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“THE MARRIAGE OF PHYSICS WITH MATHEMATICS”

Francis Bacon on Measurement, Mathematics, and the Construction of a Mathematical Physics

DANA JALOBEANU

CONSIDERATIONS ON THE NATURE OF SCIENCE played a very important role in Francis Bacon’s project for a Great Instauration. On a general level, his approach was foundational; knowledge was required to grow like a pyramid, on a solid basis of natural history, sustaining physics and metaphysics: “For knowledges are as pyramides, whereof history is the basis: so of Natural Philosophy the basis is Natural History; the stage next the basis is Physic; the stage next to the vertical point is Metaphysic.”1

Furthermore, Bacon preserved the pyramidal model in his detailed investigations into the nature of particular sciences. His claim is that in order for something to become scientia, it has to be constructed on a properly organized natural and experimental history.2 Given the importance and centrality of this concept, it is not surprising that Bacon wrote extensively about the nature, characteristics, and ways of composing such a well-organized and properly recorded natural and experimental history.3 In one such methodological text on the subject, one can find the following precept:4 “everything to do with natural phenomena, be they bodies or virtues, should (as far as possible) be set down, counted, weighed, measured and defined. For we are after works, not speculations, and, indeed, a good marriage of Physics with Mathematics begets practice [Physica autem et Mathematica bene commistae, generant Practicam]” (OFB, 6:465–66).

This is surely a puzzling set of requirements. First, there is a quantitative requirement; a proper natural and experimental history would contain quantitative descriptions of bodies and virtues. The resulting natural history will not be simply a selection of facts, but a well-ordered collection of measured quantities.5 As Graham Rees and Cesare Pastorino have already shown, measurement is a very important feature of Bacon’s natural history; it is what distinguishes it from its humanist predecessors (Rees 1985; 1986; Pastorino 2011a). In theory, at least, Bacon envisaged natural and experimental histories composed of numerical results, or tables, obtained through the careful “weighing” of experimental and instrumental results.6 In practice, however, Bacon’s natural histories are rarely openly quantitative; and with few notable exceptions they do not contain numerical tables. This makes even more puzzling Bacon’s reference to the “good marriage,” the good mixture of mathematics and physics necessary in order to generate practice. The producing of “works” belongs, for Bacon, to the realm of operative science; and it is only possible once the “science” itself has taken off the ground. A well-measured and ordered natural history becomes, in this case, a prerequisite for, and not the result of the “marriage” in question. Can this striking claim refer, therefore, to a peculiar, Baconian form of mathematical physics? What does Bacon mean here by “physics,” “mathematics,” and their “marriage”?

In a slightly later methodological text, dealing this time with the place of mathematics in a general tree of science, one can find a similar phrase: “Physics and Mathematics produce Practice or Mechanics” (SEH, 4:369, 1:576 [Physicam et Mathematicam generare Practicam sive Mechanicam]). This time, however, the phrase is attributed to Aristotle and given as a quote. In fact, Bacon refers here to a key passage from a well-known and highly debated treatise, the pseudo-Aristotelian Mechanica.7 The passage itself was subject to many interpretations (and different translations) in the second part of the sixteenth century, mainly because it addresses directly the possibility of using mathematics to treat problems of physics, such as “the moving of heavy bodies by art, for human benefit” (Berryman 2009, 106).8 The passage refers, more precisely, to the intermediary status of mechanical problems that “share in both physical and mathematical speculations,”9 and invites reflection upon the status of mechanics as a mixed-mathematical science. It portrays mechanics as resulting from a “consent” or “collaboration” of physics and mathematics. The precise nature of such a consent and collaboration was the subject of heated debates involving mathematicians, natural philosophers, and practitioners of mechanical arts.

Thus, when Bacon announced a discussion on the “marriage of Physics and Mathematics” he was most probably taking a stand in an ongoing debate, relative to what we call today the “mathematization of nature.” The purpose of this chapter is to reconstruct Bacon’s position: his original and somewhat idiosyncratic views on “physics,” ”mathematics,” and “mechanics,” and his particular way of describing the consent and collaboration between the two disciplines. My claims are the following: (1) Bacon did take a position in the wider debate over the status of mixed-mathematical sciences and the possibility of a mathematical physics; (2) his position was characterized by an instrumental and practical understanding of mathematics; and (3) he saw the “marriage of physics and mathematics” as a prerequisite to the emergence of a quantitative science of nature. In other words, I claim that there is a Baconian version of “mathematical-physics,” which becomes evident if we reconstruct more carefully what Bacon and his contemporaries meant by “physics” and “mathematics” and by the various forms of collaboration and consent between the two disciplines.

BACON ON MATHEMATICS, MEASUREMENT, AND EXPERIMENTAL PRACTICE: THE FOUR IDOLS OF BACONIAN SCHOLARSHIP

Any attempt to write about Francis Bacon’s views on the “marriage” of physics and mathematics faces serious historical and historiographical problems. For a long time, the history of early modern thought has been dominated by the famous Kuhnian divide between proper, “mathematical sciences” and the “Baconian sciences.” Presumably originating with Bacon, Baconian science was supposed to have a qualitative, preparadigmatic, and essentially nonmathematical character. Although this divide has been repeatedly refuted in the last two decades, its lingering echoes pose numerous historiographical and terminological problems. Both “physics”10 and “mathematics”11 had a fluid and changeable meaning until the mid-seventeenth century; the landscape of late sixteenth-century theoretical and practical mathematics is full of subtle complexities of which the historian of early modern thought should be aware.12 It is only recently that the subject has reached outside the borders of the history of mathematics itself, into the wider community of people interested in early modern science.13 On the other hand, Bacon’s thoughts on mathematics and his project to use physics and mathematics conjointly in order to explore the labyrinth of nature are, perhaps, the least explored of his many unfinished projects. This is only partly surprising. After all, the subject has long been obliterated by the persistence of what I have called elsewhere the four idols of Baconian scholarship (Jalobeanu 2013a; 2013b; 2015), namely recurrent general judgments of obscure origin and remarkable persistence, impermeable to refutation.14

One of the oldest and most entrenched idols of Baconian scholarship can be exemplified by the repeated claims that Bacon disliked and distrusted mathematics.15 Such claims have many features of an idol of the tribe:16 they are useful and widespread simplifications, based on common received views and essentialist historiographical presuppositions. They implicitly attribute to “mathematics” some atemporal essentialist nature, disregarding the historical character and the evolution of mathematical knowledge and mathematical disciplines in the sixteenth century. They have also displayed a remarkable persistence and resistance to refutation.17 They are partly responsible for the fact that so little work has been done, to date, to unearth the sources of Francis Bacon’s attempts to reform astronomy, astrology, and the mechanical arts. A second, related category of idols originates in a too literal interpretation of Bacon’s metaphors, especially his celebrated metaphor of the “alphabet of nature.” Since these idols relate to language and interpretation, they display common features with Bacon’s idols of the market. This category of idols illustrates the power of words over interpretations. Indeed, Bacon repeatedly claims that nature is a labyrinth and the explorer of nature a hunter; that everything in nature results from endlessly active and invisible combinations of appetites and motions (i.e., letters of the alphabet) that one needs to “become like a child” and learn the abecedarium of nature in order to be able to perform the work of interpretation. He also claims that the results of investigation need to be written down, that “experience itself has to be taught how to read and write,” that is, to become literate. It is extremely tempting to give such claims a quasiliteral interpretation, transforming Bacon’s project of an experimental investigation of nature into a form of literary pursuit.

A third category of idols can be recognized in the repeated claims that Bacon’s science is purely speculative; that Bacon never did experiments but only mimicked the language of experimental practice in order to argue for what was fundamentally a purely speculative system (Jalobeanu 2013b, 7–8). Of the same kind are the claims according to which Bacon’s natural history was a collection of recipes and phenomena borrowed from others and never tested, or tried, in practice. There are many versions of this claim, which are recurrent and remarkably persistent to refutation. Their persistence explains why relatively little work has been done to explore Bacon’s natural histories and to trace their sources and evaluate their originality.18

Last but not least, a fourth category of idols can be recognized in the repeated claims that Bacon rejected the physico-mathematics of Galileo and the mechanics of his contemporaries; that he was isolated among his more scientifically minded contemporaries, “writing philosophy like a lord Chancellor.”19

What the four categories of idols have in common is a series of assumptions about the nature of mathematics, the nature of scientific enterprise, and Francis Bacon’s “isolation” in the scientific and philosophical community of his day. Each of these assumptions has been individually refuted more than once. This chapter does not propose another refutation. I would rather use the framework of the four idols of Baconian scholarship as a background for a more precise reconstruction of Francis Bacon’s peculiar form of mathematical physics.

FRANCIS BACON ON MATHEMATICS AND MEASUREMENT: AGAINST THE FIRST IDOL

Francis Bacon gave mathematics a very important role in his restoration of sciences. Mathematics are said to be “the great appendix of natural philosophy, both speculative and operative” (SEH, 4:369)20 they are “of so much importance both in Physics and Metaphysics and Mechanics and Magic” (370). As appendices, arithmetic and geometry should function as “handmaids” of physics and metaphysics. They have an intermediate but essential role in the instauration of sciences.

Meanwhile, since arithmetic and geometry are also “sciences,” and parts of philosophy, Bacon argues that they should be constructed according to the proper method, on a natural historical foundation. Hence it is not surprising that in the list of natural histories Bacon appended to the end of his Instauratio Magna, one can find two natural histories of “mathematics”: a history of “the natures and powers of numbers” and a history of “the powers and natures of figures” (OFB, 11:485). What would the two histories consist of? One cannot infer much from their titles. However, taken together with the other things Bacon has to say about mathematics, a certain amount of reconstruction is possible. If, moreover, we read such claims in the wider context of the late sixteenth-century debates over the nature of mathematics, Bacon’s position becomes clearer. Here is an important element in this reconstruction: in the posthumous New Atlantis, among the laboratories and houses of sciences of Solomon’s House, Bacon lists a “mathematical house, where are represented all instruments, as well of geometry as astronomy, exquisitely made” (SEH, 3:164). Mathematics, therefore, deals with instruments; a collection of instruments is required both to construct mathematical histories and to provide tools for other sciences, particularly astronomy. Note that for Bacon instruments are not only the tools of mixed mathematical sciences, such as the science of astronomy, but also of pure mathematics, such as geometry. Such an instrumental and practical view on mathematics squares with its role as a “handmaid” for physics, and with Bacon’s further insistence on the importance of mathematics for measurement and calculus. In De augmentis scientiarum (DA), Bacon mentions a couple of important and yet unsolved problems of his days. In arithmetic, they are the discovery of “formulas for the abridgment of computation sufficiently various and convenient, especially with regard to progressions.” To these, Bacon claims, “there is no slight use in Physics” (SEH, 4:370–71). In geometry, he also lists a problem of measurement, computation, and calculus: “the doctrine of solids” (i.e., the problem of calculating areas and volume of solids).

Bacon’s views on mathematics are by no means singular at the beginning of the seventeenth century. In fact, in late sixteenth-century England it was quite common to define mathematics in instrumental terms—as a science of measurement and calculation. Equally common was to argue for the universal propaedeutic value of mathematics and for its practical applications in every other science. Such views are widespread among the “mathematical practitioners”21 of the late sixteenth century; but they can also be found in the works of more traditional mathematicians, especially in the case of vernacular mathematics.22 The first English translation of Euclid produced by Henry Billingsley (with a preface and textual additions by John Dee) is well stocked with instructions for constructing geometrical figures, with explanations of geometrical instruments and their uses, and with strategies for calculating areas and volumes (sometimes in practice, and with approximation). This instrumental presentation of geometry receives full-length justification in John Dee’s accompanying preface. Although Dee emphasizes the divine nature of arithmetic and geometry, his preface argues at length for their use in commerce, navigation, architecture, the art of assaying, mining and so on (Billingsley and Dee 1570). He also distinguishes between several uses of “vulgar geometry” for measuring, approximation, and, more generally, for describing the world of physics. A slightly later English Elements of Geometry, translating Ramus’s geometrical books from Schola Mathematicae, begins with the following definitions:

1. Geometrie is the Arte of measuring well
2. The thing propounded to the wel measured, is Magnitude
3. Magnitude is a continual quantitie
4. A terme is the end of a magnitude

Therefore a magnitude is both infinitely made, continued, and divided by these things wherewith it is termed (Ramus and Hood 1590).23

Ramus’s definitions sketch an instrumental, constructivist, and inductive approach to geometry; the book also offers constructional strategies for drawing basic geometrical figures, followed by instructions on how to compose, decompose, and measure various figures.24 This attitude to geometry is reinforced in Thomas Hood’s “advice to the auditors” of his mathematical lectures,25 in which geometry is presented as an essential introduction in “mathematical sciences” (mixed mathematics) but also in “anie human knowledge” (Ramus and Hood 1590).

Similar views on mathematics can be found in the more complete and less idiosyncratic French edition of Euclid, translated by Pierre Forcadel and published in 1564.26 Forcadel’s preface argues for the importance of mathematics using the same strategy: on the one hand, the object of mathematics is seen as superior to the objects of any other science; on the other hand, Forcadel argues that mathematics is, in fact, astronomy: “Mathematics treats mainly of celestial business; the motions of the sky, the course of the Sun, the Moon and other planets.”27 Again, mathematics is said to have universal practical value for any theoretical and practical knowledge (including government).

This universal practical value makes mathematics a useful tool for the treatment of physical problems as well. This, however, does not mean that mechanics, astronomy, and optics become, in any way, mathematical physics. They are mixed-mathematical sciences, classified under the general heading of geometry.28 There are, however, other ways of dealing with the “consent” between mathematics and physics, one step further along the road toward a “good marriage of physics with mathematics.” One quite striking example of consent and collaboration between mathematics and natural philosophy can be found in a composite and peculiar treatise published in 1571 under the name A Geometrical Practical Treatise Named Pantometria. The treatise was published by Thomas Digges (1571) and dedicated to Francis Bacon’s father, Nicholas. In the dedicatory letter, the treatise is said to contain: “mathematical demonstrations, and some such other rare experiments and practical conclusions.” The said “experiments” cover the construction of instruments and technologies for the production of geometrical and topographical instruments, devices for calculating lengths, surfaces, and volumes, etc. For example, the third book, Stereometria, contains “rules to measure the Superficies and Crasitude of Solide bodies, whereof, although an infinite rote of different kindes might be imagines, yet shall I only entreate such as are both usually requisite to be moten, and also many sufficiently induce the ingenious to the mensuration of all other Solides, what forme or figure so ever they beare” (Digges and Digges 1571, 81).

The book is a weird mixture of classical Euclidean geometry with practical problems of measurement; it is also a composite work of two English mathematicians, father and son.29 The “practical” part relating to measurement belongs to Leonard Digges and was left unpublished for the young and talented Thomas Digges to complete, among other things, with a “doctrine of … the five Platonicall solids.”30 Thomas Digges’s preface to the reader emphasizes not only the practical utility of geometry but also its experimental character: geometry is a science of measurement. Due to geometry: “man notwithstanding be here imprisoned in a mortal carkasse, and thereby detained in this most inferior and vilest portion of the universall world, fardest distant from that passing pleasant and beautifull frame of celestial orgbes, yet his divine minde ayded with this science of Gemoetrical mensurations, founde out the Quantities, Distances, courses, and strange intricate miraculous motions of these responded heavenly Globes of the Sunne, Moone, Planets and Starres fixed, leaving the precept hereof to his posteritie.”31

This science of measurement is seen as having universal applications: it can be used to create instruments, to measure the land, to navigate, to perfect ballistics and fortification, to discuss architecture, or the storage of goods (Digges and Digges 1571).32 What is more important, however, is that this science is seen as leading to the development of more sophisticated measuring instruments and techniques, which, in turn, can perfect the art itself. Last but not least, the science of measurement extends beyond the traditional list of mixed-mathematical sciences, into natural philosophy itself. Thomas Digges’s preface claims that one cannot understand Aristotle without being a good mathematician, because “in sundrie of his works also of naturall Philosophie, as the Physickes, Meteores, de Caelo & Mundo, &c. yee shall finde sundry Demonstratins, that without Geometrie may not possibly be understanded.” In other words, this kind of mathematics would advance the sciences traditionally considered to belong to “mixed mathematics,” but also other sciences, whose mathematical character has not been apparent so far.

Francis Bacon’s arguments for the importance of mixed mathematics are similar. He begins with a general definition: “Mixed Mathematic has for its subject some axioms [Axiomata] and parts of natural philosophy [portiones physica], and considers quantity in so far as it assists to explain, demonstrate, and actuate these.” Following the definition, Bacon draws a list of mixed-mathematical sciences, presenting them as “parts of nature” that cannot be comprehended without the aid of mathematics: “For many parts of nature can neither be invented [comprehended] with sufficient subtlety [nec satis subtiliter comprehendi], not demonstrated with sufficient perspicuity, nor accommodated to use with sufficient dexterity, without the aid and the intervention of Mathematic: of which sort are Perspective, Music, Astronomy, Cosmography, Architecture, Machinery and some others” (SEH, 4:371).

If Bacon’s list of mixed-mathematical sciences is quite traditional, his view on the role of mathematics in physics is not quite so, although it is strikingly similar to Digges’s views on the matter. Bacon claims that there should be a dynamic interplay between physics and mathematics; that without this no progress can be made. Moreover, he envisages the situation in which new parts of physics will require fresh assistance from mathematics. “In Mixed Mathematics I do not find any entire parts now deficient, but I predict that hereafter there will be more kinds of them, if men be not idle. For as Physics advances farther and farther every day and develops new axioms, it will require fresh assistance from Mathematic in many things, and so the parts of mathematics will be more numerous [eo Mathematicae opera nova in multis indigebit, et plures demum fient Mathematicae Mixtae]” (SEH, 4:371).

In what way does physics require the assistance of mathematics? Bacon’s general answer to this question is “measurement.” In order to construct a natural history on which one can build a sound physics, one has to measure, count, and weigh instances. This involves instruments and experimental techniques; it also involves mathematical instruments and techniques of calculus. All these are used in “measuring, weighing and counting”; observations and experiments further recorded in natural history and useful for the construction of physics. But this “measuring of nature” is, for Bacon, a more general concern. For example, he states that “in every inquiry into nature we must note the Quantity or, as it were, the dose of body needed to produce a given effect, and add a dash of guidance concerning Too Much or Too Little” (OFB, 11:383).

It is clear from such statements that Bacon also has methodological and theoretical concerns associated with the very process of measurement. This is supplemented, in the second book of the Novum Organum, by a selection of instances destined to provide “general and catholic observations” (OFB, 11:419) and more general guidance for the investigation of nature. Such are, for example, what he calls mathematical instances or instances of measure (OFB, 11:367). They are explicitly introduced in order to solve the problem of “inaccurate determination and measurement of the powers and actions of bodies.” They underline a general theory of measurement: “Now the powers and actions of bodies are circumscribed and measured either by point in space, moment of time, concentration of quantity, or ascendency of virtue, and unless these four have been well and carefully weighed up, the sciences will perhaps be pretty as speculation, but fall flat in practice” (OFB, 11:367).

There are four such instances, and they are extremely diverse: some are operations or operational procedures governing the investigation of a given nature; some are experiments destined to circumscribe and measure the range of a given phenomenon; others are attempts to set down instruments or techniques for measuring distances, time, or the range of a certain virtue or quality. Among the mathematical instances, some are even called “instances of quantity” or “doses of nature” (OFB, 11:381–83). All these instances provide techniques and examples of measuring; together, they seem to delineate a general theory of measurement of the “powers and actions of bodies.” What they measure, however, is a bit more complicated, and relates to the peculiarities of Bacon’s physics. Disentangling the meaning and structure of Bacon’s physics is a prerequisite to understanding what he meant by the “marriage of physics and mathematics.” This is the subject of the next section.

PHYSICS AND THE “ALPHABET OF NATURE”: BRINGING PHYSICS CLOSER TO MATHEMATICS AND THE SECOND IDOL OF BACONIAN SCHOLARSHIP

For Bacon, physics deals with three large domains of reality: “the principles of things,” “the structure of the universe,” and the realm of phenomena, that is, “all the varieties and lesser sums of things” (SEH, 4:347). His investigation focuses especially on this third domain, called “diffused physics” [Physicam Sparsam, sive de Varietate Rerum] (SEH, 1:551) and said to be “a gloss or paraphrase attending upon the text of natural history” (SEH, 4:347; see also OFB, 4:83). This “gloss” has two parts: the concrete and the abstract physics.33 The first is barely distinguishable from natural history. Indeed, Bacon’s concrete physics has the same objects and the same structure as his natural history.34 It deals with “the heavens or meteors, or the globe of earth and sea, or the greater colleges, which they call the elements, or the lesser colleges or species, as also with pretergeneration and mechanics” (SEH, 4:347).

Concrete physics is a “gloss” on natural history because it simply adds causal explanations to the “facts” of natural history. In some cases, such as mechanics, concrete physics is already a mixed-mathematical science. What about the other parts of concrete physics? Interestingly, Bacon illustrates the requirements of concrete physics with his own project of reforming the “science” of the heavens. He claims that: “Among these parts of Physic that which inquires concerning the heavenly bodies is altogether imperfect and defective,” and has been ill handled in more than one way. On the one hand, astronomy—although built on phenomena—has transformed the science of heavens into a study of an abstract and simplified “system of machinery arbitrarily devised and arranged to produce [motions]” (SEH, 4:349).35 In addition, astronomers have done a “lax and careless job” (OFB, 6:167)36 when they made observations and supplemented the lack of data with ad hoc assumptions, dogmas, and theories (ibid.). By contrast, Bacon’s proposal is to reform both astronomy and astrology in order to build a proper physica coelestis. A concrete physics of heavens would have to study the “substance, motion and influence of the heavenly bodies as they really are,” and also their “physical reasons” (SEH, 4:348).37 It would be built on a proper natural history of the heavens, carefully compiled and properly measured.38 Such a natural history would not limit its inquiries to the “exterior” of the heavenly bodies, but would also inquire into their “interiors” [viscera]. Bacon recommends a general inquisition, covering “physical causes, as well of the substance of the heavens both stellar and interstellar, as of the relative velocity and slowness of the heavenly bodies; of the different velocity of motion in the same planet;… of their progressions, stationary positions, and retrogressions; of the elevation and fall of motions in apogee and perigee; of the obliquity of motions, either by spirals … or by the curves which they call Dragons; of the poles of rotation” (SEH, 4:348).

Bacon claims that such a general inquiry has never yet been performed, mainly because the received astronomy has replaced the real problems of a celestial physics with simplified problems of calculus and prediction. On the other hand, he seems to be aware of some of the difficulties involved in such an investigation; for example, how difficult it is to get precise astronomical observations.39 He recommends “estimates” [aestimativas] (OFB, 11:466) and “comparative measurements”40 when “precise proportions are not available” (467; see also 6:167, 169). He mentions the methods of distance measurement currently in use in astronomy and the need for “other aids to be devised for this matter, which human industry may contrive” (OFB, 6:169). Such difficulties of measurement become even greater if one takes into consideration the part of celestial physics that deals with the nature and composition of heavenly matter, namely the “substance of the heavenly bodies and every sort of quality, power and influx … what is found in the bowels of nature and is actually and really true” (OFB, 6:111).41 What is found in the “bowels” and “viscera” of nature are the appetites and motions of matter and what Bacon calls “configurations,” or “schematisms” of matter; and these are precisely the elements of his abstract physics.42 Bacon calls them the letters of the alphabet of nature,43 “by which all that variety of effects and changes which we see in the works of nature and art is made up and brought about” (SEH, 5:425).

How can such primordial, constitutive elements of the universe be subject to measurement and quantitative natural history? Bacon clearly states that these “letters” of the alphabet of nature take place in the “recesses of nature” (SEH, 4:356); that they are “imperceptible” and “intangible” (OFB, 11:351); and that all information about this level of reality “comes via reduction” [per Deductionem procedit] (ibid.). On the other hand, the purpose of abstract physics is also precise measurement. The explorer of nature should find ways to “call upon nature to render her account” (SEH, 5:427). The inquisition into simple motions, sums of motions, configurations, and the other elements of the abstract physics should be pursued in such a way that “they are not diffuse [vagae], lacking in rigour, and in manner intellectually satisfying, but useless in practice. This is why we must get closer to the mathematics, namely to measures and scales of motions [Quamobrem accedendum propius ad mathematica, sive mensuras & scalam motuum], without which, well counted and weighed and defined the doctrine of motions may falter and not be reliable translated into practice” (OFB, 13:210–11).44

How is the measurement possible if the appetites, motions, and configurations of nature are not accessible to the senses? This is where Bacon’s program seems to break down, a wide gap separating his ideal of “getting closer to mathematics” from his metaphysics of simple and compound motions. Furthermore, Bacon gives no composition rules for his language. Even if we know the letters, we do not have a grammar to put the words together.

This is precisely the point at which the influence of the second idol of Baconian scholarship was persuasive and long lasting. To date, there is a wide gap between scholars interested in Bacon’s speculative metaphysics,45 those focusing on Bacon’s method, and those mainly intrigued by the structure and composition of Baconian natural histories. The three groups tend to focus on different texts, exploiting the corresponding apparent divide in Bacon’s writings between speculative metaphysics of matter and seemingly experimental natural history. However, a less literal reading of Bacon’s metaphorical language about the alphabet and the language of nature might show us that the break results in a far smaller gap than it had appeared to so far.

In fact, Bacon does have an answer to the problem of bringing abstract physics closer to mathematics. The answer is in the adoption of a mixed strategy: on the one hand, he proposes an experimental approach and particular kinds of “reductive experiments” [experimenta deductoria] (OFB, 13:215) that are able (in principle) to establish connections between the invisible causes and the visible aspects of the phenomena. On the other hand, he elaborates a reductionist strategy based on a minimal list of simple motions, or appetites. The list is provisional and subject to further corrections.46 There can be other simple motions, or there can be less simple motions; or some simple motions might be different than initially stated. This clearly happens due to some form of empirical and experimental input. Bacon’s reductionist strategy is quite sketchy; but we have enough of it to see that for him the problem of inferring per Deductionem and the theory of measurement were closely intertwined; and that they have a strong experimental component. It remains to explain what these “reductive experiments” [experimenta deductoria] are and in what way can they serve the project of bringing physics “closer” to mathematics.

MEASUREMENT IN PRACTICE: EXPERIMENTS, INSTRUMENTS, AND LEVELS OF PRECISION

A characteristic feature of Baconian experimentation is the way in which Bacon borrows observations and experiments of more traditional natural histories and turns them into experimental series aiming at precise quantitative measurements.47 He can begin, for example, with Pliny’s observation that sailors once obtained freshwater on a ship from fleeces of wool hung around the sides of the ship at night. He “tries” this by simply putting a “woolen fleece” on the ground “for a long while” and observing that it gains weight “which could not happen unless something pneumatic had condensed into something with weight” (OFB, 13:141). However, such a trial is, for Bacon, just the beginning of an entire series of experiments one can find scattered through his writings, all involving the study of the same phenomena: the unusual capacity of the porous fibers of wool to condense air into water. One such experiment is re-creating the conditions on Pliny’s ship by hanging a pack of wool in a deep well, just above the level of the water. Here is how Bacon recorded the result: “I have found that in the course of one night the wool increased to five ounces and one dram; and the evident drops of water clung to the outside of the wool, so that one could as it were wash or moisten one’s hand. Now I tried this time and time again and, although the weight varied, it always increased mildly” (OFB, 13:141).

We have, therefore, a measurement and a quantitative result. Mark that Bacon seemed to have been aware of several practical problems of measurement, such as the slight variations of the results in repeated trials and the importance of giving an estimate as the mean value and the quantitative result of the experimental series. He also seemed to have been particularly interested by what happens if one varies the experimental conditions of one’s experiment. Bacon records repeated trials with the wool placed on the ground, hung in a well at various distances from the water, and placed on the top of a closed wooden vessel containing vinegar. In each case, wool is instrumental in producing (or perhaps facilitating) a process of condensation of air (or vapors) into water (or liquid). In other words, Bacon has transformed an ancient, natural historical observation into a technology. Furthermore, by measuring different quantitative results according to different external conditions, Bacon points out how such a technology can be further developed into an instrument. One can use the capacity of wool to condense vapors in order to create an instrument measuring the properties of the surrounding air. Air is more prone to condensation in closed spaces, in cold, near the water, under certain influences of the stars, and so on. The “woolen instrument” can measure the dispositions of air to condensate by translating them into units of weight. In Century IX of the posthumous Sylva sylvarum, such experiments are used in a large-scale program of measuring the qualities of the air in a given spatial region. In order to find out, for example, which part of a property has the healthier air, one has to place weather glasses and packs of wool in various locations throughout the property, record simultaneous results, and draw tables of these properties of air. It is important to note that this experimental research program is developed before one knows exactly what one is measuring—weather glasses and packs of wool are simply used conjointly to find out more about properties of air in a given region without knowing precisely what these properties are (whether what we measure is the temperature and humidity or whether we measure the way air captures and transmits the influences and radiations of the stars).

Such experiments are precisely what Bacon calls experimenta deductoria or instantias deductoria (OFB, 11:350): they figure, for example, under the heading Summonsing Instances in the second part of Novum Organum. Summonsing Instances are experiments capable of reducing the imperceptible to the perceptible [deducunt Non-Sensibile ad Sensibile]. A particular kind of such reductions occurs precisely in situations such as those described earlier: “It is evident that air, spirit and suchlike things which are fine and subtle in their entirety cannot be seen or felt so that reductions are absolutely necessary when inquiring into them.”48 Similarly, “subtler textures and configurations of things … are imperceptible and intangible. The consequence is that information about these also come via reduction [per Deductionem]” (OFB, 11:350–51). The way in which such invisible effects are made visible is through instruments that reduce the variations of measured properties to variations of visible properties: weight (in the case of the pack of wool) or length (in the case of the weather glass).49 Repeated experiments under different external conditions and simultaneous experiments amount to calibrating the instrument. One can argue that this is exactly what Bacon is doing when placing the pack of wool into the deep well, into a situation of maximum humidity, and recording the increase in weight in this situation. He is thus determining the limits, or the range of variation, of a given property or phenomenon. Furthermore, by requiring the performance of simultaneous measurements with packs of wools and weather glasses, Bacon seems to go one step further into the creation and calibration of his instruments.50

The practical process of calibrating instruments is doubled, in Bacon’s case, by theoretical concerns regarding the accuracy and precision of such instruments. We have seen already that Bacon distinguishes (in the case of astronomical observations) between precise measurements and rough estimates or comparative measurements: “where precise proportions are not available to us we must for sure fall back on rough estimates and comparisons, as, for instance (if we happen to distrust the astronomers’ calculations of distances) that the Moon stands within the Earth’s shadow; that Mercury is above the Moon, and the like” (OFB, 11:467). A third category is introduced for practical purposes: the “setting down the extremes … where average proportions are not available, let us set down the extremes; for instance that a weaker loadstone will lift so much weight relative to the weight of the stone itself, whereas one with the greatest virtue will lift sixty times its own weight” (OFB, 11:467). What we have here is an attempt to determine the range of the virtue under observation. In the case of the woolen instrument, this will determine the maximum and minimum of a scale. In the case of magnetic virtue, setting down the extremes is equivalent with determining the orb of virtue of a particular magnet and the proportion between magnetic virtue and size (weight). More generally, Bacon indicates as a major mode of operation [modus operandi] for every experiment, the determination of “how much or dose in nature: what of distance, which is not unfitly called the orb of virtue or activity; what of rapidity or slowness; what of short or long delay; what of the force or dullness of the thing; what of the stimulus of surrounding things” (SEH, 4:357). When moving from the “main effect” to the “stimulus of surrounding things” we are already in deep waters in terms of theory of measurement, because this involves distinguishing between the variation of what is currently called the major parameter and the additional influences of the external conditions. In some cases, these are small or can be minimized. But Bacon is concerned with all sorts of situations in which the supplementary influences coming from external conditions are present and cannot be neglected. Much remains to be done before Bacon’s experimental research programs will be seriously investigated. Substantial work in this area has been constantly jeopardized by the third idol of Baconian scholarship, that is, the strong entrenched belief in the qualitative nature of Bacon’s physics. As the examples discussed so far clearly indicate, there is little truth attached to such claims. Quite on the contrary: Bacon’s experiments show a permanent preoccupation for obtaining measurements of increasing precision and tables of quantitative results.

A THEORY OF MEASUREMENT: MEAGER TABLES, COMPLEX TABLES, AND SUCCESSFUL APPROXIMATIONS

In what way does all this process of experimentation and measuring bring physics closer to mathematics? Is Bacon’s “marriage of mathematics with physics” a mere attempt to obtain precise, numerical results? The purpose of the final section of this chapter is to show that Bacon’s theory of measurement is more complex than that; that it has techniques for transforming limits into estimates and estimates into precise measurements. In order to do this, I will turn to another example of “reductive experiment,” Bacon’s much discussed table of densities.51

The experiment is designed in the following manner: two identical silver boxes of a cubic shape are filled respectively with equal volumes of various substances. They are subsequently weighed with a balance to determine their relative densities. The etalon in the experiment is gold—so, the first measurement amounts to measuring, with a balance, the relative weight of a given substance with respect to the same volume of gold. By filling the second cube with every possible substance able to fill a cubic space, Bacon is able to draw a table of densities expressed numerically, in grains. In principle, as has been said, this table of densities is not only a quantitative (or, rather, numerical) experiment, but it might also be given as an example of a good natural history; that is, facts carefully weighed, measured, and recorded (Pastorino 2011b). However, this is not what Bacon claims. He claims that the table reveals “many unexpected things” and shows the limits of our knowledge (OFB, 11:353). He also claims the table is “meager” [indigestissima]. It is incomplete (lacking entries for numerous substances that cannot be reduced to the cubical volume). It is also full of unexplained gaps (results are not ranged in a progressive series, etc.). Bacon also seems to have recognized that such a table is highly unreliable. For example, the substances entered in the table on a particular position can exist in more than one state. Metals, for example, can be solid blocks, but can also be grinded in powder, can rust and produce a special kind of powder through oxidation, can be melted and mixed with other metals, and so on. In addition, all tangible bodies might be, in themselves, subject to condensation under certain conditions: will their place in the table change in this case? Bacon lists in Sylva sylvarum and Historia densi et rari (HDR) numerous examples in which condensation seems to take place (rusting metals, swollen leaden statues, condensations by fire, etc.). He comes up with another table that compares the weight of similar substances occupying the same volume in two cases: when substances are in their “natural state” and when they are reduced to powder, rust, or distilled solutions. By contrast with the first table, the second is much more complex and introduces into discussion at least one more parameter; namely, the way in which powders and solutions were produced, in the first place. Obtaining equal volumes of crude powder by grinding, powder obtained through the rusting of a metal, or powder obtained through more complex chemical procedures will change the numerical values in the final table. So the second table is a very complex object, merely sketched, and never fully developed. Furthermore, Bacon claims that both tables (the first, simpler one, and the second, more complex, one) are “pretty meager”:

The only precise table of bodies and their openings [dilatation] would be one which displayed the weight of the individual bodies whole first, then of their crude powders, next of their ashes, limes and rusts; next of their amalgamations, then of their vitrification (in those capable of vitrification) then of their distillations (once the weight of the water they are dissolved in was taken away) and of all other alterations of the same bodies; so that in this manner a judgment might be formed of the openings of bodies and very close-knot connections of the nature in its whole state. (OFB, 13:59)

What Bacon seems to be saying is that both tables are rough estimates; neither is fine-grained enough to allow an ordering of substances according to the relative density in such a way that the results are “precise enough” to answer the questions raised thus far. Putting them together, however, might result in a finer grained table/ordering of substances according to their densities. The resulting table will have more entries, filling some of the gaps found in the first table. So, for the purpose of representing with the help of a table the wide range of densities in nature, the two tables taken together can be considered as more “precise” (fine-grained) than the first one alone. This degree of precision can also be increased. Indeed, Bacon sketches a full experimental program that results in a multiplication of tables: the experiment will amount to measuring the relative weights of equal volumes of substances in all the above-mentioned states of powder and distilled liquors, obtained through consistently applying the same experimental procedures. The envisaged result will be a precise (fine-grained enough) table of bodies and their openings (their various states of aggregation) and a more fine-grained representation of the general scale of densities in the universe.

CONCLUSION

The purpose of this chapter was to reconstruct Bacon’s original and somewhat idiosyncratic views on the consent and collaboration between physics and mathematics. I have shown first that his instrumental and practical conception of mathematics was not singular at the beginning of the seventeenth century, and that his attempts to extend the collaboration of mathematics and physics beyond the received realm of mechanical practices, into more general questions of a celestial physics, for example, were shared by many of his contemporaries. I also have shown that by defining mathematics as a science of quantity and defending the preeminence of arithmetic as a science of quantity Bacon was also in good company. Perhaps we can even see his views on mathematics as a way of taking a stand in a wider contemporary debate over the nature of physics, mathematics, and mechanics. For this, however, a wider contextual reconstruction would have been required. I have shown in the third and fourth sections of this chapter that Bacon saw the “marriage of physics and mathematics” as a prerequisite to the emergence of a quantitative science of nature. He was not simply interested in measuring, computing, and registering properties of visible phenomena. His experimental program shows awareness of the role of estimates, approximations, and, more generally, of the need for a theory of measurement in physics. I have shown how, for Bacon, experiments were used to devise technologies, instruments, and experimental research programs destined to bring natural phenomena into a form that will make the marriage of mathematics and physics possible. Bacon devised a methodical way of getting “closer to mathematics” through experimental procedures destined to discover new instruments and “reductive experiments.” Such experiments are used, in turn, to devise a complex program of measurement of increasing precision: from “setting down de extremes” to comparative measurement and eventually to precise proportion. At each step along the way, new experiments are added to the series in order to compare and refine the previous results. The ideal, final result of these procedures would be a complex table with many columns, each column containing numerical results. To the untrained eye, the complete tables resulting from the example discussed here would be indistinguishable from astronomical tables, for example. In other words, the results of such experimental research programs look very much like mixed mathematics. Without being Galilean science, we have a mathematical inquiry into nature in which physics has not been transformed into mathematics, but it has been put under the form of mixed mathematics. This, I claim, is what Bacon means by the marriage of mathematics and physics. This is also, I think, the very purpose of Baconian science, the driving force behind his painstaking efforts to measure, weigh, and experiment with nature.

ABBREVIATIONS

NOTES

1. (AL, 85). In the later DA Bacon revised the passage in a significant manner: he placed at the basis “history and experience” and claimed that “sciences” in general should follow this pyramidal pattern (SEH, 4:361; SEH, 1:561 [Sunt enim Scientiae instar pyramidum, quibus Historia et Experientia tanquam basis unica substernuntur; ac proinde basis Naturalis Philosophiae est Historia Naturalis. Tabulatum primum a basi est Physica; vertici proximum Metaphysica]).

2. Bacon uses the term scientia in a fairly traditional manner, to mean demonstrative, universal knowledge, as opposed to historia, which treats of “individuals.” He claims that natural history should serve as the basic material or “prime matter” of philosophy (see SEH, 1:501–2; OFB, 11:37, 39; SEH, 5:510–11; and SEH, 1:494ff).

3. The nature, structure, and principles of organization of Baconian natural history are currently the subject of contention and debate among scholars. It is perhaps fair to say that after a long period of neglect, in which it was considered a mere collection of facts, Bacon’s natural history has become again the focus of scholarly debates. For a recent survey of the field, see the articles in Corneanu, Giglioni, and Jalobeanu (2012). See also Jalobeanu (2010; 2012) and Manzo (2009).

4. Bacon has a number of methodological texts containing precepts on how to write natural history. Among them, the most relevant are Parasceve (OFB, 11), the introductory preface to HNE (OFB, 12) and DGI (SEH, 5). See Jalobeanu (2012).

5. Bacon’s instruments of ordering natural histories are lists and tables as it will become clear from the third and fourth sections of this chapter.

6. According to Rees, this principle of quantification was at least in part motivated by the interest in generating a practical and productive philosophy, and it was not fully substantiated by Bacon’s ways of recording natural histories, which were often lacking proper quantitative data (Rees 1985, 33). Rees also claims that only for two of Bacon’s natural histories, HVM and HDR, are “quantitative measurements … fundamental” and that Bacon’s posthumously published Sylva sylvarum “contains very little research conducted on quantitative lines.” I think that this is not correct; on the contrary, attempts to measure are fundamental for Bacon’s natural historical approach. What can vary is the degree of precision.

7. Mechanica is the earliest surviving text of mechanics found in the Aristotelian corpus, written, most probably, by a member of the early Peripatetic school. It has a number of striking features that have puzzled many scholars since its recovery. The text was included in the Aristotelian corpus published by Aldus in 1495–98 and had a very interesting sixteenth-century circulation and posterity among humanists, natural philosophers, and mathematicians. It was still extremely popular at the end of the sixteenth century and became part of the university curriculum in more than one Italian university. On the reception of Mechanica in the sixteenth century see Rose and Drake (1971). Both the question of authorship and the nature and philosophical relevance of “mechanics” in this text are still subject to discussion among historians of mathematics (see Berryman 2009). For a recent survey of the question of authorship see Coxhead (2012).

8. Duhem has already remarked that Mechanica represents a quite striking attempt to unify the action of a number of different devices under a single analysis, and to offer a mathematical account of their action. Berryman’s characterization of the text is that “it seems to be an attempt to make philosophical sense of the ‘law of the lever’ and its operation in various situations” (114).

9. The text refers to the status of mechanical problems, which “are not altogether identical with physical problems, not entirely separated from them, but they have a share in both mathematical and physical speculations” (847a; Aristotle and Hett 1936, 332).

10. In his 1605 The Advancement of Learning Bacon introduces “physicke” with the following specification: “taking it according to the derivation, & not according to our Idiome, for Medicine” (OFB, 4:82).

11. On the multiple meanings and traditions of renaissance mathematics see Goulding (2010), Cifoletti (1990), and Axworthy (2009).

12. This is the reason why, until very recently, historians of philosophy could speak of “mathematizing nature” as if geometry had a unique and essentialist meaning throughout the sixteenth and seventeenth centuries. In fact, geometry and arithmetic were evolving subjects with multiple meanings and often belonging to very different traditions.

13. For example, through the interesting works on expertise and experts in early modern Europe, or through more general discussions on mechanics and mechanical practitioners in early modern Europe (see Ash 2004; Long 2001).

14. For Bacon, the idols are “the deepest fallacies of the human mind,” originating in a “corrupt … predisposition of the mind” and able to “infect all the anticipations of the intellect” (SEH, 4:431).

15. Although fully articulated only in the twentieth century, in the works of historians and philosophers of science, this idol actually originated in the seventeenth century and can often be identified in the works of Bacon’s followers. See the discussion in Giglioni (2013). In a seminal article, Kuhn made this evaluative judgment on Bacon’s dislike of mathematics the very basis of a general classification of early modern sciences into “mathematical” and “Baconian” (see Kuhn 1977).

16. Bacon’s idols of the tribe originate “from the evenness of the substance of the human spirit, or from its preconceptions, its narrowness, its restlessness, contamination by the affections, the inadequacy of the senses, or mode of impression” (OFB, 11:89). They are “rooted in human nature itself” or in the “race” or “tribe” of men (79–80). Under this label Bacon discusses various forms of simplifications and generalizations. Of relevance for us now are the mind’s tendency to accept a limited number of simplifications and affirmatives and then to “pull everything into line and agreement with them,” or to suppose that “there is more order and equality in things than it actually finds” (OFB, 11:83).

17. Repeated refutations came from two directions in the past decades. On the one hand, some Bacon students have shown the importance of mathematics (particularly arithmetic) for Bacon’s natural historical enterprise. On the other hand, important work has been done to disclose the peculiar character of sixteenth-century “mathematics” and the various traditions within the large field of mathematics (see for example Feingold 1984; Goulding 2010; Pumfrey 2011; Ash 2004). On Bacon and mathematics see Rees (1986).

18. Although this situation is about to change due to the efforts of Graham Rees and his Oxford Francis Bacon team, it is still easy to see how little attention and research has been devoted to Bacon’s natural historical writings by contrast with his philosophical or literary output. A number of relatively recent works have tried to refute the particular claim that Bacon’s natural histories consist of large collections of random data about nature (see for example Corneanu, Giglioni, and Jalobeanu 2012; see also Anstey and Hunter 2008; Hunter 2007).

19. I have discussed the implication of this idol for the field of Baconian studies in Jalobeanu (2013b).

20. It is worth noting that in the earlier AL, Bacon classified mathematics as a branch of metaphysics. In DA he moves it from metaphysics to this intermediate, auxiliary, and very important place as an appendix of both physics and metaphysics (and their corresponding operative sciences, mechanics and magic).

21. The term “mathematical practitioner” is admittedly vague and was subject to numerous debates and attempts of refinement and replacement. I am using it here in a general sense, to designate a practical approach to mathematical problems.

22. Giovanna Cifoletti has explored the “vernacular scientific project” of late sixteenth-century France. A similar project of vernacular geometry can be found in England toward the end of the century, in the works of John Dee, and Leonard and Thomas Digges. On vernacular mathematics and vernacular geometry see Cifoletti (1990) and Taylor (2011).

23. For a description of this translation and its context see Johnston (1991, 335).

24. On the peculiarities of Ramus’s views on Euclid, his edition of Euclid’s Geometry and associated texts, see Goulding (2010).

25. As the dedicatory letter emphasizes, the book is the result of Hood’s two years of teaching as a “mathematical lecturer of the city of London.” Hood held this position from its establishment in 1588 until 1592.

26. This is followed by a second volume, containing books seven to nine, published in 1565. On the context of this translation, see Cifoletti (1990).

27. “les Mathematiques traitent principalement les negoces celestes, les mouvements des cieux, les cours du Soleil & de la Lune & des autres planettes” (Forcadel 1564).

28. A slightly more complicated case is that of John Dee’s conception of mathematics. In his Mathematical Preface he is certainly arguing for a form of putting together mathematics and physics. However, in the classification, all the ensuing new mathematical sciences are still classified as mathematics (see Rampling 2011; Johnston 2012).

29. Historians tend to say that Pantometria was largely written by Leonard Digges and published posthumously by Thomas, with minimal additions. It is however fair to say that these additions were fully investigated. Indeed, the whole treatise has never been thoroughly explored and to date there is no modern edition of it (Bennett 1991; Taylor 1967; Taylor 2011).

30. It is worth mentioning that in the preface Thomas Digges mentions explicitly that the doctrine of the five solid bodies is added “not to discourse of their secrete or mystical appearances to the Elementall regions and frame of Coelestial Spheres, as things remote and far distant from the method, nature and certaintie of Geometrical demonstration” but to determine their properties, including superficies and volumes (see Digges and Digges 1571, 97).

31. Preface to the reader.

32. On Digges’s views on mathematics see also Pumfrey (2011).

33. “Physic diffused, which touches on the variety and particularity of things, I will again divide into two parts: Physic concerning the things Concrete, and Physic concerning things Abstract; or Physics concerning Creatures, and Physic concerning Natures … But as all Physics lies in a middle term between Natural History and Metaphysic, the former part (if you observe it rightly) comes nearer to Natural history, the latter to Metaphysic. Concrete Physics is subject to the same division as Natural history; being conversant either with the heavens or meteors, or the globe of earth and sea, or the greater colleges, which they call the elements, or the lesser colleges or species, as also with pretergenerations and mechanics. For in all these Natural History investigates and relates the fact, whereas Physic likewise examines the causes” (SEH, 4:347).

34. Bacon’s concrete physics has a part dealing with celestial bodies, one dealing with meteors, a third one dealing with the “greater masses” or the elements, another one dealing with “lesser masses,” one dealing with pretergenerations; to these, Bacon adds mechanics. This addition of mechanics is highly relevant to the kind of mixed-mathematical inquiry Bacon had in mind for his physics. For a slightly different interpretation on philosophical mechanics see Weeks (2008).

35. See also SEH (4:348) with the claim that astronomy is solely interested in “mathematical observations and demonstrations,” without paying attention to what happens in the “interiors” [viscera] of the heavens. What Bacon describes here is the theorica planetarum—a two-sphere system and the calculus necessary to compute relative motions and positions of objects on the two spheres. For a more general discussion see Barker and Goldstein (1998) and Westman (1980).

36. In more detailed criticisms of astronomy Bacon delineates three different reasons for why traditional astronomy is not only false but also idolatrous. One is that by assuming more order in the universe than there really is, astronomy instantiates one of the most common idols of the tribe. The second is that astronomers patched up their lack of data and inability of calculus with ad hoc assumptions and hypotheses. The third is that such procedures stand in the way of collecting true and accurate data.

37. Bacon compares astronomy with the stuffed ox offered by Prometheus as a sacrificial victim, instead of a real ox; similarly, astronomy is an empty science, “stuffed” with seemingly complex mathematical calculations in order to seem real. His own, reformed science will be, by contrast, a Living Astronomy. Bacon was seriously interested in building up this project. The posthumous Sylva sylvarum mentions twice a special section on heavenly bodies. Such a section was never published, but can be found in the draft manuscript of Sylva sylvarum and it has been discussed by Graham Rees (1981). On Bacon’s cosmology see also Rees (1975, 101, 161–73).

38. In his projected natural history of the heavens, Bacon argues for an instrumental and experimental approach. He quotes approvingly “the industry of mechanics and the eagerness and enthusiasm of certain learned men, that by means as it were of the skiffs and boats of optical instruments have begun … to do new trade with the celestial phenomena” (OFB, 6:115). DGI also quotes Galileo’s astronomical discoveries and proves Bacon’s familiarity with both Sidereus nuncius and the discovery of sunspots.

39. In DGI, Bacon mentions the distortions of the atmosphere and the differences between naked eye observations and telescopic observations (OFB, 6:155, 157); he discusses apparent magnitudes and real magnitudes (167), and emphasizes the need of a reformation of “optical calculations” (ibid.). He enumerates the standard methods of measurement in astronomy, including parallax calculations (169). In NO, further problems of astronomical measurement are enumerated, such as, for example, the need for reliable clocks. Bacon even mentions the possibility of a limited speed of light, which would imply the introduction of an “apparent time” beside the “real time” (see OFB, 11:376–77).

40. On comparative measurements see also NO (OFB, 11:379–80).

41. The passage belongs to a longer excerpt of the same project, Bacon’s description of a proper natural history of the heavens, from the unfinished Descriptio globi intellectualis.

42. The abstract physics has two parts: a doctrine concerning the “configurations of matter” and a doctrine concerning the “appetites and motions.” The latter is subsequently divided into simple motions and compound motions, or processes (see SEH, 4:355–56).

43. Surely as the words or terms of all languages, in an immense variety, are composed of few simple letters, so all the actions and powers of things are formed by a few natures and original elements of simple motions (SEH, 5:426). The metaphor of the alphabet is persistent in Bacon’s writings and one of his later works bears the very title Abecedarium novum naturae.

44. My emphasis and with a slightly amended translation; what we have here is the same equivalence of mathematics with measurement we have seen in the previous section of this chapter.

45. On Bacon’s appetitive metaphysics see Giglioni (2010; 2011; 2013). See also Weeks (2007a; 2007b).

46. The list of simple motions from Novum organum ends with the following specification: “I do not deny that other species could perhaps be added, or that the divisions set out could be shifted the better to match the truer veins of things, or that, lastly, that their number could be reduced” (OFB, 11:413).

47. I have given more examples of this strategy in my 2016 publication.

48. Patet quod Aer & Spiritus, & huiusmodi res, quae sunt toto corpore tenues & subtiles, nec cerni nec tangi possint. Quare in Inquisitione circa huiusmodi corpora, Deductionibus omnino est opus (OFB, 11:346–47).

49. The weather glass is also on Bacon’s list of Instantia Deductoria; it makes the invisible “degrees of Heat or Cold” visible. In the weather glass “air expanded pushes the water down and contracted draws it up, and in that way the reduction to what can be seen takes place, and not before or in any other way” (OFB, 11:355).

50. Simultaneous measurements of this kind require not only identical instruments, but also good clocks and many researchers to do the job; this gives us some hints as to the complexity of Bacon’s project for measuring the properties of the air.

51. Bacon’s table of densities appears, with slight variations in PHU, NO, and HDR. It is said to be a summonsing instance (OFB, 1:353). In the subsequent reconstruction I am indebted to the following conceptual and contextual reconstructions: Pastorino (2011a; 2011b) Jalobeanu (2011; 2013c; 2015).

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