Mathematics and nature in the chemical texts of the Renaissance

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FEBRUARY, 1968 Vol. XV AMBIX t The Journal of the Society for the Study of Alchemy and Early Chemistry CONTENTS page MATHEMATICS AND NATURE IN THE CHEMICAL TEXTS OF THE RENAISSANCE. By ALLEN G. DEBUS • 1 "MATTER IN A NUT-SHELL": NEWTON'S OPTICKS AND EIGHTEENTH CENTURY CHEMISTRY By ARNOLD THACKRAY 29 THE HARTLIB CHEMISTRY. 54 REVIEWS PAPERS AND SEVENTEENTH-CENTURY By RONALD STERNE WILKINSON • PUBLICATIONS • RECEIVED w. HEFFER • • • • & SONS LTD. CAMBRIDGE, ENGLAND • •

Ambix The ] ournal of the Society for the Study of Aichen1Y and Early Chen1istry A/embers ojCouncil Dr. W. H. BROCK F. GREENAWAY,Esq. (Han Secretary) Dr. A. G. DEBUS G. HEYM, Esq. DENIS I. DUVEEN, Esq. Dr. W. A. SMEATON(Han. Treasurer) Prof. R. J. FORBES Dr. E. ASHWORTHUNDERWOOD D. GEOGHEGAN,Esq. (Han. Editor) Dr. T. I. WILLIAMS (Chairman) VOL. XV MATHEMATICS FEBRUARY, No. 1968 AND NATURE IN THE CHEMICAL OF THE RENAISSANCE I TEXTS By ALLEN G. DEBUS· By the late seventeenth century the triumph of our modern mathematicized science would seem to have been fully conlirmed. One need only look at the first sentence of the preface of the Principia Mathematica to flavour the main theme of the new scienceSince the ancients (as we are told by Pappus) esteemed the science of mechanics of greatest importance. in the investigation of natural things, and the moderns, rejecting substantial forms and occult qualities, have endeavoured to subject the phenomen of nature to the laws of mathematics, I have in this treatise cultivated mathematics as far as it relates to philosophy.l • Associate Professor of the History of Science, the University of Chicago, Chicago, Illinois 60637, U.S.A. I Isaac Newton, Mathematical Principles of Natural PlIilnxophy, trans. Andrew Motte (1729), revised with historical and explanatory appendix by Florian Cajori (Berkeley: Univ. of California Press, 1934), p. xvii.

Even fifty years earlier Galileo had insisted that the power of mathematical demonstrations "prepares the mind to understand and ascertain other facts without need of recourse to experiment".2 The history of the past three centuries shows an ever increasing emphasis on mathematics and quantification in the study of nature. The significance of the emergence of a quantified science has never been ignored by historians of science, but for the most part non-modern views of the role mathematics should play have been neglectecl.3 In an important monograph Pauli called our attention to the Kepler- Fludd debate as a crucial chapter in the rise of the new science in which the "new, quantitative, scientifically mathematical way of thinking collided with the alchemical tradition expressed in qualitative, symbolical pictures".4 Yet, Fludd himself had argued no less passionately than Kepler that we cannot understand nature without mathematics as a guide. In the sixteenth and early seventeenth centuries there was surely no argument over the necessity of mathematics in sciencethere was, however, considerable disagreement as to how it might best be applied. Since the Paracelsians offered a universal view of nature which clashed with that of the rising mechanical philosophers,6 and since the place of mathematics in science was a point of contention between the two schools, it seems not inappropriate to devote this paper to the place allotted to mathematics in the study of nature as seen by Renaissance iatrochemists. \Ve shall fmd that the mathematics of Paracelsus and Robert Fludd bears a close affinity with the work of the mystical nco-Platonists, Pythagoreans and natural magicians of this period. At the same time we sense a reaction against mathematical abstraction in science-a feeling which finds its culmination in the work of van Helmont who would have happily limited the intrusion of this subject into nature to calculations based 011 weight determinations. Both of these views 2 Galileo Galilei, Dialogues Concerning Two New Sciences, trans. Henry Crew and Alfonso De Salvio with an introduction by Antonio Favaro (New York: Macmillan, 1914), p. 276. 3 The reader is particularly directed to the June, 1961, issue of Isis (vol. 52), which includes the papers given at the "Conference on the History of Quantification in the Sciences" held in Nov., 1959. 'Wolfgang Pauli, "The Influence of Archetypal Ideas on the Scientific Theories of Kepler", trans. Priscilla SHz in C. G. Jung and W. Pauli, The Interpretation of Nature a1~d the Psyche (London: Routledge and Kegan Paul, 1955), p. 205. liOn the chemical philosophy of the Paracelsians see Allen G. Debus, The English Paracelsians (London: Oldbourne Press, 1965; New York: Franklin Watts, 1966); Allen G. Debus, "Renaissance Chemistry and the Work of Robert Fludd", included in Alchemy atld Chemistry in the Seventeenth Century (William Andrews Clark Memorial Library: lJniv. of California, Los Angeles, 1966), pp. 1-29; ibid., (reprinted in a slightly revised version), A mbix, 14 (1967), pp. 42-59.

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE may be found in the writings of Nicholas Cusanus, and they both carry religious overtones. It is to be hoped that the present study will contribute to our understanding of the background of the Scientific Revolution in pointing out that a significant proportion of the scientific community was moving away from the modem concept of mathematical abstraction in natural philosophy at precisely the time when that concept was beginning to show its power in the hands of the mechanical philosophers. A. THE BACKGROUND OF THE PROBLEM The fourteenth century scholastic analysis of Aristotelian mechanics is today rightly considered to be one of the well-springs of modern science. The scholars of Merton College and Paris had founded a critical tradition which was still alive in the sixteenth century and which influenced the young Galileo. Yet, while this may be readily admitted, it is true also that scholasticism was under heavy attack in the Scientific Renaissance. The Theology and Philosophy of the schools seemed to many to have become enmeshed in arid verbal controversies, and an ever increasing number of scholars were demanding something new to replace a system which they could no longer tolerate. One answer to this call may be found in a revived interest in the neo-Platonic and Hermetic writings-sources which were currently thought to have strong and valid connections with Christian thought and teaching.' With some this path of inquiry led to a search for truth in an alchemical cosmology, but, for nearly all, it resulted as well in a new emphasis on mathematics as a key to the universe. No better example of this may be found than in the work of Nicholas Cusanus (1401-64). For him both philosophical truth and Holy Writ lead us to the same conclusion, that a knowledge of mundane and divine truth may only be had through a knowledge of numbers. In Wisdom II: 17 we read that God created "all things in number, weight and measure".7 The Creation itself then may be interpreted as a mathematical process since arithmetic, geometry, music and astronomy are the "same sciences God employed when He made the world".8 One could hardly argue that such • See Frances A. Yates, Giordano Bruno and the Hermetic Tradition (Chicago: University of Chicago Press, 1964). 7 Nicolas Cusanus, The Idiot in Four Books. The first and second of Wisdome. The third of the Minde. The fourth of statick Experiments, 01' Experiments of the Ballance (London: William Leake, 1650), p 172. Nicolas Cusanus, Of ~earned Ignorance, trans. Fr. Germain Heron, O.F.M., Ph.D. (London: Routledge and Kegan Paul, 1954), p. 119. I Cusanus, Of Learned Ignorance, p. 118. "With arithmetic the Creator adjusted the World to unity, with geometry he balanced the design to give it stability and controlled movement while with music its parts were so allotted that there should be no more earth in the earth than water in the water, than air in the air or than fire in the fire, so that no element could be wholly transmuted into another: whence it comes that the physical system cannot sink into chaos".

a process could not be studied profitably in a mathematical way, and that from it we might learn of our Creator, since we would be using his own method of Creation as our guide. Yet, for Cusanus more than Biblical truth leads us to this conclusion. Philosophers agree that the visible universe is a faithful reflection of the invisible one and we may surely expect to rise to a firm knowledge of our Creator if we properly study the creatures and objects of our world.9 But a study of the multiple beings in nature can only lead to our belief in an underlying unity pervading all things. We don't know why, but we know for a fact that all things stand in some sort of relation to one another, that, in virtue of this inter-relation, all the individuals constitute one universe and that in the one Absolute the multiplicity of beings is unity itself.10 One may, then, use experimental science as an approach to truth, and surely Cusanus did just this in his book of static experiments. But no amount of experimental knowledge can give us complete truth since the things of this world are not perfect. If possible we must begin our search in the perfect world of ideas and here we-like so many philosophers in the past-may rely on mathematics which starts with the finite, but allows us to progress confidently to the simple infinite. It is quite proper to use an image in our search for reality -provided that we can be certain of its validity. In this guise we may confidently rely on mathematics as the perfect example of abstract truth. That explains why philosophers so readily turned to mathematics for examples of the things which the intellect had to investigate; and none of the masters of old, when solving a difficulty used other than mathematical illustrations, so that Boethius, the most learned of the Romans, went so far as to say that knowledge of things divine was impossible without some knowledge of mathematics.ll Cusanus could call with confidence on time-honoured authority for support. Not only might the reader tum to Boethius, he would also learn of the indispensable nature of mathematics in the writings of the Pythagoreans, St. Augustine and Plato. Even Aristotle although he had disagreed with his master had proceeded to use mathematical examples.a • Ibid., p. 25. For a comparison of man with nature, Cusanus turned to Plato-"The earth, as Plato says, is like some vast animal whose veins are rivers and whose hairs are the trees; and the animals that feed among the hairs of the earth are as the vermin to be found in the hair of beasts". Ibid., p. 119. 10 Ibid., p. 25. 11 Ibid., p. 26. Ibid.

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE 5 There is no question that the Pythagoreans were thought to be the ultimate source for those who believed in mathematics and number as the true key to nature. Aristotle had complained that their system really was not mathematics or the legitimate study of number. Rather, he had explained, all their discussions and interests concern nature: for they explain the generation of the heavens and carefully note the events, changes, and revolutions in the celestial regions. Thus, they employ all the principles and explanations, as if they agreed with the physicists that all being is perceptible and is contained under the vault of heaven. But their explanations and principles, as we have said, are suitable for even most exalted beings, more suited to them, in fact, than to nature.13 Indeed, Aristotle understood the Pythagorean Creation in terms of material numbers which formed the elements and all other substances.a Although the concept of material numbers had at best a limited vogue, the cosmological significance of mathematics was strongly championed by Plato. For him this science surely seeks eternal truth rather than the knowledge of perishable and transient matter.u; True astronomers (mathematicians) properly lift up their thoughts to divinity in their studies. It is they who see the Creation in all its aspects as a mathematically perfect structure.16 It is understandable that these views were echoed by many neo-Platonic and neo-Pythagorean authors in late antiquity. Many examples of this might be offered, but few would be better than the remarks of Nicomachus of Gerasa (second century A.D.) in his !ntroductio Arithmetica: All that has by nature with systematic method been arranged in the universe seems both in part and as a whole to have been determined and ordered in accordance with number by the forethought and the mind of him that created all things; for the pattern was fixed, like a 18 Aristotle, Metaphysics, Newly trans. as a postscript to natural science with an analytical index of technical terms, trans. Richard Hope (New York; Columbia V.P., 1952), p. 25 (989b). U Ibid., pp. 15 f. (986a,b). It need hardly be said that the purpose of referring briefly to the Pythagoreans, Plato, Nicomachus and other pre-Renaissance authors is only to point out the persistent interest in cosmic harmony and the divine goal of mathematics. It is not our intent to examine in detail the views of any of these authors. 16 Plato, The Republic, in The Dialogues of Plato, trans. with Analyses and Introductions (4 vols., Oxford: Clarendon Press, 1871), 2, p. 362 (Book 7-526). 16 Ibid., 2, p. 365 (Book 7-530). The concept of numerical perfection runs throughout the Platonic dialogues. The Timaeus is particularly important for this concept in reference to the universe. One Platonic theme which recurs regularly in the Renaissance chemical literature is the insistence that two extremes cannot exist without a mean. In the Timaeus this argument is used to show the necessity of a third element between Fire and Earth (3IB-32C). This "argument of the mean" was used by the Paracelsians in proof of their three principles. Debus, English Paracelsians, pp. 93-94 and passim.

preliminary sketch, by the domination of number preexistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and eternal essence, so that with reference to it, as to an artistic plan, should be created all these things, time, motion, the heavens, the stars, all sorts of revolutionsP Neo-Platonic thought was particularly persuasive to many early fathers of the Church. For St Augustine, Platonism was the philosophy which came closest to Christianity in truth,I8 and it was not insignificant that Cusanus had called on his readers to read this Father for proof of the need of mathematics in the search for divine truth. At the same time, in his account of the Creation, Augustine, like other contemporary Platonists, referred specifically to the perfect qualities of various numbers, and he insisted that it had not been without reason that the world had been created in terms of number, weight and measure.IO The evident similarities between Platonism and Christianity recognized by Augustine and others had impressed elements of this philosophy on Christianity in late antiquity. As a result, neo-Platonism was given a degree of respectability not shared by other ancient systems of thought. Number mysticism is found in the works of Macrobius and Martianus Capella which enjoyed great popularity in the early medieval period, and the Pythagorean De Institutione Aritllmetica and De Institutione Musica of Bocthius were basic texts which continued to be frequently cited into the Renaissance.2o Crombie has drawn our attention to Thierry of Chartres (twelfth century) who suggested that God had formed the universe in accordance with the mathematical ideas in His mind. Thus, only by mastering the mathematics of the quadrivium might a scholar properly understand the Creation account in Genesis.21 Similarly, the views of Roger Bacon fit in well with the earlier development of the subject. Of the four great sciences "the gate and key is mathematics, which the saints discovered 17 Nicomachus of Gerasa, Introduction to Arithmetic, trans. Martin Luther D'Ooge, with studies in Greek Arithmetic by F. E. Robbins and L. C. Karpinski (New York: Macmillan Co., 1926), p. 189 (Book I, chap. 4). 18 St. Augustine, The City o/God (Book 7, chap. 9). I have used the translation by John Healey with an introduction by Ernest Barker (3 vols., London and Toronto: J. M. Dent and Sons, Ltd., 1931), 2, pp. 46 fi. 11 Ibid., 2, p. 215. 10 Boethius is regularly referred to by authors interested in cosmic mathematics. The influence of both Cusanus and Boethius is to be found in the work of Jacobus Faber Stapulensis. An interesting example of this is his Epitome/compendiosq introductio in libros Arithmeticos diui Severini Boetij: adiecto /amiliari commentario dilucidata ... Insuper Astronomicon (Paris: Volphgangus hopilius et Henricus Stephanus, 1503). 11 A. C. Crombie, "Quantification in Medieval Physics", Isis, 52 (1961), 145.

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE at the beginning of the world".22 For Bacon there are two types of mathematics corresponding to the two great divisions of the universe. "Terrestrial things will not be known without a knowlcllge of the celestial, and the latter cannot be known without mathematics".23 Hut matter itself may be subjected to a geometrical investigation since by this method we may learn of the efficient and generating causes of those things which exist.24 Above all-we hear once again-true mathematics stirs us to reverence the Creator, and since there is no comparison in magnitude of things below with things celestial, the knowledge of things below will have no comparison with that of things above as regards their end, which is the praise and reverence for the Creator.25 In short, even a most limited group of examples in the period prior to the Renaissance shows that the Pythagorean- Platonic-Christian influence emphasized the importance of mathematics as a key to the Creation epic and consequently, to all nature. Cusanus rightly called on the testimony of earlier philosophers, for even if he might disagree with them over details, his concept of the goal of mathematics differed from theirs. B. OCCULT MATHEMATICS IN THE SIXTEENTH CENTURY Although the observational-experimental implications of Wisdom II: 17 were eventually to become predominant, the mystical-theological goal of mathematics continued to be a matter of overriding concern to fifteenth- and sixteenth-century scholars. All might agree that mathematics had manifold practical values for the businessman, the surveyor and the artisan, but at the same time few would argue with Plato that the beauty of higher mathematical proofs could 'only serve to prepare the mind for greater and more divine truths. Yet, if mathematics truly dealt with the Creation of the world, the generation and corruption of matter and other topics beyond the ken of the ordinary scholar, how might it best be treated? The neo-Platonic and neo-Pythagorean texts of late antiquity had been filled with accounts of angels, demons and magic. Because of this a distrust of this science had developed early. Roger Bacon had found it necessary to defend the true mathematics (which he derived from mathesis with a short middle syllable-and meaning knowledge) from the false mathematics or magic (derived from mathesi with a long middle syllable 22 Roger Bacon, Opus Majus, trans. Robert Belle Burke (2 vols., Philadelphia, 1928), I, p. II6. 18 Ibid., Ip, 129. 24 Ibid., I, p. 165. Ibid., I, pp. 200-1.

-meaning divination).26 The revived interest in neo-Platonic thought in the fifteenth century made this a question of vital interest. No one denied the existence of evil and demonic magic, but surely all forces in the universe must be natural in their operation since they all derive from the omnipotent Creator. It was only the goal of the magician which might be questioned. Most natural magicians of the Renaissance answered their opponents by affirming that their task was not to prepare spells, charms and incantations. Rather, they explored the wonders of God's created universe so that they might better comprehend their Creator. Since this was the meaning of the true magic, its practitioners could take pride in their goals and accomplishments. Pico della Mirandola insisted that his only interest was in this type of magic.27 But, essential to this is the true mathematics which is the formal part of physics.28 Here kabbalistic number mysticism-a popular study of Renaissance scholars-played as strong a role as the more formal Pythagorean tradition.29 We may be certain that with a knowledge of numbers we have Ibid., I, p. 261. Such derivations from similar words with radically different meanings Another example may be seen in Sir Christopher Heydon, A Defence of Iudiciall Astrologie, In Answer to a Treatise lately published by M. Iohn Chamber (Cambridge: John Legat, 1603), pp. 10-11. 17 Ioannis Pici, Mirandulae Concordiaeque Comitis, Opera quae extant omnia (2 vols., Basil: Sebastianum Henricpetri, 1601), 2, p. 114. 21 are common. 28 Ibid. Fifteenth and sixteenth century Christian scholars found a new stimulus for mystical numerical and geometrical studies in their fascination with the Hebrew Kabbalah and the Lullian system. In the kabbalistic tradition one may point to Reuchlin, Agrippa of Nettesheym and Georgius Venetus ("Zorzi") besides Pico. Because these studies were quickly applied to alchemical thought, they are of importance for an understanding of Paracelsian mathematics. Thus J. A. Pantheus (1518) discussed the art and theory of metallic transmutation in a kabbalistic system which assigned numbers to Latin, Greek and Hebrew letters. In a discussion of the four clements he gives each letter a number according to its place in the alphabet, adds the total, and then multiplies by eight. See J. A. Pantheus, A rs et Theoria Transmutationis M etallicae cum Voarchadumia, proportiollibus, 11umeris, b iconibus rei accomodis illustrata in Theatrum Chemicum, ed. Zetzner (6 vols., Argentorati: Zetzner, 1659-61),2, pp. 459-549, especially p. 472. The Renaissance symbolic and kabbalistic thought is well treated in Desiree Hirst, Hidden Riches (London: Eyre & Spottiswoode, 1963), and of special significance is F, Secret, L'Astrologie et les I(abbalistes CMetiens a la Renaissance (La Tour St.-Jacques, 1956). Ramon Lull claimed that encyclopedic knowledge would be at one's fingertips by means of tables, movable circles and geometrical patterns. Although Lull himself had little patience \ with alchemy, his disciples wrote numerous alchemical tracts which are filled with Lullian geometrical figures to aid the scholar in his understanding of divine secrets. The authorship of these tracts was-in typical alchemical fashion-assigned to Lull. An example of this is the widely known Testamentum Theoria b Practica printed in the Theatrum Chemicum, 4, pp. I-170. Here typical Lullian figures appear throughout the work. Giordano Bruno,

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE 9 a key to the proper understanding of everything which may be learned by man.30 Indeed, arising from mathematical books is a contemplative philosophy requisite for our further studies in theology.31 And since mathematics includes in its sphere the whole universe-things celestial and divine as well as things terrestrial and human--earlier philosophers were right in placing astronomy and astrology within its fold.32 The equating of astrology, astronomy and mathematics is a persistent theme in Renaissance texts and one which is referred to by authors of widely differing views. For some, mathematics really meant iatromathematics, the use of astrology in medicine.33 Leonard Digges opened a practical treatise on surveying with a defence of astronomy--or rather astrology-and mathematical sciences.34 On the other hand, Porta, in his Natural Magicl?, explained that magic is essentially the search for wisdom and seeks nothing else but -the "survey of the whole course of nature" .35 In this legitimate and worthy search the Godly magus is told that he "must know the Mathematical Sciences, and especially Astrologie".36 Similar cosmic views are to be found in Agrippa's De Occulta Philosophia. Here magic is depicted as the most perfect science, the consummation of philosophy.37 A magician, however, can do nothing without mathematics since all things were made and continue to be ruled over by number, weight, measure, harmony, motion and light. These are mathematical studies which are necessary for natural philosophy. vVhen they are mastered they may be put into practice with the mechanical arts with the an enthusiastic Lullist, was to find a strong Lullist strain in the work of Paracelsus. Walter Pagel, Parace/sus. An bttroduction tv Philosophical ~[edicille in the Era of the Renaissance (Basel: S. Karger, 1958). p. 245. Indispensable here is Frances A. Yates, "The Art of Ramon Lull: An approach to it through Lull's Theory of the Elements", J. Warburg Inst., 17 (1954), pp. 115-73. 30 Pico. As quoted by John Dee in his "Mathematicall Preface" to The Elements of Geometric of the most aUHcient Philosopher Euclide of Megara, trans. H. Billingsley (London: John Daye, 1570). sig. *i verso. 31 Pico, op. cit., 2, p. 285. 32 Ibid., 2, p. 488• 33 On the history of this subject sec Karl SlIdhoff, Iatromathematiker vor nehmlich im 15· und 16. Jahrhundert (Breslau, J. U. Kern, 1902). 3. Leonard Digges, A Prognostication of Right Good Effect, fructfully augmented, contayninge, playne, briefe, pleasant, chosen rules. to iudge the wether for euer, by the Sunne, Moone Sterres, Cometes, Raynbowe, Thunder, Cloudes, with other Extraordinarie tokens, not omitting the Aspectes of Planets, with a brefe Iudgemente for euer, of Plentie, Lache, Sichnes, Death, Warres be .... (London: Thomas Gemini, 1555). sig. A4 recto. 35 John Baptista Porta, Natural Magiek (London: Thomas Young and Samuel Speed. 1658), p. 2. as Ibid., p. 3. Henricus Cornelius Agrippa, De Oeculta Philosophia. Libri Tres [(Cologne), 1533], p. 2.

production of marvels and wonders.38 In addition, man himself might fruitfully be subjected to a numerical analysis. He is the most nearly perfect product of God's Creation and in him will be found the same composition according to weight, number and measure that we know exists in the macrocosm. Therefore, the proportions and hannonies we establish in the heavens and the earth we know must also be found by valid analogy in our own bodies.3u For Agrippa, these truths may easily-and very properly-be related to kabbalistic studies. There are few authors, however, whose work is more illustrative of Henaissance mathematics than John Dee. In his renowned "Mathematicall Preface" to the first English translation of Euclid, Dee proceeded to divide and subdiYide the many disciplines which he felt belonged within the realm of mathematics. Modern historians of science have praised him for his description of Archelll£lstrie which Hisnamed of some Scientia Experimcntalis. The Experimelltall Science" .40 Dee was certainly not unaware of recent publications of significance. Rattansi, for instance, has pointed to Dee's recognition of the work of Benedetti who, through his knowledge of the mathematical science of Statille, had shown the error in the general belief that heavier bodies fall faster than lighter ones.u Dee has been lauded also for his practical writings on astronomy, navigation and mathematics which seem to give an early indication of the significance and interrelation of technology and science. If, however, we read his works in greater detail, we find a repetition of the now familiar Platonic arguments. Mathematics is a divine science because we know the Creation was essentially a mathematical process. The Creator's Ibid., pp. 99-101. Ibid., p. 160. 40 Dee, "Mathematicall Preface", sig. Aiii recto. Archemastrie "teacheth to bryng to actuall experience sensible all worthy conclusions by all the Artcs Mathematicall purposed, & by true Naturall Philosophic concluded: & both addeth to them a farder scope, in the termes of the same Artes, & also by hys propre Method, and in peculiar termes, procedeth, with helpe of the foresayd Artes, to the performance of -eomplet ExperHkes, which of no particular Art, are hable (Formally) to be challenged.... And bycause it procedeth by Experiences, and searcheth forth the causes of Conclusions, by Experiences: and also putteth the Conclusions them selves, in Experience, it is named of some Scientia Experimentalis. The Experimentall Science". 41 Ibid., sig. of ci recto. "By these verities, great Errors may be reformed, in Opinion of the Naturall Motion of thinges, Light and Reauy. Which errors, are in Naturall Philosophie (almost) of all me allowed: to much trusting to Authority: and false Suppositions. As, Of any two bodyes, the heauyer, to moue downward faster then the lighter. This error, is not first by me, Noted: but by one Iohn Baptist de Benedictus. The chief of his propositions, is this: which seemeth a Paradox. If there be two bodyes of one forme, and of one kynde, aequall in quantitie or unequall, they will moue by aequall space, in aequall tyme: So that both theyr mouynges be in ayre, of both in water: or in anyone Middle".

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE II "numbryng ... was his Creatyng of all thinges, And his Continuall Numbryng, of all thinges, is the Conseruation of them in being".42 We find, therefore, that "The constant law of numbers ... is planted in thyngs Naturall and Supernatura11: and is prescribed to all Creatures, inviolably to be kept" :13 The mathematical sciences then rightly become a means for us to learn of our Creator. Arithmetic is, indeed, a subject little less useful than Theology,44 while the primary use of Geometry, as Plato teaches us, is to conceiuc, discourse, and conclude o( things, Int('llectual, Spirituall, aeternall, and such as conccrne our Blisse cucrlasting: which otherwise (without Speciall priuiledge of Illumination, or Reuelation fro heauen) No mortall mans wyt (naturally) is hable to reach unto, or to Compasse.46 Christian scholars can-and should-study numbers so that we may both winde and draw our sclues into the inward and deepe search and vew, of all creatures distinct vertues, natures, properties, and Formes: And also, farder, arise, clime, ascend, and mount up (with Speculative winges) in spirit, to behold in the Glas of Creation, the Forme of Formes, the Exemplar Number of all things Numerable: both visible and invisible: mortaH and immortall, Corporall and Spiri tuaU.46 Surely with this admitted, No man ... can doute, but towards the atteyning of knowledge incomparable, and Heauenly Wisdome: Mathematicall Speculations, both of Numbers and Magnitudes: are means, aydes, and guides: ready, certain, and necessary.47 Dee was a sincere alchemist and astrologer, and with this broader understanding of the primary purpose of mathematics his views in the "Mathematica11 Preface" may be seen to fit in closely with his Monas HieroglYPhica (1564), which is a mystical alchemical text, dedicated to the penetration of divine mysteries through an occult explanation of geometrical figures.48 Similarly Dee's description of astronomy relates this science to its major goal. 41 Ibid., sig. *i verso. n Ibid. Ibid., sig. ai verso. Ibid., sig. alii recto. 48 Ibid., sig. *i recto and verso. 47 Ibid., sig. aiii recto. 41 See the recent translation and commentary on the "Monas Hieroglyphica" by C. H. Josten, Ambix, 12 (1964), pp. 84-220. The first edition of this work appeared at Antwerp (1564) and the second at Frankfort (1591). Number mysticism in early alchemical texts has been discussed by H. E. Stapleton, "The Antiquity of Alchemy", Ambix, 5 (1953), 1-43.

It is true that astronomy teaches us the distances, magnitudes and motions of the stars and planets, but more important, we learn hereby of our Creator so that we may better glorify hi111.49 As a science it may well be used "for Consideratio of Sacred Prophesies, accomplished in due time, foretold: as for high Mysticall Solemnites holding".60 And if we pass on to a few of Dee's other mathematical sciences we see the familiar macrocosm-microcosm relationship develop. Astrology is such a science, for it treats quite simply of the "operations and efiectes, of the naturall beames of light, and secret influence: of the Sterres and Planets: in euery element and elementall body" .51 It is thus concerned with the actual connection of the celestial and terrestrial worlds. Anthropography, in tum, "is the description of the Number, Measure, Waight, figure, Situation, and colour of euery diverse thing, conteyned in the perfect body of MAN".52 This is the lesser world or microcosm which deserves a mathematical description no less than the greater world. Finally, natural magic called thaumaturgike by Dee, is devoted to the construction of "strange workes, of the sense to be perceiued, and of men greatly to be wondred at".63 Dee's views here are not dissimilar to those of Roger Bacon, Agrippa and Porta. Natural magic is truly a mathematical science pursued by men who "seketh ... in the Creatures Properties, and wonderfull vertues, to finde juste cause, to glorifie the Aeternall, and Almightie Creator".M It is a wicked and blasphemous thing for the ignorant to call these pious men conjurers-a term he had been called himself. C. THE MATHEMATICS OF THE PARACELSIANS Paracelsian thought may be considered as a special case of the neo-PlatonicHermetic philosophy of the Renaissance. With most Paracelsian authors the created universe remained mathematically inspired, but the major key to our understanding becomes chemistry, since the Creation is now considered primarily in terms of a chemical separation. With the divine mathematical Creation the scholar might progress in knowledge toward his Creator through a mystical concept of numbers. With the divine chemical Creation the Paracelsist might attain the same goal, but through his observations of natural objects and his chemical investigations by fire. The call for new experimental knowledge is common to most Paracelsians in the sixteenth and the seventeenth centuries. It was a plea which had its roots buried deeply in the past. In the Dee, "Mathematicall Preface", sig. bii recto. Ibid., sig. bii verso. 61 Ibid., sig. biii recto. 61 Ibid., sig. ciiii recto. IS Ibid., sig. Ai recto. 1i4 Ibid., sig. Ai verso.

MATHEMATICS AND NATURE IN CHEMICAL Hermetic texts the true philosopher of divine worship- TEXTS OF THE RENAISSANCE 13 is told to seek out knowledge as a form ... the student of philosophy undefiled, which is dependent on devotion to God, and 011 that alone, ought to direct his attcntion to the other sciences only so far as he may thereby learn to see and marvel how the returns of the heavenly bodies to their former places, their halts in pre-ordained positions, and the variations of their movements, are true to the reckonings of number; only so far as, learning the measurements of the earth, the depth of the sea, (the ... of air), the force of fire, and the properties, magnitudes, workings, and natures of all material things, he may be led to revere, adore, and praise God's skill and wisdom.55 Alchemical theorists of the Middle Ages relied heavily on such references, and in addition they called on familiar Biblical quotations. Thomas Norton, for instance, writing at about the same time as Nicholas Cusanus, rhymed For God made all things, and set it sure, In Number Ponder and in Measure.56 In short, traditional alchemy retained as its goal the glorification of the Creator, but because of its laboratory' orientation there was less interest on the part of alchemists in the mathematical concept of Creation. The shift in emphasis becomes apparent quickly when we turn to the works of Paracelsus. Here was a man who had no special interest in measurement and numbers. Yet, as Pagel has noted, Paracelslls relied on r'Visdom II: 17 when he insisted that "disease stands on weight, numher and measure", and we find in Paracelsian texts some of the earliest quantitative analyses of urine.57 Similarly, Paracelsus frequently uses the terms "Kabbalah" and 'kabbalistic", indicating a general acquaintance with this literature even though specific references to the mystical numerical interpretation of letters are lacking in his work.58 Nevertheless, it is primarily in the Astro1tomia M a~1'la that the l'ole of mathematics in philosophy is discussed in detail. Here we read that "Mathematica est Magica, Nigroma1ttica, Nectroma1ttica Astrologia, Signata, Artium incertarum, medicinae adeptae, Pltilosophia adeptae".59 If mathematics III Hermes Trismegistus, A sclepius in Herme/ica, ed., trans. and notes hy \Valtcr Scott (vol. 4 with A. S. Ferguson) (4 vols., Oxford, Clarendon Press, 1924, 1C)25, H)20, 1936), I, p. 311. III Thomas Norton, OrdinaU of AlcIJimy in Elias Ashmolc (cd.), Theatm1lt Chemicum Britannicum (London, 1(52). pp. 57-8. 17 Walter Pagel, Paracelsus. A 1t Introduction to Philosophical Medicine in the Era of the Renaissance (Basel: S. Karger, 1958), pp. 190-4, 281-2. 18 Ibid., p. 213. II Paracc1sus, Opera Omnia (3 vols. in two, Geneva: Joan. Alltonii & Samuclis De Tournes, 1(58), 2, p. 549.

is magic, true magic, in turn, may be equated with nature.60 The godly magus may concentrate in himself celestial virtues which are the hidden powers of nature. He may then use these powers to work wonders and learn of his Creator.61 We must distinguish, Paracelsus continues, between elementary mathematics and the mathematics of the adept since only the latter may be called magic. Elementary mathematics is concerned with visible and tangible objects such as the growth of a tree, the motion of the sun and the phases of the moon. The mathematics of the adept goes beyond this and its adepts are enabled to study visible, but non-palpable, objects such as the image of the sun in a river or the moon in a well. Also called sidereal mathematics, this subject examines generation and creation as well as essences and archidoxies which the practitioner may then associate with their proper geometrical figures according to their height, profundity, amplitude and angles. There is thus a real division between the two mathematics. The sidereal mathematician is the master of a subject which would hardly be given that name today. He can show that paper is nothing but wood, discover the real substance of the sun, and learn the differences of type and species. Such studies are possible only for the "Adepta Mathematica, Philosophia & Medicina".62 When late sixteenth-century Paracelsians m<,'ntionc<1mathematics other than in terms of weight measurements it was generally in these terms. Bostocke (1585) wrote that "the true and auncient phisicke" may be sought in nature "and is collected out of Mathematica11 and supernaturall precepts". This is a sacred art properly called Chymia Which sheweth foorth the compositions of all maner bodies, and their dissolutions, their natures & properties by labour by the fire, following Nature diligently. So that Philosophie naturall and supernaturall, the Mathematicals Chimia and Medicina be so combined together, that one of them can not be without the other.63 Mathematics is truly a mystical science for Bostocke, and as the goals of the sidereal mathematics of Paracelsus would indicate, it may be easily equated with the mystical chemistry of the philosophical alchemists. Similarly Thomas Ibid., 2, p. 558. Ibid., 2, p. 555. Nihil autem est alius, quam potentia, virtutes coelestes in Medium inducendi, & in illo operationis proficiendi. Medium hoc est centrum. Centrum est homo. Sic ergo per hominen vis coelestis in hominen induci potest, its, ut in homine tali virtus, constellationi illi congrua, inveniatur. 82 Ibid., 2, pp. 577-8. 81 R. B. Esq. (R. Bostocke), The difference betwene the aucient Phisicke, first taught by the godly forefathfrs, consisting in unity peace and concord: and the latter Phisicke. (London: Robert Walley, 1585), sig. Bi recto.

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE Tymme (1605) pictures the divine Creation as a universal harmony which has been produced in terms of weight, number and measure. Yet, knowing the author is a Paracelsian, the reader is hardly surprised to see all of this described as "Gods Halchymie".64 Among the early Paracelsians few authors were more concerned with numerical studies than Gerard Dorn. In his description of the Creation, and in his Physics of Trithemius, he continually referred to the power of mystical numerical analysis,65 while his Monarchia Triadis, £n Unitate, Soli Deo Sacra (1577) is a complex tract which attempts to show and relate the triune nature of divinity with the universe as a whole.56 Beyond tllis, Dorn finds a correspondence between divine unity, the Paracelsian principles, and the medicine of the Paracelsians. On the other hand, the evil of a binary system is related to the devil and finds its expression in the corrupt medicine of the Galenists. Although elements of the thought of Cusanus are present,67 the progression of geometrical figures which illustrate this tract are strongly reminiscent of John Dee's mystical Monas Hieroglyphica. Dorn's mathematical/geometrical theme is one which is recurrent among the Paracelsian authors.68 With no author did the alchemical universe find greater support than with Robert Fludd (1574-1637). An Oxford trained physician, Fludd was an honoured member of the Royal College of Physicians and a friend of \Villiam Harvey and Mark Ridley. Yet, he was far from being a "modern". Fludd wrote with sincerity about the mystical alchemical Creation, and he considered kabbalistic analysis to be a proper method in the scholar's search for divine secrets. His publications were controversial in tone and widely discussed. As such, they engaged the attention of some of the best minds in Europe-a group which included Kepler, Mersenne and Gassendi.69 In his first work, an apology for the Rosicrucians (1616), Fludd made a plea for a new learning. Nowhere did he feel that there was a more useless knowledge being spewed forth than at the Universities. These were the •• Josephus Quersitanus, The Practice of Chymicall, and Hermeticall Physiche, for the preseruation of health, trans. Thomas Timme (London: Thomas Creede, 1605), sig. A3 verso. IS The Physica Genesi and the Physica Trithemii form parts of the Liber de Natura luce Physica, ex Genesi desumpta which is printed in the Theatrum Chemicum, I. Similarly see Dorn's preoccupation with the sphere and other geometrical figures in the Clavis totius philosophiae Chemisticae per quam potissima philosophorum dicta reserantur in Ibid., I, pp. 192-361, p. 335· ea Gerard Dom, lvIonarchia Triadis, in Vnitate, Soli Deo Sacra in Paracelsus, Avrora Thesavrvsqve Philosophorvm (Basel: Palma Guarini, 1577). pp. 65-127. 17 Pagel, Paracelsus, p. 135. • 8 The incompatibility of the "medicine temarii" (or unity) with the "medicine binarii" was a major theme in Bostocke's Paracelsian apology of 1585 (see above, note 63). U On Fludd see the references cited above in footnote 5.

strongholds of the classical scholars who felt that truth might only be found in the writings of the ancients. One could only shudder at their reliance on Aristotle in philosphy and Galen in medicine, for in the ancient world none set forth doctrines more opposed to Christianity than they. Aristotle amI Galen had been heathens, and their followers were no better. The Universities must be reformed so that the divine light of Christian teachings could flourish. In a chapter on natural philosophy, medicine and alchemy, Fludd tells us that although innumerable authors have written on natural philosophy, they have presented to us only a shade of truth.70 They fill whole volumes with definitions, descriptions and divisions, and they drone on about the four causes. They lecture on motion, the continuum, the continguum-of termini, loci, of space, vacui, time and number. This for them is the basis of physics. They go forth then to generation, corruption and descriptive accounts of the heavens and the universe.71 "But, good God, how superficial and equivocal this all is".72 It is impossible for anyone to attain the highest knowledge of natural philosophy without a thorough training in the occult sciences.73 One must turn first to medicine, the most perfect science of all. There should not be understood by this term a simple description of diseases and the workings of the human body. Rather, it is the very basis on which natural philosophy must rest.74 Our knowledge of the microcosm will teach us of the great world, and this, in turn, will lead us to our Creator. Similarly, the more we learn of the universe, the more we will be rewarded with a perfect knowledge of ourselves. The vulgar medicine of the Galenist, the Astrologer, and the Paracelsist fails since it treats only of descriptive material of little value-of principles, elements, humors, the body and its parts, temperaments, virtues and the motion of the pulse.75 Yet, Hippocrates makes mention obscurely and mystically of a single medicine which cures all ills. From Pliny, Lull, Arnold, Rupescissa and others we learn that this medicine unites in peace the warring elements in the body.76 It is this medicine which must be sought in our reformed science. as the Apologia conzpendiaria, Fraternitatem de Rosea cruce maculis aspersam, veritatis quasi Fluctibus abluens et abstergens (Leyden, I616), I have used the enlarged second edition, Tractatus Apologeticus Integritatem Societatis De Rosea Cruce defendens (Lugduni Datavorum: Godfridum Dasson, 1617). Here see p. 91. A German translation is ayailable: Schutzschrijt fur die A echtheit der Rosenkreutzergesellschaft, trans. AdaMah Dooz (Leipzig: A. F. Bohme, 1782). 71 Fludd, Tractatus Apologeticus, pp. 91-3. 72 Ibid., p. 93. 73 Ibid. 74 Ibid., p. 89. 76 Ibid., pp. 95-7. 7' Ibid., pp. 97-9. 70 Originally suspicionis published et infamiae

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE 17 Even the art of alchemy from which we might expect so much, aids us little. Alchemists like Andreas Libavius do not call for a refonn but rather describe calcination, separation, conjunction, putrefaction and all the other operations by which men are deceived and persuaded to part with their money.77 True chemical authors are of a different sort and write of occult secrets. We need not vulgar, but natural fire, not artificial, but natural vessels. No miracles are performed when true alchemy is practised, only the work of nature.78 Real wisdom may be found in the writings of the natural magicians, men who are in truth, mathematicians.79 But if Fludd was distressed about the state of natural philosophy, medicine and alchemy in his day, he felt even more strongly about mathematics.80 The texts of arithmetic were filled with definitions, principles and discussions of theoretical operations. We learn of addition, subtraction, multiplication, division, golden numbers, fractions, square roots and the extraction of cubes.81 But to the old wisdom and doctrines of the Pythagoreans the Arithmeticians pay little heed. These men of old were silent about the arcane arithmetic, and like mystical alchemists they hid their profound mysteries from the vulgar crowd.82 And, Fludd continues, if we go beyond arithmetic to other mathematical studies, we find the same superficiality. Those who are really profound in these arts are considered impious and magicians.83 Indeed, we can learn more from the silent wisdom of the Pythagoreans than we can from the useless books of the philosophers. The disciples of Pythagoras reached a certainty of belief in God through their profound study of numbers and their ratios. In the same way, we may learn of the unity of Godin trinity, a theme continually stressed by these authors-and more, of the very fabric of the worlcl.84 But we look in vain for these secrets in the written records since there are no true remains of the Pythagorean teachings. It is a science which must be wholly restored.85 And what of music? This is a subject which properly forms a division of Ibid., pp. 100-1. Ibid., p. 101. 7t Ibid., p. 23. 80 Ibid., p. 103. 81 Ibid. 81 Ibid., p. 105. 81 Ibid. HI Ibid., p. 107. Fludd would have rejoiced in the publication in the same year of a book on mystical geometry by Fortunatus, Decas Elementorum Mysticae Geometrae Quibus Praecipua Divinitatis Arcana Explicantur (Padua: Peter Paul Tozzi, 1617). I am indebted to Walter Pagel for this reference. 8li Fludd, Tractatus Apologeticus, p. 108.

mathematics. Arc we to hear forever of harmonic systems, trite melodies, and sing ad irtjinitum, the musical scales ?86 Good God, what does this have to do with the true and profound music of the wise, a subject through which the proportions of natural things are investigated, and the harmonic concensus and properties of the whole world are revealed. . . .87 This science really deals with the joining of the clements, the proportions of light and weight in the stars, and their influence on our terrestrial world. Here we shall learn of the spiritual body of the sun and why the influence of Mars brings misfortune.8s An exact knowledge of these things may be had only from speculation and revelation in the occult science of music which shows how the microcosmic palace agrees in celestial harmony with the macrocosm.S9 Therefore happy will be he who is well versed in such mysteries of occult music since without a knowledge of these things it is impossible for anyone to know himself. And without this he will be unable to reach a perfect knowledge of God, for he who understands himself truly and intrinsically perceives in himself the idea of the divine Trinity .... 90 Fludd's proposals for the renovation of the remaining branches of the mathematical quadrivium may be passed over fairly rapidly since his object is already quite clear. In geometry he was aghast to find the subject dominated by the theorems, axioms and propositions of Euclid. Surely this is all far removed from the true and perfect geometry which is concerned with hidden secrets.91 Like many of his contemporaries whom we today point to as forerunners of modem science, Fludd calls on Archimedes as our guide to a new science. Why?-because of his wondrous machines.92 In effect Archimedes is to be considered as the archetype of the perfect natural magician. \Vho Ibid., p. log. Ibid., p. 109. Sed Deus bone quid hoc ad vcram & profundam sapicntis musicam, qua rerum naturalium proportiones investigantur, harmonicus consensus & totius proprietates revelantur .... 88 Ibid., p. 110. Similiter cur Mars etiam pro infortunio habetur, nisi propter extensionem suae consonantiae Diapason ad elementii suae naturae contrarium, videlicet Aquam. 88 Ibid., p. III. 80 Ibid. Foelix igitur erit qui in talibus Musices occultae misterijs bene est versatus, quippe sine quorum cognitione impossible est, ut quis seipsum cognoscat; quod quidem nisi fiat ad perfectam Dei cognitionem attingere non potest; nam qui seipsum ver~ et intrinsecus intelligit, ideam in se divinae Trinitatis percipiet ... 81 Ibid., p. 113. 81 Ibid., p. 114. For an example of Fludd's application of Arithmetic to nature see his demonstration of the mystery of the world's Creatic.n "by way of an Arithmeticall progression" in the Mosaicall Philosophy (London: Humphrey Moseley, 1659). pp. 73-4.

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE 19 today, he says, could build the masterworks of the past-the Colossus of Rhodes, the speaking birds and animals made of wood described by ancient authors, or even the brazen head of Roger Bacon? Similarly, in optics we now read descriptions of the membranes of the eye, the direction of rays and the refraction of colours. Instead we should be seeking the secrets of the mirrors which might destroy the Turks.93 Such wonders are described in the writings of Roger Bacon, but the technique of their manufacture would appear to be irredeemably lost. In short, the state of geometry is so bad that we are faced with a problem of not simply correcting errors, but totally restoring the lost treasures of this subject.94 The mathematical science of astronomy is really Astrology, but only pseudoastrology is being cultivated today because the practitioners of the art are ignorant of the arcana of nature. They tell us of the motions of the stars, the divisions of the earth, and they prepare vast astronomical tables. They must learn instead of the arcane effect of the stars.90 The celestial bodies are indeed the hieroglyphics of nature which man must learn to read.96 Mystical alchemists embellish their works with symbolic drawings. How much clearer this is than descriptions with words or numbers which have no hidden meaning. 97 Yet, if scholars might once learn to properly read the message of the stars which has been written for us by the Creator himself, they will have a new and irrefutable guide to nature. One can have no doubt that Fludd believed fervently in the use of mathematics for man's proper understanding of the universe. His meaning, however, derives ultimately from those mystical Pythagorean-Platonic and kabbalistic writings in which the subject assumes its major importance only when viewed as a key to cosmic mysteries. In this respect it is understandable that Fludd came into conflict with Kepler-a dispute which has been described so well by Wolfgang Pauli98 and which forms no part of this paper. It is true that Kepler too was a spiritual descendent of the Pythagoreans.99 He, too, seeks cosmic beauty in divine proportions, and his Mysterium Cosmographicum (1596) is famed for its relation of the planetary orbits to the regular solids. Pauli goes on to state that his ideas "reveal quite unmistakedly the influence of Paracelsus and his pUpilS".lOOHowever, in spite of these similarities, it is quite n Fludd, Tractatus Apologeticus, pp. Ibid., p. 114. t6 Ibid., pp. 1I6-8. " Ibid., pp. 32-3. 87 Ibid., p. 1I8. 18 Cited above in footnote 4. 98 Pauli, op. cit., p. 156. 100 Ibid., p. 157.

understandable that the true meaning of mathematics for Kepler was something quite different than it was for Fludd. The latter sought mysteries in symbols according to a preconceived belief in a cosmic plan. His proportions and harmonies were forced to fit these symbols. Kepler, perhaps just as obsessed with his symbolic spherical picture of the world, insisted that his hypotheses be founded on quantitative, mathematically demonstrable premises.10l If his hypothesis could not accommodate his observations, he was willing to alter it. These two views were so separated from one another that the two men could not really understand each other. Fludd f:t that he could honestly call Kepler one of the worst sort of mathematicians, one of the vulgar crowd who concern themselves with quantitative shadows; the alchemists and Hermetic philosophers, however, comprehend the true core of the natural bodies.l02 D. THE IATROCHEMICAL REACTION: THE WORK OF VAN HELMONT If the lofty and occult mathematics of the divine cosmology represents one stream of Renaissance thought which may be found in the writings of the Paracelsians, the influence of Wisdom II: 17 and other Biblical quotations led to other more mundane studies. In The Idiot Cusanus had additionally cited Proverbs 16: II ("The weight and the Ballance are the judgements of that Lord, who hath created") and Proverbs 8: 28 ("Who weighed the fountaines of waters, and the greatnesse of the Earth, in a Ballance, as the wise man saith").l03 And although such "quantification" could hardly compare with the search for the inner meaning of a divine mathematics, these Scriptural references alone were sufficient to indicate a divine sanction for such studies. Cusanus suggested that Although nothing in this world can reach precision, yet wee finde by experience, the judgement of the Ballance, one of the truest things amongst us .... 104 Ibid., p. 194. Ibid., p. 196. loa Cusanus, The Idiot, p. 172. From the Vulgate: Provo 16: II. Pondus et statera iudicia Domini sunt: et opera eius omnes lapides sacculi. Provo 8: 28 Quando aethera firmabat sursum, et librabat fontes aquarum: Provo 8: 29 Quando circumdabat mad terminum suum, et legem ponebat aquis, ne transirent fines suos: quando appendebat fundamenta terrae. The neo-Platonic and Paracelsian background to Harvey's quantification (Cusanus, the Paracelsians, van Helmont) has been discussed by Walter Pagel, William Harvey's Biological Ideas. Selected Aspects and Historical Background (Basel/New York: S. Karger, 1967), pp. 73-82. Here Pagel shows that for Harvey also, mathematics is to be considered basically a tool for observation. 1M Ibid., p. 171•

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE With this justification he then proceeded to propose the analysis of spring waters, blood and urine by weight comparisons. The natural philosopher might attempt to weigh different samples of air, and similarly test the statements of the alchemists. Most famous of all, he is told to plant seeds in a weighed earthern pot. After they have grown, hee woulde finde the earth but very little diminished, when he came to weigh it againe: by which he might gather, that all the aforesaid herbs had their weight from the water.105 This type of quantification seemed in no :way to conflict with Cusanus' grander scheme which involved a more direct search for divinity through the divine mathematical ideas. One senses far less confidence among Paracelsians in the sixteenth and seventeenth centuries that cosmic mathematics should playas vital a role in science as Cusanus had suggested. As we have noted, mathematics was given a subservient role to chemistry in the writings of Paracelsus. In general, his· followers-with the notable exception of Dorneus and Fludd-did not concentrate on the doctrine of the certainty of mathematical proof-even as they interpreted it. The chemical philosophers placed greater emphasis on weight determinations which were part of alchemical tradition and could be supported by Scriptural references in the fashion of Cusanus. Thus, we find a real interest on the part of sixteenth century iatrochemists in the weight analysis of mineral waters and urine,106and van Helmont was to carry out the plant experiment suggested by Cusanus with a tree. This mood may be seen in the work of the well known iatrochemist, Daniel Sennert, who paid little attention to the study of mathematics in his comparison of Aristotelian, Galenic and Paracelsian thought (1619),107and discussed the subject primarily as a practical art to be used whenever quantity is considered in his Epitome N aturalis Scientiae (1618).108 Surely some chemical philosophers were reacting against an excessive use of theoretical mathematics in scientific investigations in the sixteenth century. We find Peter Severinus making this a major point in his rejection of Galenic medicine (1571).109 Describing the decline of medical studies in antiquity, Severinus pictures Galen as a compiler faced with the task of putting order into the work of his predecessors. Seeking a unifying principle with laws and demonstrations, he at length came upon the writings of the geometricians. Ibid., pp. 188-<). G. Debus, "Solution Analyscs Prior to Robert Boylc", Chymia, 8 (1962), 41-61. Pagel, Paracelsus, pp. 190-4, 281-2. 107 Daniel Sennert, De Chymicorum cum A ristotelicis et Galenicis Concensu ac Dissensu Libet' (3rd cd., Paris, 1633). 108 Daniel Sennert. Epitome Naturalis Scientiae (Paris, 1633), pp. 6-7. 1011 Petrus Severinus, Idea Medicinae Philosophicae (3rd ed., Hagae-Comitis. 1660), p. 2. lOt Allen

Seduced by their exquisite demonstrations, he attempted to make medicine a part of geometry with its own principles, axioms, and mathematical explanations. So influential had been his work that only in recent years had the fallacy of his approach been demonstrated. When Fernel tried Galen's methods and laws of cure he had found that they failed in the treatment of those new diseases which were ravaging the continent. We now know, Severinus continues, that the only true medicine is that to be found in the writings of Paracelsus,11° and far from being mathematically inspired, this science finds its roots in the experiments of the chemists.lll Severinus had been the most important Paracelsian theorist in the sixteenth century. The work of van Helmont in the first half of the next century represents an even stronger protest against mathematical abstraction in theoretical science. This may be seen in 1113 attack on Galen, Aristotle and Paracclsus. At one time strongly attracted lo the work of Paracclsus, van Helmont eventually rejected many specific points of Paracelsian thought. Significantly he turned away from the macrocosm-microcosm universe in his later writings.1l2 With this scheme discarded, and with a conviction-because of his study of Copernicus-that astronomy "wa" not worthy of the time bestowed on it",113 much of the allure of the search for divine mathematical harmonies was lost. Van Helmont's disillusion with Galen and Aristotle may be traced at least in part to his views on mathematics. Like Severinus, van Helmont writes of Galen's codification of the works of his predecessors and of his attempt to frame the whole science on the four elements.lI4 He, too, laboured over logic and the principles of Euclid which seemed to contain truth, but when he sought theorems and axioms in medicine, he found this quest quite hopeless.llo Medicine is a divine rather than an ordinary science because Scripture teaches us that the Lorq created physicians.lI8 Rational, mathematically inspired Ibid., p. 3. p. 21. 112 J. B. van Helmont, Oriatrike or Pkysick Refined, trans. John Chandler (London, 1662), p. 323. "The name therefore of Microcosm or little World is Poetical, heathenish, and metaphorical, but not natural or true. It is likewise a phantastical, hypochondriacal, and mad thing, to have brought all the properties, and species of the Universe into man, and the art of healing." On van Helmont see Walter Pagel's studies, J . Bapt. van Helmont Einfukrung i'J die Pkilosophiscke Medizin des Barock (Berlin: Julius Springer, 1930); "Religious Motives in the Medical Biology of the XVIIth Century", Bulletin of the Institute of the History of Medicine, 3 (1935), 97-128, 213-31, 265-312; "The Religious and Philosophical Aspects of van Helmont's Science and Medicine", Supp. Bull. Hist. Med., NO.2 (Baltimore, The Johns Hopkins Press, 1944). 113 van Helmont, Oriatrike, p. 12. See also Ibid., p. 126. m Ibid., p. 3. 115 Ibid., p. 13. m Ibid.,

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE investigations may aid us in the study of physics, but not the chief goal of natural philosophy, medicine, for "to understand and favour these things from the spring or first cause is granted to none without the special favour of Christ the Lord",ll7 Neither the cosmic mathematics of the mystical alchemists and Pythagoreans nor the logical-mathematical method of the Galenist may help us here. And if we turn to the mathematical physics of the Aristotelians we shall do no better than before. The teachings of Aristotle dominate our schools and turn the minds of our scholars into erroneous paths. Yet, if we examine his writings we shall unmask him. It is true that Aristotle had a most persuasive manner in arranging rules and maxims. In a later-and more lazy age-students willingly accepted these as a guide to truth while they prostrated themselves to him almost in worship.ll8 But the evidence shows, van Helmont adds, that Aristotle surely had little ability as a scientific observer. Therefore he relied heavily on mathematics in .preparing his vast system simply because he was "far more skilful in this, than in Nature". And since this man had subdued nature under the rules of mathematics, our scholars are now cursed with an improper approach to science.llo In his examination of Aristotle's works, van Helmont's attention was arrested by the eight books of Natural Instructions. Here corporal matter is incorrectly combined with mathematical abstraction. Similarly, his books discuss the vacuum and the infinite-subjects which clearly do not belong to the knowledge of nature. And "they bring in locall motion, as it serves to Science Mathematical or Learning by demonstration, alike foolishly, and with an undistinct indiscretion, into nature".120 One reads the blasphemy which lUI Ibid., p. 4. See Ecclesiasticus 38: I "Honour the physician for the need thou hast of him: for the most high hath created him". Similarly, Paracelsus, stated "It was not the constellations that made me a physician; God made me.... " Paracelsus, Samtliche Werke, ed. Karl Sudhoff and Wilhelm Matthiessen (15 vols., Munich and Berlin, 1922-33), 8, pp. 63-5. From the Paragranwn. 117 van Helmont, op. cit., p. 4. On the same page we read the whole faculty of natural Phylosophy is committed to man; and therefore this ought to respect both his life immediately, and all his defects. Therefore all natural Phylosophy is limited to the use of life, the finding out of causes, the Disease and Remedies". 111 Ibid., p. 33. m Ibid. 110 Ibid., p. 45. Van Helmont might have quoted Aristotle from the :Metaphysics"The minute accuracy of mathematics is not to be demanded in all cases, but only in the case of things which have no matter. Hence its method is not that of natural science; for presumably the whole of nature has matter". Metaphysics. Book 2, Chap. 3, as trans. by W. D. Ross and printed in Great Books of the Western World, ed. Robert Maynard Hutchins (54 volumes, Chicago/London/Toronto/Geneva: Encyclopedia Britannica, Inc., 1952), 8, p. 513.

"the Schools have taught, That in every locall motion, a first unmoveable Mover is of necessity to be appointed: Which thing I neither find true by Art, nor Nature" .121 Really this "is a Paganish Doctrine drawn from Science Mathematical, which necessitates the first Mover to a perpetual unmoveableness of himself, that without ceasing he may move all things".122 We find, however, that the "most glorious Mover hath given powers to things whereby they of themselves, and by an absolute force may move themselves, or other things. Indeed, it is impertinent to run back to God the Mover to demonstrate the natural motion of Bodyes".I23 Looking at nature with the eyes of a physician and vitalist, van Helmont sees a far different approach to local motion than the physicist. Sublunary bodies contain within themselves something which permits them to move. "They are moved by a certain will and pleasure or precept of nature, and have their own natural necessities, and ends: even as is seen in the beating of the Heart, Arteries, expelling of many superfluities, &C.".124 Thus, although we may find the maxims of Aristotle serviceable in the description of certain aspects of local motion and mathematical science, they have been given far greater weight than they deserve, and we may be certain that "the Rules of the Mathematicks, or Learning by demonstration do ill square to Nature. For man cloth not measure Nature; but she him".125 Thoroughly sickened by the logical-mathematical methods of the Schools, van Helmont could only hope-as did so many of his generation-for a new learning based upon a total reform. In his early work on the magnetic cure of wounds we see van Helmont's acceptance of natural magic as lithe most profound inbred knowledge of things", and if he was to reject the most occult aspects of Paracelsian thought, we still find his emphasis on experiment and chemistry in his scheme for educational reform. No longer should young men waste their youth on Aristotelian thought-i.e. mathematical thought. In a seven year programme they may devote three years to arithmetic, mathematical science, the Elements of Euclid, and geography (which is to include the circumstances of seas, rivers, springs, mountains, provinces and minerals). In addition-in this three year period-they are to study the properties and customs of nations, waters, plants, living creatures, minerals, and the ring and astrolabe. Only then, van Helmont continues ... let them come to the Study of Nature, let them learn to know and separate the first Beginnings of Bodies. I say, by working, to have known their fixedness, volatility or swiftness, with their separation, p. 176. Ibid. 123 Ibid. 124 Ibid., p. 177. 125 Ibid. 121 Ibid.,

MATHEMATICS AND NATURE IN CHEMICAL TEXTS OF THE RENAISSANCE 25 life, death, interchangeable course, defects, alteration, weakness, corruption, transplanting, solution, coagulation or co-thickning, resolving. Let the History of extractions, divi<1ings, conjoynings, ripenesses, promotions, hinderances, consequences, lastly, of losse and profit, be added. Let them also be taught, the Beginnings of Seeds, Ferments, Spirits, and Tinctures, with every flowing, digesting, changing, motion, and disturbance of things to be altered. And all those things, not indeed by a naked description of discourse, but by handicraft demonstration of the lire. For truly, nature measureth her works by distilling, moystening, drying, calcining, resolving, plainly by the same mcancs, whereby glasses <10accomplish those same operations. And so the Artificer, by changing the operations of nature, obtains the properties and knowledge of the same. For however natural a wit, and sharpness of judgement the Philosopher may have, yet he is never admitted to the Root, or radical knowledge of natural things, without the fire. And so everyone is deluded with a thousand thoughts or doubts, the which he unfoldeth Hilt to himsdf, hut hy the help of the fire. Therefore I confess, nothing doth more fully bring a man that is greedy of knowing, to the knowledgcs of all things knO\\'able, than the fire. Therefore a young man at length, returning out of these Schooles, truly it is a wonder to see, how much he shall ascend above the Phylosophers of the University, and the vain reasoning of the Schooles.128 This is the Helmontian-iatrochemical programme for a reform in education. Students are to spend the bulk of their time learning the true facts of nature through observation and chemical operations with the fire. The purely intellectual pursuit of truth with mathematics here completely loses its value. Such mathematics is connected with Euclidean geometry, Aristotelian physics ~specially the study of motion-and the theology of the Middle Ages. Truth may be attained only by disassociating ourselves from these errors of the past. In contrast, the mathematics of real usc for the natural philosopher is connected with experiments. Thus, it may be shown "mechanically, and mathematically that all solid Bodies are oncly of water", and he suggests that urine should be examined by weight (as had Cusanus, Dorn, Paracelsus, and Thurneisser). Similarly, his willow tree experiment and his estimate of the weight of gas produced by burning 62 pounds of coal (a weight by difference based on the coal ashes) underline his belief in the value of weight determinations.127 With van Helmont the more sublime mathematical road to divinity suggested by Cllsanus has been completely discarded. Ibid., p. 45. Ibid., pp. 106, 109, 167, 1056. The quantitative analysis of urine descri1>ed in the Anatomia C01'po1'umadhuc viventium and ascribed to Paracelsus (printed in the Aurora Thesau1'usque Philosophorum Theoph1'asti Pa1'acelsi accessit M onarcllia Plzysica per Ge1'ardum Dorneum (Basel, 1577), pp. 12~1) was probably authored by Dorn. This is described in detail along with other early quantitative analyses of urine in Pagel, Paracelsus,

CONCLUSION Cusanus had told the scholar to learn of his Creator through mathcmatics. Two paths were suggested by him-the first and most important being the contemplation of the divine mathematical ideas. These lift the seeker from particulars through an ascending scale to thc divine infinite. This road to truth is in the tradition of the mystical neo-Platonic and Pythagorean sages. At the same time, Cusanus, in The Idiot, proposed a lesser road to truth in nature by quantitative weight measurements and the nse of the balance. This, too, may be supported by appeal to Scripture. Quantification and mathematical speculations were of considerable importance to somc-if not all-iatrochemists in the sixteenth and seventeenth centuries and their views are worth study because of the contemporary general interest in thcir approach to naturc. Within the ranks of these men we find advocates of both paths to truth. This becomes evident in the views on mathematics expressed by the contemporaries Robert Fludd (1574-1637) and J. B. van Belmont (1580-1644). Fludd, perhaps the most renowned mystical alchemist of his generation, was a Pythagorean, kabbalist and Hermeticist who viewed the Creation in tenus of a. divine alchemy. He insisted that it is not the study of nature, hut the search for divinp mysteries which must claim our utmost attention. For him the greater and lesser worlds exist in a state of Platonic perfection which may be interpreted through musical harmonies and an alchemical-geometrical SYmbolism. All of this bears little connection with our concept of the use of mathematics in science, and even Kepler, influenced by many of the same themes, could not understand thc approach of the English physician. Van Helmont also based his work on rcligious convictions. Like Fludd, he insisted that the writings of Aristotle and Galen may be largely ignored because of their theological implications. Not only were their works to be rejected, even their logical-mathematical approach to nature was to be discarded and branded as blasphemous. Yet, van Helmont went further than this. Rejecting the mystical macrocosm-microcosm universe of his iatrochemical colleagues, he found no valid ground for the search for divinity in the numerical harmonies and SYmbols of heaven and earth. The true scholar will learn from his own observations and experiments. Here the second path of Cusanus is acceptable-measurements by weight applied to specific experiments. Beyond this the honest student cannot go, for the application of mathematical abstraction to nature is an unholy method which can only be associated with the Peripatetics in the Universities, not with the new science of the future. The importance of all this may be seen less in the actual details of

MATHEMATICS AND NATURE IN CHE1\UCAL TEXTS OF THE RENAISSANCE 27 iatrochemical mathematics than in the insight it gives us regarding the still unsettled role of mathematics in sixteenth and seventeenth century science. If the mystical alchemists and the Helmontians were of no influence, we would be justified in ignoring their views. We fillll, however, if we may judge by the number of books published, that their influence reached a high point in the mid-seventeenth century. We have already referred to the notice taken of Fludd's system by Kepler, Mersenne and Gassendi. The debate over Fludd's works was widespread and could not have helped but define the difference between the mystical alchemists and the Galileans on this point. In England in the decade of the 1650S we find more Paracelsian and alchemical translations than from the whole preceding century. It was just at this time that Robert Fludd's major work, the M osaicall Philosophy, appeared in English translation (1659), and it was then, too, that John \Vebster proposed a reform of the Universities in terms which recall Fludd's l{osicrucian Apology of r()I7.128 Far more widespread was the appeal of van Helmont. Here was an approach to nature and divinity based on fresh observations and experiments which placed the study of man at the apex of natural knowledge. I t is little wonder that physicians were attracted to this path to truth. Van Hclmont offered a new experimental science founded on chemistry which allegedly was stripped of the mystical fantasies of his chemical predecessors. It is understandable that by the mid-seventeenth century his views could command a far greater audience than those of the more mystical Fludd. Yet, the rejection of the value of mathematical abstraction in the study of nature by van Helmont -as well as by Fludd and the Paracelsians-occurs precisely at the time when the power of mathematical analyses was being recognized by Galileo and his followers.129 The rapid decline of the Paracelso-Helmontians in the period after 1670 may be attributed at least partially to the success of the mathematicized science which found no place in their schemes. This was an issue 128 John Webster, Academiarttm Examen, or the Examillation of Academies (London: Giles Calvert, 1654). Webster praised the mystical anatomy of Fludd (p. 74), and suggested a reform of the mathematical sciences in l'luddean terms (pp. 40 fl.). He, too, strongly praised the Rosicrucian approach to nature (pp. 26 ff.). This text was strongly attacked by John Wilkins and Seth Ward. 121 There is some similarity between the views of van Helmont and Francis Bacon on the extension of mathematics in the study of nature. The latter disliked the deductive nature of the subject, and he felt that some, by applying it to nature, were relying far too much on reason rather than on observation. Mathematical systems have been more acutely inquired into than other matters, because it is "the nature of the human mind, certainly to the extreme prejudice of knowledge, to delight in the open plains of generalities rather than in the woods and inclosures of particulars", with the result that logic and mathematics, which shoul<.1 be the handmaidens of science, exercise dominion over it. Francis Bacon, Works. Collected and ed. by J. Spedding, R. L. Ellis, D. 1>. Heath (14 vols., London: Longman & Co. etc., 1857-74), 4, p. 370. From the De Augmentis.

which had been decided by 1700, but it had not been decided half a century earlier. In the first half of the century it could still be seriously argued that the role of mathematics in science shou1<l he limited to the search for divine mathematical harmonies or calculations based on experimental measurements of weight. ACKNOWLEDGMENTS The present study was completed during the tenure of an Overseas Fellowship at Churchill College, Cambridge. In addition, both a Guggenheim Fellowship and a research grant from the National Institutes of Health (LM 00046) have made it possible for him to continue his research on Renaissance Science. The author also wishes to thank \Valter Pagel for his aid in the clarification of many specific points discussed in this paper.

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