Science and mathematics from the Renaissance to Descartes

Science and mathematics from the Renaissance to Descartes George Molland Early in the nineteenth century John Playfair wrote for the Encyclopaedia Britannica a long article entitled ‘Dissertation; exhibiting a General View of the Progress of Mathematics and Physical Science, since the Revival of Letters in Europe’.¹ Ever since the Renaissance’s invention of its own self, there has been a persistent belief that, during a general rebirth of learning, the natural and mathematical sciences made advances that effectively eclipsed what William Whewell later called the ‘Stationary Period of Science’.² No wonder that this myth triggered a ‘revolt of the medievalists’,³ who in this century have done much to redress the balance in favour of their own period. But like all myths this one contains truth as well as falsehood, and this chapter will dwell more on the former than the latter, and so concentrate on areas of innovation. But the revolt still reminds us that innovation was not the norm: for most people (both educated and uneducated) the traditional wisdom, together with its non-trivial modifications, was a more important former of consciousness than any radical new developments, and Aristotelian natural philosophy remained firmly ensconced in the universities until well into the second half of the seventeenth century, retaining in many cases a strong vitality of its own. Besides its bias in favour of innovation, this chapter will exhibit other, perhaps more insidious, forms of selectivity. It will neglect almost completely many important areas, especially in the life sciences, in order to give prominence both in coverage and mode of treatment to those areas that may be ‘philosophically’ more illuminating. (The inverted commas are intended to emphasize that, for this period, to distinguish rigidly between philosophy and science would be grossly anachronistic, and add even more to the historiographical distortion introduced by the policy of selectivity.) The chapter will comprise just three sections, dealing respectively with general ideas of advancement; a new picture of the heavens; a new mechanics embedded in a new view of nature. TECHNOLOGY / PROGRESS / METHOD It will be useful in the manner of Alexandre Koyré to distinguish technics from technology4 and so allow ourselves to retain an older meaning of the English ‘technology’—as ‘the scientific study of the practical or industrial arts’.⁵ It is now commonplace to refer to the great abundance of technical activity and of technical advances in the Latin Middle Ages, but it is more problematic how far the latter were recognized as constituting a progressive movement. Many, and particularly those emphasizing the effects of Christian doctrine and monastic discipline, have seen a conscious thrust in the direction of improvement, but the evidence is sparse, especially compared with that to be found in the Renaissance and later periods. Then, a frequently met symbolic triad to emphasize the technical superiority of the moderns over the ancients was that constituted by printing, gunpowder and the magnetic compass, although ironically all of these can in one form or another be traced back to China, in whose civilizations few symptoms of a general idea of progress have been located.⁶ We should add the example of clocks. In China, and also in the Muslim countries, there had been a penchant for producing very elaborate water-driven clocks. Around 1300, mechanical clocks appeared in Europe, and very soon cathedrals and cities were vying with each other to produce ever more ornate devices, which, by their very public display, had a better chance than the other three of infecting the populace with the ideal of progress. Interestingly enough, Giovanni Dondi, the fourteenthcentury constructor of a particularly impressive astronomical clock, was at pains to disparage modern achievements in comparison with those of the ancients.⁷ But Giovanni was a university man, associated with the nascent humanism, and particularly that represented by Petrarch. His show of modesty may not have been shared by his less learned contemporaries: it certainly was not by their successors, and very soon humanists also were singing to the same tune.⁸ In a seminal article published almost half a century ago Edgar Zilsel9 saw the effective genesis of the ideal of scientific progress as located among the ‘superior artisans’ of the sixteenth century, and with qualifications this thesis has proved remarkably resilient. One necessary modification was to bring contemporary scholars who systematically examined the crafts more centrally into the picture, and, together with this, I think that we should emphasize more than has sometimes been done a somewhat speculative aspect of the indisputably important role of printing. This is in shifting the image of knowledge. In the Middle Ages, knowledge was viewed as predominantly an individual affair, a habitus ingrained into a person’s mind by education. In acquiring such a habitus the individual progressed, but there was no very potent image of a general increase of knowledge. With the advent of printing, books multiplied in overall physical volume far more than ever before, and it is plausible to envisage with this the image of a concomitant (although not proportional) increase in their contents, so that the sum total of knowledge itself appeared to have increased. To put it in contemporary terms, we have progress in Sir Karl Popper’s ‘Third World’ of objective knowledge, conceived as existing independently of the knowing subject.¹⁰ But, whatever the causative influences, the idea of technical progress certainly became prominent in the sixteenth century, and was associated with a search for more strictly scientific progress. This was due partly to superior artisans, such as Leonardo da Vinci, looking at the theoretical bases of their arts as a source for improvement, and partly to scholars publishing surveys of craft techniques, for example the De re metallica of Georg Bauer (Georgius Agricola) in which the author gave a systematic account of current mining and metallurgical practices. Many such techniques had remained unchanged since Antiquity, and were simply passed from master to apprentice by word of mouth. Bringing them out into the cold light of print could suggest modes of improvement, and later became formalized in the injunction to produce histories of trades—what can be, and sometimes were, called ‘technologies’. These and many other factors provoked a strong urge towards improvement, and a conviction that systematic intellectual activity was at least as important as trial and error or reliance upon tradition. And this applied not only to the more banausic areas of technics, but also to more rarefied and seemingly ‘impractical’ fields. But a desire for progress is one thing: achieving it is another. And here we meet more and more frequently with the term ‘method’ and its relations. The search for this had roots in several fields, mathematical, philosophical, medical, magical…. We consider briefly the first two. The mathematical revival of the sixteenth century is characterized particularly by the greater availability of important ancient works, through both the production of new translations and the dissemination of these in print. This gave rise to two reflections. First, it became evident that many significant works were (probably irretrievably) lost, but second, sufficient evidence was often available about their contents for plausible attempts at reconstruction to be made. This did not directly imply method, but it did make for mathematical progress, for restoration was to be achieved not by philology alone but by trying to do mathematics in the Greek spirit. As an attempt at exact replication this failed, but it did produce new mathematics, and so we have a kind of surreptitious progress in which efforts at reviving the old produced developments that were radically new. Others were more open in their endeavours, and indeed accused the ancients of being clandestine. From Antiquity onwards it had been realized that some of the most impressive results of Greek geometry resembled a beautiful building from which all trace of scaffolding and other accessories had been removed. Hence Plutarch on Archimedes: It is not possible to find in geometry more difficult and weighty questions treated in simpler and purer terms. Some attribute this to the natural endowments of the man, others think it was the result of exceeding labour that everything done by him appeared to have been done without labour and with ease. For although by his own efforts no one could discover the proof, yet as soon as he learns it, he takes credit that he could have discovered it: so smooth and rapid is the path by which he leads to the conclusion.¹¹ The paradox suggested that the ancients had knowledge of a particularly fruitful way of discovering new mathematical truths. As Descartes put it, ‘We perceive sufficiently that the ancient Geometricians made use of a certain analysis which they extended to the resolution of all problems, though they grudged the secret to posterity.’¹² Such suspicions were partially vindicated at the beginning of this century (although without evidence of a grudge) by the discovery of a lost work by Archimedes, known as the Method (Ephodos), in which the author showed how to use theoretical mechanics and considerations of indivisibles in order to discover new theorems about equating areas and volumes between different plane and solid figures. These theorems were then open to proof by more rigorous geometrical methods, particularly the reductiones ad absurdum involved in the so-called method of exhaustion, in which inequalities of the relevant areas and volumes were shown to lead to contradictions. There were a few brief and vague ancient references to Archimedes’ ‘method’, but rather more to the procedures of ‘analysis and synthesis’, whose exact interpretation have caused much scholarly perplexity. In a famous passage Pappus wrote: Analysis is the path from what one is seeking, as if it were established, by way of its consequences, to something that is established by synthesis. That is to say, in analysis we assume what is sought as if it has been achieved, and look for the thing from which it follows, and again what comes before that, until by regressing in this way we come upon some one of the things that are already known, or that occupy the rank of a first principle. We call this kind of method ‘analysis’, as if to say anapalin lysis (reduction backward). In synthesis, by reversal, we assume what was obtained last in the analysis to have been achieved already, and, setting now in natural order, as precedents, what before were following, and fitting them to each other, we attain the end of the construction of what was sought. This is what we call ‘synthesis’.¹³ Pappus went on to distinguish theorematic (zetetikos) and problematic (poristikos) analysis. The general thrust of the passages is clear. In theorematic analysis we work backwards towards the first principles from which a theorem follows, and in problematic analysis our goal is the solution of a problem, say the finding of a figure whose area or other features will meet certain conditions, while in synthesis we in some sense prove our results. But when we descend towards the logical niceties a host of difficulties appear,¹⁴ and these may themselves have made the subject a particularly suitable candidate for seventeenth-century transformation. An especially important figure in this process was François Viète, who drew on both the Diophantine ‘arithmetical’ tradition and more general algebraic traditions, as well as those deriving from primarily geometrical works. Viète’s most striking innovation in assuming a problem solved was to name the ‘unknown’ quantity or quantities with letters of the alphabet, as well as those that were ‘known’, and then to operate on both in the same way. This produced formulae that looked far more like equations in the modern sense than anything that had gone before, and this trend is even more accentuated in the work of Descartes, where we often have no more than minor and accidental features of notation to remind us that we are not in the twentieth century. All this may be seen as reflecting a general psychological trend in mathematics to focus more on written symbols than on what they are meant to symbolize. It is this movement that has made it seem plausible to speak of Descartes as the founder of ‘analytical geometry’, meaning thereby a geometry in which curves may be substituted by the equations representing them. But this ignores the extent to which Descartes still demanded explicitly geometrical constructions for solving what we would regard as purely algebraical problems. Nevertheless it does draw attention to how geometry and algebra could seem to proceed in tandem. This trend was to continue into the work of Newton, and can be seen as leading towards an eventual reduction of the former to the latter. With hindsight it is possible to see ripening other fruits of mathematical method, such as those associated with the infinitesimal calculus, but until well past the time of Descartes these are decidedly muted in Comparison with the algebraic results. As may be expected, philosophers were more explicit than mathematicians in their discussions of method, although one head was often capable of wearing two hats, as in the case of Descartes. In the Aristotelian tradition we frequently meet with the methods of analysis and synthesis in the Latinate guise of resolution and composition. As in mathematics, the terms are a little slippery, but resolution was basically a form of argument from effects to causes, whereas composition was demonstration of effects from their causes. The development of these ideas by medieval and Renaissance commentators on Aristotle (and also on Galen’s methodological writings) have led some scholars, for instance John Hermann Randall, Alistair C.Crombie and William A. Wallace, to place strong, and perhaps extravagant, emphasis on the positive role of Aristotelianism in furthering the emergence of modern science. One thinker who would not have been convinced by this was Francis Bacon. Although Bacon had a grudging respect for aspects of Aristotle’s own thought, he saw medieval scholasticism as the embodiment of sterility and futile contentiousness. He therefore sought a more fruitful way of eliciting knowledge from nature, of a kind that would eventually prove useful in practice. In this he displayed some affinities with the magical and alchemical traditions, but their secrecy and apparent obscurantism were antithetical to his programme for which he sought more public, methodical and ‘democratic’ procedures. In this a central role was played by his idea of induction, the careful collection and tabulation of facts (although in practice often derived from nonetoo-reliable reports), and then the cautious ascent to higher levels of generalization, with only a very wary use of hypotheses, strictly controlled by the use of experiments. Although usefulness was the remote aim, this could not properly be achieved without first seeking the truths of nature: as opposed to what he saw as the alchemists’ habits, luciferous experiments must always precede lucriferous ones. Francis Bacon very soon became a hero of science, and especially of British science, but it remains controversial how much substantive as opposed to rhetorical influence he exerted upon its development. One would be very hard pushed to find anyone successfully following the Baconian method to the letter, and some notable writers denigrated his scientific importance. For instance, it is reported that William Harvey, the discoverer of the circulation of the blood, ‘esteemed [Bacon] much for his witt and style, but would not allow him to be a great Philosopher. Said he to me, “He writes philosophy like a Lord Chancellor,” speaking in derision.’¹⁵ And in the more mathematical sciences, with which the remainder of this chapter will be mainly concerned, it is less easy to make out a case for his influence than in the biological and descriptive sciences. This is largely due to his generally acknowledged blind spot as regards the potential of mathematics in forwarding the development of science. THE ASTRONOMICAL REVOLUTION Developments in cosmology must play a leading, if not the dominating, role in any consideration of the science of this period. It all started very quietly. The first really public event occurred in the year 1543, but it was not accompanied with the stir appropriate to what many used to regard as the inaugural date of the Scientific Revolution (the other symbolic event being the publication of Vesalius’s De humani corporis fabrica). The title of Copernicus’s book of that year, De revolutionibus orbium caelestium, incorporated what can now, but not then, be regarded as a pun on the word ‘revolution’, and also a point of disputed translation concerning the word orbis. The book had been a long time in the making, but probably not for the devious reasons sometimes proposed: it simply took a long time to prepare and write. The initial conception had come some thirty years earlier, if not before, by which time Copernicus had completed a good scholastic education at the flourishing university of Cracow, together with a tour around the universities of Renaissance Italy, before settling into a canonry at the Cathedral of Frauenberg (Frombork). In about 1514 he completed and circulated in manuscript a short work, now usually known as the Commentariolus, outlining a heliocentric system of the world, with the Earth rotating daily on its axis and as a planet orbiting the now stationary Sun once a year. His reasons for the change have been a matter of controversy. Here I shall not undertake the delicate, and possibly tedious, task of assaying the various hypotheses, but present what appears a likely story, and one which seems both in accord with the majority scholarly view and with Copernicus’s explicit statements as to why he proceeded as he did. The conventional astronomical wisdom at Copernicus’s time derived from Ptolemy’s Almagest, which provided sophisticated mathematical models for showing where at any particular time a planet (the list included the Sun and Moon) would appear from the vantage point of a stationary Earth against the background of the fixed stars; or, more precisely, in a co-ordinate system determined by the great circles of the ecliptic (the path of the Sun through the sky) and the celestial equator. The devices worked by combining circular motions. Usually we have a deferent circle whose centre is at some distance from the Earth, and an epicycle, a smaller circle whose centre moves around the circumference of the deferent, while the planet is imagined to move about the circumference of the epicycle. Speeds are regulated in terms of an equant point, situated at the same distance from the centre of the deferent as the Earth but on the opposite side, so that the centre of the epicycle moves with uniform angular velocity about this point. The models for Mercury and the Moon were more complicated than this description may suggest, and that for the Sun simpler, but all incorporated sufficient flexibility in choice of parameters to make them good predictive devices. At least, they were good enough for Copernicus, but the status of the equant point was another matter. [The theories] were not sufficient unless there were imagined also certain equant circles, by which it will appear that the star is moved with ever uniform speed neither in its deferent orb nor about its proper centre, on which account a theory of this kind seems neither sufficiently absolute nor sufficiently pleasing to the mind.¹⁶ This famous passage has led some to speak of Copernicus’s Pythagorean obsession with uniform circular motion, with the implied suggestion that this was beyond the bounds of rationality. But in fact Aristotle is a better target. In Copernicus’s time, as it had been from Antiquity onwards, astronomy was regarded as a branch of mathematics, what Aristotle had referred to as one of its more physical branches; in the later Middle Ages these were often called middle sciences, as lying between mathematics and physics. Whatever Ptolemy may have thought, it was not generally seen as a hallmark of mathematics to look at causes: that was the province of the physicist. For Aristotle the heavens were made of the fifth element or aether, to which it was natural to move eternally with a uniform rotation. In his more detailed astronomical picture, for which he borrowed from the mathematicians (astronomers) of his time, Aristotle had some fifty-five spheres, centred on the Earth, of which nine carried planets. Any deviation from uniformity of motion would need an interfering cause, which could not seem plausible in the perfection of Aristotle’s celestial realm. Copernicus did not maintain the Aristotelian distinction between celestial and elementary regions, but he did demand a cause for deviation from uniformity, and none seemed available: hence, if for no other reason, farewell the equant. Here we must emphasize that Copernicus, although more mutedly than Kepler, was much concerned with causes, and also, in the context of an often bitter recent controversy, that he almost certainly ascribed a fair degree of reality to the orbs (be they spheres, circles, orbits, hoops or whatever) to which the planets were conceived as attached or along which they moved. A causal and harmonious structure for the universe was also theologically supported, for a good Creator God would surely have proceeded according to a rational plan. This attitude clearly lay behind Copernicus’s criticism of astronomers in the Ptolemaic tradition for producing fine parts which could by no means be fitted together in a tidy fashion. It was with them exactly as if someone had taken from different places, in no way mutually corresponding, hands, feet, head and other members, all excellently depicted but not in relation to a single body, so that a monster was composed from them rather than a man.¹⁷ Copernicus’s own system as portrayed in the famous diagram included in Chapter 10 of Book I of the De revolutionibus (with an accompanying panegyric on the Sun) appears admirably simple and rational, but important cracks lie very near to the surface. In the first place the system was obviously physical nonsense. If the Earth is really moving with such huge speed, we should surely experience it by such phenomena as constant high winds and stones that refused to fall at the bottom of the towers from which they are dropped; also its whole body should fly apart, like the materials on a potter’s wheel that is spun too fast. Copernicus was not blind to such objections, and answered them in summary fashion, but in terms which we may find easier to accept than did his own contemporaries. And after all, was it worth destroying important foundations of the well-tried edifice of Aristotelian physics at the behest of a mere astronomer? All this relates to Book I of the De revolutionibus, understandably the only part that is read by more than a few specialists in mathematical astronomy. But the later five books destroy most of the simplicity of this book in order to make the system work, that is, to ‘save the phenomena’, or provide a relatively good predictive device for planetary positions, for in doing this Copernicus, although rejecting the equant point, retained most of the techniques of earlier astronomy for explaining the planetary movements by combinations of circular motions, and in so doing produced a system that was arguably as complicated as Ptolemy’s. This was not arrant conservatism but a natural use of procedures with which he had been brought up, and which if rejected would almost certainly have prevented him from doing any work worthy of serious astronomical recognition. With these points made, the result is not surprising. The book did not fall on dead ground, but neither did it win unconditional acclaim. Among the general public it was seen to propose a pleasing or unpleasing paradox, perhaps interesting for idle conversation, but only to be taken seriously by eccentrics or fanatics. For instance, even before the book was published, Luther remarked that, Whoever wants to be clever must agree with nothing that others esteem. He must do something of his own. This is what that fellow does who wishes to turn the whole of astronomy upside down. Even in those things that are thrown into disorder I believe the Holy Scriptures, for Joshua commanded the Sun to stand still, and not the Earth.¹⁸ Among professional astronomers the work was greatly admired, but for its astronomy, not its physics. A characteristic posture was to adopt what Robert S.Westman19 has called the Wittenberg interpretation of the theory: to take a connoisseur’s delight in Copernicus’s mathematical techniques and employ them about the astronomer’s proper business, such as the construction of tables, but to reject as utterly mistaken the idea of a moving Earth. Paradoxically this instrumentalist outlook received support from Copernicus’s own volume, for Andreas Osiander, the Lutheran pastor who saw it through the press, inserted an anonymous preface, possibly to ward off theological criticisms, which gave a low truth status to astronomical theories. It is proper to an astronomer to bring together the history of the celestial motions by careful and skilful observation, and then to think up and invent such causes for them, or hypotheses (since he can by no reasoning attain the true causes), by which being assumed their motions can be correctly calculated from the principles of geometry for the future as well as for the past. The present author has eminently excelled in both these tasks, for it is not necessary that the hypotheses be true nor indeed probable, but this one thing suffices, that they exhibit an account (calculus) consistent with the observations.²⁰ And some careless readers of the work were misled into thinking that this was Copernicus’s own view. Instrumentalism has rarely been a satisfactory psychological stance for working scientists, and so it is not surprising that soon schemes were brought forward that gave a more realist slant to the Wittenberg interpretation. The most famous of these, and one which he guarded jealously as his very own intellectual property, was by the great Danish astronomer Tycho Brahe. In this the Moon and the Sun orbited the stationary Earth, while the other planets circled the Sun and accompanied it on its annual journey about the Earth. This had all the astronomical advantages of the Copernican system and none of what were perceived as its physical disadvantages. It also had no need to accommodate the fact that no parallax (apparent relative motion among the fixed stars) had been observed, which would have been expected if the Earth were moving. Tycho laid particular emphasis on this difficulty for Copernicanism, and calculated that, if the Earth were moving, then even a star that had only a moderate apparent size would have to be so far away that it would in fact be as big as the whole of the Earth’s orbit. Not surprisingly, when Copernicanism came under ecclesiastical fire, the Tychonic compromise emerged as the favourite system of the Jesuits, who were themselves strong pioneers of scientific advance. But Tycho’s theorizing was of less scientific importance than his practice. He was unusual among men of science in being of aristocratic birth, and this made substantial patronage easier to obtain. The King of Denmark granted him the island of Hveen (situated between Copenhagen and Elsinore), on which he built a magnificent observatory called Uraniborg. Up to that time it could only have been rivalled by Islamic or Mongol observatories, such as those at Maragha and Samarkand. Not only was the hardware, so to speak, superb, but Tycho had it manned by a group of able assistants whom he at least tried to rule with a rod of iron. The result was an incomparably accurate collection of observations of stellar positions, which historically was to play a far more significant role than his famous demonstrations that the New Star (to us a supernova) of 1572 and the comet of 1577 were supralunary, and hence symptoms of change in supposedly immutable regions, and that the latter would have to be passing through the solid spheres of Aristotelian cosmology. Nevertheless these made important dents in the old world picture, although ironically Galileo denied the validity of his arguments with regard to the comet. There is a waggish yet revealing quip that Tycho’s most important discovery of all was that of a person, namely Johann Kepler. Kepler was born into a Lutheran family, and was himself heading for the Lutheran ministry when he entered the University of Tubingen in 1587.²¹ However, his course was deflected by a growing interest in astronomy, fostered by one of his teachers, Michael Maestlin, who happened to be one of the few convinced Copernicans of the era, and in 1594 with Maestlin’s encouragement Kepler accepted the post of District Mathematician at Graz. His duties involved some elementary mathematical teaching and the drawing up of astrological prognostications, but he also pursued his own theoretical interests in astronomy, and in 1596 published a small book entitled Mysterium Cosmographicum. It could be tempting to pass this book over as merely ‘quaint’ were it not for Kepler’s much later claim that, Just as if it had been literally dictated to me, an oracle fallen from heaven, all the principal chapters of the published booklet were immediately recognised as most true by the discerning (which is the wont of God’s manifest works), and have these twenty-five years carried before me more than a single torch in accomplishing the design (initiated by the most celebrated astronomer Tycho Brahe of the Danish nobility) of the restoration of astronomy, and, moreover, almost anything of the books of astronomy that I have produced since that time could be referred to one or other of the chapters set forth in this booklet, of which it would contain either an illustration or a completion.²² And an attentive reading shows that much of this was indeed the case. In the book Kepler reveals himself as one who would out-Copernicize Copernicus in his belief in the physical reality of a heliocentric system, and this attitude is reinforced by a commitment to asking why, and answering it in terms of both geometrical and physical causes. An important example is the question of why there are six and only six (primary) planets, Mercury, Venus, Earth, Mars, Jupiter, Saturn. (This question is quite rational, if it is assumed that the world was created much as it is now some finite time ago.) Georg Joachim Rheticus, Copernicus’s first champion, had answered it arithmetologically by saying that six was a perfect number. This had a precise meaning, for 6’s factors 1, 2, 3, when added together, produce 6 itself, a property shared by relatively few numbers, the next example being 28. Kepler would have none of this mystica numerorum, and firmly believed in geometry’s priority to arithmetic, so that it, rather than arithmetic as Boethius had held, provided God with the archetype for the creation of the world. Kepler found the required linkage with geometry in the remarkable fact that there are five and only five regular solids. He then discovered even more remarkably, and we would say coincidentally, that these solids and the spherical shells enclosing the planetary orbits could be fitted together in a sort of Chinese box arrangement, so that, if an octahedron was circumscribed about the sphere of Mercury, it was almost exactly inscribed in the sphere of Venus, and so on, giving the order, Mercury, octahedron, Venus, icosahedron, Earth, dodecahedron, Mars, tetrahedron, Jupiter, cube, Saturn. This idea understandably so excited Kepler that he planned a model for presentation to the Duke of Württemberg. Another preoccupation was with what moved the planets, for Kepler remained in the tradition in which each motion demanded an efficient cause. His answer was that there was a single ‘moving soul’ (later to be depersonalized to ‘force’) located in the Sun. This had the natural consequence that the planetary orbits lay in planes passing through the Sun, which in turn virtually removed the messy problem of latitudes (the deviations of the planetary paths from the plane of the ecliptic) from mathematical astronomy. Kepler circulated copies of his book to other astronomers, includ ing Tycho Brahe, who, perhaps surprisingly, was favourably impressed, although complaining that Kepler had too great a tendency to argue a priori rather than in the a posteriori fashion more appropriate to astronomy. He invited Kepler to visit him, but this did not come to fruition until, after a series of disputes, Tycho moved from Denmark and came under the patronage of the Emperor Rudolf II, who granted him a castle near Prague for his observatory. Kepler went to see him there and soon became a member of his team. The relationship between the rumbustious and domineering Tycho and the quieter but determinedly independent Kepler was not an easy one, but it did not last too long, for Tycho died in 1601 and was succeeded by Kepler in his post as Imperial Mathematician. On joining Tycho, Kepler was set to work on Mars, whose movements had been proving particularly recalcitrant to mathematical treatment. We may count this a fortunate choice, for, to speak with hindsight, its orbit is the most elliptical of the then known planets. But it was one of Kepler’s important innovations to seek for the actual orbit of the planet rather than for the combination of uniform circular motions that would give rise to the observed appearances. However, he was traditional in at first seeking for a circular orbit and reintroducing an equant point, although not necessarily at the same distance as the Sun from the centre of the orbit. In this manner he formed a developed theory, which he later called his ‘vicarious hypothesis’; it became especially famous for a crucial deviation of eight minutes of arc from observational evidence, which would previously have been undetectable. Since the divine goodness has given to us in Tycho Brahe a most careful observer, from whose observations the error of 8 minutes in the Ptolemaic account (calculus) is argued in Mars, it is fitting that with grateful mind we should recognise and cultivate this gift of God…. For if I had treated these 8 minutes of longitude as negligible, I should already have sufficiently corrected the hypothesis…. But because they could not be neglected, these eight minutes alone have led the way to reforming the whole of astronomy, and have been made the matter for a great part of this work.²³ With the vicarious hypothesis rejected, Kepler embarked on a bewildering variety of sophisticated procedures. One strategy was to place the Earth more on a par with the other planets; for it provided the moving platform from which we observed, but hitherto its orbit had lacked an equant point. To this end Kepler found it useful to imagine that he was on Mars and observing the Earth from there. Another strategy was that of quantifying the causes of the planetary motions. A force emanating from the Sun, and inversely proportional to the distance from the Sun, was conceived as pushing the planets around. This eventually led to what we know as Kepler’s second law of planetary motion, that the radius vector from the Sun to a planet sweeps out equal areas in equal times. But this force did not explain a planet’s varying distance from the Sun. For this purpose a quasi-magnetic push-pull force was introduced with allusion to William Gilbert’s De magnete which had been published in 1600. This caused a libration of the planet on an epicycle’s diameter directed towards the Sun. With this theoretical apparatus Kepler proceeded to seek the actual path of Mars. For this he experimented with a variety of egg-shaped orbits. Readers of Gulliver’s Travels will remember that eggs have big ends and little ends, but Kepler did use ellipses as calculating devices, and eventually came with a start to a realization that the orbit itself was an ellipse, with the Sun at one focus (the ‘first law’). O, how ridiculous of me! As if the libration in the diameter could not be a way to the ellipse. I have become thoroughly convinced that the ellipse stands together with the libration, as will be evident in the next chapter, where at the same time it will be demonstrated that no figure remains for the orbit of the planet other than a perfectly elliptical one.²⁴ Kepler’s first two laws (which were soon generalized to the other planets) were published, with a detailed account of his procedures, in 1609 in his Astronomia Nova, whose full title is particularly evocative: New Astronomy, Reasoned from the Causes, or Celestial Physics, Delivered Up by Considerations of the Motions of the Star Mars, From the Observations of the Great Tycho Brahe.²⁵ What we call the third law linked together the various planets in stating that the square of the periodic time of a planet was proportional to the cube of its mean distance from the Sun. It appeared in print in Kepler’s Harmonice Mundi of 1619. As the title indicates, the principal aim of this work was in a very literal way the search for musical harmonies in the heavens, and it is illustrated with scales appropriate to the various planets. This has misleadingly encouraged the anachronistic, and now thankfully outdated, attitude that Kepler can properly be split into two distinct halves, the mystical and the scientific. Another consequence of this attitude was to cause some unnecessary puzzlement about the subsequent fate of Kepler’s laws. To a superficial modern eye, it can seem that Kepler had now definitively established the facts about planetary motion (in a commodious description) which only awaited a Newton in order to explain them, and indeed the Kepler-Newton motif played a marked role in later scientific rhetoric, as with Ampère and Maxwell. But this interpretation raises the question of why Kepler’s ‘laws’ seem to have been so neglected between their formulation and the time of Newton. Certainly Kepler’s writings are not easy to penetrate, and the laws themselves are not so clearly sign-posted as a modern reader may expect. Also Kepler’s second law in particular was not easy to calculate with, and some astronomers, such as Seth Ward and Ismael Boulliau, found it easier to combine the ellipse with an equant point at the focus not occupied by the Sun. In general it seems that knowledgeable astronomers were well aware of the laws but did not accept them with quite the alacrity that we might think appropriate, so that even Newton could comment that, ‘Kepler knew ye Orb to be not circular but oval & guest it to be Elliptical’.²⁶ The situation has been much clarified in important articles by Curtis Wilson. The validity of Kepler’s laws did not rest solely on observational evidence, with the quasi-animist forces (his ‘mystical’ side) being mere psychological scaffolding that could be cleared away once the building was erected. These laws depended on theoretical support as well, for the former was insufficient by itself. But Kepler’s theory, his system of forces, was very much of his own making, and did not transfer easily to other workers, especially in an age in which both conservatives and more mechanistically minded radicals were wary of any suspicion of unmediated action at a distance, and when the latter were moving towards a new type of inertial physics in which the continuance of a motion did not demand a continually acting force to explain it. Thus, before Newton, it was quite rational to regard Kepler as having provided an ingenious and useful, but only approximate, account of the planetary motions. When Newton showed how his inverse square law could be derived from Kepler’s second and third laws, and then the first law deduced from this, the ‘laws’ were back in business with new theoretical support, but, despite the use of distance-related forces, this was very different from that provided by Kepler. It nevertheless fitted well with a new general system of mechanics, and gave good licence for Kepler’s laws to be named as such. But this was for the future. For the time being popular educated and less educated interest in the new astronomy was to centre on a more accessible figure. One of the people to whom Kepler sent a copy of his Mysterium Cosmographicum was Galileo Galilei. This was natural, for Galileo, then in his early thirties, was the occupant of the Chair of Mathematics at one of the foremost scientific centres in Europe, the University of Padua. Galileo replied immediately, saying how much he was looking forward to perusing it, since he had been of Copernicus’s opinion for many years. But, remarkably, this document itself is one of the few pieces of evidence for Galileo’s own opinion until some thirteen years later, and the change depended upon a new observational instrument. It is probably better to think in terms of the emergence of the telescope rather than its invention, but we may take 1608 as a symbolic year, when the Dutchman Hans Lippershey presented a spyglass to Prince Maurice, and also applied for a patent, thereby indicating the passage of the device from the realm of fairground attraction for producing illusions to that of something potentially useful. Whoever should be given the prime credit for the invention, the news spread rapidly, and reached Italy by the following year. Galileo seized on it avidly and constructed glasses for himself, but in his reports probably exaggerated the extent to which he had employed optical theory. Then, like Thomas Harriot in England at almost exactly the same time, he turned his telescope to the skies, but unlike Harriot he quickly published his findings, in a booklet of 1610 entitled Sidereus Nuncius. This caused a sensation, not only because of the new facts themselves, but on account of their possible implications for rival cosmological systems. We isolate three discoveries. The first concerns the Moon. The Man in the Moon was, so to speak, an old friend, and scholastic discussions had frequently touched on the reason for this feature always pointing towards us. The telescope revealed to Galileo that the man was far more pock-marked than hitherto thought, and by observing the changing configuration of the spots he was able plausibly to infer the existence of shadows caused by mountains and chasms on the Moon’s surface. This made the Moon more like the Earth, and was a great help in breaking down the gulf separating the perfect celestial regions from the imperfect elementary ones, and as such offered indirect support to heliocentric cosmology. As regards the fixed stars, Galileo observed far more of them than had been done previously, and plausibly argued that the Milky Way was composed of stars too numerous to separate from one another with the naked eye. But a more important discovery concerned their magnification, for they were not increased in apparent size as much as would be expected from observation of nearer objects. Galileo therefore concluded that much of the observed size was to be attributed not to the body of the star itself, but to twinkle, or, as he called it, adventitious light, and he later confirmed this with experiments using naked eye observations. This had the important implication that the stars could be a vast distance away without needing to be of the enormous size that Tycho thought would have been necessary to explain the absence of observed parallax on the hypothesis of a moving Earth. But for Galileo the most exciting discovery related to Jupiter. He recounts how in early 1610 he observed three small but bright stars near it, which changed their relative positions without straying far from the planet itself. He was later to add a fourth, and reasonably concluded that all four were satellites of Jupiter, ‘four PLANETS never seen from the creation of the world up to our own time’.²⁷ With a piece of calculated flattery he named them the Medicean stars, after Cosimo de’ Medici, the Grand Duke of Tuscany. This paid off, for soon Galileo procured appointment as Chief Mathematician and Philosopher to the Grand Duke, with the phrase ‘and Philosopher’ being added at his own insistence to emphasize that his interests were not merely mathematical but intimately concerned with the structure of the physical universe. And Galileo was quick to show how his discovery could support (but again only indirectly) the Copernican system of the world. Here we have a fine and elegant argument for quieting the doubts of those who, while accepting with tranquil mind the revolutions of the planets about the Sun in the Copernican system, are mightily disturbed to have the Moon alone revolve about the Earth and accompany it on an annual rotation about the Sun. Some have believed that this structure of the universe should be rejected as impossible. But now we have not just one planet rotating about another while both run through a great orbit about the Sun; our own eyes show us four stars which wander around Jupiter as does the Moon around the Earth, while all together trace out a grand revolution about the Sun in the space of twelve years.²⁸ The initial reaction to Galileo’s book, including that from the Jesuit College in Rome, was favourable, although for some this meant attempts to incorporate the new evidence within a traditional cosmological framework. There were a few who were reported to have refused (probably half jokingly) to look through the telescope on the grounds that they would not trust what they saw there; this was not completely unreasonable, given the previous reputation of optical devices for making things appear as they were not, and also given the difficulty of properly manipulating the instrument to show what was in fact there. Meanwhile Galileo continued observing, and also frequently allowed his disputatious temperament to lead him into behaviour which in retrospect we can see as unfortunately tactless. One of his preoccupations was with sunspots, about which he carried on an adversarious correspondence through an intermediary (Mark Welser) with the Jesuit Christopher Scheiner, writing under the pseudonym of Apelles; although Welser informed him that Apelles did not read Italian Galileo persisted in writing in that language rather than the mutually accessible Latin, and the result of Galileo’s side of the exchange was published in 1613 as Istoria e Dimostrazioni intorno allé Macchie Solari. Besides a rather fruitless dispute about priority, the authors disagreed about the nature of sunspots, with Galileo favouring the view that they were like clouds around the Sun, thus emphasizing the theme of mutability of the heavens. But other matters were discussed, the most important of which was undoubtedly the phases of Venus. It is a notable consequence of Ptolemaic theory that the two inner planets, Mercury and Venus, never deviate far from the Sun in celestial longitude—hence the position of Venus as both morning and evening star. Also, it was almost universally held that they were nearer to us than the Sun. There accordingly arose the question of how they received the light to make them visible, for on the common assumption that this came from the Sun they would be almost entirely illuminated from behind, and at most we should see a small sliver. One opinion had it that they were possessed of their own light and another that they were translucent, but basically the problem remained unresolved, or, to use T.S.Kuhn’s phraseology, was an anomaly within the Ptolemaic paradigm. On the Copernican hypothesis the predictions were different, for, if these planets were orbiting the Sun, they should display phases in the manner of the Moon. And this is what Galileo observed in the case of Venus, and triumphantly reported in the Letters on Sunspots. These things leave no room for doubt about the orbit of Venus. With absolute necessity we shall conclude, in agreement with the theories of the Pythagoreans and of Copernicus, that Venus revolves about the Sun just as do all the other planets.²⁹ So far so good, but what Galileo fails to mention, and what remained a thorn in his flesh, was that these observations were also perfectly compatible with the Tychonic system of the world, which, as mentioned above, soon became a favourite with Jesuit astronomers. Until the 1610s the Copernican system had aroused little religious discussion, apart from a few casual references to scriptural passages that seemed to contradict it, such as the command to the Sun to stand still over Gibeon in order to lengthen the day, so that Joshua could have time to finish a battle. With the new popularity that Galileo had brought to the issue, the religious implications became of major concern, and opposition to the system was probably egged on by mere conservatism masquerading as high principle. Galileo was himself drawn into the controversy, and wrote a long letter on the relation of Copernicanism to the Scriptures. This closely reasoned piece made relatively liberal use of the principle of accommodation in biblical interpretation. The Bible was addressed to ignorant people, and appealed to the common understanding of the time; it was not intended as a textbook in astronomy. In the epigram of one Cardinal Baronius, which Galileo gleefully quoted, ‘The intention of the Holy Ghost is to teach us how one goes to heaven, not how heaven goes.’³⁰ A tougher line, although also closely reasoned and susceptible of its own nuances of interpretation, was taken by Cardinal Bellarmine in a letter to Paolo Foscarini, a priest who had espoused Copernicanism. The Council [of Trent] prohibits interpreting Scripture against the common consensus of the Holy Fathers; and if Your Paternity wants to read not only the Holy Fathers, but also the modern commentaries on Genesis, the Psalms, Ecclesiastes, and Joshua, you will find all agreeing in the literal interpretation that the Sun is in the heavens and turns around the Earth with great speed, and that the Earth is very far from heaven and sits motionless at the center of the world.³¹ In 1616 the matter had become sufficiently serious for Rome to take a hand, and the Theologians to the Inquisition, after what may seem indecently hasty deliberations, reported that the proposition that ‘The Sun is the centre of the world and completely devoid of local motion’ was ‘foolish and absurd in philosophy, and formally heretical’, and that the proposition that ‘The Earth is not the centre of the world nor motionless, but it moves as a whole and also with diurnal motion’ should receive ‘the same judgement in philosophy and that in regard to theological truth it is at least erroneous in faith’.³² What precisely happened next as regards Galileo himself is debatable, but at the least he was at the Pope’s behest officially informed of the judgement and acquiesced therein. And so the matter rested for several years. Galileo did not exactly refrain from controversy, and in fact carried on a bitter dispute centring on the nature of comets but taking in many aspects of what constituted proper scientific procedure with the Jesuit Horatio Grassi, but he kept quiet on the question of the motion of the Earth. In 1623 there was a change of Pope, and Maffeo Barberini, an old friend and supporter of Galileo’s, ascended to the Throne of St Peter with the title of Urban VIII. Galileo visited Rome and was granted several audiences, from which he seems to have come away with the impression that he could say what he liked about the Copernican system provided that he treated it as hypothetical and did not bring in scriptural arguments (which should be left to the theologians). He then set to work on one of his most important books, which, after a little sharp practice in getting it past the censor, was published in 1632 as Dialogo…sopra i Due Massimi Sistemi Del Mondo Tolemaico, E Copernicano. This was in the form of a dialogue lasting for four days between three friends, Salviati, Sagredo and Simplicio. Salviati can usually be taken as acting as spokesman for Galileo, Sagredo as an intelligent layman and Simplicio as the Aristotelian, but not one that is too stupid, for after all there is no honour in winning arguments over fools. The burden of the first three days is basically to show that everything would appear to happen the same whether or not the Earth was moving, and if the book had stopped there (and the Preface had been strongly modified), there would probably have been no trouble. But Galileo was not content with showing that the Copernican system was possible: he wanted to show that it actually was the case. As Salviati says early in the Fourth Day, ‘Up to this point the indications of [the Earth’s] mobility have been taken from celestial phenomena, seeing that nothing which takes place on the Earth has been powerful enough to establish the one position any more than the other.’ He then continued, Among all sublunary things it is only in the element of water (as something which is very vast and is not joined and linked with the terrestrial globe as are all its solid parts, but is rather, because of its fluidity, free and separate and a law unto itself) that we may recognise some trace or indication of the Earth’s behaviour in regard to motion and rest.³³ This provided the cue for Salviati to expound, but not in highly developed form, Galileo’s notorious doctrine of the tides. This, which attributed the tides to a ‘sloshing around’ of the seas caused by the Earth’s twofold motion of translation and rotation, was based on a phoney argument, even on Galileo’s own terms, and Galileo should have known it, but apparently genuinely did not. But historically its main importance is that Galileo thought that he had found a particularly weighty argument for establishing the Copernican system. To be sure, it could not be regarded as utterly conclusive, but the statement to that effect was put at the end of the Day in the mouth of Simplicio, who had been regularly losing all the arguments. As to the discourses we have held, and especially this last one concerning the reasons for the ebbing and flowing of the ocean, I am really not entirely convinced; but from such feeble ideas of the matter as I have formed, I admit that your thoughts seem to me more ingenious than many others I have heard. I do not therefore consider them true and conclusive; indeed, keeping always before my mind’s eye a most solid doctrine that I once heard from a most eminent and learned person, and before which one must fall silent, I know that if asked whether God in His infinite wisdom could have conferred upon the watery element its observed reciprocating motion using some other means than moving its containing vessels, both of you would reply that He could have, and that He would have known how to do this in many ways which are unthinkable to our minds. From this I forthwith conclude that, this being so, it would be excessive boldness for anyone to restrict the Divine power and wisdom to some particular fancy of his own. The reference to the Pope was unmistakable, and the offence was heavily compounded by Salviati’s ironic comment. An admirable and angelic doctrine, and well in accord with another one, also Divine, which, while it grants to us the right to argue about the constitution of the universe (perhaps in order that the working of the human mind shall not be curtailed or made lazy) adds that we cannot discover the work of His hands. Let us, then, exercise these activities permitted to us and ordained by God, that we may recognize and thereby so much the more admire His greatness, however much less fit we may find ourselves to penetrate the profound depths of His infinite wisdom.³⁴ There has been much discussion as to exactly why Galileo himself was condemned, but it seems clear that this thinly veiled insult to the Pope together with his open flaunting (in all but the letter) of the injunction to treat the Copernican system as no more than hypothetical would in themselves have provided ample reason. In any case he was called before the Inquisition in the following year, made to recant, and spent the rest of his life under house arrest. As we shall see, this did not prevent Galileo from preparing and having published another book of outstanding importance, but he naturally refrained from making any statements about the motion of the Earth. And for a while the events of 1632–3 did put a damper on discussions of the Copernican system, especially in Catholic countries. Descartes, for instance, had finished his Le Monde at about this time, but suppressed it because I learned that people to whom I defer and whose authority over my own actions can hardly be less than is that of my own reason over my own thoughts, had disapproved an opinion on physics published a little before by someone else.³⁵ But in general the condemnation probably had less lasting effect on the development than did natural inertia, and in both England and Scotland it was well into the second half of the seventeenth century before Aristotelian cosmology was displaced from university teaching. In the eighteenth century the end of the old cosmology was symbolized by the curt note inserted by the minimite friars Le Seur and Jacquier in their standard edition of Newton’s Principia. At the beginning of Book III, they wrote: In this Third Book Newton assumes the hypothesis of the motion of the Earth. The author’s propositions could only be explicated by our making the same hypothesis. Hence we are driven to don an alien persona. For the rest we promise to obey the decrees borne against the motion of the Earth by the high pontiffs.³⁶ And they then proceeded to elucidate the work with no further mention of the matter. MOTION AND MECHANICAL PHILOSOPHY The new cosmology necessitated a new theory of motion, for, as had been obvious from the time of Copernicus and even before, Aristotelian ‘mechanics’ could not accommodate a moving Earth. Some have even seen cosmological reform as providing the prime motivation for reform in mechanics, but the old system also had many strains of its own. Aristotle made a sharp distinction between the celestial and sublunary regions. The former, which we have already considered in summary fashion, was the province of the fifth element and of universal circular motions, but the latter was far more chaotic; as also was Aristotle’s account of it, seeing that his usual technique was to start from the situation in front of him and try to impose some semblance of order on it, rather than develop a new science axiomatically from first principles. All bodies were composed of a mixture of the four elements, earth, water, air and fire, and their basic behaviour was dominated by the doctrines of natural places and natural motions. The natural place of earth was at the centre of the universe and that of fire at the periphery of the elementary regions, with the other two elements being in between. All bodies aspired towards their natural places, so that a heavy, or predominantly earthy, body would tend to move downwards and a predominantly fiery one upwards. These motions were conceived mainly in terms of their final causes, and less attention was paid to the question of their efficient causes. This was not the case with violent motions, in which a body was moving against its own nature. If I am lifting a heavy body, I am clearly the efficient cause of its motion, but if I throw it upwards the situation is more difficult, since there is no obvious mover once it has parted company with my hand. Aristotle proved himself a model of consistency. The projectile has nothing in contact with itself except the air surrounding it: therefore the air must be the mover, and in the process of throwing it I must have communicated a power of continuing the motion to the air, which would then, besides moving the projectile, pass the power on to the succeeding parts of itself. Despite the internal coherence of this scheme, it understandably drew much criticism from different cultural areas, and we find many thought experiments, such as those concerning the efficacy of shooting arrows by means of flapping the air behind them. Along with other Italians, the young Galileo saw these considerations as providing a fine stick with which to beat Aristotle. Like many of their predecessors these ‘radicals’ replaced the power communicated to the air with an internal moving force communicated to the projectile itself, which in the later Western Middle Ages was often known as ‘impetus’. But Galileo was different in that he came to realize that, even though he was giving anti-Aristotelian answers, he was still asking Aristotelian questions, and this led to his imposing a self-denying ordinance whereby he did not consider causes in his discussions of motion. This new attitude took root from around the beginning of the century, and received its most mature public expression in what is arguably his greatest work, the Discorsi e dimostrazioni matematiche, intorno à due nuove scienze of 1638, another dialogue between the three friends Salviati, Sagredo and Simplicio, but this time steering well clear of the dangerous question of the motion of the Earth. Galileo’s strategy in considering local motions (that is, motions according to place rather than to quality or quantity) was to split them into two components, horizontal and vertical. In an amusing exchange in the Dialogo, the unwary Simplicio is trapped by Salviati into admitting that a perfectly shaped ball rolling on a perfectly smooth horizontal surface would continue its motion indefinitely and with uniform speed. The context is the question of the behaviour of stones let drop from the mast of a moving ship (relevant of course to arguments about Copernicus). The Aristotelian Simplicio had demanded experiment, but Salviati was adamant that without experiment he could demonstrate that they would fall at the bottom of the mast and not be left behind by the motion of the ship. For this purpose he made use of an ingenious thought experiment with inclined planes. Passages like this led Alexandre Koyré and others to lay great stress on what they saw as Galileo’s Platonic streak, and heated controversy continues concerning the importance of experiment in Galileo’s work as regards both the context of discovery and the context of justification, but no serious scholar would now go to Koyré’s extremes in denigrating its role. Although Salviati demonstrated to his companions’ satisfaction the uniform speed of unimpeded horizontal motion, a complication remained, for such a motion is in fact in a circle about the centre of the Earth. In the Dialogo Galileo made some play with this to illustrate the circle’s superiority over the straight line, but in the Discorsi he quickly approximates this with a straight line, and in doing so appeals to Archimedes (a particular hero of his), who in his On Floating Bodies had said that verticals could be treated as parallel even though in fact they all pointed towards the Earth’s centre. Vertical motion needed two considerations. In Day 1 of the Discorsi Salviati argued that in a vacuum all bodies, whatever their density, would fall with the same speed. A vacuum was unrealizable in practice, and so his main strategy employed thought experiments concerning ever rarer media, but he does cite somewhat exaggeratedly some actual experimentation with pendulums: we certainly do not need to consider here objects dropped from the Leaning Tower of Pisa. His more famous, and much discussed, argument about falling bodies occurs in Day 3, and concerns the acceleration of falling bodies. That they do accelerate had been known from time immemorial, but since at least the time of Aristotle most discussion had centred on why they did so. Salviati would have none of this. The present does not seem to me to be an opportune time to enter into an investigation of the cause of the acceleration of natural motion, concerning which various philosophers have produced various opinions, some reducing it to approach to the centre, others to there remaining successively less parts of the medium to be divided, others to a certain extrusion of the ambient medium which, in being rejoined at the rear of the mobile, is continually pressing and pushing it; which fantasies and others like them it would be appropriate to examine and resolve but with little gain. For now, it suffices our Author that we understand that he wishes to investigate and demonstrate to us some properties of a motion accelerated (whatever be the cause of its acceleration) in such a way that….³⁷ The structure of Day 3 is that Salviati reads aloud a Latin treatise by Galileo, which the friends concurrently discuss in Italian. The form of Galileo’s Latin text is mathematical. After a brief introduction, there follow (without interruption for comment) one definition, four axioms and six theorems concerning uniform motion. We then move on to accelerated motion, for which ‘it is appropriate to search for and explicate a definition that above all agrees with what nature employs’.³⁸ The basic criterion of choice was simplicity. When therefore I observe a stone falling from rest from on high to acquire successively new increments of speed, why should I not believe these increments to be made in the simplest way and that most accessible to everyone? And, if we consider attentively, we shall find no addition and no increment simpler than that which is applied always in the same way…. And so it seems in no way discordant with right reason if we accept that intensification of speed is made according to extension of time, from which the definition of the motion that will be our concern can be put thus: I call an equably or uniformly accelerated motion one which proceeding from rest adds to itself equal moments of swiftness in equal times.³⁹ An obvious point, and one that is made by Sagredo, is that it might be clearer to say that speed increased proportionally with distance rather than with time. Salviati reports that Galileo himself had once held this view, but that it was in fact impossible, and indeed we may say that, if falling bodies did behave in this way, they could not start naturally from rest but would remain suspended as it were by skyhooks until given a push. Galileo adds to the definition a principle that will allow him to draw into his discussion of falling bodies the behaviour of balls rolled down inclined planes. ‘I accept that the degrees of speed acquired by the same mobile on different inclinations of planes are equal when the elevations of the planes are equal.’ After Salviati has adduced some experimental evidence from pendulums for this, he proceeds to quote Galileo’s first two theorems on accelerated motion. The first states that The time in which a space is traversed by a mobile in uniformly accelerated transference from rest is equal to the time in which the same space would be traversed by the same mobile carried with a uniform motion whose degree of speed was half of the last and highest degree of speed of the former uniformly accelerated motion.⁴⁰ The medieval version of this theorem is now usually referred to as the Merton rule, after its probable origins in Merton College, Oxford, in the first half of the fourteenth century. There has been much discussion of the possibility of medieval influence on Galileo in this regard, but it seems ill-advised to look for anything more specific than the important step of (implicitly or explicitly) representing the intensities of speeds by segments of straight lines. The second theorem and its corollary descend from speeds, or their intensities, to distances. The theorem shows that in a uniformly accelerated motion the distances traversed are as the squares of the times, and the corollary that the successive distances traversed in equal times, starting from rest, are as the successive odd numbers 1, 3, 5, 7,…. We are now reaching directly measurable quantities, and so the stage is set for Simplicio to ask for experimental evidence to show whether this is in fact the mode of acceleration employed by nature in the case of falling bodies. Salviati willingly complies. As a true man of science [scienziato] you make a very reasonable demand, and one that is customary and appropriate in the sciences which apply mathematical demonstrations to physical conclusions, as is seen with perspectivists, with astronomers, with mechanicians, with musicians, and others who confirm with sensory experiences their principles, which are the foundations of the whole succeeding structure.⁴¹ The experiments involved rolling well-prepared bronze balls down different lengths of an extremely smooth channel arranged at different inclinations to the horizontal, and measuring the times of their descents by water running out of a hole in a pail of water. One’s initial impression may be that the set-up smacks of Heath Robinson (or Rube Goldberg), but repetitions of the experiments in recent years have shown that they can be surprisingly accurate, and so we may say that they did indeed give Galileo strong support for his ‘law’ of falling bodies. The burden of the Fourth Day of the Discorsi is to combine the horizontal and the vertical, and so produce a general description of the behaviour of unimpeded ‘natural’ motion. The result is the famous parabolic path for projectiles. This was admirable mathematically but less satisfactory empirically for, as gunners and others were quick to point out, the maximum horizontal trajectory was not to be achieved by firing at an elevation of 45°, nor did actual cannon balls follow a neat mathematical parabola.⁴² All this goes to show the extent to which, despite his practical rhetoric, such as that provided by setting the Discorsi in the Arsenal at Venice, the success of Galileo’s mechanics depended on his focusing on ideal situations and ignoring many of the messy complexities of the actual physical world. The extent of Galileo’s reliance upon mathematics, sometimes to the neglect of exact correspondence with empirical facts, may have worried a good Baconian, but to his younger contemporary René Descartes the principal fault lay in a different direction. Writing to Marin Mersenne in 1638, shortly after the publication of the Discorsi, he gave the opinion that, although Galileo ‘philosophized much better than most, yet he has only sought the reasons of certain particular effects without considering the first causes of nature, and so has built without foundation’.⁴³ This was to be seen as contrasting with Descartes himself, whose science of mechanics depended intimately upon both his method and metaphysics, and whose Discours de la Méthode, together with the Diotrique, Météores and Géométrie, had been published in the preceding year. And before that Descartes had almost completed his Le Monde, a major work on natural philosophy, which, as noted above, he then suppressed because of Galileo’s condemnation in 1633. After Descartes had in a familiar manner proved to his own satisfaction the existence of himself, of God, and of an external physical world, he was in a position to consider more exactly the nature of the last of these. And a very austere picture it was that he had of it. As he put it in the Principia Philosophiae (published in 1644), if we attend to the intellect, rather than the senses, We shall easily admit that it is the same extension that constitutes the nature of body and the nature of space, nor do these two differ from each other more than the nature of the genus or species differs from the nature of the individual. If while attending to the idea which we have of a body, for example a stone, we reject from it all that we recognize as not required for the nature of body, let us certainly first reject hardness, for if a stone is liquefied or divided into the minutest particles of dust, it will lose this, but will not on that account cease to be body; let us also reject colour, for we often see stones so transparent as there were no colour in them; let us reject heaviness, for although fire is so light, we do not the less think it to be body; and then finally let us reject cold and heat, and all other qualities, because either they are not considered to be in the stone, or, if they are changed, the stone is not on that account thought to have lost the nature of body. We shall then be aware that nothing plainly remains in the idea of it other than that it is something extended in length, breadth and depth, and the same is contained in the idea of space, not only full of bodies, but also that which is called a vacuum.⁴⁴ In this way the physical universe is reduced to characterless matter swirling around in the famous Cartesian vortices, and by its actions on our sense organs producing our perceptions of all the different qualities. Descartes’s physical universe has certain similarities with that of the ancient atomists, but there were important differences. In the first place, matter for Descartes was not composed of indivisible atoms moving in void, but constituted a continuous plenum without empty spaces, although for most scientific purposes one could focus on the three different sorts of particles that were separated out, in a manner similar to relatively self-subsistent eddies, by the vortical motion. These were in a way a replacement for the Aristotelian elements. The first comprised extremely subtle and mobile matter, the second small spherical globules, and the third larger particles which were less apt for motion. And from these three we show that all the bodies of this visible world are composed, the Sun and the fixed stars from the first, the heavens from the second, and the Earth together with the planets’ and the comets from the third.⁴⁵ In this way one is relieved from the almost impossible task of thinking all the time starkly in terms of the motions of fundamentally undifferentiated continuous matter. A perhaps more important difference from ancient atomism was that the motions of matter were by no means random, and Descartes’s system was also more effectively rule-governed than those of Empedocles, with his guiding principles of Love and Strife, and of Anaxagoras with that of Mind. Descartes believed that any system of matter created by God would obey certain laws of motion that followed from God’s unchangeableness. The first is that every body would remain always in the same state unless this was changed by the action of external forces, and Descartes was adamant that there be included in this (contrary to the Aristotelian tradition) a body’s state of motion. The second46 is that the motion of any part of matter always tends to be in a straight line, and this when added to the first gives a fair approximation to Newton’s first law of motion, the principle of rectilinear inertia, which may rightly be regarded as the foundation stone of classical mechanics. The third law is rather more complicated: it concerns the collisions of bodies, and asserts inter alia that in these the amount of motion is conserved. From it Descartes deduces some hideously incorrect laws of impact, which older fashioned histories of mechanics used often to crow over, but this was to neglect the fact that, as regards historical influence, the form of Descartes’s discussion was far more important than its exact content. Descartes’s first two laws in particular led to some novel questions about forces. For instance, the acceleration of falling bodies implied the existence of a force to bring about this change from its preferred state of uniform rectilinear motion, and similarly did the deviation of the planetary orbits from this state. Armed with hindsight, the reader will recall how important these questions were for Newton on his journey towards the principle of universal gravitation, but this serves to show up important differences between the two men. Whereas Newton asked typically how big the forces were (at least as a preliminary to deeper explanation), Descartes sought for their causes, and in both cases specified these elaborately, but still vaguely, in terms of what we may call differential pressures from the vortices. Compared with Newton on this and other issues, Descartes appears a qualitative rather than a quantitative scientist, and this ties in with another important methodological difference. Newton, at least rhetorically within the context of justification, was a strong inductivist. In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.⁴⁷ For Descartes and his followers, however, hypotheses played a crucial role. The general structure of matter and the general laws of motion could be reached purely by deduction, but this process could not proceed unaided to unique explanations of particular phenomena. When I wished to descend to [effects] that were more particular, so many different ones were presented to me that I did not think it possible for the human mind to distinguish the forms or species of bodies that were on Earth from an infinity of others which could have been there if it had been the will of God to put them there, nor consequently to relate them to our use without coming to the causes by means of the effects and employing several particular experiences. Following this, in passing my mind again over all the objects which were ever presented to my sense, I indeed dare to say that I have not remarked there anything that I could not explain suitably enough by the principles that I have found. But I must also admit that Nature’s power is so ample and so vast and that my principles are so simple and so general that I hardly remark any particular effect without immediately recognising that it can be deduced in many different fashions, and that my greatest difficulty is usually in finding upon which of these fashions it does depend. And for this I know no other expedient than to seek once again some experiences which are such that their outcome will not be the same if it is in one of the fashions that one must explain it as it will be if it is in the other.⁴⁸ In this way Descartes gives a reasonably clear, if not unproblematic, expression of what is often referred to as a hypothetico-deductive methodology, of a kind which was employed by many notable scientists of the later seventeenth century. Descartes was not alone in producing a mechanical philosophy— one need only think of the work of Pierre Gassendi and of Thomas Hobbes—but it was his system together with later developments and modifications that was most generally influential. And when Aristotelian natural philosophy began at last to be displaced from the universities it was usually replaced by a version of Cartesianism, although this enjoyed a relatively short reign before succumbing to the incursions of Newtonianism. We may thus conveniently regard Descartes’s work as representing the culmination of the first phase of the Scientific Revolution. NOTES 1 The Works of John Play fair, Esq…. with a Memoir of the Author (Edinburgh, Archibald Constable, 1822), vol. 2. This was being completed at the time of his death in 1819; cf. vol. 1, pp. lxi-lxii, vol. 2, pp. 3–4. For helpful comments on an earlier draft of this chapter I am very grateful to Steven J.Livesey and Jamil Ragep. 2 W.Whewell, History of the Inductive Sciences, from the Earliest to the Present Time (London, Parker, 3 vols, 3rd edn, 1857), vol. 1, p. 181. 3 cf.Ferguson [3.10], ch. 11. 4 cf. A.Koyré, Etudes d’Histoire de la Pensée Philosophique (Paris, Armand Colin, 1961), pp. 279–309. 5 Oxford English Dictionary, s.v. 6 But cf. J.Needham, The Grand Titration: Science and Society in East and West (London, Allen & Unwin, 1969), pp. 276–85. 7 cf. N.W.Gilbert, ‘A Letter of Giovanni Dondi dall’Orologia to Fra’Guglielmo Centueri; A Fourteenth-Century Episode in the Quarrel of the Ancients and the Moderns’, Viator 8 (1977) 299–346. 8 See for example A. Keller, ‘A Renaissance Humanist Looks at “New” Inventions: The Article “Horologium” in Giovanni Tortelli’s De Orthographia’, Technology and Culture 11 (1970) 345–65. 9 Zilsel [3.56]. 10 cf. K.R.Popper, Objective Knowledge: An Evolutionary Approach (Oxford, Clarendon, 1972), ch. 3; Molland [3.51]. 11 I.B.Thomas, Selections Illustrating the History of Greek Mathematics (London, Heinemann, 1939), vol. 2, p. 31. 12 Descartes [3.36], vol. 10, p. 373. 13 Pappus of Alexandria, Book 7 of the Collection, ed. A.Jones (New York, Springer, 1986), p. 82. 14 See, for instance, R.Robinson, ‘Analysis in Greek Geometry’, in R.Robinson, Essays in Greek Philosophy (Oxford, Clarendon, 1969), pp. 1–15; M.S. Mahoney, ‘Another Look at Greek Geometrical Analysis’, Archive for History of Exact Sciences 5 (1968–9) 318–48; J.Hintikka and U.Remes, The Method of Analysis: Its Geometrical Origin and General Significance (Dordrecht, Reidel, 1974). 15 Aubrey’s Brief Lives, ed. O.L.Dick (London, Seeker & Warburg, 1949), p. 130. 16 L.Prowe, Nicolaus Coppernicus (Berlin, Weidmannsche Buchhandlung, 1883–4), vol. 2, p. 185. 17 Copernicus [3.33], sig. iii.v. 18 Luther’s Worksy Volume 54: Table Talk, ed. and trans. Theodore G.Tappert (Philadelphia, Penn., Fortress Press, 1967), p. 359. 19 Westman [3.72]. 20 Copernicus [3.33], sig. i.v. 21 This is the date of his matriculation; he did not actually move to Tubingen until 1589. See the article on him by Owen Gingerich in the Dictionary of Scientific Biography. 22 Kepler [3.46], vol. 8, p. 9; Kepler [3.47], 38–9. 23 Kepler [3.46], vol. 3, p. 178. 24 Kepler [3.46], vol. 3, p. 366. 25 Astronomia Nova , seu Physica Coelestis, tradita commentaniis De Motibus Stellae Martis, Ex Observationibus G.V.Tychonis Brahe. 26 The Correspondence of Isaac Newton, ed. H.W.Turnbull et al. (Cambridge, Cambridge University Press, 1959–77), vol. 2, p. 436. 27 Galileo [3.43], 50–1. 28 Galileo [3.43], 57. 29 Galileo [3.43], 93–4. 30 Galileo [3.43], 186. 31 Finocchiaro [3.65], 67–8. 32 Finocchiaro [3.65], 146. 33 Galileo [3.44], 416–17. 34 Galileo [3.44], 464. 35 Descartes [3.36], vol. 6, p. 60. 36 Isaac Newton, Philosophiae Naturalis Principia Mathematica…. Perpetuis Commentariis illustrata, communi studio pp. Thomae Le Seur & Francisci Jacquier Ex Gallicana Minimorum Familia, Matheseos Professorum (Geneva, Barrillot, 1739–42), vol. 3. 37 Galileo [3.40], vol. 8, p. 202. In producing my own translations from the Discorsi, I have made use of those by Stillman Drake in Galileo [3.45], whose volume includes the page numbers from the Edizione Nazionale. 38 Galileo [3.40], vol. 8, p. 197. 39 Galileo [3.40], vol. 8, p. 198. 40 Galileo [3.40], vol. 8, p. 208. 41 Galileo [3.40], vol. 8, p. 212. 42 cf. A.R.Hall, Ballistics in the Seventeenth Century: A Study in the Relations of Science and War with reference particularly to England (Cambridge, Cambridge University Press, 1952), and M.Segre, ‘Torricelli’s Correspondence on Ballistics’, Annals of Science 40 (1983) 489–99. 43 Descartes [3.36], vol. 2, p. 380. 44 Principia Philosophiae II. 11, in Descartes [3.36], vol. 8–1, p. 46. 45 Principia Philosophiae III. 52, in Descartes [3.36], vol. 8–1, p. 105. 46 The numbering is from the Principia Philosophiae (II. 37–40); it was different in Le Monde, where the laws are deduced rather more vividly from God’s creation of an imaginary world. 47 Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and his System of the World, trans. Andrew Motte, revised by Florian Cajori (Berkeley, Calif., University of California Press, 1934), p. 400. 48 Descartes [3.36], vol. 6, pp. 64–5. BIBLIOGRAPHY General and background works 3.1 Boas, M. 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