From the Table I., or by quadrature of the curve in fig.
Thus he devoted some attention to the quadrature of surfaces and the cubature of solids, which he accomplished, in some of the simpler cases, by an original method which he called the "Method of Indivisibles"; but he lost much of the credit of the discovery as he kept his method for his own use,while Bonaventura Cavalieri published a similar method which he himself had invented.
Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.
Among his later poems may be mentioned La Divine Tragedie (1916) and La Quadrature de l'Amour (1920).
If a fourth integral is obtainable, the solution is reducible to quadrature, but this is not possible except in a limited series of cases, investigated by H.
Quadrator, squarer), in mathematics, a curve having ordinates which are a measure of the area (or quadrature) of another curve.
The intercept on the axis of y is 2a/7r; therefore, if it were possible to accurately construct the curve, the quadrature of the circle would be effected.
His chief works are: Astronomical History of Observations of Heavenly Motions and Appearances (1634); Ecliptica prognostica (1634); Controversy with Longomontanus concerning the Quadrature of the Circle (1646?); An Idea of the Mathematics, 12m0 (1650); A Table of Ten Thousand Square Numbers (fol.; 1672).
Mensuration, then, is mainly concerned with quadratureformulae and cubature formulae, and, to a not very clearly defined extent, with the methods of obtaining such formulae; a quadrature-formula being a formula for calculating the numerical representation of an area, and a cubature-formula being a formula for calculating the numerical representation of a volume, in terms, in each case, of the numerical representations of particular data which determine the area or the volume.
On the other hand, it is worth noticing that the words " quadrature " and " cubature " are originally due to geometrical rather than numerical considerations; the former implying the construction of a square whose area shall be equal to that of a given surface, and the latter the construction of a cube whose volume shall be equal to that of a given solid.
Quadrature-formulae or cubature-formulae may sometimes be conveniently replaced by formulae giving the mean ordinate or mean section.
Thus a quadrature-formula is a formula for expressing [A x .24] or fudx in terms of a series of given values of u, while a cubature-formula is a formula for expressing [[Vx, 0 .
The value of S 13 _ 1 has to be found by a quadrature-formula.
One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied.
The use of quadrature-formulae is important in actuarial work, where the fundamental tables are based on experience, and the formulae applying these tables involve the use of the tabulated values and their differences.
He extended the "law of continuity" as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the parabola y=xm, where m is a positive integer, the area of the curves when m is negative or fractional.
A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy=const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log bla.
The first attempts to solve the purely geometrical problem appear to have been made by the Greeks (Anaxagoras, &c.) 2, one of whom, Hippocrates, doubtless raised hopes of a solution by his quadrature of the so-called meniscoi or lune.3 [The Greeks were in possession of several relations pertaining to the quadrature of the lune.
" Quadrature of the Circle, '"in' English Cyclop.; Glaisher, Mess.
5 For minute and lengthy details regarding the quadrature of the circle in the Low Countries, see de Haan, " Bouwstoffen voor de geschiedenis, &c.," in Versl.
Essentially, therefore, Descartes's process is that known later as the process of isoperimeters, and often attributed wholly to Schwab.2 In 16J5 appeared the Arithmetica Infinitorum of John Wallis, where numerous problems of quadrature are dealt with, the curves being now represented in Cartesian co-ordinates, and algebra playing an important part.
Montucla, successful attempt to show that quadrature of the circle by a Euclidean construction was impossible.'
A much less wise class than the 7r-computers of modern times are the pseudo-circle-squarers, or circle-squarers technically so called, that is to say, persons who, having obtained by illegitimate means a Euclidean construction for the quadrature or a finitely expressible value for 7r, insist on using faulty reasoning and defective mathematics to establish their assertions.
Such persons have flourished at all times in the history of mathematics; but the interest attaching to them is more psychological than mathematical.2 It is of recent years that the most important advances in the theory of circle-quadrature have been made.
In 1873 Charles Hermite proved that the base of the Napierian logarithms cannot be a root of a rational algebraical equation of any degree.3 To prove the same proposition regarding 7r is to prove that a Euclidean construction for circle-quadrature is impossible.
The area of the surface included between this curve and its base is found - the first known instance of a quadrature of a curved surface.
The second includes a "Method for the Quadrature of Parabolas," and a treatise "on Maxima and Minima, on Tangents, and on Centres of Gravity," containing the same solutions of a variety of problems as were afterwards incorporated into the more extensive method of fluxions by Newton and Leibnitz.
He studied the properties of the cycloid, and attempted the problem of its quadrature; and in the "infinitesimals," which he was one of the first to introduce into geometrical demonstrations, was contained the fruitful germ of the differential calculus.
Equation of the centre and evection are, at quadrature 6.29° sin g+I 27° sin g= 7.56° sin g.
When the satellite is in quadrature the convergence of the lines of attraction toward the centre of the sun tends to bring the two bodies together.
The first contains an explanation of the doctrine of fluxions, and of its application to the quadrature of curves; the second, a classification of seventy-two curves of the third order, with an account of their properties.
In 1 754 he published an anonymous treatise entitled Histoire des recherches sur la quadrature du cercle, and in 1758 the first part of his great work, Histoire des mathdmatiques, the first history of mathematics worthy of the name.
Pascal solved the hitherto refractory problem of the general quadrature of the cycloid, and proposed and solved a variety of others relating to the centre of gravity of the curve and its segments, and to the volume and centre of gravity of solids of revolution generated in various ways by means of it.
I.) to appear in 1656, take care to remove some of the worst mistakes exposed by Wallis, and, while leaving out all the references to Vindex, now profess to make, in altered form, a series of mere " attempts " at quadrature; but he was far from yielding the ground to the enemy.