6

RATIOS, QUOTIENTS, AND THE LANGUAGE OF NATURE

DOUGLAS JESSEPH

IN HIS 1623 ESSAY “The Assayer,” Galileo notoriously claimed that the “book of nature” was written in the language of mathematics.1 Yet when we consider his actual formulation of the laws of nature (most notably the law of free fall in the Two New Sciences) it becomes apparent that he took the language of mathematics to be something rather different than the mathematical formulations we typically use today. As is well known, Galileo used the Euclidean-Eudoxian language of proportions to express the law of free fall, formulating it in terms of ratios between distances and the squares of elapsed times, rather than as a second-degree equation linking distance covered to elapsed time.2 Because the 1637 publication of Descartes’ Géométrie marked the first appearance of analytic geometry, it is no surprise that Galileo did not employ its techniques to state his results. Nevertheless, the Galilean preference for the traditional language of ratios and proportions reminds us that the mathematics employed by seventeenth-century natural philosophers is, in many cases at least, firmly rooted in classical Greek doctrines. Even Newton, who developed his calculus of fluxions some two decades before the publication of the Principia, chose to develop his celestial mechanics in the classical language of ratios and proportions drawn from book 5 of the Euclidean Elements.3

My purpose here is to investigate a seventeenth-century dispute over how best to interpret the classical doctrine of ratios and to link these differences to alternative programs for mechanics. In particular, I wish to focus on the doctrines of Isaac Barrow (first Lucasian Professor of Mathematics at Cambridge) and John Wallis (Savilian Professor of Geometry at Oxford from 1649 until his death in 1703). These two professors took quite different approaches to the account of ratios developed in the Euclidean Elements. Wallis identified ratios with quotients arising from division, seeking thereby to place the theory of ratios within a very general algebraic theory that he identified as the mathesis universalis, or universal mathematics.4 Barrow, in contrast, insisted that the doctrine of ratios could only be properly understood when it was taken as grounded in essentially geometric concepts with no algebraic content. A significant part of this dispute dealt with the prospects of applying the theory of ratios to the study of the behavior of bodies in motion. In Part I of his 1670 Mechanica, Wallis insisted that the only way to develop a truly general mechanics was to follow the algebraic approach. In contrast, Barrow insisted that such physico-geometrical concepts as space, body, and motion were the only appropriate foundation for a mathematics that could be applied to nature. I begin with a general overview of the classical account of ratio and proportion, then turn to a consideration of Wallis’s and Barrow’s interpretations of the theory. I close by examining the connection between the theory of ratios and the foundations of mechanics, focusing primarily on Wallis’s Mechanica.

THE CLASSICAL THEORY OF RATIOS

Seventeenth-century treatments of the theory of ratio and proportion all arise from the interpretation of a series of definitions in Euclid’s Elements. The relevant definitions from book 5, which introduced the concepts of ratio and proportion, are these:

3.  A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

4.  Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

5.  Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

6.  Let magnitudes which have the same ratio be called proportional. (Elements, bk. 5, defs. 3–6)5

Thus understood, a ratio is not a quotient formed by the division of one number by another, but rather a relation that holds between geometric magnitudes. Magnitudes are grouped into species or kinds, and the third and fourth definitions guarantee that it is only within species that a ratio can be constructed or magnitudes compared.

To take an example: lines, angles, surfaces, and solids are fundamentally distinct kinds of magnitudes, and there is no way to compare directly the magnitude of one kind (a line, say) with the magnitude belonging to another (such as an angle). This is because no number of lines could ever exceed an angle, as required by the fourth definition. To inquire into how many lines might amount to an angle is a nonsense question, on par with seeking to determine how many potatoes could equal a symphony. Thus, there is no “relation in respect of size” holding between heterogeneous magnitudes. Nevertheless, definitions 5 and 6 do permit the comparison of ratios across species of magnitude in a proportion, so it makes sense to say that the ratio L1 : L2 between the length of two lines is the same as the ratio V1 : V2 between the volumes of two spheres. In other words, the proportion L1 : L2 : : V1 : V2 is legitimate, even though the magnitudes V1 or V2 cannot be directly compared with L1 or L2. Likewise, the definition of equality of ratios (definition 5) does not assert that α : β : : γ : δ whenever α × δ = γ × β because the relevant magnitudes may be heterogeneous and incapable of being multiplied together. Instead, sameness of ratio is defined in definition 5 by the preservation of order relations under arbitrary equimultiples.

Although the classical theory of ratios had an impeccable Euclidean pedigree and was often put forward as a paradigm of rigorous mathematics, a great many seventeenth-century authors sought to introduce an alternative understanding of ratios. Much of the motivation for moving beyond the Euclidean scheme arose from concerns about the status of definition 5: it seemed too prolix and intricate to be a true first principle of geometry, and although its truth was never challenged it was thought that there must be simpler and more elegant principles from which the theory of ratios could be developed.6 This alternative approach to ratio and proportion can be usefully termed the “numerical” treatment of ratios, in contrast with the classical “relational” theory.

The fundamental difference between these two approaches can be brought to light by asking whether ratios themselves are quantities; that is, things that can be greater or less. According to the relational theory, the answer is no: ratios are not quantities, but rather relations that hold between two quantities. Just as it would be nonsense to assert that such numerical relations as “greater than” or “divisible by” are themselves some sort of number or magnitude, the relational theory of ratios holds that a ratio is radically distinct from the quantities that stand in a ratio. From the standpoint of the numerical theory, however, it makes perfect sense to say that one ratio could be greater than another. On the numerical understanding, each ratio is taken to have a size (or “denomination” or “exponent”), and the sameness of two ratios amounts to their having the same size, denomination, or exponent. This approach assimilates ratios into a general domain of magnitudes, and it avoids the complex Euclidean definition of the sameness of ratio in terms of the preservation of order relations under arbitrary equimultiples.

In point of fact, the Euclidean doctrine admits the comparison of ratios as to greater and less, which makes it seem plausible that ratios themselves should count as quantities. In the seventh definition of Elements book 5, Euclid states, “When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second magnitude, but the multiple of the third magnitude does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.” So, the ratio 5:3 exceeds the ratio 7:8, because by multiplying the first and third terms by 2 and the second and fourth terms by 3, we discover that (5 × 2) > (3 × 3) while (7 × 2) < (8 × 3). But if anything capable of a “greater than” comparison is a quantity in its own right, there seems to be a solid case for attributing quantities to ratios.

Notwithstanding its appealing simplicity, the numerical theory nevertheless faces its own difficulties. It is natural to assume that the criterion for sameness of ratio in the numerical theory should be expressed in the principle that the ratio α : β is the same as the ratio γ : δ just in case α × δ = γ × β. However, if the quantities α and δ are different species of magnitudes, there is no clear sense to be made of the notion that they could be multiplied together. Indeed, this is precisely why the Euclidean definition requires that two magnitudes have a ratio to one another only if each can exceed the other by being multiplied.

The principal consequence here is that the numerical theory requires all ratios to be homogeneous, or capable of direct comparison with one another. One natural way to do this would be to characterize the denomination of a ratio as a quotient formed by dividing the antecedent of the ratio by its consequent. But doing this raises the difficulty of understanding how the quotient of two incommensurable magnitudes can be understood. Classically conceived, the quotient is a fraction that arises from the division of integers—a fact reflected in the etymological observation that the root of the term is the Latin quoties, or “how many.” In effect, quotients are simply rational numbers that express how many common units of the denominator are contained in the numerator. Incommensurable magnitudes cannot, of course, be understood as quotients in this sense, so the numerical theory of ratios seems committed to expanding the traditional concept of a quotient to include quotients of irrational magnitudes, that is, to making sense of expressions such as . As a result, the development of the numerical theory of ratios required a fundamental reconsideration of the traditional concept of number, namely one that expands the traditional Greek notion of number (άριτημόζ), conceived as a collection of units, to include all magnitudes in an abstract general theory of magnitudes that is fundamentally algebraic.7

The differences between these two approaches to the Euclidean theory of ratios came into sharper focus in the thorny issue of compounding ratios—a much-disputed point that traces back to a pseudo-Euclidean definition that appeared in some editions of book 6. The (now generally regarded as spurious) fifth definition of book 6 reads, “A ratio is said to be compounded of ratios when the sizes [πηλιϰότητεζ] of the ratios multiplied together make some (ratio, or size).” The definition is unusual for its apparent reference to the ‘sizes’ of the compounded ratios, as well as its employment of the arithmetical operation of multiplication to construct a ratio from the sizes of two given ratios. None of this makes any sense on the relational theory, because a ratio is not a quantity and there is no sense in which it might have a size. For these reasons the definition is now regarded as a late emendation to the Euclidean text, although it was still regarded as canonical in the seventeenth century.8 This fact led to some creative interpretations of the text. Barrow noted that “ratios, as they lack all quantity, can neither be added nor multiplied,”9 and took this to indicate that the Greek term πηλιϰότητεζ should be understood as indicating the quantities contained in the compounded ratios rather than any quantities pertaining to the ratios themselves. A further oddity with this definition is the fact that it is never used in the Elements, even in the one place where Euclid speaks of compounded ratios (Elements 6 23).

Whatever conceptual problems the definition may present for the relational theory of ratios, it makes perfect sense on the numerical theory of ratios. If ratios have sizes, or if they are identified with quotients, then there is no obstacle to accepting the notion that multiplication of two ratios can form a third. For instance, if the ratios 3 : 8 and 9 : 11 are compounded, the new ratio will arise from the multiplication of the quotients 3/8 and 9/11, yielding 27 : 88 as the compounded ratio.

The two competing accounts of the nature of ratios did not originate in the seventeenth century; in fact, the differences between the relational and numerical theory were discussed among medieval authors.10 For my purposes, however, the most important exchanges over this topic occurred in the seventeenth century in the works of John Wallis and Isaac Barrow. It is to an analysis of their doctrines that I now turn.

WALLIS AND THE NUMERICAL THEORY OF RATIOS

Wallis was one of the most prominent advocates of the numerical theory of ratios. He argued for it at length in his 1657 treatise Mathesis Universalis, which originated as Savilian lectures and is largely devoted to making the case that the principles of geometry are subordinate to those of arithmetic.11 Wallis was, in fact, a proponent of a view I term “algebraic foundationalism,” according to which all of geometry can and should be developed from arithmetical principles, which in turn can be shown to be special cases of more fundamental principles of algebra, or the “arithmetic of species.”12 In other words, Wallis held that algebraic theory is the proper foundation for all of mathematics.

In the Mathesis Universalis Wallis argued that geometrical results can be achieved more perspicuously and naturally by the use of arithmetical arguments. In service of this goal he devoted the twenty-third chapter to a series of “arithmetical” demonstrations of results from the second book of Euclid’s Elements, an enterprise he took to illustrate his contention that the important results in geometry are ultimately based on arithmetical principles. He argued:

Because some take the geometric elements for the basis of all mathematics, they even think that all of arithmetic is to be reduced to geometry, and that there is no better way to show the truth of arithmetical theorems than by proving them from geometry. But in fact arithemtical truths are of a higher and more abstract nature than those of geometry. For example, it is not because a two foot line added to a two foot line makes a four foot line that two and two are four, but rather because the latter is true, the former follows. (MU 11; OM 1 53)

This led Wallis to conclude:

The close affinity of arithmetic and geometry comes about, rather, because geometry is as it were subordinate to arithmetic, and applies universal principles of arithmetic to its special objects. For, if someone asserts that a line of three feet added to a line of two feet makes a line five feet long, he asserts this because the numbers two and three added together make five; yet this calculation is not therefore geometrical, but clearly arithmetical, although it is used in geometric measurement. For the assertion of the equality of the number five with the numbers two and three taken together is a general assertion, applicable to any other kinds of things whatever, no less than to geometrical objects. For also two angels and three angels make five angels. And the very same reasoning holds of all arithmetical and especially algebraic operations, which proceed from principles more general than those in geometry, which are restricted to measure. (MU 11; OM 1 156)

The remark that “especially algebraic operations” are abstract and apply to “any kinds of things whatever” indicates Wallis’s notion of algebra as a highly general science of quantity with no specific connection to any specific kind of number, magnitude, or measure.

This doctrine leads quite naturally to the numerical theory of ratios. At the very least, the project of interpreting all of mathematics as essentially algebraic is helped along by reducing the entire theory of ratios to a special case of arithmetic, which in turn happens to be a special case of algebra. As Wallis saw the matter, the comparison of magnitudes in ratios renders all ratios homogeneous. In his words: “Where a comparison of quantities according to ratio is made, it frequently happens that the ratio of the compared quantities leaves the genus of magnitude of the compared quantities and is transferred into the genus of number, whatever that genus of the compared quantities may be … And this is the principal reason I affirm that the doctrine of ratios belongs rather to the speculations of arithmetic than geometry” (MU 25; OM 1 136). Wallis’s reasoning can be summarized as follows: when we construct a proportion between two pairs of magnitudes, we have established that the two ratios are of the same size. But the only way to compare things together in regard to their size is to have a common measure of their sizes. Therefore, there must be some common measure for all ratios, which requires that they be instances of a very general concept of number, more abstract than the traditional Euclidean definition of a number (άριτημόζ) as simply a collection of units.

In fact, Wallis argued that Euclid’s treatment of ratios in the fifth book of the Elements should be demonstrated “arithmetically,” and in chapter 35 of the Mathesis Universalis he undertook precisely this task. Not surprisingly, Wallis took Euclid’s definition of the sameness of ratios to be defective, and declared:

We have thought it fit to omit this definition from our demonstrations, although it is indeed true and well enough suited to Euclid’s purpose, nor do we examine proportionals according to this criterion. And indeed it seems somewhat complex, and perhaps not perspicuous enough—especially to learners—nor indeed does it immediately respect the nature of proportionals, but rather some remote affection of them. But for us, who earlier judged ratios by how much, it seems sufficient to prove the identity or equality of ratios if there is an equality or identity of quotients. So, for instance, if a/α = b/β, then a : α : : b : β , and vice versa. (MU 35; OM 1 184)

Wallis’s thoroughgoing identification of ratios with quotients is equally apparent in his approach to the question of compounding ratios. He held that the obscurities surrounding the fifth definition of book 6 could be set aside by showing that Euclid himself accepted the numerical theory of ratios. In a Savilian lecture from 1663 (later published in the second volume of his Opera Mathematica of 1693) Wallis undertook to make this case. He first argued that the Euclidean definition of the term ‘ratio’ must be given a slightly different interpretation than the tradition had accorded it. In particular, he held that Euclid’s “sort of relation in respect of size between two magnitudes of the same kind” (Elements, book 5, def. 3) must be rephrased as “a ratio is that relation or habitude of homogeneous magnitudes to one another in which it is shown how the one is to the other, considered according to quantuplicity” (OM 2 665).

The neologism ‘quantuplicity’ is Wallis’s term for how much one magnitude is in comparison with another, or how many times the one is contained in the other. Specifically, he intended to allow relations of quantuplicity that cannot be expressed as ratios of integers, so that the diameter of a circle would be the 1/π th quantuple part of the periphery. Given this understanding of ratios, the composition of ratios is a simple matter: each ratio has an “exponent” that indicates its quantity, which is the quotient arising from the division of the antecedent by the consequent. Thus, according to Wallis, definition 5 of book 6 should be understood to say “a ratio is said to be compounded of ratios when the exponents of the ratios multiplied together make the exponent of that ratio” (OM 2 666). Having considered Wallis’s exposition of the numerical theory of ratios, we can now turn to Barrow’s response.

BARROW IN DEFENSE OF THE RELATIONAL THEORY

Barrow termed the theory of ratios “the very soul of mathematics,” because he saw in it a doctrine “on which nearly everything remarkable and abstruse demonstrated in mathematics ultimately depends” (LM 16; MW 1 252). His Lectiones Mathematicae originated as Lucasian lectures in the 1660s and were principally concerned with defending Euclid’s account of ratios. In point of fact Barrow understood that his defense of the classical doctrine is an essential part of a broader program to see geometry established as the one true foundation for all of mathematics. Barrow’s announced purpose in vindicating Euclid was to show that “there is nothing in the whole work of the Elements more subtly found out, more solidly established, or more accurately treated than this whole doctrine of proportions” (LM 23; MW 1 378). In the end, I think that this defense of the classical approach to ratios is due in large part to Barrow’s conception of geometric demonstration as founded in the consideration of true causes that are best understood by attending to the motions by which geometric magnitudes are produced.

Barrow viewed any departure from Greek tradition with suspicion, and he spent many pages of the Lectiones Mathematicae defending the relational doctrine of ratios against its modern rivals.13 Indeed, he asked his audience to “pardon my contentiousness, and not hold it against me that I have been led by a certain piety to undertake to vindicate the father and prince of geometry from the undeserved reproaches that are heaped upon him from every side” (LM 18; MW 1 283).

Barrow’s response to Wallis was to turn the tables on his opponent and mount a case for the primacy of geometry over arithmetic. One of the main themes of the Lectiones Mathematicae is Barrow’s argument for what I call “geometric foundationalism,” or the view that all of mathematics is ultimately based on geometric concepts and principles. Barrow argued that arithmetic lacked the kind of determinate content necessary to found a true science. “Any number at all,” he declared, “may with equal right denote and denominate any quantity” (LM 3; MW 1 51). Barrow’s point here can be illustrated as follows: a given line may be deemed one, one hundred, or one thousand, depending on whether we divide it into meters, centimeters, or millimeters.

Barrow responded to Wallis’s argument that the arithmetical fact that 2 + 2 = 4 is too general to be based solely on geometry, from which he had concluded that geometry must be founded on arithmetical or algebraic concepts. To this reasoning, Barrow retorted:

I respond by asking, How does it happen that a line of two feet added to a line of two palms does not make a line of four feet, four palms, or four of any denomination, if it is abstractly, i.e. universally and absolutely true that two plus two makes four? You will say, This is because the numbers are not applied to the same matter or measure. And I would say the same thing, from which I conclude that it is not from the abstract ratio of numbers that two and two make four, but from the condition of the matter to which they are applied. This is because any magnitude denominated by the name two added to a magnitude denominated two of the same kind will make a magnitude whose denomination will be four. Nor indeed can anything more absurd be imagined than to affirm that the proportions of magnitudes to one another depend upon the relations of the numbers by which they are expressed. (LM 3; MW 1 53)

Consequently, in Barrow’s view, there is no arithmetical fact without the specification of a unit, but such a specification is too arbitrary to be the basis of a proper science. This led him to conclude that “mathematical number is not some thing having existence proper to itself, and really distinct from the magnitude which it denominates, but is only a kind of note or sign of magnitude considered in a certain manner” (LM 3, MW 1 56). The result is that “number (at least that which the mathematician contemplates) does not differ in the least from that quantity which is called continuous, but is formed wholly to express and declare it. And neither are arithmetic and geometry conversant about diverse matters, but equally demonstrate properties common to one and the same subject, and from this it will follow that many and great advantages derive to the republic of mathematics” (LM 3; MW 1 47). Barrow took this argument for geometrical foundationalism to the extreme of denying that algebra is a mathematical science at all. In his estimation algebra fails to qualify as an independent science because it is at best a fragment of logic, and at worst a collection of purely formal rules for the manipulating symbols. So, where Wallis and others took algebra to be a highly abstract science that took for its object quantity in general, Barrow dismissed it as unscientific “because it has no object distinct and proper to itself, but only presents a kind of artifice, founded on geometry (or arithmetic), in which magnitudes and numbers are designated by certain notes or symbols, and in which their sums and differences are collected and compared” (LM 2; MW 1 46).

These considerations support Barrow’s central objection to the numerical theory of ratios, namely that Wallis is guilty of a kind of category mistake in thinking that ratios are quantities that can be studied by an abstract, algebraic science of quantity. Because a ratio is a “pure, perfect relation,” it cannot “pass into another category and become a genus of quantity” (LM 20; MW 1 318). In other words, to treat a ratio itself as a quantity is to confuse a relation with one or another of its relata—the ratio is a way for two quantities to be compared, but it cannot itself be a quantity. Barrow admitted that the classical theory of ratios permits such locutions as “the ratio α : β exceeds the ratio γ : δ” (in accordance with definition seven of book 5 of the Elements). However, he held that this can be understood without requiring ratios to be quantities. Rather, such expressions arise whenever the antecedent of one ratio exceeds the antecedent of another, provided that the ratios have common consequents. He explained: “Whatever is commonly attributed to ratios, only truly and properly agrees with the denominators of ratios, that is, to their antecedents reduced to a common consequent. The quantity that others assign to ratios is nothing other than the quantity and ratio of the denominators, and when they think they add or subtract ratios themselves, they only add or subtract these denominators, and this is the same thing when they multiply or compound, divide or resolve them” (LM 20; MW 1 315). The methodological picture that emerges from these considerations is fairly straightforward. Geometry is the foundational science for all of mathematics, in the sense that every mathematical truth is ultimately analyzable as a statement about the properties and relations of continuous magnitudes. The correct method for investigating these properties and relations is by constructions carried out in accordance with the definitions, axioms, and postulates of Euclidean geometry. Such constructions will typically aim to establish ratios and proportions, which constitute the “very soul” of mathematics. Further, these constructions will constitute demonstrations that proceed from true causes, and Barrow devoted the sixth of his Lectiones Mathematicae to establishing the “causality” of geometric demonstrations. The causality he has in mind “can be called formal causality, from the fact that from one property first taken as given, the remaining affections [of a geometric object] arise as from a form” (LM 6; MW 1 93).14 There is no significant role for algebra in this scheme, since all of the demonstrative work is accomplished by constructions that trace back to first principles that articulate the essential form or nature of geometric objects.

FROM RATIOS TO MECHANICS

The differences of opinion separating Wallis and Barrow were based in divergent conceptions of how best to interpret Euclidean geometry, and particularly the theory of ratios. Yet these differences were not confined to the realm of pure mathematics. Wallis’s algebraic foundationalism led him to take the science of mechanics as an application of the very general algebra or “arithmetic of species” that he regarded as the true mathesis universalis. Barrow, in contrast, held that mechanics was essentially a branch of geometry proper, where such concepts as space, time, and motion turn out to be both the foundations of pure geometry and the basis for a science of material bodies.

Wallis developed his account of the connection between the theory of ratios and the science of motion in the Mechanica, which begins by declaring mechanics to be “that part of geometry that treats of motion, and investigates through geometric reasonings and demonstrations, by what force any motion is effected” (Mechanica 1; OM 1 575). The “geometric reasonings” in the Mechanica are taken from the theory of ratios, so that after the definitions of key terms, the first propositions deal with ratios and their composition. Thus, Proposition II reads “When a ratio is composed of two or more [ratios], given the components, the composite is given. That is to say, having multiplied the exponents of the compounds into one another, the exponent of the composite may be determined” (Mechanica 1, prop. 2; OM 1 580). In the scholium to this proposition, Wallis explains that “Euclid calls the indices or exponents of compounded ratios πηλιϰότητεζ, which his interpreters call ‘quantities,’ but I prefer ‘quotients’, for it means that which arises from the division of the antecedent term by the consequent” (Mechanica 1, prop. 2; OM 1 580).

Although the initial propositions in Mechanica are “taken from the doctrine of ratios,” Wallis explained that their demonstrations are so contrived that they apply both to the general algebra of species and to the specific case of geometric lines and figures. Stating and demonstrating such principles algebraically renders them “more general,” so as to apply to any magnitudes whatever, thereby making ratios of geometric figures “only a single case among many, that are contained within a universal proposition” (Mechanica 1, prop. 6, Scholium; OM 1 583). The consequence of Wallis’s taking the theory of ratios as part of a universal algebra is that it can deliver results that extend beyond geometry and enable the doctrine of motion to be studied mathematically (i.e., algebraically).

The cornerstone of Wallis’s approach is Proposition VII of the Mechanica: “Effects are proportional to their adequate causes.” This proposition, which says nothing directly about lines, figures, or other geometric objects, permits us to reason about the relation between causes and effects, and specifically to investigate the causes of motions by considering motion as an effect of some motive cause. As Wallis announced, “I have reckoned that this universal proposition should be set out at the beginning, since it opens the way by which, from purely mathematical speculation, one may move on to physical [speculation], or rather that the one is connected to the other” (Mechanica 1, prop. 7, Scholium; OM 1 584).

Barrow left no systematic treatise on mechanics, but his views on the subject are easily enough reconstructed from his approach to foundational questions in geometry and his remarks on the nature of “mixed mathematics.” He proposed a highly kinematic conception of the origin of geometry in which magnitudes such as lines, angles, and surfaces are generated by motions. Thus, he conceived a line or curve as the path traced by a point in motion through space, while a circle is characterized as something produced by the revolution of a line about one of its endpoints. In his Lectiones Geometricae (which were assembled some years after the Lectiones Mathematicae) he explained that “among the ways of generating magnitudes, the primary and chief is that performed by local motion, which all [others] must in some sort suppose, because without motion nothing can be generated or produced” (LG 2; MW 2 159). Thus, in Barrow’s view, the basic concepts of geometry include space, time, and motion. Further, when a geometrical object such as a curve or surface is defined in terms of the motions that produce it, the definition expresses the true formal cause of the object and allows the deduction of necessary properties of the object.

One notable consequence of this view is that “because local motion in general can scarcely be judged as regards its duration, impetus, intension, direction, or any other of its properties, either in itself or compared with another motion, except by the spaces (that is straight or circular lines) that it can describe or pass through, it follows that most parts of physics … are to be judged part of mathematics” (LM 2; MW 1 44). Indeed, Barrow concluded that “mathematics, as it is commonly taken, is so to speak coextended and made equal with physics itself” (LM 2; MW 1 44). But, given Barrow’s identification of geometry as the mathematical science par excellence, it follows that the whole of physics is to be understood geometrically. Thus, as Barrow conceived of the issue, the science of mechanics is concerned with nothing distinct from the continuous quantities of geometry, and these are to be investigated by attending to the properties of such magnitudes as expressed in the motions that generate them.

As a result, the space of geometry is identical with the space of physics, and we understand the properties of such magnitudes by attending to the motions with which they are produced. Significantly, Barrow used the comparison of compound motions in the Lectiones Geometricae to effect the construction of tangents and determination of areas, both of which are essential to any treatment of mechanics. The key to his method was to treat a curve as traced by “composite motions” of a point, and then to decompose the composite motion into two instantaneous rectilinear motions, from which the determination of various properties followed fairly naturally. One particular problem is of interest here, namely the determination of properties of the parabola, and specifically the parabolic arcs of bodies in free fall. Speaking of the success of his method in relation to this problem (which he had succeeded in generalizing), Barrow remarked “I believe that not only this but many other propositions of Galileo connected to this one and related to the matter, howsoever they are demonstrated, can also be rendered more general or extended to all sorts of other curves” (LG 4; MW 2 199). In other words, the study of bodies subject to motive forces (i.e., mechanics) is a branch of geometry that proceeds by determining the ratios of the component motions arising from such forces.

If Barrow himself did not leave a systematic treatise on mechanics, it is arguable that his most successful student did. That student is, of course, Newton, and his Principia is developed in precisely the style that Barrow held to be necessary for any proper study of the physics of moving bodies. Rather than expressing his results in terms of algebraic equations (in the style of Wallis), Newton constructed ratios and proportions derived from a consideration of the motions of point masses and their trajectories that arise from the application of forces, all the while avoiding anything that might seem overly algebraic or disconnected from the consideration of continuous magnitudes in physical space.15

CONCLUSION

If the account I have been developing is anywhere near the truth, we should read the history of seventeenth-century physics against the background of disputes over the nature of ratios. The two traditions I have identified, namely the relational and numerical accounts of ratios and proportions, are associated with two different ways of constructing the mathematical language to be used in investigating the properties of bodies, and more specifically their mechanical properties. The numerical theory lent itself to a highly algebraic treatment of mechanics in Wallis’s Mechanica, and was further developed by Leibniz and physicists in the Leibnizian tradition. In contrast, the relational doctrine of ratios was tied to a much more geometric treatment of mechanics that avoided the apparatus of algebra in favor of the more traditional language of ratios and proportions. It is an odd irony of history that what is today taught as Newtonian mechanics uses the conceptual and mathematical apparatus of algebra and the associated notion of real-valued functions, which belong to a tradition that neither Barrow nor Newton would recognize as appropriate for expressing the fundamental principles of mechanics. How that came to pass is, however, a matter for another day.

ABBREVIATIONS

NOTES

1. Galilei 1890–1909, 6:232.

2. Machamer (1998, 65) notes that “Galileo used a comparative, relativized geometry of ratios as the language of proof and mechanics, which was the language in which the book of nature was written. This is very different from what will follow in the eighteenth century and from the way we think of science today.”

3. See Guicciardini (1999) on the mathematics behind Newton’s formulation of his mechanics and the debates it engendered among his eighteenth-century interpreters.

4. On the concept of a mathesis universalis, its origins, and its role in seventeenth-century thought, see Crappuli (1969) and Rabouin (2009).

5. My references to the Euclidean Elements are to Euclid ([1925] 1956), in the translation of Heath. References are given in the text to book number and definition or proposition number.

6. See Palmieri (2001) on various attempts to rework the classical theory of ratios, notably those by Galileo and Christopher Clavius.

7. See Whiteside (1960, section 2) on the development of a broader algebraic conception of algebraic number in the seventeenth century, as well as Klein (1968).

8. See Heath’s introductory note to book 6 of the Elements (Euclid [1925] 1956, 2:189–90) for a summary of the evidence for regarding the definition as an interpolation.

9. This remark appears in the twentieth of Barrow’s Lectiones Mathematicae. Henceforth, I will give citations to Barrow parenthetically in the text, using the abbreviations LM and LG for his Lectiones Mathematicae and Lectiones Geometricae, with a reference to the relevant lecture number. I also supply page and volume references to Barrow’s Mathematical Works (Barrow 1860).

10. The medieval history of the definition and its role in seventeenth-century mathematics is studied in Sylla (1984).

11. My references to the Mathesis Universalis are given parenthetically in the text, using the abbreviation MU and the relevant chapter number. I add a parallel citation to Wallis’s Opera Mathematica (Wallis 1693–99). References to Wallis’s Mechanica are to chapter and definition or proposition number, with a parallel citation to OM.

12. I contrast algebraic and geometric foundationalism in seventeenth-century philosophy of mathematics in Jesseph (2010).

13. See Mahoney (1990) on Barrow’s mathematics and its odd mixture of innovation and methodological conservatism.

14. Barrow denied that the causality characteristic of mathematics could be construed as efficient or final causality. This is because “the connection (at least such as can be understood by us) of an external cause (for instance, of an efficient cause) with its effect cannot be such that, the cause having been posited, the effect must necessarily be granted; nor from a posited effect may some determinate cause, strictly speaking, be shown” (LM 6; MW 1 91–92). This view is a consequence of Barrow’s theological voluntarism, which requires that in the case of efficient causality the connection between a cause and its effect “depend upon the most free will and omnipotence of Almighty God, who at his pleasure can prevent the influx and efficacy of any [efficient] cause” (LM 6; MW 1 92). For more on the connection between Barrow’s voluntarism, his nominalism, and his approach to mathematics, see Malet (1997) and Sepkowski (2005).

15. See Guicciardini (2009, chapter 13, “Geometry and Mechanics”) for details on Newton’s account of the relationship between geometry and mechanics.

REFERENCES

Barrow, I. 1860. The Mathematical Works. Ed. W. Whewell. 2 vols. Cambridge: Cambridge University Press.

Crappuli, G. 1969. Mathesis Universalis: Genesi di un’Idea nel XVI Secolo. Rome: Edizioni dell’Ateneo.

Euclid. [1925] 1956. The Thirteen Books of Euclid’s “Elements” Translated from the Text of Heiberg. Ed. and trans. T. L. Heath. 3 vols. New York: Dover.

Galilei, G. 1890–1909. Le Opere de Galileo. Ed. A. Favaro. 20 vols. Florence: Barbéra.

Guicciardini, N. 1999. Reading the “Principia”: The Debate on Newton’s Mathematical Methods for Natural Philosophy from 1687 to 1736. Cambridge: Cambridge University Press.

______. 2009. Isaac Newton on Mathematical Certainty and Method. Cambridge, Mass.: MIT Press.

Jesseph, D. M. 2010. “The ‘Merely Mechanical’ vs. the ‘Scab of Symbols’: Seventeenth-century Disputes over the Criteria for Mathematical Rigor.” In Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics, ed. A. Heeffer and M. Van Dyck, 273–88. Studies in Logic, vol. 26. London: College Publications.

Klein, J. 1968. Greek Mathematical Thought and the Origin of Algebra. Trans. E. Brann. Cambridge, Mass.: MIT Press.

Machamer, P. 1998. “Galileo’s Machines, His Mathematics, and His Experiments.” In The Cambridge Companion to Galileo, ed. P. Machamer, 53–79. Cambridge: Cambridge University Press.

Mahoney, M. S. 1990. “Barrow’s Mathematics: Between Ancients and Moderns.” In Before Newton: The Life and Times of Isaac Barrow, ed. M. Feingold, 179–249. Cambridge: Cambridge University Press.

Malet, A. 1997. “Isaac Barrow on the Mathematization of Nature: Theological Voluntarism and the Rise of Geometrical Optics.” Journal of the History of Ideas 58: 265–87.

Palmieri, P. 2001. “The Obscurity of the Equimultiples: Clavius’ and Galileo’s Foundational Studies of Euclid’s Theory of Proportion.” Archive for History of the Exact Sciences 55: 555–97.

Rabouin, D. 2009. “Mathesis Universalis”: L’idée de mathématique universelle d’Aristote à Descartes. Paris: Presses Universitaires de France.

Sepkowski, D. 2005. “Nominalism and Constructivism in Seventeenth-Century Mathematical Philosophy.” Historia Mathematica 32: 33–59.

Sylla, E. 1984. “Compounding Ratios: Bradwardine, Oresme, and the First Edition of Newton’s Principia.” In Transformation and Tradition in the Sciences, ed. E. Mendelsohn, 11–43. Cambridge: Cambridge University Press.

Wallis, J. 1657. Mathesis Universalis. Oxford: Litchfield.

______. 1693–99. Opera Mathematica. 3 vols. Oxford: Oxford University Press.

Whiteside, D. T. 1960–62. “Patterns of Mathematical Thought in the Later Seventeenth Century.” Archive for History of the Exact Sciences 1: 179–338.