neither ancient nor modern: wallis and barrow on the composition of continua. part two: the...

Neither Ancient nor Modern: Wallis and Barrow on the Composition of Continua. Part Two:The Seventeenth-Century Context: The Struggle between Ancient and ModernAuthor(s): Katherine HillSource: Notes and Records of the Royal Society of London, Vol. 51, No. 1 (Jan., 1997), pp. 13-22Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/532032 .

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Notes Rec. R. Soc. Lond. 51 (1), 13-22 (1997)

NEITHER ANCIENT NOR MODERN: WALLIS AND BARROW

ON THE COMPOSITION OF CONTINUA.

PART TWO: THE SEVENTEENTH-CENTURY CONTEXT:

THE STRUGGLE BETWEEN ANCIENT AND MODERN

by

KATHERINE HILL

Institute for the History and Philosophy of Science and Technology, University of Toronto, Room 316, Victoria College, Toronto, Canada M5S 1K7

In the the first part of this article* it was shown that neither John Wallis nor Isaac Barrow fits precisely into the modem or traditional categories commonly used by historians. Yet other kinds of tension between tradition and innovation did exist in the wider intellectual community of early modem England that might have some

bearing on the state of mathematics. We therefore need to explore how the

opposition between ancient and modem was expressed in the seventeenth century. First, there was the religiously based belief in the decay of nature; supporters of this belief regarded traditional methods to be superior. The defenders of the moderns, on the other hand, did not consider the Bible to rule out contemporary improvements on ancient techniques. But the very idea that knowledge could be advanced, particularly beyond the classical works, was still new and strange in this period.' Second, the dispute surrounding educational reform included the rejection of traditional educational methods. Social and economic pressures led reformers to

propose an increased concern with utility and practical methods, which might increase employment. Once the context of the tension between the supporters of the ancients and the supporters of the moders has been explored, we can ascertain whether Wallis and Barrow stood on different sides in the conflict, and how this may have have influenced their mathematics.

ANCIENTS AND MODERNS

One of the most important and popular seventeenth-century texts comparing the achievements of ancients and moders was a book by the protestant minister

Godfrey Goodman: The Fall of Man, or the Corruption of Nature, Proved by the

Light of our Naturall Reason.2 Goodman argued for the superiority of the ancients,

*Notes Rec. R. Soc. Lond. 50 (2), 165-178

13

© 1997 The Royal Society



Katherine Hill

especially of Aristotle, whom he considered to be the father of all human knowledge. But his main objective was to support his Christian faith by the use of reason.3 He claimed that nature in general, and man in particular, had been declining ever since the fall from grace in the garden of Eden. He considered the world to be in its old age, and mankind to be in a state of decay; even the sun was lower in the sky than it once had been. He believed that:

Man (as all other creatures) being immediatlie created by God, as he comes nearer and nearer the first mould, so is he more and more perfect, and according to the degrees of his distance, so he incurres the more imperfection and weakenesse.4

Thus men in ancient times were stronger, lived longer, and had greater stature and more 'ripenesse of wit'. Nature's decay in itself was not his main concern, but was used instead to support the Christian religion by verifying the Biblical story of man's fall. Goodman's ideas were widely popular and contributed to the veneration of ancient ideas, influencing the fields of natural philosophy and mathematics.5

George Hakewill, the Archdeacon of Surrey, wrote his Apologie in 1627 in reaction against Goodman's work.6 For him, the idea of nature's decay was not supported by the Bible. He was 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'.7 The idea of the decay of man in particular led, he believed, to a lack of moder achievement and a sense of despair. Therefore, he declared instead that the moders were the equals of the ancients; modem efforts would be rewarded with successes. Any lack of achievement, he felt, could be explained by idleness and negligence. He could not believe that 'Adam's fault' led to a decline in man's abilities or altered the stars in the heavens and the elements on Earth.8

Hakewill replaced the idea of nature's decay with the concept of circular progress; all things have their birth, growth, flourishing and fading, and then, within a short while, their resurrection and reflourishing. In order to show that the modems were not in universal decline he compared their accomplishments to those of the ancients. Mathematics and astronomy were both areas where he asserted moder superiority. He began by maintaining that, in mathematics, Ramus, the famous sixteenth-century pedagogue, could safely be compared with the best of the ancients. Moreover, he listed as improvements such modem accomplishments as the quadrature of the circle, and the 'excellent' invention of 'geometrical engines and proportions' by Simon Stevin. A short essay by Henry Briggs, Professor of Geometry at Oxford, which catalogued other mathematical and astronomical advances, was also included in the text. In particular, Briggs only briefly mentioned algebra's advantages, especially in dealing with lines and curves of irrational magnitudes. Both men also discussed the improvements made in astronomy by Copernicus, Galileo and Kepler.9

Joseph Glanvill also took the modem side of this dispute in his Plus Ultra or The Progress and Advancement of Knowledge since the Days of Aristotle. Although this work is mainly intended to support the Royal Society, Glanvill included a list of

14



Neither ancient nor modem: Wallis and Barrow

areas in which the modems had surpassed the ancients' achievements. An entire section of the book is devoted to mathematics, since they 'are mighty helps to practical and useful knowledge'.10 Logarithms and decimal arithmetic were again seen to be particularly important modem advances." Algebra is praised in that it is useful for practical purposes, such as measuring and taking distances. He mentioned that algebra is considered to be an ancient technique, but claimed that the modems had at least improved it by moving past quadratic equations.12

The ideas of Goodman, Hakewill and Glanvill illustrate for us the predominant seventeenth-century meanings of ancient and modem. Support for the ancients seems to have meant in this religious and intellectual context simply a preference for classical learning and a suspicion of recent developments. Applying Goodman's 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, superior to any modem contributions. Thus if a mathematician supported Goodman's conclusions, he would have been inclined to adhere to classical mathematical concepts and techniques. Wallis seems to have had great respect for ancient precepts; he wished at least to maintain classical terminology, although he did not let the Aristotelian foundation for numbers interfere with his actual mathematical practice.

But Barrow 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 Ramian innovations.13 Ramus had rejected almost every aspect of Aristotelian thought, and wanted to reform the curriculum and teaching methods of universities.'4 Duport went as far as to publish poems attacking the new science in his 1676 Musae Subseciuae."5 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'.16 Moreover, Barrow was certainly traditional in the

seventeenth-century sense 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. Supporters of the modems, such as Hakewill, were concerned that belief in the decay of nature and the veneration of the ancients was dangerously widespread.17 Many later authors also felt that these ideas were an impediment to progress.18 Wallis was familiar with the arguments for recent accomplishments through his friendship with Samuel Hartlib, who was a supporter of the modems.19 Hence, although Wallis venerated the ancients, at the same time he 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.

But we must keep in mind that Hakewill's defense of moder mathematics, including Briggs's comments, was only one and a half pages long. Glanvill's account is more detailed, but it provides the same picture of moder mathematical achievements. It is clear, however, that their concept of what should be considered

15



Katherine Hill

modem in mathematics differed from the definitions given by historians. Algebra was not mentioned by Hakewill as a modern improvement. Likewise, Barrow, Wallis and Glanvill all considered algebra to be an ancient technique that had been improved somewhat by the modems. Only a few sentences in Briggs's essay were devoted to algebra. No mention is ever made of the superior heuristic value of modem techniques. So the two features, algebra and heuristics, which are now seen to be signs of modem mathematics, were not viewed as being particularly novel in the seventeenth century. Modem in this context seems to have simply meant recently discovered mathematics, such as logarithms and the new mathematical instruments. Wallis might have been considered modem from this point of view as 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.

EDUCATIONAL REFORM

But there was another group of men concerned with educational reform in this period for whom modem seems to have had yet another meaning.20 The movement for educational reform did not always directly address the dispute between the supporters of the ancients and the modems. But the reformers found traditional practices unacceptable and pushed for new techniques and for the overall reform of the universities. As Wallis and Barrow were both professors of mathematics, respectively at Oxford and Cambridge, the aspects of this dispute that dealt with mathematical education might have influenced their work. John Durie's work, The Reformed School, which first appeared in 1649, does not seem to have personally influenced either Wallis's or Barrow's education. Yet Durie's book suggests what kind of transformation was being sought.

Durie desired that 'nothing is to be taught but that which is useful in itself to the Society of Mankind'.21 Languages, for example, were only useful insofar as they relayed the 'Reall Truths in Science'; Durie also advocated the abandonment of Peripatetic principles. This stance was certainly in opposition to the received educational practices of the period. In mathematics, he suggested concentrating mainly upon its 'practicall parts'. The rules of addition, subtraction, multiplication, division, the reduction of fractions, and the rules of proportion should be taught to the younger boys.22 Indeed, even for the older students the emphasis was to be on such practical things as the 'experimental ways of Measuring Land' and the use of mathematical instruments, as well as the 'Art of Dialing' and the keeping of accounts.

Other reformers, such as John Webster, also focused on the question of utility, but it was not their primary motivation. Webster rejected the State Church, opposed the paying of tithes, and rejected the need for ministers to have a university education.23 In the 1640s an important factor in the educational debates was whether or not

16



Neither ancient nor modem: Wallis and Barrow

unordained men could preach, regardless of their educational status.24 Even the most radical of the reformers did not want to destroy the universities: they wished to secularize them and end the universities' role as a 'factory of divines'.25 Thus the introduction of more science into the universities and the reform of mathematical education were actually secondary considerations for most of the reformers.

Yet Webster did seem to be very interested in mathematics. He asked,

Can the Mathematical Sciences, the most noble, useful, and of the greatest certitude of all the rest, serve for no more profitable end, than speculatively and abstractly to be considered of? ... Is the admireable knowledge that Arethmetick affords worthy of nothing but a supine and silent speculation? Let the Merchant, Astronomer, Mariner, Mechanick and all speak whether its greatest glory stand not principally in the practik part?26

Thus part of the reformers' programme was the promotion of practical mathematics. Geometry itself was not scorned, as it was considered useful for surveying land and measuring. Likewise, algebra was only praised as it related to utilitarian pursuits. And mathematical instruments, which are seldom mentioned by historians in this context, were believed to be a truly valuable modem innovation.

One reason that supporters of educational reform stressed utility was because they believed in the betterment of the poor. In terms of poverty, the period 1590-1640 was perhaps the worst in the whole of English history.27 Economic depressions had plagued England in the 1630s. Likewise, unemployment increased in the 1640s, and the political crisis of 1648-49 caused another depression. The harvest failed five consecutive times in the years 1646-50, causing food prices to fluctuate rapidly. Fear of popular disorder, and especially of the London crowds, created anxiety about the poor on the part of the better off. The existing educational programs seemed totally unsuited to raising the standard of living for the lower classes: logic, rhetoric, grammar and philosophy were gentlemanly studies that had no wider applicability to the problems of the real world.28 The reformers felt that education should instead focus on discovering better practices in husbandry and manufacturing that would improve the welfare of the nation. John Durie went as far as proclaiming that anyone who 'doth Teach or Learn any Science for any other end but this doth ... pervert the Truth either of the Science, or of the Method.'29

Yet even those who supported some moder innovations might recoil from a total overhaul of the entire university system. Men usually labelled 'Moderate Puritans', such as John Wilkens and Seth Ward, for example, approved of the universities' role in the training of divines. They accused Webster of being part of a 'gang of the vulgar Levellers.'30 In other words, they feared that his suggestions would bring about too much social change; they especially feared an increase in social mobility. Ward also did not wholly approve of Webster's desire to make mathematical education more utilitarian.31 He was anxious that, if too much emphasis were placed on utility, the theoretical foundations for solving problems would be overlooked. He also seemed to doubt that Webster's programme of practical applications would appeal to certain students, particularly upper class students, who were not interested in utility:

17



Katherine Hill

Of those very great numbers of youth, which come to our Universities, how few are there, whose designe is to be absolute in Natural Philosophy? Which of the Nobility or Gentry, desire when they send their Sonnes hither, that they should be set to Chemistry, or Agriculture, or Mechanicks?32

Yet Ward was proud of the advances in mathematics recently made in the universities, especially Wallis's recent work on conic sections, algebra and indivisibles.33 He felt that no changes in mathematical education were necessary because the universities were already teaching all the modem techniques. Thus Ward himself was neither wholly traditional nor modern: he approved of the recent mathematical advances, but for a variety of reasons resisted modem innovations in education.

It is difficult to ascertain Barrow's views about educational reform. But none of his mathematical lectures seem to be at all concerned with utility. Instead, he lectured on such topics as the foundations of mathematics.34 And although he discussed at great length the nature of numbers, he never presented practical examples or solutions to problems. Yet Barrow was unsatisfied with the level of his mathematical lectures. He seems to have wished his students were capable of learning actual mathematical techniques. Indeed, he stops in the middle of his Cambridge mathematical lectures and breaks off to lament,

But shall I never extricate myself from these Quirks and Trifles? Shall I always spend my time in examining what is of no value? ... Shall I grow old in these outer Courts of general Matters? Shall I perpetually tarry in the Entrance of the Sciences? Shall I always stick in the Threshold? Shall I only knock at the Doors of Mathematics? ... But I am afraid the dry Subtilty of the particular Demonstrations, which the inward Parts of Mathematics abound with, their extreme Rigor requiring the closest attention, and their way of arguing, will somewhat offend and deter the most. These subjects ... are in my opinion not so convenient for publick Lectures of this kind.35

The lectures themselves demonstrate the justice of Barrow's complaint. Modern

techniques, such as analysis (or algebra) are sometimes mentioned, but no examples are given. Yet, although Barrow seems to wish he could give lectures on a more advanced level by including a description of the method of indivisibles and the

analysis of Vieta and Descartes, there is no sign that he approved of Webster's

proposal to make mathematical education more utilitarian. He wanted to teach the latest mathematical methods, not the use of mathematical instruments or practical skills like surveying. A factor that might have influenced him against the reforms

proposed by Durie and Webster was his violent disapproval of their religious and

political beliefs: Barrow hated the idea of predestination (the belief of the puritan reformers) and felt that the importance of accepting authority had been shown by the 'anarchy of the Civil War'.36

We can gain an understanding of the style and contents of Wallis's lectures by studying his 1657 work, Mathesis Universalis.37 His lectures met the reformers' requirements for practicality insofar as they treated Hindu-Arabic numerals and elementary operations, the requested subjects, whereas Barrow's lectures had not.

18



Neither ancient nor modern: Wallis and Barrow

Such skills as addition, subtraction, multiplication and division were all explained, and particular numerical examples are given. Unlike Barrow, Wallis also gave a complete treatment of the evolution of algebraic notation.38 And, although the lectures were not given at a particularly advanced level, he did give an overview of the current algebraic techniques. In many respects, a student would have been better prepared for practical mathematical activities after attending Wallis's lectures than after Barrow's.

It is not really surprising that Wallis's lectures were in closer accord with the ideas of the reformers than were Barrow's. After all, Wallis had been rewarded with the Savilian Professorship of Geometry for his practical service to Parliament: he had deciphered Royalist codes throughout the Civil War.39 He was also acquainted with Wilkens and Ward in both London and Oxford, and was familiar with the arguments surrounding educational reform. Barrow, on the other hand, was first elected Regius Professor of Greek after the Restoration in 1660. His inaugural oration made it clear that he was delighted with the restored monarchy and that he had sided with the universities against their attackers.40 Or perhaps Barrow's closer familiarity with classical literature increased his respect for traditional educational methods. After all, he began his career as a professor of Greek, so he had a deep knowledge and appreciation of classical works.

CONCLUSION

Several points become clearer after an examination of seventeenth-century notions of traditional and modem. The idea of modernity in mathematics was not associated with any particular techniques: algebra was not considered to be a modem development, and was believed to have been known by the ancients. In fact, logarithms and mathematical instruments were seen to be the important recent improvements. In the context of the struggle between supporters of the ancients and the modems, modem merely meant mathematical techniques that were believed to be newly developed. The educational reformers, although they were also dissatisfied with traditional techniques, felt that utility was the important issue. Traditional methods that were practical, such as the use of geometry in surveying, were appreciated alongside the new mathematical instruments and algebraic techniques for accounting. Although Wallis did view himself as differing from the classical authors in having a greater concern for methods of discovery, he did not consider himself to be uninterested in rigour. Wallis was, however, willing to include in his lectures methods that met the reformers' requirements for practicality. Thus the classification system that utilizes a preference for algebraic and heuristic techniques to define moder ignores the educational reformers' demands for utility, and is somewhat anachronistic.

Classification systems that utilize characteristics seen to be important only in hindsight are particularly misleading when used to analyse developments in periods

19



Katherine Hill

of transition. Although Wallis wavered over the appropriateness of labelling the rationals and irrationals numbers, he did not hesitate to employ them in his work. Moreover, though Barrow himself did not use numerical techniques, he conceived the notion of making numbers 'notes or signs' of magnitude, which gave his contemporaries' mathematical practice a firmer foundation. Decimal expansions, logarithms and infinite series all treated numbers as continuous rather than as discrete. Yet there was no formal theoretical basis for such practices. Thus much of the mathematical practice in early modem England was torn between a desire to use the new techniques and a fear that these techniques lacked the rigorous justification of Greek mathematics. This tension was, to a certain extent, carried over from the wider struggle between the supporters of the ancients and the moderns, and disputes over educational reform. 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.

NOTES

1 Mark Curtis, Oxford and Cambridge in Transition 1558-1642: An Essay on Changing Relations between the English Universities and English Society, p. 277 (Oxford, 1959).

2 Godfrey Goodman, The Fall of Man, or the Corruption of Nature, Proved by the Light of our Naturall Reason. Which being the First Ground and Occasion of our Christian Faith and Religion, may likewise serve for the first step and degree of naturall mans conversion (London, 1616).

3 There were other works before Goodman's that gave accounts for the decay of nature, but Goodman's book was the most popular and provoked more responses. For example, Francis Shakelton in his 1580 work A Blazying Starre (London, 1580) gave an earlier account of the decay of nature. But his main concern was to disprove Aristotle's idea that the world is eternal in order to support the Christian religion.

4 Goodman, op. cit. (2), p. 349. 5 Richard Jones, Ancients and Moderns: A Study of the Rise of the Scientific Movement in the

Seventeenth-Century England, pp. 26-29 (St Louis, 1961). William Ashworth discusses some of the same material and is critical of Jones's interpretation. See William Asworth Jr, The Sense of the Past in English Scientific Theory of the Early Seventeenth Century: The Impact of the Historical Revolution, (dissertation, University of Wisconsin, 1975).

6 George Hakewill, An Apologie or Declaration of the Power and the Providence of God in the Government of the World Consisting in an examination and censure of the Common error touching Natures perpetuall and Universall Decay, 3rd edn (Oxford, 1635).

7 Ibid., p. 1. 8 Ibid., p. 81. 9 Ibid., pp. 301-302.

10 Joseph Glanvill, Plus Ultra or The Progress and Advancement of Knowledge since the Days of Aristotle, p. 20 (London, 1668).

11 Ibid., pp. 22-23. 12 Ibid.,p. 31. 13 Curtis, op. cit. (1), p. 116. 14 Hugh Kearney, Scholars and Gentlemen: Universities and Society in Pre-Industrial Britain

1500-1700, pp. 49-53 (London, 1970).

20



Neither ancient nor modern: Wallis and Barrow

15 Mordechai Feingold, The Mathematicians Apprenticeship: Science, Universities and

Society in England, 1560-1640, p. 68 (Cambridge, 1984). Feingold, however, believes that

Duport's conservatism has been overrated. 16 Percey Osmond, Isaac Barrow: His Life and Times, p. 94 (London, 1944). 17 Hakewill, op. cit. (6), p. 1. 18 Jones, op. cit. (5), pp. 39-49. 19 Richard L. Greaves, The Puritan Revolution and Educational Thought: Background for

Reform, p. 31 (New Brunswick, 1969). 20 Jones, op. cit. (5) includes an extended discussion of literature on this subject. For a

seminal investigation of the various reform movements and their impact on science, medicine and mathematics see Charles Webster, The Great Instauration: Science, Medicine and Reform 1626-1660, (London, 1975).

21 John Durie, The Reformed School and the Reformed Librarie-Keeper, p. 18 (London, 1651).

22 Ibid., p. 56. 23 Christopher Hill, Change and Continuity in 17th-Century England, pp. 131-132 (New

Haven, 1991). 24 Greaves, op. cit (19), p. 17. 25 Hill, op. cit. (23), p. 132. 26 John Webster, Academiarum Examen, or the Examination of Academies, Wherein is

discussed and examined the Matter Method and Customes of Academick and Scholastick

Learning, and the insufficiency thereof discovered and laid open; As also some Expedients proposed for the Reforming of Schools, and the perfecting and promating of all kind of Science. Offered to the judgement of all those that love the proficiencie of Arts and Sciences, and the advancement of Learning, pp. 19-20 (London, 1653).

27 Ann Hughes, The Causes of the English Civil War, p. 130 (London, 1991). 28 Greaves, op. cit. (19), p. 48. 29 Durie, op. cit. (21), p. 41. 30 John Wilkens and Seth Ward, Vindicae Academiarum containing, Some briefe

Animadversions upon Mr Websters Book, Stiled, The Examination of Academies, p. 6 (Oxford, 1654).

31 Ibid., p. 15. 32 Ibid., p. 50. 33 Ibid., p. 30. 34 I agree with Michael Mahoney ('Barrow's mathematics: between ancients and modems',

in Before Newton: The Life and Times of Isaac Barrow (ed. Mordechai Feingold), pp. 202-203) that the Geometrical Lectures were never delivered at Cambridge in their

published form. Barrow's Optical Lectures might have been read to the students at

Cambridge, but it is not clear in what form, he certainly kept revising them until their

publication in 1669. See Alan Shapiro, 'The Optical Lectures and the foundations of the

theory of optical imagery', in Before Newton: The Life and Times of Isaac Barrow (ed. M.

Feingold), p. 110 (Cambridge, 1990). 35 Barrow, The Usefulness of Mathematical Learning Explained and Demonstrated: Being

Mathematical Lectures Read in the Publick School at the University of Cambridge (tr. John

Kirby, London), pp. 235-241 (London, 1734). 36 John Gascoigne, 'Isaac Barrow's academic milieu: Interregnum and Restoration

Cambridge' in Before Newton: The Life and Times of Isaac Barrow, (ed. Mordechai

Feingold), pp. 260-261 (Cambridge, 1990). 37 J.F. Scott, The Mathematical Work Of John Wallis D.D., FR.S. (1616-1703), p. 68

(London, 1938). Scot claims that Mathesis Universalis is a collection of Wallis's lectures. 38 John Wallis, Opera Mathematica, vol. 1, p. 68 (Oxford, 1695).

21



22 Katherine Hill

39 Christoph J. Scriba, 'The autobiography of John Wallis, F.R.S.', Notes Rec. R. Soc. Lond. 25, 37-40 (1970).

40 Mordechai Feingold, 'Isaac Barrow: divine, scholar, mathematician', in Before Newton: The Life and Times of Isaac Barrow, (ed. Mordechai Feingold), p. 55 (Cambridge, 1990).



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