John Wallis

"The true "principle of number," for Wallis as for Stevin, is the "nought". It is the sole numerical analogue of the geometric point (just as the instant is the temporary analogue... Wallis expressly rejects the accusation that he is relinquishing the unanimous opinion of the ancients and the moderns, who all saw the unit as the element of number. ...the traditional opinion can be brought into accord with his own if the following distinction is taken account of: Something can be a "principle" of something (1) which is the "first which is such" (primum quod sic) as to be of the same nature as the thing itself and (2) which is the last which is not" (ultimum quod non) such as to be of the same nature of the thing itself. In the first sense the unit may indeed be called the "principle of number," while the nought is a "principle" in the second sense. ...The ancients... overlooked the fact that the analogy which exists is not between the "point" and the "unit," but between the point and the "nought." For this reason they were able to develop their algebra only for "geometric magnitudes"..."

"Paralleling what happened in France, an English group centered about John Wallis began in 1645 to hold meetings in Gresham College, London, These men emphasized mathematics and astronomy. The group was given a formal charter by Charles II in 1662 and adopted the name of the Royal Society of London for the Promotion of Natural Knowledge."

"Before Newton and Leibniz, the man who did most to introduce analytical methods in the calculus was John Wallis. Though he did not begin to learn mathematics until he was about twenty—his university education at Cambridge was devoted to theology—he became professor of geometry at Oxford and the ablest British mathematician of the century, next to Newton. In his Arithmetica Infinitorum (1655), he applied analysis and the method of indivisibles to effect many quadratures and obtain broad and useful results."

"[E]arly analytic geometers—Descartes in particular—did not accept that geometry could be based on numbers or algebra. Perhaps the first to take the idea of arithmetizing geometry seriously was Wallis... [(1657) Mathesis universalis. Opera 1, 11-228.] Chs. XXIII and XXV, gave the first arithmetic treatment of Euclid's Books II and V, and he had earlier given purely algebraic treatment of s [(1655) De sectionibus conicus. Opera 1, 291-364.]. He initially derived equations from classical definitions by sections of the cone but then proceeded to derive their properties from the equations, "without the embranglings of the cone," as he put it."

"The greatest of modern have been so far from adding any thing of importance to the discoveries of ancient mathematicians, that some of their most splendid inventions are either wholly erroneous or remarkable instances of the possibility of deducing true conclusions from unscientific and false principles. Strange, however as this assertion may seem, the following elementary treatise demonstrates it to be true; by showing that all the leading propositions of the Arithmetic of Infinites of Dr. Wallis are false, and that the Doctrine of Fluxions is a baseless fabric, and in the language of the ingenious Bishop Berkley, "must be considered only as a presumption, as a knack, an art, or rather an artifice, but not a scientific demonstration."

"Wallis, whether by his own efforts or not, acquired sufficient mathematics at Cambridge to be ranked as the equal of mathematicians such as Descartes, Pascal, and Fermat."

"There was then no professorship in mathematics and no opening for a mathematician to a career at Cambridge; and so Wallis reluctantly left the university. In 1649 he was appointed to the Savilian chair of geometry at Oxford, where he lived until his death on Oct. 28, 1703. It was there that all his mathematical works were published. Besides those he wrote on theology, logic, and philosophy; and was the first to devise a system for teaching deaf mutes."

"The most notable of these [his mathematical works] was his Arithmetica infinitorum, which was published in 1656. It is prefaced by a short tract on conic sections which was subsequently expanded into a separate treatise. He then established the law of indices, and shewed that x^{-n} stood for the reciprocal of x^n and that x^\frac{p}{q} stood for the q^{th} root of x^p. He next proceeded to find by the method of indivisibles the area enclosed between the curve y = x^m, the axis of x, and any ordinate x = h; and he proved that this was to the parallelogram on the same base and of the same altitude in the ratio 1:m + 1. He apparently assumed that the same result would also be true for the curve y = ax^m, where a is any constant. In this result m may be any number positive or negative, and he considered in particular 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 shewed that similar results might be written down for any curve of the form y = \sum{ax^m}; so that if the ordinate y of a curve could be expanded in powers of the abscissa x, its quadrature could be determined. Thus he said that if the equation of a curve was y = x^0 + x^1 + x^2 +... its area would be y = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 +... He then applied this to the quadrature of the curves y = (1 - x^2)^0, y = (1 - x^2)^1, y = (1 - x^2)^2, y = (1 - x^2)^3, &c. taken between the limits x = 0 and x = 1: and shewed that the areas are respectively1,\quad \frac{2}{3},\quad \frac{8}{15},\quad \frac{16}{35},\quad \&c."

"He next considered curves of the form y = x^\frac{1}{m} and established the theorem that the area bounded by the curve, the axis of x, and the ordinate x = 1 is to the area of the rectangle on the same base and of the same altitude as m:m + 1. This is equivalent to finding the value of \int_{0}^{1}x^\frac{1}{m}dx. He illustrated this by the parabola in which m = 2. He stated but did not prove the corresponding result for a curve of the form y = x^\frac{p}{q}."

"As he was unacquainted with the he could not effect the quadrature of the circle, whose equation is y = (1 - x^2)^\frac{1}{2}, since he was unable to expand this in powers of x. He laid down however the principle of interpolation. He argued that as the ordinate of the circle is the geometrical mean between the ordinates of the curves y = (1 - x^2)^0 and y = (1 - x^2)^1, so as an approximation its area might be taken as the geometrical mean between 1 and \frac{2}{3}. This is equivalent to taking 4\sqrt{\frac{2}{3}} or 3.26... as the value of \pi. But, he continued, we have in fact a series 1, \frac{2}{3}, \frac{8}{15}, \frac{16}{35},... and thus the term interpolated between 1 and \frac{2}{3} ought to be so chosen as to obey the law of this series. This by an elaborate method leads to a value for the interpolated term which is equivalent to making\pi = 2\frac{2\cdot2\cdot4\cdot4\cdot6\cdot6\cdot8\cdot8...}{1\cdot3\cdot3\cdot5\cdot5\cdot7\cdot7\cdot9...}The subsequent mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct algebraic analysis."

"The Arithmetica infinitorum was followed in 1656 by a tract on the angle of contact; in 1657 by the Mathesis universalis; in 1658 by a correspondence with Fermat; and by a long controversy with Hobbes on the quadrature of the circle."

"In 1659 Wallis published a tract on s in which incidentally he explained how the principles laid down in his Arithmetica infinitorum could be applied to the rectification of s: and in the following year one of his pupils, by name William Neil, applied the rule to rectify the x^3 = ay^2. This was the first case in which the length of a curved line was determined by mathematics, and as all attempts to rectify the ellipse and hyperbola had (necessarily) been ineffectual, it had previously been generally supposed that no curves could be rectified."

"In 1665 Wallis published the first systematic treatise on Analytical conic sections. Analytical geometry was invented by Descartes and the first exposition of it was given in 1637: that exposition was both difficult and obscure, and to most of his contemporaries, to whom the method was new, it must have been incomprehensible. Wallis made the method intelligible to all mathematicians. This is the first book in which these curves are considered and defined as curves of the second degree and not as sections of a cone."

"In 1668 he laid down the principles for determining the effects of the collision of 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."

"In 1686 Wallis published an Algebra, preceded by a historical account of the development of the subject which contains a great deal of valuable information... This algebra is noteworthy as containing the first systematic use of formulae."

"A particle moving with a uniform velocity would be denoted by Wallis by the formula s = vt, ...while previous writers would have denoted the same relation by stating what is equivalent to the proposition s1 : s2 = v1t1 : v2t2 (see e.g. Newton's Principia, bk. I. sect. I., lemma 10 or 11)."

"Wallis rejected as absurd and inconceivable the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity."

"The writings of Wallis published between 1655 and 1665 revealed and explained to all students the principles of those new methods which distinguish modern from classical mathematics. His reputation has been somewhat overshadowed by that of Newton, but his work was absolutely first class in quality. Under his influence a brilliant mathematical school arose at Oxford. In particular I may mention Wren, Hooke, and Halley as among the most eminent of his pupils. But the movement was shortlived, and there were no successors of equal ability to take up their work."

"The foundations of the new analysis were laid in the second half of the seventeenth century when Newton... and Leibnitz... founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernoulli... and John Bernoulli... the ideas and methods of the Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new analysis was to replace the old geometrical or semi-geometrical methods of calculating \pi by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of \pi to any assigned degree of approximation."

"The first result of this kind was due to John Wallis... undergraduate at Emmanuel College, Fellow of Queen's College, and afterwards Savilian Professor of Geometry at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of analysis. Wallis gave in his Arithmetica Infinitorum the expression\frac{\pi}{2} = \frac {2}{1}\cdot\frac {2}{3}\cdot\frac {4}{3}\cdot\frac {4}{5}\cdot\frac {6}{5}\cdot\frac {6}{7}\cdot\frac {8}{7}\cdot\frac {8}{9}\cdotsfor \pi as an infinite product, and he shewed that the approximation obtained at stopping at any fraction in the expression on the right is in defect or in excess of the value \frac{\pi}{2} according as the fraction is proper or improper. This expression was obtained by an ingenious method depending on the expression for \frac{\pi}{8} the area of a semi-circle of diameter 1 as the definite integral \int\limits_{0}^{1}\sqrt{x-x^2}dx. The expression has the advantage over that of Vieta that the operations required are all rational ones."

"Lord Brounckner... communicated without proof to Wallis the [] expression\frac{4}{\pi} = 1 + \frac {1}{2 +} \frac {9}{2 +} \frac {25}{2 +} \frac {49}{2 +}\cdots,a proof of which was given by Wallis in his Arithmetica Infinitorum. It was afterwards shewn by Euler that Wallis' formula could be obtained from the development of the sine and cosine in infinite products, and that Brounckner's expression is a particular case of much more general theorems."

"Of the contemporaries of Newton one of the most prominent was John Wallis. ...Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics... He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it in the quadrature of such curves as y=xn, y=x1/n, and y=x0 + x1 + x2 +... He failed in his attempts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. He also obtained the equivalent of ds = \!dx \sqrt{1+(\frac{dy}{dx})^2} for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature."

"In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of classical literature of the science. This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning."

"Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates."

"His writings include works on mechanics, sound, astronomy, the tides, the laws of motion, the Torricellian tube, botany, physiology, music, the calendar (in opposition to the Gregorian reform), geology, and the compass,—a range too wide to allow of the greatest success in any of the lines of his activity. He was also an ingenious cryptologist and assisted the government in deciphering diplomatic messages."

"Among his interesting discoveries was the relation \frac{4}{\pi} = \frac32\cdot\frac34\cdot\frac54\cdot\frac56\cdot\frac76\cdot\frac78\cdots one of the early values of π involving infinite products."

"In his 1657 Mathesis universalis... Wallis printed geometric representations of the first 10 propositions of book II of the Elements alongside symbolic 'arithmetical' demonstrations and worked examples in numbers; he believed that the symbolic treatment was more general."

"Wallis’s 1656 Arithmetic of infinities was a crucial text for Newton’s development of calculus, a fact that Newton... related to Gottfried Leibniz in a... letter."

"Wallis... denigrat[ed] synthetic demonstration for its failure to bring clarity to mathematical investigation."

"The association between geometric synthesis and 'common' understanding was not Newton’s innovation: many of Newton’s predecessors, including Wallis, endorsed it."

"Seventeenth-century proponents of symbols frequently advanced the fecundity of their approach as evidence for its superiority over classical mathematics: Wallis’s Arithmetic of infinities displayed results 'neither discovered by nor known to others', and René Descartes’s 1637 Geometry announced its general solution to a family of problems that the Greeks had left mostly unsolved. ...Newton, inspired ...by these ...texts ...used symbols to invent an algorithmic version of the calculus."

"Algebra appealed to Newton’s contemporaries... for its fecundity... [and] the... presentational qualities... In... his... Treatise on conic sections [dedication], Wallis contrasted his novel treatment of the s with the diagrammatic treatment in Apollonius’s Conics... 'neglected beyond measure' by Wallis’s contemporaries 'as though it were insurmountable and full of troublesome madness'. Wallis implied that geometers read it... superficially since they feared that it would drive them mad. He contrasted his own figures with Apollonius’s... to use 'schemata as simple as possible, lest intricate leadings of lines bring in confusion'."

"Wallis saw Greek diagrams as intricate and confused. His own... clearly displayed the conic sections and the key lines characterizing... their ‘essential affections’... More importantly, he used the same diagram as often as possible and, when not... changed the diagram only minimally. ...[He] claim[ed]... his 1685 Treatise on algebra... considered the sections 'abstractly as Figures in plano, without the embranglings of the Cone'."

"By regularizing... symbols Wallis circumvented the typical role of diagrams in geometric argument. ...Wallis’s new lettering strategy meant that the reader would not have to constantly look back at the diagram to understand the argument... the reader might recognize from the letters which parts of the cone were signified."

"Newton opposed entangled and tedious algebraic calculations to simple, elegant geometric constructions; Wallis opposed difficult, embrangled geometric diagrams to simplified, rationally lettered diagrams and the symbolic expressions they enabled."

"[I]n On conic sections, Wallis... claimed that algebra would enable an 'absolute contemplation' of the conic sections by directly expressing... 'primary and essential affections' from which secondary affections could be 'deduced by calculation'. ...Both the use of symbols to express essences and their manipulation to obtain easy results characterized Wallis..."

"Based on a pattern he observed in the characters of the first few s, Wallis conceived a procedure for generating further characters. ...Wallis explained how to use figurate numbers to measure areas and volumes and, eventually, to square the circle. ...[H]e demphasized traditional forms of mathematical demonstration ...taking ...analysis as a way of finding. He believed he could promote mathematics more by exposing his own investigations than through sterile demonstrations."

"Wallis used Arithmetic of infinities to produce genuine understanding rather than brutal persuasion."