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Preliminary matters
CHAPTER 1
Preliminary matters 1.1. Purpose and scope of the present work My intention in this work is to highlight a neglected theme in the early history of the calculus, namely what I call the problem of transcendental curves (to be defined in Section 2.1). It is a theme intertwined with foundational concerns long since abandoned, and hence primed for neglect among historians of whiggish inclinations. It is a theme often drawing implicitly on a once shared but now largely forgotten background knowledge, making it easy to miss when looking at some works in isolation. Yet, for all its bashfulness, the problem of transcendental curves permeates the early development of the calculus, and once the mathematical works of the period are studied as a cogent corpus in their own terms its prime importance emerges and becomes undeniable. Once brought into the light, one can see the unmistakeable iron grip that this implicit framework held on the early development of the Leibnizian calculus. My goal in this study, then, is to lay bare this implicit logic and its prominent and consistent influence on the direction of the mathematical research of the period. This book is by no means intended as a complete history of the early Leibnizian calculus. But I have aspired to give an essentially complete history of the problem of transcendental curves, by which I mean treating all material pertaining to this problem in the published works and correspondence of Leibniz and his contemporary affiliates. Leibniz is undoubtedly the main protagonist, as the problem of transcendental curves was to him the guiding star for the better part of his mathematical works throughout his life. Jacob and Johann Bernoulli are also prominent figures; they too attacked the problem with much persistence and zeal, though they inherited it from Leibniz and probably appreciated it in a more narrow sense, more for its mathematical fertility than for its epistemological implications. For these main figures I have aimed for a sympathetic and holistic understanding of their work on this problem. In addition, I have aimed to include all material with a direct bearing on this problem from other contemporaries, notably Huygens and Newton, as well as a few other minor figures. But, as we shall see, these people did not always appreciate the Leibnizian vision of the problem of transcendental curves. Therefore I use them primarily as a “contrast class” to elucidate the Leibnizian vision, rather than as figures of focus in their own right. As chronological markers of the study one may use Leibniz’s first Transcendental Curves in the Leibnizian Calculus http://dx.doi.org/10.1016/B978-0-12-813237-1.50001-6 Copyright © 2017 Elsevier Inc. All rights reserved.
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paper on the calculus in 1684 and his death in 1716 as proximate boundaries, with the bulk of the activity focused in the 1690s. This work is based entirely on sources that are now published. In particular, I have not consulted Leibniz’s unpublished Nachlass, which is bound to contain voluminous materials relating to my topic. However, the story I will tell has great cohesion and evidence saturation, and covers essentially everything on the matter that Leibniz deemed worthy of communicating to others in any form. Therefore I consider it highly unlikely that a study of the remaining manuscripts would necessitate any substantial revisions of my story. It is in any case sound historical method to study an author’s published works thoroughly and systematically before delving into his manuscripts—and for Leibniz’s crucial work on the problem of transcendental curves nothing of the sort has been attempted previously.
1.2. Previous scholarship related to the present work This work is in effect a sequel volume to Henk Bos’s masterpiece Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction (2001). As expressed in that work (p. 6), Bos originally hoped to complete a second volume himself, which was to treat the themes of geometrical exactness and construction in the generations following Descartes. However, Bos has since retired and, regrettably, found himself unable to fulfil this intention. The honour and privilege of taking up the project with his blessing thus fell upon me. Bos outlined a vision for the main themes of the post-Cartesian part of his research programme in broad strokes in various places.1 In addition, Bos discussed many of the relevant mathematical case studies that we shall also be concerned with. My intention, however, has been to make the present work complete in and of itself without assuming any familiarity with these previous studies. For the reader who wishes to consult Bos’s accounts in parallel with mine, I give an exhaustive correspondence table in the notes.2 By and large I have no major disagreements with Bos on any matter.3 But I would say that my discussions generally differ in especially two respects. Firstly, I place these episodes in the broader context of a systematic account of the problem of transcendental curves as a whole. Secondly, and relatedly, I tend to focus on the underlying rationale of the reasoning of the historical actors in each of these cases. By contrast, Bos tends to be more concerned with reporting and classifying than with rationalising. An indication of this is for example his scheme for classifying extramathematical arguments,4 which is more phenomenological than my approach. Guicciardini (2009) is an important recent study and useful introduction to the literature on Newton’s philosophy of mathematics, which treats much of the same material as I do on that topic but from a different perspective.
Preliminary matters
The philosophy of mathematics of my main character, Leibniz, has not previously been the subject of a concerted study comparable with these two major studies of Descartes and Newton. No other sustained studies have dealt with the problem of transcendental curves directly, though some aspects of Leibniz’s attempts at a justification of the transcendental are touched upon in works such as Breger (1986) and Knobloch (2006). Also, several of the particular episodes we shall discuss have been discussed in some detail from other perspectives. Notably, Truesdell (1960) includes good accounts of the history of the catenary and the elastica from the point of view of physics, and Tournès (2009) discusses the tractional constructions of our Chapter 5 in the context of his history of such constructions, which soon turned into a technical problem without the philosophical import of the original context of the problem of transcendental curves. Loria (1902) gives a comprehensive modern mathematical treatment of all prominent special transcendental curves, including historical notes.
1.3. What is new in the present work I shall now indicate, to the best of my knowledge, what is new in the present work, i.e., what has not before been discussed in the secondary literature (my own work excluded). I incorporate in this work over one hundred substantial quotations from the primary sources that have never before been translated into English or discussed in the secondary literature. All quotations from Leibniz and his contemporaries for which I give no reference to a previous translation in the corresponding footnote are of this type. A number of the mathematical topics I discuss have never before been covered in the secondary literature. To my knowledge this includes: much of the material from Johann Bernoulli’s lectures on the calculus (Section 4.4); Johann Bernoulli’s “crawling curves” (Section 5.6); Leibniz’s recipe for determining logarithms from the catenary (Section 6.3.2); some points regarding the early history of the use of the logarithm function (Section 6.3.4); Newton’s result the rectification of quadratures (Section 7.3.4); Jacob Bernoulli’s derivation of the differential equation for the elastica (Section 8.2); the Johann Bernoullistyle derivation of the differential equation for the paracentric isochrone (Section 8.3); Lagrange’s treatment of the catenary (Section 9.5). When I discuss mathematical episodes that have been treated in previous studies, the mathematical exposition in this work is nevertheless my own, based on the primary sources. My goal has been to make the mathematical ideas as clear as possible, which means that my discussions typically differ substantially in terms of presentation from other accounts in both the primary and secondary literature. In Section 2.3, I outline several prevalent standard views of the history of this period,
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which I maintain are refuted by the picture emerging from my study. In Chapter 3, I give a big-picture characterisation of 17th-century philosophy of mathematics for which I am not aware of any clear precedents in the literature. This includes the following points: the cohesion of Descartes’s and Leibniz’s general philosophies and their philosophies of mathematics, and the way in which both are modelled closely on the Greek tradition; the contrast of this “Continental rationalist” view with the “British empiricist” tradition, and the characterisation of this split as stemming from divergent extrapolations of the geometrical method; the attribution of the corresponding clash in the philosophy of physics as stemming from the same source; a number of parallels with the history of Greek geometry, which are illuminating in both directions. To be sure a number of these points are to some extent explicit in the primary sources, as I show, so aspects of them have not gone unnoticed. For instance, Bos (2001) certainly noted the influence of the Greek tradition on Descartes, corresponding roughly to my Section 3.3.2. Nevertheless, I am aware of nothing like a systematic and synthesising account similar to my own. Throughout I strive to not only report 17th-century choices and preferences regarding the representation of transcendental curves, but also to explain why these choices were always rational.5 This is a perspective rarely found in other studies, and never with anything near the systematic synthesis I give. Big-picture overviews to this effect are notably: the new interpretative framework of Section 3.3.5.2 and its use notably in Section 7.3.2; Section 4.4.7; the introductions to Chapters 5 through 8; and the concluding Chapter 10. More specific points pertaining to the same purpose are raised throughout, often leading to new perspectives even on matters discussed by previous scholars (as in Sections 4.3, 5.3, 5.4.2, 5.5, 6.2, 6.3, 7.2, 7.3.2). Also new, to my knowledge, is my explanation (in Chapters 9 and 10) of why the transition from a geometrical to an analytical paradigm occurred (and became rational) only with the generation of Euler, even though the technical prerequisites were there long before.
1.4. Conventions adopted in this work 1.4.1. Policy on the presentation of mathematical arguments My story will hinge on a clear and coherent understanding of the key mathematical ideas involved. I have therefore opted to make judicious use of slightly modernised notation and phraseology when I found this beneficial for expository purposes, pointing out anachronisms only when they are relevant to the matter at hand. This should not be mistaken for a lack of commitment to the highest standards of historiography. I remain absolutely committed to understanding the thought of historical mathematicians in their own right rather than through an anachronistic lens. In my opinion, my slight and occasional deviations
Preliminary matters
from their mode of expression never compromises this commitment but is rather a natural concomitant of it.6 Indeed, the early history of the calculus affords many opportunities to observe that idiosyncrasies of surface form are often quite incidental, the most obvious case being the fact that the differences in mode of expression used by British and Continental mathematicians did not prevent them from seeing themselves as doing the exact same thing and even accusing each other of plagiarism.7 17th-century calculus differs from its modern equivalent in a number of often superficial respects: it is focussed on curves rather than functions; on differentials and concrete geometrical characteristics such as lengths of tangents rather than derivatives; on geometrical measurements of lengths and areas of circles and conic sections rather than the now canonised arsenal of transcendental functions such as arcsin(x), log(x), etc. Novices may consult Bos (1974a), sections 1–2, for a good overview of these kinds of conceptual differences of the Leibnizian and the modern calculi. I occasionally ignore these kinds of differences when expedient and harmless. I also take the liberty of altering notations, changing orientations of coordinate systems, setting generic constants to unity, and so on, when this can be done in a way that does no harm to the faithful understanding of the historical sources. By way of illustration we may consider Leibniz’s equation for the cycloid discussed in Section 4.2.2. I give this equation as Z 2 p dt 2 √ x = 2y − y + 2t − t2 y whereas what Leibniz actually wrote is
A number of superficial differences are immediately apparent, none of which are of any consequence for our purposes. In particular, the modern way of indicating bounds of integration was never used in the 17th century, but it is clear that Leibniz meant the exact equivalent of what I have written. I believe the kinds of liberties of paraphrase that I have outlined here are sound. Unconditional refusal to allow such liberties, which some advocate, would only double the bulk of the book and provide nothing but a massive distraction from the ideas that actually matter for the purposes of my argument. To those who demand complete faithfulness to the original sources in these kinds of regards there is a simple solution: read the original. My goal is to contribute to knowledge and understanding by writing a clear and synthesising account, not to slavishly reproduce what is already written for all to read in the sources themselves.
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In the interest of clarity, I have strived to use currently standard notations whenever possible, without stopping to define it in each instance. Thus, unless otherwise noted, x, y are rectilinear coordinates, t is time, s is arc length, a, b, c, . . . are constants, log(x) is the natural logarithm and e its base. Of course, 17th-century calculus also makes use of infinitesimal reasoning in a manner that would be considered very informal by modern standards. I follow this 17th-century manner of reasoning. For a textbook introduction to this way of doing calculus, readers may consult Blåsjö (2016a).
1.4.2. Translation and referencing conventions In my translations I have strived to remain as literal as possible, with a few systematic exceptions. In translating from the Latin, I have often used “curve,” “integral,” and “variable,” where the more literal translations would be “line,” “sum,” and “indeterminate letter” respectively. This is the most faithful translation into modern terminology, I would argue, even though it sacrifices strict etymological continuity. I have also occasionally inserted without warning occasional words which are strictly implied grammatically; especially the noun “line” or “curve” is often omitted in the Latin, with the writer saying, e.g., simply “transcendental” where “transcendental curve” is meant. I have also taken the liberty to alter capitalisations, including capitalising the first letter when I start a quotation midsentence. I have also often standardised references to persons, thus writing, e.g., simply “Leibniz” instead of “Mr. Leibnitz” and so on. I have often opted against giving page references for letters and short articles, since these are in any case only a few pages long and typically printed in multiple editions with different paginations.
1.5. Some key terms It will be useful to clarify upfront some terms and concepts that will feature prominently in our story. First we may consider the principal labels by which curves are classified. An algebraic curve is a curve that can be expressed by a polynomial equation in rectilinear coordinates x and y. The algebraic and coordinate-system methods of Descartes’s Géométrie are in a sense coextensive with this class of curves: they treat such curves exhaustively, and other curves not at all. According to Descartes’s vision of geometry this was most appropriate, for he argued that the set of all algebraic curves is precisely the set of all curves knowable with geometric rigour. To signify this he used the term geometric curve, which means precisely the same thing as algebraic. Curves that were not “geometric” Descartes called mechanical, the implication being that these curves were not susceptible to a truly mathematical and exact treatment. The term can be confusing. As we shall see in Section 3.3.2, one must not be misled by this terminology into thinking that Descartes rejected any
Preliminary matters
association of geometry with mechanisms; on the contrary, his proposed foundations for “geometrical” curves are based on rulers and pegs and motions in a manner many a modern mind would be inclined to call very “mechanical” indeed. When Leibniz set out to extend mathematics beyond these Cartesian bounds, he naturally did not care for Descartes’s terminology, since he rejected the foundational assumptions Descartes had built into the very words. He therefore started referring to Descartes’s geometric curves as algebraic, and Descartes’s mechanical curves as transcendental, since they “transcend all algebraic equations.”8 This excellent terminology is exactly the one still in use today. It applies to numbers as well as curves (or functions), as Leibniz himself often pointed out. Leibniz advocated for this terminology innumerable times in print and correspondence, and with good reason, since it is both objectively better than Descartes’s and furthermore removes a terminological bias against Leibniz’s new transcendental mathematics. In principle, all leading mathematicians of Leibniz’s generation agreed with his point that the foundational assumptions embedded in Descartes’s terminology were obsolete. Nevertheless, through historical inertia, Descartes’s terminology remained in widespread use for many years still, even by mathematicians who unequivocally rejected the foundational connotations they came with at their inception. Indeed, Cartesian geometry was so entrenched in mathematical consciousness that the geometry of algebraic curves was often called “common” or “ordinary” geometry. As a flip side, the Leibnizian calculus was sometimes referred to as the “transcendental calculus,” since one of its defining characteristics was that it went beyond the “common geometry” of algebraic curves. It is often said that the transcendence of π and e was not formally proved until the 19th century, but we must not be misled into thinking that 17th-century mathematicians were not conscious of the finer points of the algebraic–transcendental distinction. On the contrary, they had a refined understanding of this issue. The leading mathematicians of Leibniz’s generation were fully convinced of the transcendence of the numbers and curves they studied as such, and they supported these beliefs sometimes with proofs and, if not that, then at least very compelling informal arguments.9 Analytic and synthetic. Throughout this work I shall use the term analytic to mean any method based on working with formulas or symbols. This by and large tracks late 17th-century usage quite well, though the history of the term is complex. In Greek times, “analysis” meant “working backwards,” i.e., assuming what one sought as known and trying to deduce from it various consequences until one struck upon something already established, at which point one could reverse one’s steps in order to obtain a demonstration of the matter. The opposite of this is synthesis, which means working “forwards” from first principles or established theorems to deduce new results. Synthesis was
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considered the only right and proper style in which to write a formal mathematical treatise, but analysis was recognised as an invaluable tool of discovery and explorative research. Where finding a direct synthetic proof required too much ingenuity, analysis would often lead to a proof using more systematic and predictable steps. When algebraic methods where introduced in the early modern period, the meaning of these terms evolved. The algebra of the moderns was in many ways identified with the analysis of the ancients. Like analysis, algebra can be seen as a methodical technique of discovery. Furthermore, in algebra, when one denotes a sought quantity by x, one in effect assumes it to be known and treats it as such, just as the ancient conception of analysis prescribes. Hence the association of “analysis” with symbolic methods, which gradually grew into the new primary sense of the term. In parallel with this, “synthetic” took on a new meaning too: since analysis had now become synonymous with using modern symbolic methods, “synthetic” came to mean geometry done in the classical style of the Greeks. This sense remains current to this day when we speak of “synthetic geometry.” Later still, as the calculus eclipsed all other analytic methods on the center stage of mathematical research, “analysis” came to be associated specifically with the calculus, as in current parlance such as “real analysis.” But this phraseology is still in the future as far as out era of focus is concerned. Yet another facet to this complicated terminology is its foundational connotations. Virtually everyone agreed that synthesis and analysis were useful techniques that each had their place in mathematics. But more ambiguous was the question of which method is the epistemologically primary one, or the one that reflected the ultimate nature of mathematical knowledge. The status of axioms in particular depends on this question. The picture of synthesis as meaning building up a complicated theory from a few basic starting points fits well with the notion of these starting points or axioms being supposedly obvious and immediate truths, for if they are not the synthetic geometer’s starting point can be accused of being arbitrary. The method of analysis, on the other hand, inasmuch as it means taking complicated things apart or breaking them down, is more amenable to agnosticism as to whether these more basic pieces need be obvious or not, for the analytic geometer can claim to have uncovered underlying principles and assumptions without having to pass judgement on them. These attitudes may be called synthetic foundationalism and analytic foundationalism, respectively. All of these metamorphoses of the terms afford ample opportunity for confusion. Thus Descartes and Leibniz were wholehearted lovers of analysis, in the sense of a method of discovery, especially in symbolic form. Yet they insisted with equal fervour on synthetic foundationalism. Newton on the other hand had these priorities precisely reversed: as far as surface form goes, his Principia is doggedly synthetic in its insistence on classical ge-
Preliminary matters
ometrical, rather than symbolic, presentation; yet this very work is at the same time the manifesto par excellence of analytic foundationalism. If one does not heed the terminological confusion, it may thus appear that each man is both the committed champion and sworn enemy of the same concept. Such is the plasticity of the terms analysis and synthesis. Construction. In our period of interest, a solution to a mathematical problem generally meant not a “formula” as on a modern calculus exam, but a construction. The precise meaning of this is elusive, however. Euclid spent much time in the Elements constructing things with ruler and compass, and these are certainly the archetype instances of constructions. Later mathematicians tried to generalise the notion and ended up with a variety of possible curves and instruments that could be construed as analogous to the ruler and compass in one sense or another. Already in Greek times there were a variety of competing proposals as to the best way of proceeding in such matters, including some that stretched the concept too thin to be accepted as genuine constructions at all.10 Such ambiguities only multiplied further in the era of our focus. Mathematicians were certain that they sought constructions but they were not certain what exactly this meant. It was an explorative, formative period; the 1690s was brainstorming decade. Various possibilities were tried out in the hope that a clear winner would emerge, but in the end this hope was not realised. Which is perhaps just as well for our purposes since it enables us to view this foundational turmoil in its true state of flux, without the distortion of a “right answer” perceived in retrospect. Quadrature and rectification. In the 17th century, finding the area of a figure is often referred to as quadrature or in more English terms squaring. This is an inheritance from the Greek paradigm of geometry, which did not deal with areas numerically but rather operationalised the concept, so that instead of speaking of finding the area of a given figure they spoke of finding a square equal to it (the square being the simplest or prototypical figure as far as area is concerned). In the same vein, lengths were to be exhibited as line segments, whence the term rectification (i.e., extension of a curve into a straight line) for the finding of an arc length. In the 17th century, these terms had to a large extent lost their literal meaning and had come to mean simply finding area and arc length respectively. Yet this gradual transition was far from a definitive break with the Greeks, for their operationally oriented mode of geometry held sway to the very end of the 17th century. Function. The importance of the notion of function has tended to be overestimated by modern authors, as far as our era of study is concerned. The centrality of this notion in later developments has led some to feel that its absence in the early calculus must have been a serious conceptual limitation. I do not agree with such an interpretation. In my view, the reason these mathematicians did not develop the notion of a function is not some invisible conceptual ceiling that they could not break through; rather they did not develop it for the
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simple reason that it would not have served any purpose to them. In general, as a matter of historiographical principle I prefer to explain the choices made by mathematicians of the past in positive terms as the rational and natural way of approaching the problems of their concern, rather than in negative terms as due to conceptual limitations in their ways of thinking. I have found that I have very rarely needed to resort to an explanation of the latter kind. When I use the term function, I mean it in a flexible, pre-formal sense. A more precise conception is not of interest to us. The term “function” was introduced casually and parenthetically in the late 17th century,11 but no one at the time made the slightest fuss about it as any kind of conceptual innovation. And that with good reason. The leaders of the early calculus understood their subject very clearly, and they constantly and seamlessly switched between geometric, analytic, and abstract-relational ways of viewing it with ease and fluency. The modern notion of function would have contributed virtually nothing of relevance to any of their concerns. By the same token, a focus on “the evolution of the function concept” would be an anachronistic perspective of little value to our story.
Preliminary matters 1.1. Purpose and scope of the present work My intention in this work is to highlight a neglected theme in the early history of the calculus, namely what I call the problem of transcendental curves (to be defined in Section 2.1). It is a theme intertwined with foundational concerns long since abandoned, and hence primed for neglect among historians of whiggish inclinations. It is a theme often drawing implicitly on a once shared but now largely forgotten background knowledge, making it easy to miss when looking at some works in isolation. Yet, for all its bashfulness, the problem of transcendental curves permeates the early development of the calculus, and once the mathematical works of the period are studied as a cogent corpus in their own terms its prime importance emerges and becomes undeniable. Once brought into the light, one can see the unmistakeable iron grip that this implicit framework held on the early development of the Leibnizian calculus. My goal in this study, then, is to lay bare this implicit logic and its prominent and consistent influence on the direction of the mathematical research of the period. This book is by no means intended as a complete history of the early Leibnizian calculus. But I have aspired to give an essentially complete history of the problem of transcendental curves, by which I mean treating all material pertaining to this problem in the published works and correspondence of Leibniz and his contemporary affiliates. Leibniz is undoubtedly the main protagonist, as the problem of transcendental curves was to him the guiding star for the better part of his mathematical works throughout his life. Jacob and Johann Bernoulli are also prominent figures; they too attacked the problem with much persistence and zeal, though they inherited it from Leibniz and probably appreciated it in a more narrow sense, more for its mathematical fertility than for its epistemological implications. For these main figures I have aimed for a sympathetic and holistic understanding of their work on this problem. In addition, I have aimed to include all material with a direct bearing on this problem from other contemporaries, notably Huygens and Newton, as well as a few other minor figures. But, as we shall see, these people did not always appreciate the Leibnizian vision of the problem of transcendental curves. Therefore I use them primarily as a “contrast class” to elucidate the Leibnizian vision, rather than as figures of focus in their own right. As chronological markers of the study one may use Leibniz’s first Transcendental Curves in the Leibnizian Calculus http://dx.doi.org/10.1016/B978-0-12-813237-1.50001-6 Copyright © 2017 Elsevier Inc. All rights reserved.
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paper on the calculus in 1684 and his death in 1716 as proximate boundaries, with the bulk of the activity focused in the 1690s. This work is based entirely on sources that are now published. In particular, I have not consulted Leibniz’s unpublished Nachlass, which is bound to contain voluminous materials relating to my topic. However, the story I will tell has great cohesion and evidence saturation, and covers essentially everything on the matter that Leibniz deemed worthy of communicating to others in any form. Therefore I consider it highly unlikely that a study of the remaining manuscripts would necessitate any substantial revisions of my story. It is in any case sound historical method to study an author’s published works thoroughly and systematically before delving into his manuscripts—and for Leibniz’s crucial work on the problem of transcendental curves nothing of the sort has been attempted previously.
1.2. Previous scholarship related to the present work This work is in effect a sequel volume to Henk Bos’s masterpiece Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction (2001). As expressed in that work (p. 6), Bos originally hoped to complete a second volume himself, which was to treat the themes of geometrical exactness and construction in the generations following Descartes. However, Bos has since retired and, regrettably, found himself unable to fulfil this intention. The honour and privilege of taking up the project with his blessing thus fell upon me. Bos outlined a vision for the main themes of the post-Cartesian part of his research programme in broad strokes in various places.1 In addition, Bos discussed many of the relevant mathematical case studies that we shall also be concerned with. My intention, however, has been to make the present work complete in and of itself without assuming any familiarity with these previous studies. For the reader who wishes to consult Bos’s accounts in parallel with mine, I give an exhaustive correspondence table in the notes.2 By and large I have no major disagreements with Bos on any matter.3 But I would say that my discussions generally differ in especially two respects. Firstly, I place these episodes in the broader context of a systematic account of the problem of transcendental curves as a whole. Secondly, and relatedly, I tend to focus on the underlying rationale of the reasoning of the historical actors in each of these cases. By contrast, Bos tends to be more concerned with reporting and classifying than with rationalising. An indication of this is for example his scheme for classifying extramathematical arguments,4 which is more phenomenological than my approach. Guicciardini (2009) is an important recent study and useful introduction to the literature on Newton’s philosophy of mathematics, which treats much of the same material as I do on that topic but from a different perspective.
Preliminary matters
The philosophy of mathematics of my main character, Leibniz, has not previously been the subject of a concerted study comparable with these two major studies of Descartes and Newton. No other sustained studies have dealt with the problem of transcendental curves directly, though some aspects of Leibniz’s attempts at a justification of the transcendental are touched upon in works such as Breger (1986) and Knobloch (2006). Also, several of the particular episodes we shall discuss have been discussed in some detail from other perspectives. Notably, Truesdell (1960) includes good accounts of the history of the catenary and the elastica from the point of view of physics, and Tournès (2009) discusses the tractional constructions of our Chapter 5 in the context of his history of such constructions, which soon turned into a technical problem without the philosophical import of the original context of the problem of transcendental curves. Loria (1902) gives a comprehensive modern mathematical treatment of all prominent special transcendental curves, including historical notes.
1.3. What is new in the present work I shall now indicate, to the best of my knowledge, what is new in the present work, i.e., what has not before been discussed in the secondary literature (my own work excluded). I incorporate in this work over one hundred substantial quotations from the primary sources that have never before been translated into English or discussed in the secondary literature. All quotations from Leibniz and his contemporaries for which I give no reference to a previous translation in the corresponding footnote are of this type. A number of the mathematical topics I discuss have never before been covered in the secondary literature. To my knowledge this includes: much of the material from Johann Bernoulli’s lectures on the calculus (Section 4.4); Johann Bernoulli’s “crawling curves” (Section 5.6); Leibniz’s recipe for determining logarithms from the catenary (Section 6.3.2); some points regarding the early history of the use of the logarithm function (Section 6.3.4); Newton’s result the rectification of quadratures (Section 7.3.4); Jacob Bernoulli’s derivation of the differential equation for the elastica (Section 8.2); the Johann Bernoullistyle derivation of the differential equation for the paracentric isochrone (Section 8.3); Lagrange’s treatment of the catenary (Section 9.5). When I discuss mathematical episodes that have been treated in previous studies, the mathematical exposition in this work is nevertheless my own, based on the primary sources. My goal has been to make the mathematical ideas as clear as possible, which means that my discussions typically differ substantially in terms of presentation from other accounts in both the primary and secondary literature. In Section 2.3, I outline several prevalent standard views of the history of this period,
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which I maintain are refuted by the picture emerging from my study. In Chapter 3, I give a big-picture characterisation of 17th-century philosophy of mathematics for which I am not aware of any clear precedents in the literature. This includes the following points: the cohesion of Descartes’s and Leibniz’s general philosophies and their philosophies of mathematics, and the way in which both are modelled closely on the Greek tradition; the contrast of this “Continental rationalist” view with the “British empiricist” tradition, and the characterisation of this split as stemming from divergent extrapolations of the geometrical method; the attribution of the corresponding clash in the philosophy of physics as stemming from the same source; a number of parallels with the history of Greek geometry, which are illuminating in both directions. To be sure a number of these points are to some extent explicit in the primary sources, as I show, so aspects of them have not gone unnoticed. For instance, Bos (2001) certainly noted the influence of the Greek tradition on Descartes, corresponding roughly to my Section 3.3.2. Nevertheless, I am aware of nothing like a systematic and synthesising account similar to my own. Throughout I strive to not only report 17th-century choices and preferences regarding the representation of transcendental curves, but also to explain why these choices were always rational.5 This is a perspective rarely found in other studies, and never with anything near the systematic synthesis I give. Big-picture overviews to this effect are notably: the new interpretative framework of Section 3.3.5.2 and its use notably in Section 7.3.2; Section 4.4.7; the introductions to Chapters 5 through 8; and the concluding Chapter 10. More specific points pertaining to the same purpose are raised throughout, often leading to new perspectives even on matters discussed by previous scholars (as in Sections 4.3, 5.3, 5.4.2, 5.5, 6.2, 6.3, 7.2, 7.3.2). Also new, to my knowledge, is my explanation (in Chapters 9 and 10) of why the transition from a geometrical to an analytical paradigm occurred (and became rational) only with the generation of Euler, even though the technical prerequisites were there long before.
1.4. Conventions adopted in this work 1.4.1. Policy on the presentation of mathematical arguments My story will hinge on a clear and coherent understanding of the key mathematical ideas involved. I have therefore opted to make judicious use of slightly modernised notation and phraseology when I found this beneficial for expository purposes, pointing out anachronisms only when they are relevant to the matter at hand. This should not be mistaken for a lack of commitment to the highest standards of historiography. I remain absolutely committed to understanding the thought of historical mathematicians in their own right rather than through an anachronistic lens. In my opinion, my slight and occasional deviations
Preliminary matters
from their mode of expression never compromises this commitment but is rather a natural concomitant of it.6 Indeed, the early history of the calculus affords many opportunities to observe that idiosyncrasies of surface form are often quite incidental, the most obvious case being the fact that the differences in mode of expression used by British and Continental mathematicians did not prevent them from seeing themselves as doing the exact same thing and even accusing each other of plagiarism.7 17th-century calculus differs from its modern equivalent in a number of often superficial respects: it is focussed on curves rather than functions; on differentials and concrete geometrical characteristics such as lengths of tangents rather than derivatives; on geometrical measurements of lengths and areas of circles and conic sections rather than the now canonised arsenal of transcendental functions such as arcsin(x), log(x), etc. Novices may consult Bos (1974a), sections 1–2, for a good overview of these kinds of conceptual differences of the Leibnizian and the modern calculi. I occasionally ignore these kinds of differences when expedient and harmless. I also take the liberty of altering notations, changing orientations of coordinate systems, setting generic constants to unity, and so on, when this can be done in a way that does no harm to the faithful understanding of the historical sources. By way of illustration we may consider Leibniz’s equation for the cycloid discussed in Section 4.2.2. I give this equation as Z 2 p dt 2 √ x = 2y − y + 2t − t2 y whereas what Leibniz actually wrote is
A number of superficial differences are immediately apparent, none of which are of any consequence for our purposes. In particular, the modern way of indicating bounds of integration was never used in the 17th century, but it is clear that Leibniz meant the exact equivalent of what I have written. I believe the kinds of liberties of paraphrase that I have outlined here are sound. Unconditional refusal to allow such liberties, which some advocate, would only double the bulk of the book and provide nothing but a massive distraction from the ideas that actually matter for the purposes of my argument. To those who demand complete faithfulness to the original sources in these kinds of regards there is a simple solution: read the original. My goal is to contribute to knowledge and understanding by writing a clear and synthesising account, not to slavishly reproduce what is already written for all to read in the sources themselves.
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In the interest of clarity, I have strived to use currently standard notations whenever possible, without stopping to define it in each instance. Thus, unless otherwise noted, x, y are rectilinear coordinates, t is time, s is arc length, a, b, c, . . . are constants, log(x) is the natural logarithm and e its base. Of course, 17th-century calculus also makes use of infinitesimal reasoning in a manner that would be considered very informal by modern standards. I follow this 17th-century manner of reasoning. For a textbook introduction to this way of doing calculus, readers may consult Blåsjö (2016a).
1.4.2. Translation and referencing conventions In my translations I have strived to remain as literal as possible, with a few systematic exceptions. In translating from the Latin, I have often used “curve,” “integral,” and “variable,” where the more literal translations would be “line,” “sum,” and “indeterminate letter” respectively. This is the most faithful translation into modern terminology, I would argue, even though it sacrifices strict etymological continuity. I have also occasionally inserted without warning occasional words which are strictly implied grammatically; especially the noun “line” or “curve” is often omitted in the Latin, with the writer saying, e.g., simply “transcendental” where “transcendental curve” is meant. I have also taken the liberty to alter capitalisations, including capitalising the first letter when I start a quotation midsentence. I have also often standardised references to persons, thus writing, e.g., simply “Leibniz” instead of “Mr. Leibnitz” and so on. I have often opted against giving page references for letters and short articles, since these are in any case only a few pages long and typically printed in multiple editions with different paginations.
1.5. Some key terms It will be useful to clarify upfront some terms and concepts that will feature prominently in our story. First we may consider the principal labels by which curves are classified. An algebraic curve is a curve that can be expressed by a polynomial equation in rectilinear coordinates x and y. The algebraic and coordinate-system methods of Descartes’s Géométrie are in a sense coextensive with this class of curves: they treat such curves exhaustively, and other curves not at all. According to Descartes’s vision of geometry this was most appropriate, for he argued that the set of all algebraic curves is precisely the set of all curves knowable with geometric rigour. To signify this he used the term geometric curve, which means precisely the same thing as algebraic. Curves that were not “geometric” Descartes called mechanical, the implication being that these curves were not susceptible to a truly mathematical and exact treatment. The term can be confusing. As we shall see in Section 3.3.2, one must not be misled by this terminology into thinking that Descartes rejected any
Preliminary matters
association of geometry with mechanisms; on the contrary, his proposed foundations for “geometrical” curves are based on rulers and pegs and motions in a manner many a modern mind would be inclined to call very “mechanical” indeed. When Leibniz set out to extend mathematics beyond these Cartesian bounds, he naturally did not care for Descartes’s terminology, since he rejected the foundational assumptions Descartes had built into the very words. He therefore started referring to Descartes’s geometric curves as algebraic, and Descartes’s mechanical curves as transcendental, since they “transcend all algebraic equations.”8 This excellent terminology is exactly the one still in use today. It applies to numbers as well as curves (or functions), as Leibniz himself often pointed out. Leibniz advocated for this terminology innumerable times in print and correspondence, and with good reason, since it is both objectively better than Descartes’s and furthermore removes a terminological bias against Leibniz’s new transcendental mathematics. In principle, all leading mathematicians of Leibniz’s generation agreed with his point that the foundational assumptions embedded in Descartes’s terminology were obsolete. Nevertheless, through historical inertia, Descartes’s terminology remained in widespread use for many years still, even by mathematicians who unequivocally rejected the foundational connotations they came with at their inception. Indeed, Cartesian geometry was so entrenched in mathematical consciousness that the geometry of algebraic curves was often called “common” or “ordinary” geometry. As a flip side, the Leibnizian calculus was sometimes referred to as the “transcendental calculus,” since one of its defining characteristics was that it went beyond the “common geometry” of algebraic curves. It is often said that the transcendence of π and e was not formally proved until the 19th century, but we must not be misled into thinking that 17th-century mathematicians were not conscious of the finer points of the algebraic–transcendental distinction. On the contrary, they had a refined understanding of this issue. The leading mathematicians of Leibniz’s generation were fully convinced of the transcendence of the numbers and curves they studied as such, and they supported these beliefs sometimes with proofs and, if not that, then at least very compelling informal arguments.9 Analytic and synthetic. Throughout this work I shall use the term analytic to mean any method based on working with formulas or symbols. This by and large tracks late 17th-century usage quite well, though the history of the term is complex. In Greek times, “analysis” meant “working backwards,” i.e., assuming what one sought as known and trying to deduce from it various consequences until one struck upon something already established, at which point one could reverse one’s steps in order to obtain a demonstration of the matter. The opposite of this is synthesis, which means working “forwards” from first principles or established theorems to deduce new results. Synthesis was
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considered the only right and proper style in which to write a formal mathematical treatise, but analysis was recognised as an invaluable tool of discovery and explorative research. Where finding a direct synthetic proof required too much ingenuity, analysis would often lead to a proof using more systematic and predictable steps. When algebraic methods where introduced in the early modern period, the meaning of these terms evolved. The algebra of the moderns was in many ways identified with the analysis of the ancients. Like analysis, algebra can be seen as a methodical technique of discovery. Furthermore, in algebra, when one denotes a sought quantity by x, one in effect assumes it to be known and treats it as such, just as the ancient conception of analysis prescribes. Hence the association of “analysis” with symbolic methods, which gradually grew into the new primary sense of the term. In parallel with this, “synthetic” took on a new meaning too: since analysis had now become synonymous with using modern symbolic methods, “synthetic” came to mean geometry done in the classical style of the Greeks. This sense remains current to this day when we speak of “synthetic geometry.” Later still, as the calculus eclipsed all other analytic methods on the center stage of mathematical research, “analysis” came to be associated specifically with the calculus, as in current parlance such as “real analysis.” But this phraseology is still in the future as far as out era of focus is concerned. Yet another facet to this complicated terminology is its foundational connotations. Virtually everyone agreed that synthesis and analysis were useful techniques that each had their place in mathematics. But more ambiguous was the question of which method is the epistemologically primary one, or the one that reflected the ultimate nature of mathematical knowledge. The status of axioms in particular depends on this question. The picture of synthesis as meaning building up a complicated theory from a few basic starting points fits well with the notion of these starting points or axioms being supposedly obvious and immediate truths, for if they are not the synthetic geometer’s starting point can be accused of being arbitrary. The method of analysis, on the other hand, inasmuch as it means taking complicated things apart or breaking them down, is more amenable to agnosticism as to whether these more basic pieces need be obvious or not, for the analytic geometer can claim to have uncovered underlying principles and assumptions without having to pass judgement on them. These attitudes may be called synthetic foundationalism and analytic foundationalism, respectively. All of these metamorphoses of the terms afford ample opportunity for confusion. Thus Descartes and Leibniz were wholehearted lovers of analysis, in the sense of a method of discovery, especially in symbolic form. Yet they insisted with equal fervour on synthetic foundationalism. Newton on the other hand had these priorities precisely reversed: as far as surface form goes, his Principia is doggedly synthetic in its insistence on classical ge-
Preliminary matters
ometrical, rather than symbolic, presentation; yet this very work is at the same time the manifesto par excellence of analytic foundationalism. If one does not heed the terminological confusion, it may thus appear that each man is both the committed champion and sworn enemy of the same concept. Such is the plasticity of the terms analysis and synthesis. Construction. In our period of interest, a solution to a mathematical problem generally meant not a “formula” as on a modern calculus exam, but a construction. The precise meaning of this is elusive, however. Euclid spent much time in the Elements constructing things with ruler and compass, and these are certainly the archetype instances of constructions. Later mathematicians tried to generalise the notion and ended up with a variety of possible curves and instruments that could be construed as analogous to the ruler and compass in one sense or another. Already in Greek times there were a variety of competing proposals as to the best way of proceeding in such matters, including some that stretched the concept too thin to be accepted as genuine constructions at all.10 Such ambiguities only multiplied further in the era of our focus. Mathematicians were certain that they sought constructions but they were not certain what exactly this meant. It was an explorative, formative period; the 1690s was brainstorming decade. Various possibilities were tried out in the hope that a clear winner would emerge, but in the end this hope was not realised. Which is perhaps just as well for our purposes since it enables us to view this foundational turmoil in its true state of flux, without the distortion of a “right answer” perceived in retrospect. Quadrature and rectification. In the 17th century, finding the area of a figure is often referred to as quadrature or in more English terms squaring. This is an inheritance from the Greek paradigm of geometry, which did not deal with areas numerically but rather operationalised the concept, so that instead of speaking of finding the area of a given figure they spoke of finding a square equal to it (the square being the simplest or prototypical figure as far as area is concerned). In the same vein, lengths were to be exhibited as line segments, whence the term rectification (i.e., extension of a curve into a straight line) for the finding of an arc length. In the 17th century, these terms had to a large extent lost their literal meaning and had come to mean simply finding area and arc length respectively. Yet this gradual transition was far from a definitive break with the Greeks, for their operationally oriented mode of geometry held sway to the very end of the 17th century. Function. The importance of the notion of function has tended to be overestimated by modern authors, as far as our era of study is concerned. The centrality of this notion in later developments has led some to feel that its absence in the early calculus must have been a serious conceptual limitation. I do not agree with such an interpretation. In my view, the reason these mathematicians did not develop the notion of a function is not some invisible conceptual ceiling that they could not break through; rather they did not develop it for the
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simple reason that it would not have served any purpose to them. In general, as a matter of historiographical principle I prefer to explain the choices made by mathematicians of the past in positive terms as the rational and natural way of approaching the problems of their concern, rather than in negative terms as due to conceptual limitations in their ways of thinking. I have found that I have very rarely needed to resort to an explanation of the latter kind. When I use the term function, I mean it in a flexible, pre-formal sense. A more precise conception is not of interest to us. The term “function” was introduced casually and parenthetically in the late 17th century,11 but no one at the time made the slightest fuss about it as any kind of conceptual innovation. And that with good reason. The leaders of the early calculus understood their subject very clearly, and they constantly and seamlessly switched between geometric, analytic, and abstract-relational ways of viewing it with ease and fluency. The modern notion of function would have contributed virtually nothing of relevance to any of their concerns. By the same token, a focus on “the evolution of the function concept” would be an anachronistic perspective of little value to our story.