PYTHAGOREANS PYXIS NAUTICA For the character and working of the Pytha gorean brotherhoods, see Grote's History of Greece, part ii. ch. xxxvii. An account of the astronomical theories of Pythagoras and his followers is given by Sir G. C. Lewis, Astronomy of the Ancients, p. 13, &c. See also Thirlwall's History of Greece, vol. ii. c. xii.; Ritter's History of Philosophy, b. iv.; Boeckh's Philolaus, &c. Pythia (Gr.). The name of the priestess of the Delphian oracle of Apollo. [ORACLE.] p. 451; Astronomy of the Ancients, 123-269); | right and left, male and female, still and but he is said finally to have fixed his abode moved, straight and curved, light and darkat Crotona, one of the Dorian colonies in the ness, good and evil, square and oblong. south of Italy. He here attached to himself a large number of youths of noble descent, whom he formed into a secret fraternity for religious and political as well as philosophical purposes; and by their assistance produced many beneficial changes in the institutions of Croton and the other Græco-Italian cities. Of the strictly philosophical tenets of the Pythagoreans very imperfect records are preserved. Many of the doctrines ordinarily imputed to them are evi- The doctrine of METEMPSYCHOSIS, or the dently the fabrication of the later Pythago- transmigration of souls through different orders reans, a class of visionaries who lived during of animal existence, is the main feature by which the decline of the Roman empire. One point the Pythagorean philosophy is popularly known. is sufficiently evident, that the Pythagoreans It is, however, by no means certain that the were the greatest mathematicians of their genuine Pythagoreans held this doctrine in a time, and that they sought in the study of literal sense. It may have been only a mythimathematical relations that solution of the cal way of communicating their belief in the principal philosophical problems for which individuality of the soul and its existence after their contemporaries, the Ionic and Eleatic death. philosophers, sought, the first in physical, the others in ontological hypotheses. The relations of space and quantity, as they are the most Pythian Games. One of the four great obvious, are also the most definite forms, in national festivals of Greece, celebrated every which the laws of the outward world can fifth year in honour of Apollo, near Delphi. present themselves to this faculty. Hence, Their institution is variously referred to Amas the atomic philosophers have endeavoured phictyon, son of Deucalion, founder of the to explain all things by a diversity in the council of Amphictyons, and Diomedes, son of figure of their ultimate parts, the Pytha- Tydeus; but the most common legend is that goreans seem to have found, in the number they were founded by Apollo himself, after he and proportions of those parts, the true essence had overcome the dragon Python. The conof the things themselves. Having proceeded thus far, they went a step farther. They perceived that the universe and its parts are obedient to certain laws, and that these laws can be expressed by numbers. By a mistake prevalent during every period of speculation, they mistook the necessary conditions of a thing's subsistence for the essence of that thing itself; and at once pronounced that numerical relations were not merely all that could be understood in outward phenomena, but were, in fact, all that was real in them. Units of number grew gradually into points in space, and these into material atoms. To every order of existence, even to many abstract conceptions, a distinct number was assigned. God is represented as the original unity; the human soul, the earth, the planets, the animal creation have each their own peculiar arithmetical essence; as have also the abstractions justice, opportunity, opinion, &c. The outlines of a duallistic scheme are discernible in a singular table of opposites (ovσTоixía), preserved to us by Aristotle, in which the two principles of the universe are successively represented under the form of limit and the unlimited, odd and even, one and many, 154 tests were the same as those at Olympia, and the victors were rewarded with apples and garlands of laurel. [DELPHI.] Python (Пúowv). In Greek Mythology, the name of the dragon slain by Apollo. [PHŒBUS; PERSEUS.] The name was interpreted by the word new, to rot, because its dead body was left to rot at Delphi; but this explanation is of no more value than that which professes to account for the name Lycaon. [RISHIS.] In Teutonic myths, Python reappears as Fafnir. [MYTHOLOGY, COMPARATIVE; EDIPUS; PERSEPHONÊ; SIGURDR.] PYTHON. In Zoology, the name of a genus of large Ophidian reptiles, having anal hooks, and a double series of sub-caudal scutæ. Pyx (Gr. Tugis, a box of box-wood). The name given to the box in which the host is kept by the Roman Catholic priesthood. Pyx, Trial of. [COINAGE.] Pyxidium (Gr. Tugídiov, dim. of Tuέls). In Botany, a fruit which divides circularly into a lower and upper half, of which the latter acts as a kind of lid, as in the Pimpernel. Pyxis Nautica. The Mariner's Compass. A constellation of the southern hemisphere formed by Lacaille. QUADRANT OF ALTITUDE Q. In all the languages in which it is used this letter is invariably followed by u, the combination being represented in English pronunciation by the letters kw, as in quote. Q is used as an abbreviation for question; Qy. for query; Q.E. D. for quod erat demonstrandum, which was to be demonstrated, &c. Quader Sandstone. The cretaceous rocks of the north of Germany chiefly consist of sandstones, called Quader sandstones. There are two divisions-the Upper Quader, corresponding nearly in geological age to the main body of the chalk in England and Europe, and the Lower Quader, which represents our upper greensand and firestone. These German beds are not without calcareous matter, but it is chiefly present as a cementing medium. They are fossiliferous towards the base. Parts of them are much used as building material, and are well adapted for this purpose. Quadragesima (Lat. fortieth). In the Calendar, a term applied to the time of Lent, because it consists of about forty days. Quadragesima Sunday is the first Sunday in Lent, and about the fortieth day before Easter. Quadrangle (Lat. quadrangulus, fourcornered). A figure with four angles and four sides; in short, a quadrilateral. This is the ordinary acceptation of the term. In modern geometry, however, a quadrangle or tetragon denotes a system of four points (angles or corners), whilst a quadrilateral or tetragram is regarded as a system of four lines. [QUADRILATERAL.] A quadrangle is regarded as having six sides or lines through two angles. Thus the broken lines in the figure are the sides of the quadrangle a, b, c, d, and a, B, y, are the three intersections of opposite sides, which latter are sometimes called the diagonal points of the complete quadrangle. One of the most important properties of the quadrangle is that the rays joining any one of these three diagonal points with the other two are harmonic conjugates with respect to the sides which pass through the first point. Thus a (a b By), B(abya), y(ad aß) are harmonic pencils. Quadrans (Lat.). A division of the Roman as, consisting of one-fourth of it, or three ounces when the as was of its full weight. [AS; FARTHING; PENNY; TERUNCIUS.] tion. The instrument is variously contrived and fitted up, according to the purpose for which it is intended; but it consists essentially of a limb or are of a circle equal to the fourth of the circumference, and divided into 90°, with subdivisions. The mural quadrant is of considerable size (six or eight feet radius, for example), the axis of which moves in a wall or solid piece of masonry. [MURAL CIRCLE.] Ptolemy, in the Almagest, describes a quadrant with which he determined the obliquity of the ecliptic. Tycho Brahe had a large mural quadrant for observing altitudes, and others which revolved on a vertical axis for measuring azimuths. Picart, in his measurement of the earth, used a quadrant for his terrestrial angles. In 1725 a mural quadrant, by Graham, was erected in the Royal Observatory at Greenwich, which, in 1750, was replaced by Bird's quadrant, with which Bradley made his celebrated observations. The quadrant has, however, of late. years been entirely superseded by the mural circle; it having been found that the circle, on account of the symmetry of its form, and the advantage which it possesses of allowing the readings to be made at different parts of the limb, is an instrument much more to be relied on. [MURAL CIRCLE.] Hadley's quadrant, in its principle and application, is the same as the sextant, by which it has been superseded. [SEXTANT.] For further information respecting the quadrant, see Lalande, Astronomie, s. 2,311; Vince's Practical Astronomy; Pearson's. Practical Astronomy; and the Penny Cyclopædia. QUADRANT. In Geometry, the fourth part of a circle; an arc of ninety degrees. QUADRANT. In Gunnery, an instrument occasionally used for regulating the elevation of pieces of ordnance. It consists of two bars of wood or brass, at right angles to each other, with an are between them divided into degrees. A plumb line hangs from the angle at which the bars meet. One of the bars being placed in the bore of the piece, the degree on the arc intersected by the plumb line shows the elevation. Quadrant of Altitude. An appendix to an artificial globe, consisting of a thin pliable slip of brass, which is applied to the globe, and used as a scale for measuring the distances between points in degrees. It is graduated into 90°, the degrees being of the same length as those on one of the great circles of the globe. At the end where the division terminates a nut is riveted on, and furnished with a screw, by which it is attached to the brass meridian of the globe at any point. This point being placed in the zenith, and the quadrant applied to the globe, its zero coincides with the horizon, and consequently the altitude of any point along its graduated edge is indicated by the corre Quadrant. A mathematical instrument, formerly much used in astronomy and naviga- sponding division. QUADRANTAL TRIANGLE QUADRATS Quadrantal Triangle. In Trigonometry, | reputed inventor, a brother of Menechmus and a spherical triangle which has one side equal to disciple of Plato. This curve is generated as a quarter of a circle, or 90°. follows: Quadratic Equation. In Algebra, an equation which involves the second, but no higher power of the unknown quantity. The most general form of a quadratic equation is In the circular quadrant C A B, suppose the radius C A to revolve uniformly about C, passing through the different positions CK, Ck, &c., till it arrives at the position CB; and that during the same time a line AL, at right angles to CA, moves parallel to itself with a uniform mo M P k B tion from the position AL, through the different positions MN, mn, &c., so as to arrive at CB at the same instant that CK coincides with CB; then the continual intersection of The solution of a pure quadratic is obvious; the revolving radius and the parallel line will its two roots are trace the quadratrix A P Q. is it From this mode of describing the curve, easy to see how it may be applied to divide an angle into any number of equal parts. Let it be required, for example, to trisect the angle AC k. Having applied the quadratrix to CA, take AM equal to a third of Am, and through M draw MN perpendicular to AC, meeting the curve in P; join CP, and the angle ACP is equal to one-third of ACk; for by the nature of the quadratrix AM: Am:: AK: AK. The application of this curve to the quadrature of the circle depends on the property that the line CQ is a third proportional to the quadrantal are A B, and the radius. Hence the arc A B of the quadrant CAB = C B3 2CQ If the quadratrix be continued beyond A, without the circle, it will consist of a series of infinite hyperbolic branches, cutting the axis CA produced, in points which are separated from each other by a space equal to 2 A C. Other curves may be formed in a similar manner, by which the quadrature of the circle would be obtained. Thus, instead of supposing the lines M N, m n to be intersected by the radiants CK, Ck, we may suppose straight lines drawn from Kk parallel to A C, intersecting MN, mn in r and s; these intersections form a different curve, which is called the Quadratrix of Tschirnhausen. Let A M = x, MP=y, and A C = a; then since a AK: AB, or, we have A K Пх Hence the equation of the quad2 a ratrix of Dinostratus is = The expression bac, under the radical sign, is called the discriminant of the equation. When it has a positive value, the roots are real and unequal; when it vanishes, the two roots are real and equal; and when it has negative value, these roots are impossible or and that of the quadratrix of Tschirnhausen is imaginary. [DISCRIMINANT.] QUADRATURE QUADRIC this number of atoms of each element. Hence the name. than the types themselves, so that they leave a' blank space on the paper when printed. An enquadrat is in thickness half the depth, an Quadric. In Algebra, a homogeneous ex en equal in thickness and depth, a two-em quadrat twice the depth, &c. They are used to fill out short lines, form white lines, &c. Quadrature (Lat. quadratum, a quartering). In Astronomy, this term denotes the position of the moon when she is 90° from the sun, or at one of the two points of her orbit equally distant from the conjunction and opposition. QUADRATURE. In Geometry, this word signifies the determination of the area of a curve, or finding an equal square. The differential element of the area of a curve referred to rectangular coordinates is yda; and since y is given in terms of r by the equation of the curve of which the area is proposed to be found, the problem of quadratures in general reduces itself to the integration of the differential Xdr, in which X is an algebraic function of x and known quantities. In the applications of the higher geometry, a problem is conceived to be resolved when it is reduced to quadratures; i.e. when the variable quantities have been separated, and its solution been made to depend on finding the values of one or more integrals of the form SXdx. The quadrature of the circle is a problem of great celebrity in the history of mathematical science. The whole circular area being equal to the rectangle under the radius, and a straight line equal to half the circumference, the quadrature would be obtained if the length of the circumference were assigned; and hence the particular object aimed at in attempting to square the circle is the determination of the ratio of the circumference to the diameter. This ratio can be expressed only by infinite series, of which many have been given that converge with great rapidity. [CIRCLE.] Pretenders to the discovery of the quadrature of the circle occasionally present themselves even at the present day. They are to be found only among those who have an imperfect knowledge of the principles of geometry; and when their reasoning happens to be intelligible, their paralogisms are in general easily detected. With a view to discourage the futile attempts so frequently made on this and similar subjects, the Academy of Sciences of Paris, in 1775, and the Royal Society shortly after, publicly announced that they would not examine in future any paper pretending to the quadrature of the circle, the trisection of an angle, the duplication of the cube, or the discovery of the perpetual motion. For the history of this famous problem, see the third supplement to the fourth volume of Montucla. Quadri-hydrocarbon. A liquid hydrocarbon of the same chemical constitution as olefiant gas, and containing eight atoms of carbon united with eight atoms of hydrogen. Formerly it was supposed to contain only half pression of the second degree in the variables or facients. [QUANTIC.] Ternary and quaternary quadrics, equated to zero, represent respectively curves and surfaces which have the property of cutting every line in the plane or in space in two points, and to which the name quadric is also applied. Plane quadrics, therefore, are identical with the conic sections, and admit of three principal forms, the ellipse, hyperbola, and parabola; subforms of which are the circle, a pair of intersecting, and a pair of coincident lines. [CONIC SECTIONS.] The ellipse is characterised as being a closed curve, the hyperbola as having two distinct points at infinity, and consequently two real asymptotes, and the parabola as having two coincident points at infinity, and therefore an infinitely distant tangent. A plane quadric may also be regarded as the locus of the intersections of corresponding rays of two homographic pencils [PENCIL], or as the envelope of the line joining corresponding points of two homographic divisions. [HOMOGRAPHIC.] The envelope in the last case can easily be shown to be of the second class. For if a and a, be two corresponding points on the homographically divided lines A and A1, and o any other point in the plane, o a and oa, will clearly be corresponding rays of two concentric pencils, which latter will, of course, have two common rays; so that of the tangents a a, to the envelope two will in general pass through an arbitrary point o. To prove that such an envelope of the second class is also a quadric or curve of the second order, it is necessary to show that two pairs of consecutive tangents intersect at two points of a given arbitrary line L. To do so, conceive any point m on L, and let the two tangents through m cut A in a and a; then as m changes, a and a will clearly determine an involution of the second order on A [INVOLUTION], which will of course possess two double points, to which will correspond on L two distinct intersections of consecutive tangents, in other words two points on the envelope. Since a plane can always be drawn through three points of a non-plane curve, it is manifest that there are no non-plane quadric curves. Quadric surfaces are classified in various ways. The central quadrics, or those which have centres, are the ellipsoid and the hyperboloids of one and two sheets, respectively, to which may be added the cone. The non-central quadrics are the elliptic and hyperbolic paraboloids, to which may be added the several cylinders, distinguished as elliptic, hyperbolic, or parabolic, according to the nature of their sections. All plane sections of the ellipsoid are ellipses, and those of the hyperboloids are either ellipses or hyperbolas. The paraboloids, besides having plane parabolic sections, have either elliptic or hyperbolic ones, and are named accordingly. Besides the cone and cylinders there are two quadric ruled surfaces, QUADRIC CONE the hyperboloid of one continuous sheet, and the hyperbolic paraboloid, each of which may be generated by a line which moves so as to rest on three rectilinear directrices which do not intersect one another. [RULED SURFACE.] If the three directrices are parallel to one and the same plane, then the generator will always remain parallel to another plane, and the generated quadric will be a hyperbolic paraboloid; in other cases it will be a hyperboloid. If the two planes, to which the directrices and generator are respectively parallel, be at right angles to each other, the paraboloid is said to be equilateral; it is in fact a conoid surface, since it may be generated by the motion of a line resting on two directrices to one of which it is always perpendicular. [CONOID.] Every plane through a generator of a quadric ruled surface meets the latter in a second line, and touches it at the point where the lines intersect each other; so that at every point of a ruled quadric a straight edge can be applied to the surface in two distinct directions, and the whole surface is filled, as it were, by two systems of lines or generators such that each generator meets no generator of its own system, but cuts every generator of the other system. The distinctive character of the paraboloid is that one generator in each system is infinitely distant. In the hyperboloid the generators are all parallel to those of a quadric cone, the asymptotic cone; in the paraboloid they are parallel to a system of two planes, the asymptotic planes. Ruled quadrics may also be regarded as the locus of the line which joins corresponding points of two homographically divided lines not in the same plane. If the lines are divided proportionally the quadric will be a paraboloid; or lastly a ruled quadric may be regarded as generated by the intersections of corresponding planes of two homographic pencils whose axes are not in the same plane. From these modes of generation it is at once evident that every plane cuts the generated surface in a quadric curve, and that the tangent planes through any point in space envelope a quadric cone; in other words, that the surface is of the second order and second class. Quadric Cone. A cone of the second order. [CONE.] Quadricorns (Lat. quatuor, four; cornu, a horn). A family of Apterous insects, comprehending those which have four antennæ. A species of Antelope with four horns is called Antilope (Tetracerus) quadricornis. Quadrifores (Lat. quatuor, and foro, I pierce). A name given by Latreille to a family of sessile Cirripeds, comprehending those in which the opercular covering of the tube is composed of four valves or calcareous pieces. Quadriga (Lat. contracted from quadrijuga, a team of four animals). In Roman Antiquities, a car or chariot drawn by four horses, which were harnessed all abreast, and not in pairs. The quadriga is often met with on the reverse of medals, which are thence termed nummi QUADRILLE quadrigati or victoriati, as exhibiting a representation of a figure of Victory holding the reins. Quadrilateral (Lat. quadrilaterus, of four sides). In Elementary Geometry, a plane figure contained by four straight lines. Such a figure has four angles or corners, and is consequently also a quadrangle. The lines joining its oppo α site corners constitute its two diagonals. In modern geometry, however, a quadrilateral or tetragon denotes a system of four lines (sides); whilst by quadrangle is usually meant a system of four points (angles). The former has six angles or points in two sides, and the latter has six sides or lines through two points. If the full lines in the figure represent any complete quadrilateral, the broken lines a a, b b1, cc, joining the three pairs of opposite angles, constitute its three diagonals. One of the most important properties of a complete quadrilateral is that each diagonal is cut by the other two in harmonic conjugates with respect to the two angles which it contains. Thus a, a1, B, y; b, b1, y, a ; c, c1, a, B, are four sets of harmonical points. Four lines in space, two of which, though not in the same plane, are intersected by each of the others, form a skew quadrilateral. QUADRILATERAL. This name has been used, in the recent struggles between the Italians and the Austrians, to denote the territory, which forms a sort of square, between the fortresses of Peschiera, Verona, Legnano, and Mantua. Quadrilaterals. The name of a tribe of crabs (Brachyurous Crustaceans), comprehending those in which the carapace or shell is more or less square-shaped. Quadrille (Fr.). A game at cards for four persons, having some resemblance to whist. It was very popular and fashionable in England some two generations back, but is now almost forgotten. It ought to be revived, for it has great merits. It demands less science, thought, and memory than whist; but still it gives ample scope for skilful play, and it is much more varied, amusing, and suitable for younger players. It is a highly original game, having some peculiar features, and therefore requires a little attention from beginners; but the peculiarities are soon mastered, and are easily remembered. Quadrille is played with a pack of forty cards, the eight, nine, and ten of each suit being rejected. The dealing and order of playing are similar to whist; except (1) that they go the contrary way round, the person at the right of |