John Wallis and Isaac Barrow: tradition and innovation and the state of mathematics

John Wallis and Isaac Barrow: tradition and innovation and the state of mathematics Katherine Hill

John Wallis and Isaac Barrow were key figures in a transitional period in the development of mathematics in early modern England: their work reveals a tension between the emerging algebraic techniques and the more traditional geometric mode of thought. Both men were among the first professional mathematicians in England.

John Wallis studied at Cambridge, deci- phered royalist codes for Parliament during the Civil War, and was one of the Secretaries to the Assembly of Divines at Westminister. He was rewarded for his support of Parliament with the Savilian Professorship of Geometry at Oxford. Isaac Barrow was also a student at Cambridge and in 1660 was appointed Regius Professor of Greek at Trinity College. He subse- quently became Professor of Geometry at Gresham College before finally becoming the Lucasian Professor of Mathematics at Cambridge. The work of both Wallis and Barrow was at the forefront of English mathematics in the second half of the seventeenth century. Additionally. each man’s work exhibited a similar tension between ‘tradition’ and ‘innovation’ - between the mathematical ideas inherited from the Greeks and the demands of the new methods and problems. Yet even though both enjoyed very similar educations and careers, their mathematical techniques were quite different. Wallis’s style can be seen as ‘algebraic’ in comparison with Barrow’s ‘geometric’ approach.

When historians attempt to describe and explain this type of difference in mathematical methods they often speak of a tension between two modes of thought, ‘modem’ and ‘traditional’. Modems. it is claimed, preferred algebra to geometry and adopted

Katherine Hill, BSc. (Hons) M.A.. Ph.D.

Studied mathematics at the Universrty of Houston, University Park. She obtained her doctorate in history of science and mathematics from the Institute for the History and Philosophy of Science and Technology, University of Toronto, in 1996. Visiting lecturer in history of science at University College of Cape Breton, Nova Scotia, in 1995, and lecturer at the Science Studies Unit at the University of Edinburgh 1996-98. In 1999 she will take up a position as a lecturer for the History of Science Department at the University of Sydney, Australia. Research interests include early modern English mathematics.

the new symbolism; they emphasized the heuristic aspects of mathematics. Traditionalists embraced geometry and synthetic proof; they valued rigour in mathematics. Moreover, modems are associated with a preference for innovation, while traditionalists are said to have remained attached to the more established classical techniques. The usual tactic is either to place early modem mathematicians into one of these categories, or to describe their work as revealing a tension between the two modes of thought. John Wallis is usually placed in the modem camp while Isaac Barrow is considered traditional.

Certain non-mathematical debates in early modem England, however. might better explain the differences between Wallis’s and Barrow’s mathematical techniques, such as the wider early 17th-century struggle between the supporters of ancients and modems, and the argument surrounding educational reform in the middle of the century. Both of these controversies involved a struggle between tradition and innovation. In the first case one faction believed that the ancients and their works were superior to the modems, while the other side felt that the modems had in some cases surpassed the accomplishments of the ancients. In the second case reformers were unsatisfied with traditional educational methods, and so they wished to transform education in many areas, including mathematics. These disagreements will show that ‘ancient’ and ‘modem’ had different meanings in the seventeenth century from the meanings given to these terms by historians. Moreover, placing Wallis and Barrow into these somewhat anachronistic categories may cause historians to overlook the subtlety of their conceptions of mathematical continua and the influence of their actual working environment,

Although Barrow’s work was rigorous and geometric in nature, he abandoned classical concepts of number, space and time. Wallis’s work, on the other hand.

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utilized heuristic techniques and was often algebraic, but he hesitated about changing ancient terminology and attempted to work within what he considered the ‘classical’ framework. In fact, Wallis considered algebra to be ancient, not modem. The details of Wallis’s and Barrow’s mathematical practice challenge the validity of the traditional versus modern classification scheme.

Ancient versus modern: mathematical styles Wallis has been regarded as a member of the modem school because his mathematical work can be characterized as algebraic, which is depicted as a new way of doing mathematics in this period. Additionally, he emphasized methods of discovery instead of proof, and he employed the new symbolism. In his 1656 Arithmetica Infinitorum, Wallis found the area under the curve !: = _V between zero and one for positive fractions and negative integers by using analytic techniques and numerical series; and in his 1657 Mathesis uni~~evsalis there is an extended discussion of algebraic notation. Barrow is placed in the traditional school because his work was usually geometric, because he was concerned with rigour, and was not consistent in his use of symbolism. Barrow’s most famous work. his 1670 Lectiones geometricne. included geometrical constructions of tangents, quadrature and rectification problems.

The problems that display Wallis’s and Barrow’s mathematical characteristics have been dealt with elsewhere in great detail’. Short examples. however. will bring out Wallis’s skill at finding numerical patterns and his intuition about continuity, as well as Barrow’s geometrical talents and adeptness in using points moving through space.

Wallis’s best-known work, Arithmetica Infinitorum. published in 1656, describes his work on the quadrature of the circle?. Wallis’s method in this work is non-rigorously

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based upon what seems to be his strong feel- ing for pattern. His techniques might be related to his work on codes during the English Civil War. Evidently the whole lay- out on the printed page corresponds with the natural way that one sets out a message for decoding. The codes used in the Civil War were numerical, and there were easily rec- ognizable frequency patterns occurring among the various number-sets used. Many of the same skills that one would develop while decoding would be useful for Wallis’s style of numerical techniquess.

In modem terms, the main focus of the Arithmetica Infinitorum was Wallis’s attempt to evaluate the expression I (l-x*)‘/2 dx between zero and one. He relied heavily on his ‘method of induction’: one looks at a certain number of individual cases, observes the emerging ratios, and compares these with one another so that a universal proposition may be established. For example, in Proposition 1 he observed,

O+l 1 -=- 1+1 2

0+1+2 1 ---=- 2+2+2 2

0+1+2+3 1 =- 3+3+3+3 2

so he concluded (0 + 1 + 2 + . . + n) I (n + n + n + . . . n) = l/2 (Ref. 4). He then went on to use this technique to find the re- quired ratios for many positive integer pow- ers, finding the consequent or correspond- ing ratio of a series n (when n is a positive integer) to be ll(n + 1). In effect, he was finding the area under the curves using numerical techniques. Wallis thought that the principle of continuity, or interpolation, was the key to fractional indices.

Wallis felt that arithmetic was more universal than geometry, as it was purer and more abstract. His tendency was to unify the mathematical sciences under the priority of algebra. Barrow, however, followed a completely different pattern of mathematical practice. Barrow’s Geometrical Lectures dealt with the geometrical characteristics of curved lines. In Lecture VI, for example, Barrow began an ‘Investigation and ready Demonstration of Tangents, without the trouble of Calculation . . . [which] have not been so fully handled or exhausted as some others...‘s. This lecture deals mainly with the properties of tonics. Except for the kine- matic aspects of the demonstrations, Barrow treats mainly the properties of curves discussed in Apollonius’s Con&. His proofs are, in general, completely geometric.

Barrow was not claiming to offer new techniques. His work differed from classical, static demonstrations, such as Apollonius’s, in that it utilized lines or points moving though space. Yet in the end his proofs often work like conventional locus theorems. Occasionally, when he sub- stitutes variables for magnitudes, his proofs

seem on the surface to be algebraic. But he was careful to keep the terms of his equations in second degree to assure geometric intelligibility. On the other hand, the problems in Lecture X demonstrate that Barrow was also familiar with analytical techniques. For example, he gave a method of finding tangents similar to Fermat’s? nevertheless, he preferred to utilize geometry. Moreover, unlike Wallis, Barrow never worked with numerical sequences or series.

These examples of Wallis’s and Barrow’s mathematical styles help to explain why historians typically label them modem and traditional respectively. Barrow was ‘traditional’, in the sense in which the word is usually used, insofar as he favoured geometry .and, ‘valued rigour in mathematics. None of his work depended upon numerical properties, and he seldom overtly used algebraic manipulations. He certainly gave geometry priority over algebra, and felt that the ancients had more prestige than the modems. Wallis, on the other hand, was ‘modem’ in that he preferred algebra to geometry. Many of his results were created by working with numerical patterns, and the results were only later applied to geometry. Wallis did not always require synthetic proofs, satisfying himself with his ‘Method of Induction’. He was nevertheless satisfied that all of his results could be confirmed by more rigorous proofs. A deeper explo- ration of Wallis’s and Barrow’s work, however, coupled with an examination of how they themselves viewed their mathematical practice, will show that the ‘traditional’ versus ‘modem’ categories are too simplistic.

Wallis’s and Barrow’s views on ‘ancient’ and ‘modern’ Wallis did not feel that his mathematical techniques were a departure from classical methods. He did not regard algebra as a modem subject. In his Treatise of Algebra, Both Historical and Practical, he said algebra ‘was in use of old among the Grecians; but studiously concealed as a Great Secret’4. It is unclear exactly what mathematical methods he was attributing to the Greeks when he said that one could find examples of ‘algebra’ in Pappus, Archimedes and Apollonius, but that they were ‘obscurely covered and disguised’4. However, he fiiy believed that the Greeks possessed a secret ‘Art of Invention’ used to discover the propositions which they then demonstrated in other ways. Indeed, he followed Vieta in believing algebra to be a rediscovery of ancient principles; it was merely improved by the new symbolism. For Wallis, to prefer algebra was simply to favour one branch of classical mathematics over another, not to make a radical break with tradition.

Furthermore, Wallis was following traditional Aristotelian dictates when he claimed that arithmetic is more abstract and universal than geometry7. Aristotle had argued that

points implied position while units do not, and that therefore units were simpler than points. Wallis agreed that arithmetic is more general than geometry because of the wholly abstract nature of numbers. The assertion that arithmetic, including algebra, is more universal and abstract than geometry certainly places Wallis in the ‘modem’ school in the sense that this term is often used by historians. He did not, however, view his ideas as constituting a break from the classical methods. Indeed, he would not have considered his propensity for algebra to be a modem characteristic at all.

Barrow did not agree with Wallis’s description of number or his placement of arithmetic within mathematics, but in fact his own opinions on these subjects were less traditional than Wallis’s. Barrow began with the assumption that ‘there is really no Quantity in Nature different from what is called Magnitude or continued Quantity’s. Magnitude alone was to be the object of mathematics, and so all mathematics was for him subsumed under geometry - arithmetic had no independent standing. Moreover, all ‘mixed mathematics’, or natural science, deals with objects that have magnitude and is therefore dependent upon geometry. He stated: ‘those which are called Mixed or Concrete Mathematical Sciences are rather so many Examples only of Geometry’g. So Barrow broke with both the classical division of mathematics into arithmetic and geometry, and the Aristotelian distinction between pure and applied mathematics. The latter division refers to the separation between pure mathematics, which deals with the intrinsic properties of number and magnitude, and applied mathematics, which includes such things as music and astronomy. Although Barrow was quite familiar with the traditional opinions, he disregarded them in order to reformulate the foundation of mathematics.

Barrow agreed with Wallis about the ancients’ method of discovery being some- thing like algebrais. But even though he con- curred with Wallis on this point, they disagreed about the proper place of algebra within mathematics. Thus, for Barrow, algebra was not a part of arithmetic or even a part of mathematics at all. Instead, he thought of it as a tool or a heuristic deviceri. However, this view was not a product of hostility to modem methods in mathematics. He greatly valued algebra as a method of discovery; as he put it, ‘the new Method of Analysis, chiefly cultivated by Vieta and Cartesius, [is] almost equal to every soluble question”*. And the mathematical problems solved in Lecture X make it clear that Barrow was capable of using the new analytic methods.

Thus neither Wallis nor Barrow would have accepted the historians’ claim that using algebraic techniques was somehow ‘modem’. Algebra was generally viewed by 17th-century mathematicians as being a mathematical tool that was available to, but hidden by, the Greeks. There was, however,

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also a belief that progress was being made in terms of notation and procedures. For instance, Vieta said of his work, ‘Behold the art which I present is new, but in truth SO old, so spoiled and defiled by barbarians, that I consider it necessary . to think out and publish a new vocabulary’L3. Moreover, Descartes and Ramus both held that ‘algebra was only a vulgar (Descartes later said “barbaric”) name for a sort of analytic mathematics that the Greeks used”“. Thus the general consensus was that for all the improvements being made to algebraic techniques. the method itself was classical.

Why did the formulations of mathematics put forward by Wallis and Barrow differ so greatly if it was not simply a question of a propensity towards traditional or modem methods? The dissimilarity may be explained in part by their conflicting opinions about mathematical objects, such as numbers. For the Greeks, the unit, or one, was not a number; it was the beginning of number, and it was used to measure a multi- tude. Numbers were merely collections of discrete units that measured some multiple. Magnitude, on the other hand, was usually described as being continuous, or being divisible into parts that are infinitely divisible. The differences in Wallis’s and Barrow’s points of view on this topic further illustrate that neither mathematician belongs entirely in the ‘traditional’ or ‘modem’ camps”. To a great extent, although Wallis utilized algebraic and numerical methods, he accepted the traditional con- ception of number. Barrow, on the other hand, abandoned the ancient distinction between discrete number and continuous magnitude. Thus if we consider ‘traditional’ to mean following classical ideas, and ‘modem’ to mean accepting innovations, then with respect to numbers Wallis was the traditionalist while Barrow was the modem.

The decay of nature Historians have also explored broader sorts of tension between ‘tradition’ and ‘innovation’ in the wider intellectual community of early modem England that might provide an alternative explanation for the stylistic differences in Wallis’s and Barrow’s mathematical methods. For instance, there was the religiously based belief in the ‘decay of nature’ whose supporters regarded traditional methods as being superior. Proponents of ‘the decay of nature’ claimed that nature in general, and man in particular, had been declining ever since the fall from grace in the garden of Eden. They considered the world to be in its old age, and mankind to be in a state of decay: the sun was even said to be lower in the sky than it once had been. Men in ancient times were stronger, lived longer and had greater stature and more ‘ripenesse of wit’. Nature’s decay in itself was really their main con- cern, but the idea was used to support the Christian religion by verifying the biblical story of man’s fall. The ‘decay of nature’

was a popular idea, and it contributed to the veneration of ancient ideas, influencing the fields of natural philosophy and mathematics16.

The defenders of the modems, on the other hand, did not consider the Bible as ruling out contemporary improvements on ancient techniques, and disagreed with the idea of the ‘decay of nature’17. They were concerned that the idea of the ‘decay of nature’ was so generally accepted ‘not only among the Vulgar, but the Learned, both Divines and others’ 18. In particular, the idea of the decay of man was believed to lead to a lack of modem achievement and a sense of despair. The opponents of the idea of the ‘decay of nature’ declared instead that the modems were the equals of the ancients; modem efforts would be rewarded with suc- cesses. Any lack of achievement, it was felt, could be explained by idleness and negli- gence. They argued that ‘Adam’s fault’ had not led to a decline in man’s abilities or altered the stars in the heavens and the elements on earth19. One tactic for showing that the modems were not in universal decline was to compare their accomplishments to those of the ancients. Mathematics and astronomy were both areas where modem superiority was asserted.

The disagreements over the ‘decay of nature’ illustrate the predominant 17th- century uses of ‘ancient’ and ‘modem’. Support for the ancients seems to have meant, in this religious and intellectual con- text, simply a preference for classical leam- ing and a suspicion of recent developments. Applying ‘traditional’ beliefs to the domain of mathematics, the ancient philosophers and mathematicians would have been seen as nearer to the ‘first mould’, and therefore more perfect. Consequently their work was. in some sense, viewed by this camp as being superior to any modem contributions. Thus if a mathematician supported the traditionalists’ conclusions, he would have been inclined to adhere to classical mathematical concepts and techniques. For instance, Wallis seems to have had great respect for ancient precepts; he wished at least to main- tain classical terminology, although he did not let the Aristotelian foundation for numbers interfere with his actual mathematical practice.

Barrow, on the other hand, was more directly influenced by the supporters of the ancients through his tutor James Duport, who was an avid follower of Aristotle and very opposed to any sort of ‘modem’ inno- vations2O. Barrow likewise greatly revered ancient notions; for instance, he called Aristotle the ‘unchallenged Prince of all who have ever been or will ever be philosophers’“. Moreover Barrow certainly was ‘traditional’, in the sense in which term was used in the 17th century, insofar as he favoured classical geometrical techniques.

Although Wallis and Barrow respected tradition, they were also familiar with the contemporary counter-arguments in support of modem achievements. Wallis was aware

of the arguments for recent accomplishments through his friendship with Samuel Hartlib, who was a supporter of the ‘modems’22. Hence, although Wallis venerated the ancients, he also at the same time seemed to enjoy pointing out improvements that had been made to traditional techniques by English mathematicians such as Thomas Harriot. Likewise, Barrow seemed to believe himself capable of improving on classical concepts of numbers and time. It is clear, however, that Barrow’s and Wallis’s notion of what should be considered ‘modem’ in mathematics differed from the defi- nitions given by historians. Algebra was not seen as a modem improvement; it was considered to be an ancient technique that had been improved somewhat by the modems. So the two features, algebra and heuristics, which are now seen to be signs of ‘modem’ mathematics, were not viewed as being par- ticularly novel in the seventeenth century. ‘Modem’ seems to have meant in this con- text simply recently discovered mathematics, such as logarithms and the new mathematical instruments. From this standpoint, Wallis might be considered ‘modem’ since he was fond of utilizing the latest techniques. Thus in some ways Wallis and Barrow were associated with different interest groups, and were on different sides of the dispute between the supporters of the ‘ancients’ and the ‘modems’, but not for the reasons usually claimed by historians.

Much of the mathematical practice in early modem England was tom between a desire to use new techniques and a fear that these techniques lacked the rigorous justifi- cation of Greek mathematics. This tension was, to a certain extent, carried over from the wider struggle between the supporters of the ‘ancients’ and the ‘modems’, and dis- putes over educational reform*J. Should society cleave to traditional beliefs, or should it embrace the exciting new developments in technology and the experimental philosophy? Neither Wallis nor Barrow was wholly in one camp or the other.

References Wallis’s and Barrow’s mathematics are both discussed in D.T. Whiteside (1960) A&z. Hist. E.mct Sci. 1, 178-388. See also C. Scriba (1966) Studim SW Mathemarik des John Wallis (1617-1703), Franz Steiner and M. Mahoney (1990) in Before Nm~ton: The Life and Times of Isaac Barrow (Feingold, M., ed.), pp. 179-249. Cambridge University Press Scott. 1. F. (1938) The Mathematical Work of John Wallis D.D.. ER.S. (16/G17031, Taylor & Francis Whiteside. D.T. (1960) Arch. His?. Exact Sci. 1, 263 Wallis, .I. (1695) Opera Mathemutica, Vol. 1, Oxford, p. 365 Barrow, I. ( 1734) Geometrical Lectures. (tram Edward Stone). p. 100 Mahoney. M. (1990) in Before Newton: The Life and Times of Isaac Barrow (Feingold, M., ed.), p. 266, Cambridge University Press

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8 9

10 11 12 13

14

This was also pointed out by Barrow (1734) in The Usefulness of Mathematical Learning Explained and Demonstrated: Being Mathematical Lectures Read in the Publick Schools at the University of Cambridge, (tram John Kirby), London, 47 Barrow, op. cit. (note 6), 20 Ibid., 27 Wallis, op. cit. (note 4), 3 Barrow, op. cit. (note 6), 28 Ibid., 243 Vieta, F. (1968) in J. Klein, Greek Mathematical Thought and the Origin of Algebra (trans Eva Brann), p. 318, MIT Press Mahoney, M.S. (1980) in Descartes’ Philosophy, Mathematics and Physics (Gaukroger, S., ed.), p. 148, Barnes &

Noble 18 Ibid., 1 15 Hill, K. (1996) Notes Rec. R. Sot. London 19 Ibid., 81

50 (2), 165-178 20 Curtis, M. (1959) Oxford and Cambridge 16 Jones, R. (1961) Ancients and Modems: A in Transition 1558-1642: An Essay on

Study of the Rise of the Scientific Changing Relations between the English Movement in Seventeenth-century Universities and English Society, 277, England, Washington University Press, Oxford University Press, p. 116 26-29. William Ashworth discusses some 21 Osmond, I’. (1944) Isaac Barrow: His Life of the same material and is critical of and Times, The Society for Promoting Jones’s interpretation. See W. Ashworth Jr Christian Knowledge, 94 (1975) diss. University of Wisconsin 22 Greaves, R.L. (1969) The Puritan

17 Hakewill, G. (1635) An Apologie or Revolution and Educational Thought: Declaration of the Power and the Background for Reform, Rutgers Providence of God in the Government of University Press, 31 the World Consisting in an examination 23 For the educational reform aspect, see K. and censure of the Common error touching Hill (1997) Notes Rec. R. Sot. London 51 Natures perpetual1 and Universal1 Decay, (l), 13-22 and K. Hill (1997) The third edition, Oxford Seventeenth Century 12,23-36

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John Wallis and Isaac Barrow: tradition and innovation and the state of mathematics