QUADRATURE

, in Astronomy, that aspect or position of the moon when she is 90° distant from the sun. Or, the Quadratures or quarters are the two middle points of the moon's orbit between the points of conjunction and opposition, viz, the points of the 1st and 3d quarters; at which times the moon's face shews half full, being dichotomized or bisected.

The moon's orbit is more convex in the Quadratures than in the syzygies, and the greater axis of her orbit passes through the Quadratures, at which points also the is most distant from the earth.—In the Quadratures, and within 35° of them, the apses of the moon go backwards, or move in antecedentia; but in the syzygies the contrary.—When the nodes are in the Quadratures, the inclination of the moon's orbit is greatest, but least when they are in the syzygies.

Quadrature Lines, or Lines of Quadrature, are two lines often placed on Gunter's sector. They are marked with the letter Q, and the figures 5, 6, 7, 8, 9, 10; of which Q denotes the side of a square, and the figures denote the sides of polygons of 5, 6, 7, &c sides. Also S denotes the semidiameter of a circle, and 90 a line equal to the quadrant or 90° in circumference.

, in Geometry, is the squaring of a figure, or reducing it to an equal square, or finding a square equal to the area of it.

The Quadrature of rectilineal figures falls under common geometry, or mensuration; as amounting to no more than the finding their areas, or superficies; which are in effect their squares: which was fully effected by Euclid.

The Quadrature of Curves, that is, the measuring of their areas, or the finding a rectilineal space equal to a proposed curvilineal one, is a matter of much deeper speculation; and makes a part of the sublime or higher geometry. The lunes of Hypocrates are the first curves that were squared, as far as we know of. The circle was attempted by Euclid and others before him: he shewed indeed the proportion of one circle to another, and gave a good method of approximating to the area of the circle, by describing a polygon between any two concentric circles, however near their circumferences might be to each other. At this time the conic sections were admitted in geometry, and Archimedes, perfectly, for the first time, squared the parabola, and he determined the relations of spheres, spheroids, and conoids, to cylinders and cones; and by pursuing the method of exhaustions, or by means of inscribed and circumscribed polygons, he approximated to the periphery and area of the circle; shewing that the diameter is to the circumference nearly as 7 to 22, and the area of the circle to the square of the diameter as 11 to 14 nearly. Archimedes likewise determined the relation between the circle and ellipse, as well as that of their similar parts: It is probable too that he attempted the hyperbola; but it is not likely that he met with any success, since approximations to its area are all that can be given by the various methods that have since been invented. Beside these figures, he left a treatise on a spiral curve; in which he determined the relation of its area to that of the circumscribed circle; as also the relation of their sectors.

Several other eminent men among the Ancients wrote upon this subject, both before and after Euclid and Archimedes; but their attempts were usually confined to particular parts of it, and made according to methods not essentially different from theirs. Among these are to be reckoned Thales, Anaxagoras, Pythagoras, Bryson, Antiphon, Hypocrates of Chios, Plato, Apollonius, Philo, and Ptolomy; most of whom wrote upon the Quadrature of the circle; and those after Archimedes, by his method, usually extended the approximation to a higher degree of accuracy.

Many of the Moderns have also prosecuted the same problem of the Quadrature of the circle, after the same methods, to still greater lengths; such are Vieta, and Metius; whose ratio between the diameter and the circumference, is that of 113 to 355, which is within about (3/10000000) of the true ratio; but above all, Ludolph van Collen, or a Ceulen, who, with an amazing degree of industry and patience, by the same methods, extended the ratio to 36 places of figures, making the ratio to be that of 1 to 3.14159,26535,89793,23846,26433,83279,50288 + or 9 -. And the same was repeated and confirmed by his editor Snellius. See Diameter, and Circle; also the Preface to my Mensuration.

Though the Quadrature, especially of the circle, be a thing which many of the principal mathematicians, among the Ancients, were very solicitous about; yet nothing of this kind has been done so considerable, as | about and since the middle of the last century; when, for example, in the year 1657, Sir Paul Neil, Lord Brouncker, and Sir Christopher Wren geometrically demonstrated the equality of some curvilineal spaces to rectilineal ones. Soon after this, other persons did the like in other curves; and not long afterwards the thing was brought under an analytical calculus, the first specimen of which ever published, was given by Mercator in 1688, in a demonstration of Lord Brouncker's Quadrature of the hyperbola, by Dr. Wallis's method of reducing an algebraical fraction into an infinite series by division.

Though, by the way, it appears that Sir Isaac Newton had discovered a method of attaining the area of all quadrable curves analytically, by his Method of Fluxions, before the year 1668. See his Fluxions, also his Analysis per Æquationes Numero Terminorum Infinitas, and his Introductio ad Quadraturam Curvarum; where the Quadratures of Curves are given by general methods.

It is contested, between Mr. Huygens and Sir Christopher Wren, which of the two first found out the Quadrature of any determinate cycloidal space. Mr. Leibnitz afterwards discovered that of another space; and Mr. Bernoulli, in 1699, found out the Quadrature of an infinity of cycloidal spaces, both segments and sectors &c.

As to the Quadrature of the Circle in particular, or the finding a square equal to a given circle, it is a problem that has employed the mathematicians of all ages, but still without the desired success. This depends on the ratio of the diameter to the circumference, which has never yet been determined in precise numbers. Many persons have approached very near this ratio; for which see Circle.

Strict geometry here failing, mathematicians have had recourse to other means, and particularly to a sort of curves called quadratices: but these being mechanical curves, instead of geometrical ones, or rather transcendental instead of algebraical ones, the problem cannot fairly be effected by them.

Hence recourse has been had to analytics. And the problem has been attempted by three kinds of algebraical calculations. The first of these gives a kind of transcendental Quadratures, by equations of indefinite degrees. The second by vulgar numbers, though irrationally such; or by the roots of common equations, which for the general Quadrature is impossible. The third by means of certain series, exhibiting the quantity of a circle by a progression of terms. See Series.

Thus, for example, the diameter of a circle being 1, it has been found that the quadrant, or one-fourth of the circumference, is equal to (1/1) - (1/3) + (1/5) - (1/7) + (1/9) &c, making an infinite series of fractions, whose common numerator is 1, and denominators the natural series of odd numbers; and all these terms alternately will be too great, and too little. This series was discovered by Leibnitz and Gregory. And the same series is also the area of the circle.

If the sum of this series could be found, it would give the Quadrature of the circle: but this is not yet done; nor is it at all probable that it ever will be done; though the impossibility has never yet been demonstrated.

To this it may be added, that as the same magnitude may be expressed by several different series, possibly the circumference of the circle may be expressed by some other series, whose sum may be found. And there are many other series, by which the quadrant, or area, to the diameter, has been expressed; though it has never been found that any one of them is actually summable. Such as this series, 1 - (1/6) - (1/40) - (1/112) &c, invented by Newton; with innumerable others.

But though a definite Quadrature of the whole circle was never yet given, nor of any aliquot part of it; yet certain other portions of it have been squared. The first partial Quadrature was given by Hippocrates of Chios; who squared a portion called, from its figure, the lune, or lunule; but this Quadrature has no dependence on that of the circle. And some modern geometricians have found out the Quadrature of any portion of the lune taken at pleasure, independently of the Quadrature of the circle; though still subject to a certain restriction, which prevents the Quadrature from being perfect, and what the geometricians call absolute and indefinite. See Lune. And for the Quadrature of the different kinds of curves, see their several particular names.

Quadratures by Fluxions.—The most general method of Quadratures yet discovered, is that of Newton, by means of Fluxions, and is as follows. AC being any curve to be squared, AB an absciss, and BC an ordinate perpendicular to it, also bc another ordinate indefinitely near to the former. Putting AB = x, and BC = y; then is Bb = x^. the fluxion of the absciss, and yx^. = Cb the fluxion of the area ABC sought. Now let the value of the ordinate y be found in terms of the absciss x, or in a function of the absciss, and let that function be called X, that is y = X; then substituting X for y in yx^., gives Xx^. the fluxion of the area; and the fluent of this, being taken, gives the area or Quadrature of ABC as required, for any curve, whatever its nature may be.

Ex. Suppose for example, AC to be a common parabola; then its equation is , where p is the parameter; which gives , the value of y in a function of x, and is what is called X above; hence then is the fluxion of the area; and the fluent of this is of the circumscribing rectangle BD; which therefore is the Quadrature of the parabola.

Again, if AC be a circle whose diameter is d; then its equation is , which gives , and the fluxion of the area . But as the fluent of this cannot be found in finite terms, the quantity √(dx - x²) is thrown into a series, and then the fluxion of the area | is ; and the fluent of this gives for the general expression of the area ABC. Now when the space becomes a semicircle, x becomes = d, and then the series above becomes for the area of the semicircle whose diameter is d.

In spirals CAR, or curves referred to a centre C; put y = any radius CR, x = BN the arc of a circle described about the centre C, at any distance CB = a, and Cnr another ray indefinitely near CNR: then , and by sim. fig. the fluxion of the area described by the revolving ray CR; then the fluent of this, for any particular case, will be the Quadrature of the spiral. So if, for instance, it be Archimedes's spiral, in which x : y in a constant ratio suppose as m : n, or my = nx, and ; hence then the fluxion of the area; the fluent of which is the general Quadrature of the spiral of Archimedes.