John Wallis

John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is also credited with introducing the symbol ∞ for infinity.

Mathematics

In 1655, Wallis published a treatise on conic sections in which they were defined analytically. This was the earliest book in which these curves are considered and defined as curves of the second degree. It helped to remove some of the perceived difficulty and obscurity of René Descartes' work on analytic geometry.

Related Topics:
René Descartes - Analytic geometry

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Arithmetica Infinitorum, the most important of Wallis's works, was published in 1656. In this treatise the methods of analysis of Descartes and Cavalieri were systematised and extended, but some ideals were open to criticism. He begins, after a short tract on conic sections, by developing the standard notation for powers, extending them from positive integers to rational numbers:

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• $x^0\; =\; 1$, $x^\{-1\}\; =\; 1/x$, $x^\{-2\}\; =\; 1/x^2$, etc.
• $x^\{1/2\}\; =$square root of $x$, $x^\{2/3\}\; =$cube root of $x^2$, etc.
• $x^\{1/n\}\; =$nth root of $x$.
• $x^\{p/q\}\; =$qth root of $x^p$.

Leaving the numerous algebraic applications of this discovery, he next proceeds to find, by the method of indivisibles, the area enclosed between the curve y = xm, the axis of x, and any ordinate x = h, and he proves that the ratio of this area to that of the parallelogram on the same base and of the same altitude is 1/(m + 1). He apparently assumed that the same result would be true also for the curve y = axm, where a is any constant, and m any number positive or negative; but he only discusses the case of the parabola in which m = 2, and that of the hyperbola in which m = −1. In the latter case, his interpretation of the result is incorrect. He then shows that similar results might be written down for any curve of the form

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: y = sum_{m}^{} ax^{m}

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and hence that, if the ordinate y of a curve can be expanded in powers of x, its area can be determined: thus he says that if the equation of the curve is y = x0 + x1 + x2 + ..., its area would be x + x2/2 + x3/3 + ... He then applies this to the quadrature of the curves y = (x − x2)0, y = (x − x2)1, y = (x − x2)2, etc., taken between the limits x = 0 and x = 1. He shows that the areas are respectively 1, 1/6, 1/30, 1/140, etc. He next considers curves of the form y = x1/m and establishes the theorem that the area bounded by this curve and the lines x = 0 and x = 1 is equal to the area of the rectangle on the same base and of the same altitude as m : m + 1. This is equivalent to computing

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:int_0^1x^{1/m},dx

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He illustrates this by the parabola, in which case m = 2. He states, but does not prove, the corresponding result for a curve of the form y = xp/q.

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Wallis showed considerable ingenuity in reducing the equations of curves to the forms given above, but, as he was unacquainted with the binomial theorem, he could not effect the quadrature of the circle, whose equation is y =, since he was unable to expand this in powers of x. He laid down, however, the principle of interpolation. Thus, as the ordinate of the circle y = is the geometrical mean between the ordinates of the curves y = and y =, it might be suppose that, as an approximation, the area of the semicircle dx which is might be taken as the geometrical mean between the values of

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dx and dx

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that is, 1 and ; this is equivalent to taking or 3.26... as the value of . But, Wallis argued, we have in fact a series 1,,,,... and therefore the term interpolated between 1 and ought to be chosen so as to obey the law of this series. This, by an elaborate method, which I need not describe in detail, leads to a value for the interpolated term which is equivalent to taking

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In this work also the formation and properties of continued fractions are discussed, the subject having been brought into prominence by Brouncker's use of these fractions.

Related Topics:
Continued fraction - Brouncker

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A few years later, in 1659, Wallis published a tract containing the solution of the problems on the cycloid which had been proposed by Blaise Pascal. In this he incidentally explained how the principles laid down in his Arithmetica Infinitorum could be used for the rectification of algebraic curves; and gave a solution of the problem to rectify (i.e. find the length of) the semi-cubical parabola x3 = ay2, which had been discovered in 1657 by his pupil William Neil. Since all attempts to rectify the ellipse and hyperbola had been (necessarily) ineffectual, it had been supposed that no curves could be rectified, as indeed Descartes had definitely asserted to be the case. The logarithmic spiral had been rectified by Torricelli, and was the first curved line (other than the circle) whose length was determined, but the extension by Neil and Wallis to an algebraic curve was novel. The cycloid was the next curve rectified; this was done by Wren in 1658.

Related Topics:
1659 - Blaise Pascal - 1657 - William Neil - Logarithmic spiral - Torricelli - Wren - 1658

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Early in 1658 a similar discovery, independent of that of Neil, was made by van Heuraët, and this was published by van Schooten in his edition of Descartes's Geometria in 1659. Van Heuraët's method is as follows. He supposes the curve to be referred to rectangular axes; if this be so, and if (x, y) be the coordinates of any point on it, and n be the length of the normal, and if another point whose coordinates are (x, ) be taken such that : h = n : y, where h is a constant; then, if ds be the element of the length of the required curve, we have by similar triangles ds : dx = n : y. Therefore h ds = dx. Hence, if the area of the locus of the point (x,) can be found, the first curve can be rectified. In this way van Heuraët effected the rectification of the curve y³ = ax² but added that the rectification of the parabola y² = ax is impossible since it requires the quadrature of the hyperbola. The solutions given by Neil and Wallis are somewhat similar to that given by van Heuraët, though no general rule is enunciated, and the analysis is clumsy. A third method was suggested by Fermat in 1660, but it is inelegant and laborious.

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The theory of the collision of bodies was propounded by the Royal Society in 1668 for the consideration of mathematicians. Wallis, Wren, and Huygens sent correct and similar solutions, all depending on what is now called the conservation of momentum; but, while Wren and Huygens confined their theory to perfectly elastic bodies, Wallis considered also imperfectly elastic bodies. This was followed in 1669 by a work on statics (centres of gravity), and in 1670 by one on dynamics: these provide a convenient synopsis of what was then known on the subject.

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In 1685 Wallis published Algebra, preceded by a historical account of the development of the subject, which contains a great deal of valuable information. The second edition, issued in 1693 and forming the second volume of his Opera, was considerably enlarged. This algebra is noteworthy as containing the first systematic use of formulae. A given magnitude is here represented by the numerical ratio which it bears to the unit of the same kind of magnitude: thus, when Wallis wants to compare two lengths he regards each as containing so many units of length. This perhaps will be made clearer by noting that the relation between the space described in any time by a particle moving with a uniform velocity is denoted by Wallis by the formula s = vt, where s is the number representing the ratio of the space described to the unit of length; while the previous writers would have denoted the same relation by stating what is equivalent to the proposition . It is curious to note that Wallis rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity.

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