The Very Soul of Mathematics — Isaac Barrow C17th — Part 1

0:00 I've gone from the 1930s to roughly 1900, and now it's the next obvious step, it's about 370 years. I've distributed a revolutionary handout for you. I'm painting a bunch of Latin and some English. This is actually the first time in my working career that I've distributed a handout like this and there's no diagrams. Even though I'm talking about something in geometry, that probably says some very deep point about the nature of geometrical plot and what we're up against. I thought to put it on PowerPoint, but my long and bitter experience with trying to get things to render properly, and then, you know, project, convince me that a talk on 17th century issues should use 17th century technology. Although Fabian informed me that apparently printing with laser printers was not a PDF file, so they're definitely out. At any rate, Isaac Barrow is the subject, the principal subject of this talk. He was the first of the occasion professor of mathematics at Cambridge, and he presents something of a paradoxical figure in early ladder-clother mathematics. On the one hand, Barrow made fundamental contributions to the development of infinitesimal methods, and I think he can properly be regarded as one of the grandfathers, if not the father, of the calculus, although his results certainly pale in comparison with those achieved by Newton or Leibniz. He made really quite fundamental contributions to the development of this theory. On the other hand, Barrow was very much a mathematical traditionalist. whose veneration for ancient methods and the sanctity of classical proof procedures set him staunchly against the mainstream of what we might call progressive mathematics in the 17th century. Michael Mahoney has characterized Barrow as poised, as he put it, between ancients and moderns, and I think this formulation captures a good deal of the flavor of Barrow's mathematical work. He simultaneously looks backward at the classical tradition and forward to the great mathematical advances of the 17th and 18th centuries. Obviously, a question crops up. How do you make that consistent? How can someone, on the one hand, venerate Euclid and classical Archimedean techniques and on Wednesdays work with infinitesimals? There seems to be something of an incongruity here. I think I've got an answer, although I postpone that,

2:30 as always, to the very end, furious hand waving, etc. My purpose is to investigate Barrow's defense in the classical Euclidean theory of ratios against innovators, as he called them, innovator for a 17th century British divine, a theologian is not a good, it's a term of abuse. Innovation is very, very bad if you're a 17th century English Church of England theologian. So, against these innovators who proposed alternative accounts of ratio and proportion, Barrow termed the theory of ratios the very soul of mathematics, because he saw in a doctrine, as he put it, on which nearly everything in remarkable and abstruse demonstrated in geometry ultimately depends. And in point of fact, Barrow understood his defense of the classical doctrine as an essential part of a broader program to see geometry established as the one true foundation for all of mathematics. So my plan is to review briefly some of the in the interpretation of the classical account of ratio and proportion, and then examine Barrow's main arguments for the superiority of the classical approach and its role in geometry. My focus will be on Barrow's replies to John Wallace, Thomas Hobbes, and Giovanni Borrelli, who are the three authors whose work I find to have elicited the most interesting replies from Barrow. I should note, however, that the mathematical lectures contain a wealth of other information on the theory of proportion in the 17th century. You can find such authors as Christopher Clavis, André Taquet, and Gretel Saint-Denisant, also the subject of Barrow's criticisms, occasionally apoplectic criticisms. I shouldn't be then thought to be giving you some sort of, you know, a full story or a comprehensive treatment ratio, as understood by Barrow in the lectures. Now, his announced purpose, says Barrow, in vindicating Euclid was to show that, as he put it, 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 proportion. And in the end, I think that this defense of the classical approach to ratios is due in significant measure to Barrow's conception of geometric demonstration as founded in the consideration of true causes, which are expressed and understood by attending to the motions by which geometric magnitudes are produced. might wonder, you know, what is it about ratios that makes them interesting? And the answer

5:00 from the geometric perspective is pretty much any result you're attempting to establish in the system to show that one quantity or one magnitude stands in a ratio to another. So the theory of ratios is in some sense foundational geometry or absolutely fundamental. The starting point for any discussion of the theory of ratios is a small number of definitions in in the fifth book of the Euclidean Elements. The relevant definitions are given there on the section one in the handout. Those are definitions three through six. The definition of ratio just says it's a kind of relation in respect of size between two magnitudes. Magnitudes will have a ratio to one another when they can be made to exceed one another by multiplication. The fifth is the most important definition, and it's, of course, the one that's the subject of enormous dispute, controversy, and interpretation down the ages. Magnitudes are said to be in the same ratio, the first to the second, the third to the fourth, when any equimultiples whatever taken of the first and third and any equimultiples whatever taken of the second and fourth, the former alike exceed or like equal to or like fall short of the latter taking respectively in corresponding order. You might notice that fails to answer most of Aristotle's requirements for a first principle. It's very and complicated, and it doesn't have the sort of punch or spice that you get with a very simple definition. The sixth definition there, magnitude in the same ratio, we call proportionals as a standard thing. I should mention that there's a linguistic problem that crops up when you're trying to deal with Latin and English authors who write about this, because the terminology is just a train wreck. The Greeks have logos and analogia for ratio and The Latins who sort of know what they're doing, people writing in Latin, will use ratio and proportion as the standard term, but some people use proportion, but others would use ratio to mean logos, and then it just gets a big mess, because the theory of proportions is the theory of identity, of ratios, and then the English come along completely unprincipled as always. Some use reason, some use ratio, some use proportion, it's just anarchy, and no wonder they needed a revolution. It's obviously obvious that no notion of doing. Anyway, I think we can bracket most of those difficulties, but it does get very complicated in trying to be quite clear about who meant what on what occasion. Now, thus conceived, a ratio is not taken to be a quotient

7:30 formed by the division of two numbers. Instead, a ratio is a special kind of relation that holds between two quantities, two geometric magnitudes. Magnitudes themselves are conceived of as falling into species or kinds. In the third and fourth definitions, they're guaranteed that it is only within species that a ratio can be constructed or magnitudes compared. Now, to take an example, then, lines or angles, surfaces, and solids are all fundamentally distinct kinds of magnitudes, and there is no way to compare directly the magnitude of one kind of a magnitude of a line, say, with the magnitude belonging to another, say, such as a surface. And this is because no number of lines could ever exceed a surface as required by the fourth definition. Consequently, there is no relation in respect of size holding between heterogeneous magnitudes. To ask, you know, how many lines are there in an angle is like asking how many potatoes make a symphony or something. I mean, it's just an all-category mistake. Nevertheless, Nevertheless, the fifth and sixth definitions do permit the comparison of ratios across species of magnitude in a proportion, so it makes sense to say that the ratio L1 to L2 between the length of two lines is the same as the ratio S1 to S2 between the volumes of two spheres, so the proportion L1 is to L2 as S1 is to S2 is a legitimate mathematical object, even though the magnitude S1 or S2 couldn't be directly compared with the magnitude L1 or L2, and the definition of equality of ratios, the complicated definition 5 there, does not assert that ratios stand in the same proportion whenever the 1 multiplied by the other equals the other multiplied by the 1, there's no multiplication there across species that's forbidden. Instead, sameness of ratio is defined by the preservation of order relations under arbitrary equimultiple so the one is taken in like order note that the classical theory of ratios reflects a finitistic standpoint characteristic of Greek geometry. The fourth and fifth definitions are understood to require only finite multiplications of finite magnitudes, and the restriction bars the introduction of infinitesimals. The extent to which infinitary reasoning is admissible in geometry is in fact a matter that will be of some consequence later on. And in particular, one common concern regarding Euclid's definition of proportions,

10:00 and particularly definition five, is that the definition requires us to consider a potential infinity of cases where equimultiples of magnitudes have to be prepared, and the thought that means, as we're scanned across an infinite number of pairs of equimultiples, that might seem to violate the finitistic standpoint. Now, although the classical theory ratios has an impeccable Euclidean pedigree, and it was frequently put forward as a paradigm of rigorous mathematics, a great many 17th introduce an alternative understanding of ratios. This alternative can be usefully termed the numerical treatment of ratios, in contrast with the classical relational theory, as I call it. And the fundamental difference between these two approaches can be brought to light by asking whether ratios themselves are quantities, i.e., is a ratio something that could be greater or less? And 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 some kind of magnitude, the relational theory holds that a ratio is radically distinct from the quantities that stand in ratio. Important point in the sort of classical understanding. On the other hand, from the standpoint of the numerical theory, the alternative introduced and studied in the 17th century, it makes perfect sense to say that one ratio could be greater than another. Given the numerical understanding, every ratio has a size, or a denomination, as it was often called, and two ratios are the same when they have the same denomination. No need for this messy definition of a Euclid. This approach then assimilates ratios into a general domain of magnitudes, and it avoids that complicated Euclidean definition of sameness of ratio, in terms of arbitrary equimultiple, saying that sameness of ratio is simply sameness of size or sameness of denomination. But despite this appealing simplicity, the numerical theory of ratios faces its own problems. It is natural to assume that the numerical theory's criterion of sameness of ratio should be expressed in the principle that the ratio of alpha to beta is the same as the ratio of gamma to delta, just in case the product of alpha times delta is equal to the product of beta times gamma. if the quantities, say alpha and delta, are different species of magnitudes. If you look at the ratio between the lengths of two lines and then the ratio between, say,

12:30 the volumes of two solids, you may have the sameness of ratio there, but you can't compare the line to the solid or the line to the surface or anything else. In that case, there's no clear sense to be made of the notion that you can multiply these ratios together. And indeed, this is precisely why the Euclidean definition requires that two magnitudes have a ratio to of one another only if each can exceed the other by multiplication, and that leads quite naturally, well maybe it's not natural, but it leads inevitably to the definition of sameness of ratio in definition 5. The salient consequence of this is that the numerical theory of ratios requires that all ratios be homogeneous, or capable of direct comparison one with another. One natural way to do this would be to characterize the denomination of the ratio as a quotient or by the division of the antecedent to 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. This is, in fact, reflected in the etymological fact that the root of the term is the Latin quoties, or how many. And in effect, quotients are simply rational numbers that express how many common units in the denominator contain the numerator. magnitudes cannot, obviously enough, 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. In other words, to making sense of expression such as, you know, pi divided by the square root of two. Now, obviously, given, you know, a decade later theory of real numbers, there's no issue there. From the point of view of the 17th century, that's actually a real question. The result, then, is that with the development of the numerical theory of ratios, you need a fundamental reconsideration of the traditional concept of number, namely one which expands the typical Greek notion of number, of arikomos, conceived as a collection of units, expands that out to include any magnitude and includes it in a kind of abstract general theory of quantity that at the end of the day turns out to be fundamentally algebraic. Now, Barrow viewed this departure from Greek tradition with suspicion. of his mathematical lectures defending the relational doctrine of ratios against its modern rivals. And indeed, at one point, he asked his audience to, quote, pardon my contentiousness and not hold it against me, that I've been led by a certain piety to undertake and vindicate

15:00 the father and prince of geometry and the undeserved reproaches that are everywhere heaped upon him. A Freudian interpretation here would say that Barrow has serious issues maybe his defense of the relational theory is so furious because he has this veneration for the father, but then he feels himself overwhelmed by some sort of edible desire to go out and develop infinitesimal mathematics you can try that, I think it's bullshit I think I've got a better answer than that, but that's the way is open, right? Anyway, let's now turn to a quick Wallace and Barrow on this. John Wallace was Oxford's civilian professor of geometry from 1649 until his death in 1703, and was probably the most prominent advocate of the numerical theory of ratios in his day. He argued for it at length in his 1657 treatise, Mateisis Universalis, which originated as civilian lectures, and is largely devoted to making the case that the principles of geometry are in fact subordinate to those of arithmetic. Wallace was a proponent view I term algebraic foundationalism, which holds that all of geometry could be developed from arithmetical principles, which in turn can be shown to be special cases of more fundamental algebraic principles. In other words, algebraic theory turns out to be the proper foundation of all mathematics. And in Matesis Universalis, Wallace argues that the geometrical results achieved by Euclid can be attained more perspicuously and more naturally by the use of arithmetical arguments. In service of this goal, he devoted the 23rd chapter of the Matesius Universalis to a series of what he termed arithmetical demonstrations of results from book two of Euclid's elements, simply rewriting them in an arithmetical form. And this he took to illustrate his contention that the important results of geometry are ultimately based on arithmetical principles. As he argued, this is the second item on your handout, which I'll give a proper translation into English. Barrow says, because some take 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, arithmetical truths are of a higher and more abstract nature than those of geometry.

17:30 For instance, 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. In other words, if the arithmetical truths are foundational, anything in geometry turns out to be just a special case. From this, Wallace concludes that the close affinity of arithmetic and geometry comes about because geometry is, as it were, subordinate to arithmetic, and it applies universal principles of arithmetic to its special objects. For if someone asserts that a line of 3 feet added to a line of 2 feet makes a line 5 feet long, he asserts this because the numbers 2 and 3 added together make 5. Yet this calculation is not, therefore, geometrical. It is clearly arithmetical, though it might be used in geometric measurement. The assertion, he says, of the equality of the number 5 with the numbers 2 and 3 taken together is a general assertion applicable to any other kinds of things, whatever, no less than to geometrical objects. The very same holds of all arithmetical and especially all algebraic operations, which proceed from principles more general than those in geometry, to measure. Now, Wallace's account of the matter leads quite naturally to his embrace of the numerical theory of ratios. And indeed, the numerical theory is tantamount to the assertion of the priority of arithmetic over geometry. I mean, at the very minimum, this idea, this project that Wallace has of interpreting all the mathematics as essentially arithmetical will be helped along significantly by reducing the entire theory of ratios to a special case of arithmetic. As Wallace saw the matter, the comparison of magnitudes in ratios, the formation of ratios, generally renders all ratios homogeneous. They all turn out to be instances of the same kind of thing. And this is what he expresses in the item three on the handout there. He says, 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 the genus of the compared quantities may have been. And this is the principal reason I affirm that the doctrine of ratios belongs rather to the speculations of arithmetic than to those of geometry. So, Wallace's reasoning can be summarized as follows. When we construct a proportion between two pairs of bank of dudes, we've established that the two ratios are the same size. But the only way to compare things together in regard to their size

20:00 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 Greek notion of number as simply a collection of units. And in fact, Wallace argued that Euclid's treatment of ratios in Book 5 of the Elements should be demonstrated, as he put it, arithmetically. And, not surprisingly, the 35th chapter of the Matesis Universalis undertakes exactly this task. there took the Euclidean definition of sameness of ratio, the complex fifth definition of book five of the elements, he takes this to be confected, and argues, this is item four there on the handout, argues more or less as follows, we've thought fit to omit this definition from our demonstrations, although it is indeed true and well enough accommodated to Euclid's purpose. Nor do we examine proportionals according to this criterion. But because it seems somewhat complex and perhaps not perspicuous enough, especially to learners, nor indeed does it immediately respect the nature of proportionals, but rather give some remote affection of them, we have dismissed it. 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 over alpha equals B over beta, then the ratio of A to alpha is the the same as the ratio of B to beta, and vice versa. It's interesting that in the course of his dispute with Thomas Hobbes, Hobbes pointed out that Wallace identified ratios with quotients and then took him to task for deviating from Euclid. Wallace replied that he'd never said such a thing at all. And here we have him actually saying precisely that. This gives you some idea of the standards of argument that employed in the 17th century exchange of polemics. At any rate, so there he is. quotients. End of story. Barrow's response to Wallace was to mount a case for the primacy of geometry over arithmetic, in effect, as turning the table on Wallace. And indeed, one of the main themes of Barrow's mathematical lectures is an argument for what I call geometric foundationalism, namely the view that all of mathematics is ultimately based on geometric considerations. Barrow argued that arithmetic lacks the kind of determinate content necessary to found a true science. Any number at all, he declared, may with equal right denote or denominate any quantity whatever. And the point here could be illustrated as follows. A given line, say,

22:30 might be deemed one or a hundred or a thousand, depending on whether we divide it into meters, centimeters, or millimeters. And in Barrow's view, then, there is no arithmetical fact until we have determined a specification of a unit. But this, he thinks, is something really be mathematical facts that are ultimately arithmetical, that are in some sense arbitrary, and Barrow then concludes that, as he put it, mathematical number is not some thing having existence proper to itself and really distinct from the magnitude it denominates. Instead, it is only a kind of note or sign of magnitude considered in a certain manner. So the consequence that Barrow draws is item 5 there. He says number, or at least that which the mathematician contemplates does not differ in the least from that quantity which is called continuous but it is formed wholly to express and declare it and neither are arithmetic and geometry conversant about diverse matters but equally demonstrate the 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 even though Barrow had no republican sentiments particularly Now, he took this argument for geometrical foundationalism to the extreme of denying that algebra is a mathematical science at all. According to Barrow, algebra is nothing more than a collection of purely formal rules for the manipulation of symbols. So, where Wallace, for example, or others had taken algebra to be a highly abstract science of quantity in general, Barrow dismissed it as unscientific because, as he put it, it has no object distinct and proper to itself. It only presents a kind of artifice founded upon geometry in which magnitudes and numbers are designated by certain notes or symbols and in which their sums and differences are collected and compared. These considerations then support Barrow's central objection to the numerical theory of ratios, namely that Wallace is guilty of a kind of category mistake in thinking that ratios are quantities dragged algebraic signs of quantity. Because a ratio is, as Barrow put it, a pure, perfect relation, it cannot pass into another category and become a genius of quantity. In the Aristotelian argument here, it's a relation, it has no quantity, it can't all of a sudden one day turn into a quantity. Now Barrow admits the somewhat embarrassing

25:00 fact that the classical theory of ratios does allow such expressions as the ratio of alpha to beta exceeds the ratio of gamma to delta in accordance with definition 5 of 05, but this, he said, could be understood without making ratios into quantities. Rather, such expressions arise whenever the antecedent of one ratio exceeds the antecedent of another, provided that both ratios have been reduced to common consequence. Once that reduction has been taking place, then the first ratio is greater than the second just means that the antecedent of the first is greater than the second. This is worked out in item 6 on but if I tear through it, it's not all that terribly significant for the moment. We can come back to it if we need to. Well, this concludes my survey of Barrow's attack on Wallace's theory of ratios. We can now turn to an investigation with reply to Hobbes. Thomas Hobbes was a mathematical interesting case, to say the least. Just one question. Yes, sorry. On Wallace. I'm sure, when I understand that it's a relation between algebraic and arithmetic. Well, algebra is ultimately, at the end of the day, the basis. But the argument initially is that geometry can be replaced by arithmetic, by some kind of reduction. Let's worry not too much about what we mean by reduction. Then he later says, well, arithmetic is just a special case of algebra. But geometry reduces to arithmetic, arithmetic reduces to algebra. So, algebra is the foundation. Now, ultimately, I mean, there's a bit of a problem. Obviously, sometimes the argument that he makes looks purely arithmetical, so you might think that he's some sort of arithmetical foundation. I think arithmetic is really doing all the work. And most of his matices can actually be read that way without much problem until you get to the end, where he says, okay, now I'm going to give you my treatment of algebra. amount of time, talk to you later. Or the proof is left as an exercise to the reader, this kind of thing. Later, his treatise of algebra, which was published not until 84, I think, 85, gives, sort of delivers the goods that had been promised, but it's a long time coming. It's a 30-some year, 20-some year

27:30 hiatus. But I mean, I think at the end of the day, he's an algebraic foundationalist, or, I mean, we could quibble over that, but I mean, certainly, in the arguments that he gives, that Barrow focuses on, it's only the reduction to arithmetic that's in the cards, and it's somewhat later that the reduction to algebra is going to be a particular, but I don't think, yeah, since there's no, you know, I mean, we have to wait for Wallace to tell us what algebra is before we can reduce to it, right, so that's why we had to wait 25 years. Anyway, let's go to Hobbes now. Hobbes gives us an account of ratios that has some important points of agreement with Barrow's, interestingly enough, although it drew sharp and dismissive condemnation from Barrow, the Lucasian professor. Like Barrow, Hobbes endorsed geometrical foundationalism, and he argued that all of mathematics must ultimately be derived from fundamental concepts such as space, motion, and body. Where Hobbes differs, and differs quite radically from Barrow, was in his embrace of a thoroughgoing materialism. Barrow held the traditional view that the objects of mathematical investigation are abstract, purely immaterial, intelligible entities with no dependence on the material world. Hobbes, in contrast, held that the only substance was material body, so the first principles of geometry have to be interpreted in strictly material terms. In Hobbes' words, there is no subject of quantity or of equality or any other accident but body. So geometry, on this way of thinking about it, becomes a very general science of material bodies. It's not just sort of that some experience with material bodies in the world gives the initial idea that it leads to geometry, but actually the very object or subject of geometrical investigation is material bodies. So Hobbes proposed to rewrite the first principles of Euclidean geometry in a way that transforms it into a generalized science of body. Hobbes is rather exceptional in this way. Just about everybody else in the tradition thinks that Euclid's definitions are fine, and I need a little interpretation here and there, but the idea of just throwing it all out and starting again from scratch, Hobbes isn't the only one who's pursued that in any detail. A point, for example, which Euclid had defined as that which has no part, is reinterpreted by Hobbes to be a body, but a material body small enough that its magnitude can be neglected in a demonstration. For those of you who care, that actually turns out to be

30:00 the physicist's definition of a particle, right? A particle of material bodies is the distance between two parts of it is irrelevant or doesn't enter into computation. A line classically defined as a length without breadth is defined by Hobbes to be the path traced by a moving point. A surface is then that which is produced by the motion of lines, right? Solids are those things produced by the motion of surfaces. angles by rotation, etc. In the case of ratios, Hobbes dismissed Euclid's definition of ratio, definition 1 on point 1 there. He says, dismisses it as a, what shall I call it, isness or someness of two magnitudes. This is pretty dismissive. He says it must be replaced by a definition in terms of bodies, and more specifically, as he puts it in item 7 there, the relation of the antecedent to the consequent according to magnitude, that is to say it's equality, is called the ratio or proportion, he just should have said ratio, but the ratio or proportion of the antecedent to the consequent, so ratio is nothing other than the equality or inequality of the antecedent compared to the consequent according to Magnet. The centerpiece of Hobbes' treatment of ratios is his alternative definition of sameness of ratio or proportionality, his alternative to definition of I from the book V of Euclid. Where Euclid had defined proportions in terms of arbitrary equimultiples or paired terms Hobbes sought a definition that would establish the theory of proportion on the basis of concepts of body and motion. These considerations led him to profound the following definition. One geometrical ratio is the same as another geometrical ratio when some cause can be assigned such that producing equal effects at equal times, it determines both ratios. Now, this idiosyncratic approach to the theory of ratios, in doing so, Hobbes attempted to steer something force between the numerical and relational treatments of ratios. In rejecting Euclid's definition of ratio as an intolerably vague bit of nonsense, Hobbes wanted to found the doctrine of ratio proportion on principles of body and motion. But he accepts the principle that ratios are, in fact, quantities, since they are capable of being greater or less than one another. And therefore, he rejects the key element in Barrow's defense of the relational theory. Hobbes also, however, opposed the fundamental thesis of the numerical theory, that ratios should be understood as a kind of quotient arising on the division of the antecedent by the consequent. Where Wallace had subsumed the doctrine of ratios within an abstract algebraic science of quantity in general

32:30 by identifying ratios of quotients, Hobbes insisted that a quotient can be formed only in the case of commensurable magnitudes. As he put it, quote, In lines incommensurable there may be the same proportion, but nevertheless there is no quotient. For setting their symbols one above the other doth not make a quotient. For quotient, there is none but in aliquot parts. It is therefore impossible to define proportion universally by comparing quotients. Hobbes explained that his own account of ratio and proportion was developed with the intent of accommodating incommensurable magnitudes within a theory that avoids what he called the obscurity of Euclid while retaining sufficient generality to permit ratios and incommensurables, all the while treating geometry as this science of body, to talking material bodies. His explanation of this is item 8 on the handout. I won't bother to work through it. It's basically just his attempt to defend his definition against criticism. It's probably no surprise that Barrow's distaste for Hobbes' doctrine was boundless. There's no ratio to capture the amount of hate he had for Hobbes on this point. Although both Barrow and Hobbes shared a commitment to geometric foundationalism, is nothing more than a set of rules for manipulating empty symbols, Barrow was horrified by Hobbes' materialism. He dismissed his account of proportionality with the remark that, quote, if anyone should read through the writings of all mathematicians, both ancient and modern, I think he would never come across anything an author undertakes to illustrate that is made more obscure and nothing laboring under more or graver errors. Hobbes' gravest error, Barrow explained, definitions are, quote, physical rather than mathematical. This, he said, is a result of Hobbes' being, quote, more intent upon those things Galileo has written concerning uniform motion, and his mind is fixed upon the contemplation of merely physical motions when he refers everything concerning magnitude and quantity to certain preconceived ideas on motion. End Hobbes, on Barrow's view, isn't really doing geometry. Hobbes is not an example of geometric thought. It's barely an example of thought. It's certainly not geometry. It's contaminated

35:00 by all of this materialism. It's clearly, it's just so far out of bounds, it's barely even worth talking about. So at the end of the day, I think Barrow's case comes down to the thesis that mathematics must not be contaminated by physical principles of the sort that Hobbes took to be fundamental to any science. Where Hobbes had insisted that concepts such as body and impact were all at the root of true scientia, Barrow insisted such concepts are insufficiently pure and abstract to serve as the foundation of geometry. Barrow and Hobbes agreed on one very significant point, namely that geometry is more basic than arithmetic, and they also shared a commitment to the notion that the best means to approach geometry was through kinematic conception of magnitudes in which lines, angles, surfaces, and solids are all generated by motion. So they're closer than Vero might have wanted to allow, although they have significant differences. Well, putting Hobbes aside, now let's take a look at Borelli's revision of Euclid. Giovanni Alfonso Borelli is best known to historians of science for his work in biomechanics, but he was actually something of a polymath who took a degree in mathematics. He was appointed and later Pisa, which I gather is moving up in Italian context. I'll leave that to others to determine. His 1650... He's on the road. Supposedly, he wanted Galileo's old Pisa chair. Galileo didn't want to have anything to do with it. That's the story. That's all I don't know Euclidean elements with particular attention paid to the doctrine of ratio and proportion. And what Borelli proposed is an alternative axiomitization of the theory of proportions. And in the third book of Euclides Restitutus, he offers it to us. This is item nine on the handout, which has the following axioms, which I'll just tear through very quickly. He says, if a first quantity measures a second and the third measures a third, the first will also measure a third. The second says that three quantities having been proposed, whatever ratio the first has to the second, the third will have to some other quantity of the same canis. Item three is almost exactly the inverse of that. Whatever ratio the first has to the second, some other quantity of the same canis will have to the third. Four is formulated as a very bizarre Latin, but what it boils down to is that if one of two equal quantities

37:30 makes some parts or is a part of a third, and the other also makes parts or is a part of the same third, then the one will be the same part as the other. The fifth one there, three terms of the least to which ratios will be contained, not a terribly interesting piece. Item six, however, is interesting. It says, if the first to four quantities is greater than the second, but the third is not greater than the fourth, then the first will have to the second a greater ratio than the third has to the fourth. Now, on the basis of these axioms and accompanying definitions In part, Borelli developed a material on ratio and proportion that we find in Euclid, but developed it in a manner somewhat different than Euclid had proposed. Specifically, Borelli's first principles do not require the Euclidean definition of same ratio, which others have found excessively complex. It should, however, be mentioned that Borelli's alternative axiomatization of the material requires him to undertake more complicated proofs than you find in Euclid. Maybe he's got more axioms than his proofs tend to be proofs by cases. He typically has to prove a result for commensurable cases and then do another result for incommensurable cases, where the Euclidean definition is powerful enough to handle all of them. So, I mean, it's the sort of thing you would expect from an alternative exposition. He gains a little bit by having somewhat simpler axioms and definitions and loses something by having more complicated proofs. He was also an extraordinarily long-winded man, No amount of, I mean, that seems to have infected the length of the demonstrations also. But Borrelli argued that his treatment of ratios was superior to the Euclidean treatment precisely because he could dispense with the definition of sameness of ratio, the typical definition Euclid had given in terms of preservation or order relations under arbitrary equimultiples. He took this to be a principle that managed his system over Euclid, and he went to some great length to argue that the Euclidean definition was inadequate. on the handout, which is an example of what happens when a 16th or 17th century Italian mathematician gains access to a printing press. It just goes on and on and on. No offense, but this guy goes on enormously. His objections basically boil down to the claim that in the case of ratios of incommensurable magnitudes, the Euclidean definition demands an infinite task, literally surveying all possible equimultiples and pairs of quantities and ratios. According

40:00 to Morelli, this use of essentially infinitary reasoning makes the definition of Euclid inadequate because even though it might be true that four quantities and proportion will in fact preserve their order relations whenever multiplied by common multiples, this is a fact demonstrable from more elementary features of equality and proportion. So it should be taken as a that ought to be proved on the basis of more elementary principles rather than assumed as a definition. Barrow took the most subtle Borelli, as he termed him, to be what he called the most bitter adversary of Euclid, one who, above all, has done this extraordinary thing, namely to produce from his own resources a new doctrine of proportionality, indeed one that is beautiful and solid, but nevertheless hardly to be preferred on any account to that of Euclid. Borelli's doctrine was very firm, as he put it, built upon a good foundation, and all of the principal results he thought were demonstrated appropriately. The only drawback to Borelli's development of the theory is that it requires more axioms and more definitions in Euclid, so that the resulting proofs of central theorems are significantly longer and more complex than we find in Euclid. Barrow, however, did strenuously oppose Borelli's objections to the Euclidean On Barrow's reading of Inflict, there is no difficulty at all involved in a definition that covers a potential infinity of cases. As Barrow explains, once the intellect has verified that a property is essential to an object, that insight is reliable in any number of cases. So if you have appropriate definitions, the intellect grasps the definition, and the application to an infinity of cases is not problematic. There's no difficulty to be found, at least as Barrow would have it. and Barrow then offered a number of responses to Borelli not all of which are of interest here but it's useful to consider a couple of them in a little bit of detail to Borelli's claim that it is uncertain whether four quantities might ever be found satisfying Euclid's definition because this would involve an infinite task of comparing it to multiples Barrow answered his fault, this is the first part of 11 on the handout he says this is not unknown because it may be shown by clear instances that there are many cases of four quantities endowed with the said condition. And if this were not so, this definition could by no means

42:30 be applied, for as often as it is applied, there are so many instances, then, of quantities having this property. It is in the same way unknown that there is such a figure as one heading equal radii, until it is made plain to the senses by generation, or to the mind by some sufficiently evident discourse. The intriguing point here is that the definition showing the generation of a geometric object can establish a result, and that evident discourse, as Barrow terms it, can yield insight that goes beyond any finite number of experienced cases to rule generally. Barrow's preferred approach to geometric proof was, in fact, to appeal to the motions whereby curves, surfaces, etc. are generated, and he spent a good deal of time in his mathematical lectures, which were published, some of his geometrical lectures later, later, arguing that geometrical definitions are causal in the appropriate sense, and that they show not only that a given result holds, but they can, in fact, construct the relevant magnitudes and show why the result has to hold. A similar exchange between Barrow and Borelli concerns the role of infinite comparisons in the criteria for rigorous demonstration. Borelli complained that Euclid's fifth definition involves infinite comparison of pairs of magnitudes, and he complains that such a definition fails to be evident and obvious enough to be judged a true first principle of demonstrative science. Barrow's reply is that there are any number of cases demonstrated where the proportionalities and questions hold. Proportionality between triangles and parallelograms of equal bases and heights among pyramids, prisms, combs, and cylinders with common bases, etc. Any number of cases can be found. The question, says Barrow, is not whether the definition is comprehensible or useful, but whether it requires an illegitimate appeal to the infinite. And Barrow is quite emphatic on that point. This is the second piece of section 11 there in the handout. Barrow says, I claim there is no infinity of comparison supposed here any more than in any general proposition where the universality of the terms supposes an infinity. For can that which is understood in the definition by the words by any multiplication be anything other than the universality of a condition set down, proved, not by viewing all other particular instances or by an inductive inference, but from a universal argumentation? It is only required that all the equimultiples and the antecedents be related in such a manner to all the equimultiples, and the consequence to be related in such a manner, I say, as to have the same kind of ratio, whether of excess, defect,

45:00 equality. If any infinity here is incomprehensible, then also all theorems of all sciences must