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

Neither Ancient nor Modern: Wallis and Barrow on the Composition of Continua. Part One:Mathematical Styles and the Composition of ContinuaAuthor(s): Katherine HillSource: Notes and Records of the Royal Society of London, Vol. 50, No. 2 (Jul., 1996), pp. 165-178Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/531905 .

Accessed: 15/06/2014 23:31

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Notes and Records ofthe Royal Society of London.

http://www.jstor.org

This content downloaded from 185.2.32.89 on Sun, 15 Jun 2014 23:31:43 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/531905?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Notes Rec. R. Soc. Lond. 50 (2), 165-178 (1996)

NEITHER ANCIENT NOR MODERN: WALLIS AND BARROW

ON THE COMPOSITION OF CONTINUA.

PART ONE: MATHEMATICAL STYLES AND

THE COMPOSITION OF CONTINUA

by

KATHERINE HILL

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

INTRODUCTION

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. Wallis studied at Cambridge, deciphered Royalist codes for Parliament during the

John Wallis, by Gerard Soest Isaac Barrow, engraving, 1676

165

© 1996 The Royal Society



Katherine Hill

Civil War, and was one of the secretaries to the Assembly of Divines at Westminster. He was rewarded for his support of Parliament with the Savilian Professorship of Geometry at Oxford. Barrow was also a student at Cambridge and, in 1660, was appointed Regius Professor of Greek at Trinity College. He subsequently 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. But even though both enjoyed very similar educations and careers, their mathematical techniques were quite different. Wallis's style is usually considered algebraic, while Barrow's is considered geometric. At the same time 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.

When historians attempt to describe and explain this phenomenon they often speak of a tension between two modes of thought - modem and traditional. Modems, it is claimed, preferred algebra to geometry, and adopted the new symbolism; they emphasized the heuristic aspects of mathematics. Traditionalists embraced geometry and synthetic proof; they valued rigor in mathematics. 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. Of the two mathematicians whose work will be explored in this paper, John Wallis is usually placed in the modem camp, while Isaac Barrow is considered traditional.

Historians give several justifications for this classification scheme. For example, this division is sometimes claimed to be a mathematical facet of a wider battle between supporters of the ancients and modems.' In contrast, other historians give an explanation in terms of a division between heuristic analysts and rigorous mathematicians.2 My first objective in this paper is to show why historians place Wallis and Barrow in the modem and traditional camps respectively. At first glance, their mathematical techniques would seem to fit this classification scheme, at least in a general way. However, in the next section of the paper I argue that, on closer inspection of their mathematical techniques, and of their own definitions of traditional and moder, these neat categories are artificial and anachronistic. 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, utilized heuristic techniques and was often algebraic, but he hesitated over changing ancient terminology, and attempted to work within what he considered the classical framework. In fact, Wallis considered algebra to be ancient, not modem. Thus it will be seen that the current classification scheme does not hold up when concepts of mathematical continua are compared.3

In the second part of the paper I suggest that certain non-mathematical debates in early modem England might better explain the differences between Wallis's and Barrow's mathematical techniques. Even if the mathematical ideas of Wallis and

166



Neither ancient nor modern: Wallis and Barrow

Barrow were not simply ancient or modem in the sense used today by historians of mathematics, there did nevertheless exist certain other tensions between tradition and innovation in the wider intellectual context of the time; such tensions may help to explain certain tendencies in their work. Two disputes in particular are discussed: the wider, early seventeenth-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 moderns, while the other side felt that the moderns 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 show that ancient and modern had different meanings in the seventeenth century from those used by historians today. Moreover, placing Wallis and Barrow into these somewhat anachronistic categories causes some historians to overlook the subtlety of their conceptions of mathematical continua, and the influence of their actual working environment.

ANCIENT VERSUS MODERN: MATHEMATICAL STYLES

Historians place Wallis in the modern school because his mathematical work can be characterized as algebraic. He also emphasized methods of discovery instead of proof, and he employed the new symbolism. In his Arithmetica Infinitorum (1656), Wallis found the area under the curve y = x" between zero and one for positive fractions and negative integers using analytic techniques and numerical series, and in his 1657 Mathesis Universalis, 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 rigor, and because he was not consistent in his use of symbolism. Barrow's most famous work, his 1670 Lectiones Geometricae, included geometrical constructions of tangents, quadrature and rectification problems.

The problems that display Wallis's and Barrow's mathematical characteristics have been dealt with in great detail in the secondary sources.4 But 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, can be

brought out through the familiar examples. Wallis's best known work, Arithmetica Infinitorum, describes his work on the

quadrature of the circle.5 Wallis's method in this work is non-rigorously based upon what seems to be his strong feeling for pattern. His techniques might be related to his work on codes during the English Civil War. Evidently, the whole layout on the printed page corresponds with the natural way one sets out a message for decoding. The codes used in the Civil War were numerical, and there were easily recognizable

167



Katherine Hill

frequency patterns occurring among the various number-sets used. Many of the skills used in decoding would also be useful in Wallis's numerical techniques.6

In modem terms, the main focus of the Arithmetica Infinitorum was Wallis's attempt to evaluate the expression J (1 - i2)12 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 that

0+1 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)/(n+n+n+...n) = 1/2.7 He used this proposition to show how the area of a triangle could be easily

determined. In Proposition 3 he says that a triangle is made up of an infinite number of parallel lines arithmetically proportional, starting from zero. Thus the sum of the lines making up the triangle is half the sum of those making up the rectangle of the same base and altitude, because the ratio of the line making up the triangle to those making up the rectangle may be written as (0+1+2+...+n)/(n+n+n+...+n).8 He then went on to use this technique to find the required ratios for many positive integer powers, finding the consequent or corresponding ratio of a series n (when n is a positive integer) to be 1/(n+l). In effect, he was finding the area under the curves using numerical techniques. Wallis felt that the principle of continuity, or interpolation, was the key to fractional indices. He believed that the principle of continuity guaranteed that one could interpolate between positive integer consequents to discover the consequent of a fractional series.

The above example demonstrates several of Wallis's mathematical characteristics. His extraordinary skill in finding numerical patterns led him to rely upon his 'method of induction' and his principle of continuity. Once he had made his discoveries using numerical methods, he did not hesitate to apply his results to

Figure 1. Proposition 14 of Barrow's lecture VI, 'A Theorem on the parabola'.

168




geometric problems. He was also comfortable using algebraic notation. As he put it, 'For though such choice of Notes do not at all influence the Demonstration, yet doth it assist the Fancy and Memory, which would otherwise be in danger of being confounded'.9

It will later be shown that 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'.'° This lecture deals mainly with the properties of conics. Except for the kinematic aspects of the demonstrations, Barrow treats mainly the properties of curves discussed in

Apollonius's Conics. Proposition 14 is fairly typical of the work done in this section (see figure 1). He let the straight lines DB and BA be given in position, and let the straight line

CX move along the straight line DB parallel to BA; at the same time he also let a straight line DY revolve around the point D so that it cut BA in E. He also supposed that there is always the same ratio between BE and DC, which he assumed was the same ratio that some quantity R had to DB. Next, he called the intersection of the right lines DE and CX the point N. He concluded that the line DNN would be a parabola. The proof is geometric.

(i) For let R : DB = DB: P

Barrow here introduces the quantity P, which is defined by the above relationship. (ii) BE: DC = DB : P

This follows from the definition of R given in the construction of the problem. (iii) DB : BE = DC: CN

This follows from the similar triangles DBE and DCN.

(iv) DB: BE + BE: DC = DC: CN + DB: P

By adding (ii) and (iii)

(v) DB: DC = DC DB: CN.. P

By multiplying (ii) and (iii),

(vi) DB -DC: DCq = DC DB: CN P

By multiplying (v) by DC/DC, on the left.

(vii) DCq = CN P

Barrow closes with 'hence it is manifest that the curve DNN is a Parabola, whose Parameter is P, Vertex D, and Diameter parallel to BA'." Step (vii), which concludes

169



Katherine Hill

the proof, is based on the ordinate properties of the parabola as defined by Apollonius.

In the previous example Barrow was not claiming to offer new techniques. He certainly avoided calculations in this proposition: he depended upon geometrical properties, such as similar triangles, to complete his proof. His work differed from classical, static demonstrations, such those of Apollonius, in that it utilized lines or points moving though space. Yet, in the end, the proof works like a conventional locus theorem. Occasionally, when he substitutes 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.12 Barrow was familiar with analytical techniques; 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 work help to explain why historians typically label them as modem and traditional respectively. Barrow was traditional in the usual sense of the word, insofar as he favoured geometry and valued rigour in mathematics. None of his work depended on 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 modern 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 his results could be confirmed by more rigorous proofs. Although these mathematical examples have shown that there is some basis for the classification scheme used by many historians, the following more detailed examination of Wallis's and Barrow's work, coupled with an examination of how they themselves viewed their mathematical practice, shows that neither man fits completely in either category.

COMPOSITION OF CONTINUA, AND WALLIS'S AND BARROW'S VIEWS ON

ANCIENT AND MODERN

Exploring Wallis's and Barrow's views about number and the nature of continua helps to elucidate their own positions on what they considered traditional or ancient versus modem. It also shows that not all the techniques considered to be modem by historians were viewed as innovations in the seventeenth century.

Wallis did not feel that his mathematical techniques were a departure from classical methods. He accepted the traditional division of pure mathematics into two disciplines, arithmetic and geometry, and accepted that arithmetic dealt with discrete or unextended quantity, whereas geometry dealt with continuous or extended

170




quantity. He stated that 'geometry is the science of magnitude insofar as it is measurable, and arithmetic the science of number insofar as it is countable'.3 Algebra was for him a part of arithmetic. Yet he did not regard algebra as a modem subject. In his Treatise of Algebra, Both Historical and Practical, he wrote that algebra 'was in use of old among the Grecians; but studiously concealed as a Great Secret'.T4 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'.'5 But he firmly 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 geometry.'6 Aristotle had argued that points implied position, whereas units do not; units, therefore, were simpler than points. Wallis agreed that arithmetic is more general than geometry because of the wholly abstract nature of numbers. He said the assertion that 2+3 = 5 is more general than a statement applying to geometrical objects because,

For two angels and three angels also make five angels. But the same argument holds for all operations, whether arithmetical or specially algebraic, which proceed from more general principles rather than being restricted to geometrical measures.'7

The assertion that arithmetic, including algebra, is more universal and abstract than geometry certainly places Wallis in the modem school in the conventional historical sense. 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 with his placement of arithmetic within mathematics, but, in fact, his own opinions on these subjects were less traditional than those of Wallis. Barrow began with the assumption that 'there is really no Quantity in Nature different from what is called Magnitude or continued Quantity'.8 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 that 'those which are called Mixed or Concrete Mathematical Sciences are rather so many Examples only of Geometry'.19 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

171



Katherine Hill

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 something like algebra.20 But even though he concurred with Wallis on this point, they disagreed about the proper place of algebra within mathematics. Barrow's opinion of algebra (which was also referred to as analysis, and which included techniques that we would consider pre-calculus) is often quoted out of context by historians who fail to take account of his other comments on the subject:

Because indeed Analysis, understood as intimating something distinct from the Rules and

Propositions of Geometry and Arithmetic, seems to belong no more to Mathematics than to

Physics, Ethics or any other Science. For this is only a Part or Species of Logic, or a certain Manner of using Reason in the Solution of Questions ... Wherefore it is not a Part or

Species of, but rather an Instrument subservient to Mathematics21

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 device. However, this view was not a product of hostility to modern methods in mathematics, as previous historians have often assumed. 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'.22 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 seventeenth-century mathematicians as being a mathematical tool that was available to, but hidden by, the Greeks. However, there was 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'.23 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'.24 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 much 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 multitude. 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 difference between Wallis's and Barrow's views on this topic further illustrate the point that neither mathematician belongs entirely in the traditional or modem camp.

Wallis's views on the nature of numbers were not presented in a systematic

172




fashion, and indeed they sometimes conflicted. In chapter LXVI of his Treatise of Algebra, during a discussion of 'Imaginary Quantities', he seems to imply that numbers are positive integers only. Negative quantities are necessary, but they are not numbers. He said, 'But it is also Impossible, that any Quantity (though it not a supposed square) can be Negative. Since it is not possible that any Magnitude can be Less than Nothing, or any Number Fewer than None'.25 Although, strictly speaking, negative quantities are not numbers, Wallis went on to explain how they may be understood; negative quantities were described in terms of an integer 'number line'.

Fractions were discussed in terms of division. Sometimes Wallis felt division to be impossible. For example, if 2 is to be divided into 3 equal parts, it is impossible because 'the nature of Numbers doth not permit' this type of operation. But, although he might not label 2/3 a number, he freely used fractions in his work, and he treated fractions as though they were numbers. Thus the metaphysical questions about the ontology of numbers did not really interfere with his mathematical practice. Incommensurable quantities were also thought not to be expressible in terms of numbers. However, he believed they could be approached as closely as one liked by continual approximation. Two ways of dealing with surd quantities, which we would call irrational numbers, are discussed. He proposed that Stevin's method of decimal fractions was very useful in cases that were not in need of mathematical exactness. A series of approximations gave him decimal representations for such numbers as 2 and <2. One could also utilize infinite series through interpolation.

Wallis found only positive integers to be numbers in his Treatise of Algebra; negative numbers, fractions and surds were considered to be quantities. But in his A Defense of the Treatise of the Angle of Contact, published as an appendix to his work on algebra, he seems to broaden his definition. In chapter V, 'Concerning the Composition of Magnitudes', he discussed the classical belief that numbers were composed of undivided units. The 'Greek Authors' had believed that arithmetic could only be applied to positive integers. The classical opinion was that numbers were multiples of indivisible units. However, Wallis was willing to extend the field of arithmetic 'not only to Fractions, but to Surds also'.26 He went as far as to admit 'it is hard to say, what bounds can be set to it [arithmetic]; or, to what it may not extend'.27 But it is difficult to know what he meant by this. By 'Arithmetick' he certainly meant algebra. Yet Wallis never clearly states that fractions and surds were numbers, merely that one could operate on them algebraically. Barrow's definition of number, in contrast, did not include these limitations.

Barrow's conception of number was decidedly untraditional in that it did not stem from classical beliefs. He was convinced that number was 'continued quantity', and, therefore, that arithmetic and geometry dealt with the same subject. Referring to remarks like Wallis's comment that numbers might refer to such things as angels, and that numbers are therefore more abstract than magnitude, Barrow separates numbers into two kinds. First, mathematical numbers are used by mathematicians to express and compare the dimensions of homogeneous magnitudes. Second,

173



Katherine Hill

transcendental or metaphysical numbers are 'indifferently' attributed to all things, even if they are heterogeneous.28 The proper object of study was the mathematical numbers. As Barrow explained, 'a Mathematical Number has no Existence proper to itself, and really distinct from the Magnitude it denominates, but is only a kind of Note or Sign of Magnitude considered after a certain Manner.'29 So Barrow, the so- called traditionalist, was willing to include as numbers fractions and surds, as well as positive integers, which the 'modem' Wallis would not.

Indeed, Barrow was passionately opposed to the exclusion of surds from the realm of numbers, and to mathematicians referring to surds as irrational, irregular or inexplicable. Barrow found the removal of surds to the domain of algebra particularly offensive because surds were arithmetic's 'noblest and most profitable member', in that when one is measuring or comparing magnitudes the results are more often irrational than rational. He believed that surds are necessary and thus have the same right to be considered numbers as integers and fractions do.30 There is some truth to Barrow's criticism of mathematical practice. It appears that Wallis had banished surds as well as fractions to the realm of algebra. Moreover, magnitude in general - which included time, weight, strength, motion, and acceleration - was for Wallis the subject of geometry.3' On this point at least the two men agreed. But for Barrow, numbers were signs denoting magnitude, and thus also part of geometry. Barrow abandoned the ancient distinction between discrete number and continuous magnitude, while Wallis maintained it. 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 and Barrow the modernist.

Another example of their ambiguous position in the ancient-modern dichotomy is how Wallis and Barrow viewed the composition of continuous magnitude. Did they accept the classical dictum of infinite divisibility? Aristotle believed that lines, for example, were not composed of points, or any other indivisibles, as this would require the points to be immediately next to each other. Thus points would have to have parts in order for their extremities to touch, and this would contradict the definition of point. In fact, Aristotle defined continuous as 'that which is divisible into parts which are divisible without limit'.32 Wallis did not deal with such issues directly, but we may investigate his views by examining his work on indivisibles. He learned of Cavalieri's methods around 1653 through a study of Torricelli's works. Wallis employed the powerful new tool of analytic geometry, combined with the method of indivisibles, in his 1656 work on the quadrature of the circle, Arithmetica Infinitorum. Mathematicians in this period realized that there were problems with Cavalieri's technique of regarding a line as made up of an infinite number of points, and surfaces as made up of an infinite number of lines. Wallis attempted to circumvent this difficulty in two ways.

First, he claimed in a later work, in which he discussed the applicability of his techniques, that the method of indivisibles was 'not, as to the substance of it, really different from the Method of Exhaustions'.33 It was merely a short cut, which could be rephrased in more rigorous terms. Thus, if one doubted a certain result, it was

174




always possible to reformulate it in terms of the method of exhaustion. Second, instead of lines having no breadth making up a surface, Wallis introduced

'small surfaces (of such a length, but very narrow)', or infinitely small parallelograms.34 By 'infinitely' he perhaps meant indefinitely small, or as small as one pleased. He was not the first to improve Cavalieri's method in this fashion. Pascal and Roberval had made this suggestion earlier. This innovation does suggest, however, that he did not think a surface could be constructed out of lines with no breadth, and it is therefore unlikely that he would have assumed that a line could be built up from points. Thus it can be deduced that Wallis implicitly adhered to the classical belief in the infinite divisibility of magnitude. He certainly gave no indication of holding any sort of atomic views.

It is simpler to ascertain Barrow's conception of magnitude because he wrote directly on foundational subjects. In his lecture IX, titled 'Of the Termination, Extension, Composition and Divisibility of Magnitudes', he discussed the controversy about whether magnitude is composed of indivisible parts or is infinitely divisible. A full exposition of both ancient and modem opinions was given because he felt that this was a very difficult subject. He considered the arguments of the ancient atomists, such as Epicurus and Lucretius, as well as modem atomists such as Galileo. But, in the end, Barrow concurs with the classic conception of magnitude as being infinitely divisible. He says this idea 'answers well to the Ideas of Men; and that most of the excellent Philosophers have supported and defended it'.35

Barrow's response to the method of indivisibles was similar to that of Wallis. He highly valued the method itself, calling it 'that excellent Method of Indivisibles, the most fruitful Mother of new Inventions in Geometry', which 'never sufficiently can be praised'.36 Yet, although he admired the technique, he felt its results could be restated rigorously through the classical method of exhaustion. And, like Wallis, he realized the problems inherent in such expressions as 'all these lines are equal to such and such a rectangle'.37 Barrow believed that such remarks should be rephrased: 'all the lines' could be replaced by a sum of parallelograms 'of a rather small ... and inconsiderable height'. At least, if one insisted on referring to lines, he felt that one must view them as an infinite collection. Barrow not only shared Wallis's belief in the infinite divisibility of magnitude, but his views on the method of indivisibles as well.

On other subjects, such as space and time, which were considered continua in the seventeenth century, it is more difficult to compare Wallis's and Barrow's ideas. Wallis's work did not directly concern itself with these topics; in fact, he did not often discuss the foundations of mathematics. Barrow, on the other hand, discussed space and time in detail, since, in this period, many mathematical curves were generated by motion. As we have seen, in his Geometrical Lectures he embedded his discussion of the properties of curves in the broader context of the generation of magnitude. A consideration of motion led Barrow to the topic of time. He believed that there exists a 'great Affinity' between space and time. Space, for him, had the same relation to magnitude as time does to motion; time was considered to be in

175



Katherine Hill

some sense the space of motion. Barrow's ideas about the nature of space and time were as untraditional as his conception of number. The following brief summary of his opinions further suggests that Barrow does not fit precisely into the classical tradition.

Barrow's interest in space stemmed from the question of whether it is distinct from magnitude, since an important attribute of geometrical magnitude is that it occupies space. When we examine the idea of space we seem to define it by its extension and indefinite capacity, which were also considered to be the properties of magnitude. Barrow realized that both the ancients (such as Aristotle) and modems (such as Descartes) believed that there was no space separate from magnitude, or indeed from matter. But he could not agree. God could, if he wished, create worlds beyond this one, thus God extends beyond matter and magnitude. Space was assumed to be the capacity or possibility of magnitude. He concluded that 'Space is a thing really distinct from Magnitude'.38

Barrow's conception of time was even less traditional than his ideas about space. Time, for Aristotle, could not elapse without change. Once again, Barrow did not accept the classical view. Instead, he posited the existence of absolute time, which is a quantity that one cannot measure, except through some constant uniform motion, such as the Sun and the stars.39 But for him, time's existence did not depend on motion. Barrow's preoccupation with motion also led him to consider the composition of time. He examined the continuous flow of a point and a moment (or instant), and decided that time was a quantum with one dimension. In other words, time is like a line, continuous in one dimension. Time may be envisioned as the 'trace of a continuously sliding instant, having some indivisibility by virtue of the instant'.40 But the infinite divisibility of time does not imply that it is composed of instants, just as the fact that a line consists of points does not imply that they compose it. Barrow considered time to have many similarities to a line, and thought that time could be conveniently represented by a line. Usually, he seemed uninterested in this distinction, and favoured using terminology that would let him get on with his work. He said that it did not matter whether one used the term 'instant' or 'indefinite particle' when referring to time, just as it did not matter 'whether we understand a line to be composed of innumerable points or of indefinitely small linelets'.41 When the traditional views conflicted with Barrow's techniques, he ignored them.

In short, then, the above examples demonstrate that neither Wallis nor Barrow was wholly traditional or modern in the senses used today. Both exhibited a certain tension between tradition and innovation. Although Wallis's propensity for numerical methods and Barrow's preference for geometric techniques give historians some justification for this classification scheme, an examination of their conception of continua has shown that this classification is not universally applicable. Masked in a conventional format, Barrow's Mathematical Lectures included a new concept of number and a new basis for comparing magnitude, which reflected the needs of his contemporaries to accommodate numbers that were the sums

176




of infinite series, or numbers that symbolized ratios and proportions among infinite

aggregates. Moreover, his view of space and time conflicted with classical precepts. Thus portions of Barrow's work broke away from traditional ideas. Wallis, on the other hand, struggled to incorporate the new symbolism and algebraic techniques into his work, while simultaneously maintaining the traditional concepts of number and

magnitude. Although much of his work was modem in the sense that it was algebraic, Wallis himself did not view this as a break with tradition. At most, he believed that he was more open than the Greeks had been about his method of discovery.

NOTES

1 Helena Pycior, 'Mathematics and philosophy: Wallis, Hobbes, Barrow and Berkeley', Journal of the History of Ideas 48, 266 (1987). The main focus of her article is the seventeenth-century roots of Berkeley's mathematical views. Thus she does not explore the details of how the dispute between ancients and moders influenced the transition to an algebraic mode of thought.

2 Chikara Sasaki, 'The acceptance of the theory of proportions in the sixteenth and seventeenth centuries: Barrow's reaction to the analytic mathematics', Historia Scientiarum 29, 83 (1985).

3 For the Greeks a number was a multitude of units. Numbers were discrete. 'One' was not a number, but the origin which was used to generate numbers. 'One' was a unit, and, as such, it was not thought to be divisible. Aristotle, Physics, 12 2209 27. Consequently, the Greeks could in no way develop a real number system. Many of the mathematical details of the transition from numbers being considered as collections of discrete units to numbers being treated as similar to continuous magnitude have been thoroughly explored in Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (New York, 1992).

4 Wallis's and Barrow's mathematics are both discussed in D.T. Whiteside, 'Patterns of mathematical thought in the later seventeenth century', Arch. Hist. Exact Sci. 1, 178-388 (1960). See also: Christoph Scriba, Studien zur Mathematik des John Wallis (1617-1703) (Wiesbaden, 1966); and Michael Mahoney, 'Barrow's mathematics: between ancients and modems' in Before Newton: The Life and Times of Isaac Barrow, ed. Mordechai Feingold, pp. 179-249 (Cambridge, 1990).

5 J.F. Scott, The Mathematical Work Of John Wallis D.D., F.R.S. (1616-1703) (London, 1938).

6 D.T. Whiteside, 'Patterns of mathematical thought in the later seventeenth century', Arch. Hist. Exact Sci. 1, 263.

7 John Wallis, Opera Mathematica, vol. 1, 365 (Oxford, 1695). 8 Ibid., p. 366. 9 John Wallis, A Treatise of Algebra, Both Historical and Practical, p. 68 (London, 1685).

10 Isaac Barrow, Geometrical Lectures, (tr. Edward Stone), p. 100 (London, 1734). 11 Ibid., p. 110. Although I have attempted to keep to the spirit of Barrow's proof, some

adaptations have been made. For example he used A.B::C.D to express A bears to B the same ratio as C to D. Here DCq, for instance, stands for (DC)2 or the square on DC. Also the justifications given for each step were not supplied by Barrow, he merely provided the equations.

12 Michael Mahoney, 'Barrow's mathematics: between ancients and moders', in Before Newton: The Life and Times of Isaac Barrow, (ed. Mordechai Feingold), p. 266 (Cambridge, 1990).

177



178 Katherine Hill

13 Quoted in op. cit. (11), p. 189. 14 Wallis, op. cit. (9). 15 Ibid. 16 This was also pointed out by Barrow in The Usefulness of Mathematical Learning

Explained and Demonstrated: Being Mathematical Lectures Read in the Publick Schools at the University of Cambridge (tr. John Kirby, London), p. 47 (London, 1734).

17 Quoted in Mahoney, op. cit. (12), p. 190. 18 Barrow, op. cit. (16), p. 20. 19 Ibid., p. 27. 20 Wallis, op. cit. (9), p. 3. 21 Barrow, op. cit. (16), p. 28. 22 Ibid., p. 243. 23 F. Vieta, 'Analytic art', in J. Klein, Greek Mathematical Thought and the Origin of Algebra

(tr. Eva Brann), p. 318 (New York, 1992). 24 Michael S. Mahoney, 'The beginning of algebraic thought in the seventeenth century', in

Descartes' Philosophy, Mathematics and Physics, (ed. Stephen Gaukroger), p. 148 (New Jersey, 1980).

25 Wallis, op. cit. (9), p. 265. 26 Ibid., p. 92. 27 Ibid. 28 Barrow, op. cit. (16), p. 37. 29 Ibid., p. 41. 30 Ibid., p. 44. 31 Wallis, op. cit. (9), p. 92. 32 Aristotle, op. cit. (3) 6 231b 15-20. 33 Wallis, op. cit. (9), p. 285. The method of exhaustion was a rigorous geometrical technique

that was perfected by Archimedes. 34 Ibid., p. 286. 35 Barrow, op. cit. (16), p. 152. 36 Ibid., p. 186. 37 Ibid., p. 198. 38 Barrow, op. cit. (10), p. 176. 39 Ibid., p. 256. 40 Quoted in Mahoney, op. cit. (12), p. 205. 41 Ibid.



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

Documents