu

π 2.4.4.6.6.8.8.10.10. &o.
4 3.3.5.5.7.7.9.9.11. &c.

which is interesting, as being the first recorded example of the determination, in a finite form, of

the value of the ratio of two infinite products.

Lord Brouncker, being consulted by Wallis as to the value of the above expression, put it

@tan. tan. 30+ tan. This was appropriated by Leibnitz, and formed perhaps the first of that audacious series of peculations from English mathematicians which have for ever dishonoured the name of a man of real genius.

T

If we notice that, by ordinary trigonometry, the arc whose tangent is unity (the arc of 45° or) 4' falls short of four times the arc whose tangent is by an angle whose tangent is, we may easily calculate to any required number of decimal places by calculating from Gregory's formula the values of the arcs corresponding to and as tangents. And it is, in fact, by a slight modification of this process (which was originally devised by Machin), that has been obtained, by independent calculators, to 600 decimal places.

It is not yet proved, and it may not be true, that the area or circumference of a circle cannot be expressed in finite terms; if it can be, these must (of course) contain irrational quantities. The integral calculus gives, among hosts of others, the following very simple expression in terms of a definite integral :

where and are irrational numbers. Such an expression, if discovered, would undoubtedly be hailed as a solution of the grand problem.

But this, we need hardly say, is not the species of solution attempted by squarers.' We could easily, from our own experience alone, give numerous instances of their helpless absurdities, but we spare the reader, and refer him, for further information on this painful yet ridiculous subject, to a recent series of papers by De Morgan in the Athenaeum; and to the very interesting work of Montucla, His toire des Recherches sur la Quadrature du Cercle.

QUADRIENNIUM U'TILÉ, in Scotch Law, means the four years after majority during which a person is entitled to reduce or set aside any deed made to his prejudice during minority. This protection was also given by the Roman law to minors, to enable them to neutralise any unfair advantage that may have been taken of their inexperience

been caused, not by an accident, but by the imprudence or negligence of themselves or of their curators. The proceeding, therefore, must be commenced before the minor attains 25, after which it is too late to seek restitution. See INFANT. QUADRI'GA. See CHARIOT.

QUADRILATERAL, in Military Language, is an expression designating a combination of four fortresses, not necessarily connected together, but mutually supporting each other; and from the fact that if one be attacked, the garrisons of the others, unless carefully observed, will harass the besiegers, rendering it necessary that a very large army should be employed to turn the combined position. As a remarkable instance, and a very

powerful one, may be cited the celebrated quadrilateral in Venetia, comprising the four strong posts of Mantua, Verona, Peschiera, and Legnago. These form a sort of outwork to the bastion which the southern mountains of the Tyrol constitute, and divide the north plain of the Po into two sections by a most powerful barrier. Napoleon III., in 1859, even after the victories of Magenta and Solferino, hesitated to attack this quadrilateral.

QUADRILLE, a dance of French origin, consisting of consecutive dance movements, generally five in number, danced by couples, or sets of couples, opposite to, and at right angles to each other. The name seems to be derived from its having been originally danced by four couples.

QUADRILLE is a card game, which, as its name cards employed is forty, the tens, nines, and eights denotes, is played by four persons. The number of being discarded from the pack. The rank and order of the cards in each suit vary according as they are or are not trumps, and are different in the black and red suits. The ace of spades, whatever suit be trumps, is always the highest trump, and is called spadille; the ace of clubs is always the third highest trump, and is known as basto; while the second highest trump, or manille, is the deuce of spades or clubs, or the seven of hearts or diamonds, according to the suit which is trumps, it being always of the trump suit. When the black suits are not trumps, the black cards rank as in whist; and when they are trumps, the order is the same, with the exception, as above mentioned, of the deuce, which then (in the trump suit only) becomes manille, the deuce of the black suit which is not trumps retaining its position as the lowest card. When the red suits are not trumps, the order of rank is as follows: