Thomas S. Kuhn
Mathematical vs. Experimental Traditions in the Development of Physical Science
Journal of Interdisciplinary History,7 (1)
Summer 1976, 1-31.
Anyone who studies the history of scientific development repeatedly encounters a question, one version of which would be, “Are the sciences one or many?” Ordinarily that question is evoked by concrete problems of narrative organization, and these become especially acute when the historian of science is asked to survey his subject in lectures or in a book of significant scope. Should he take up the sciences one by one, beginning, for example, with mathematics, proceeding to astronomy, then to physics, to chemistry, to anatomy, physiology, botany, and so on? Or should he reject the notion that his object is a composite account of individual fields and take it instead to be knowledge of nature tout court? In that case, he is bound, insofar as possible, to consider all scientific subject matters together, to examine what men knew about nature at each period of time, and to trace the manner in which changes in method, in philosophical climate, or in society at large have affected the body of scientific knowledge conceived as one.
Given a more nuanced description, both approaches can be recognized as long-traditional and generally non-communicating historiographic modes.  The first, which treats science as at most a loose linked congeries of separate sciences, is also characterized by its practitioners’ insistence on examining closely the technical content, both
Thomas S. Kuhn is the M. Taylor Pyne Professor of
This essay is the revised and extended version of a
George Sarton Memorial Lecture, delivered in
1. For a somewhat more extended discussion of these two approaches, see Kuhn, “History of Science” in the International Encyclopedia of the Social Sciences, XIV (New York, 1968), 74-83. Note also the way in which distinguishing between them both deepens and obscures the now far better known distinction between internalist and externalist approaches to the history of science. Virtually all the authors now regarded [as internalists address themselves to the evolution of a single science or of a closely related set of scientific ideas; the externalists fall almost invariably into the group that has treated the sciences as one. But the labels “internalist” and “externalist” then no longer quite fit. Those who have concentrated primarily on individual sciences, e.g., Alexandre Koyré, have not hesitated to attribute a significant role in scientific development to extra-scientific ideas. What they have resisted primarily is attention to socioeconomic and institutional factors as treated by such writers as B. Hessen, G. N. Clark, and R. K. Merton. But these non-intellectual factors have not always been much valued by those who took the sciences to be one. The “internalist-externalist debate” is thus frequently about issues different from the ones its name suggests, and the resulting confusion is sometimes damaging.]
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experimental and theoretical, of past versions of the particular specialty being considered. That is a considerable merit, for the sciences are technical, and a history which neglects their content often deals with another enterprise entirely, sometimes fabricating it for the purpose. On the other hand, historians who have aimed to write the history of a technical specialty have ordinarily taken the bounds of their topic to be those prescribed by recent textbooks in the corresponding field. If, for example, their subject is electricity, then their definition of an electrical effect often closely resembles the one provided by modern physics. With it in hand, they may search ancient, medieval, and early modern sources for appropriate references, and an impressive record of gradually accumulating knowledge of nature sometimes results. But that record is drawn from scattered books and manuscripts ordinarily described as works of philosophy, literature, history, scripture, or mythology. Narratives in this genre thus characteristically obscure the fact that most items they group as “electrical” - e.g., lightning, the amber effect, and the torpedo (electric eel) - were not, during the period from which their descriptions are drawn, ordinarily taken to be related. One may read them carefully without discovering that the phenomena now called “electrical” did not constitute a subject matter before the seventeenth century and without finding even scattered hints about what then brought the field into existence. If a historian must deal with enterprises that did exist in the periods that concern him, then traditional accounts of the development of individual sciences are often profoundly unhistorical.
No similar criticism may be directed at the other main historiographic tradition, the one which treats science as a single enterprise. Even if attention is restricted to a selected century or nation, the subject matter of that putative enterprise proves too vast, too dependent on technical detail, and, collectively, too diffuse to be illuminated by
historical analysis. Despite ceremonial bows to classics like
To an understanding of that relationship, the tradition which takes science to be one can in principle contribute nothing, for it bars by presupposition access to phenomena upon which the development of such understanding must depend. Social and philosophical commitments that fostered the development of a particular field at one period of time have sometimes hampered it at another; if the period of concern is specified, then conditions that promoted advance in one science often seem to have been inimical to others.  Under these circumstances, historians who wish to illuminate actual scientific development will need to occupy a difficult middle ground between the two traditional alternatives. They may not, that is, assume science to be one, for it clearly is not. But neither may they take for granted the subdivisions of subject matter embodied in contemporary science texts and in the organization of contemporary university departments.
Textbooks and institutional organization are useful indices of the natural divisions the historian must seek, but they should be those of the period he studies. Together with other materials, they can then provide at least a preliminary roster of the various fields of scientific practice at a given time. Assembling such a roster is, however, only the beginning of the historian’s task, for he needs also to know something about the relations between the areas of activity it names, asking, for example, about the extent of interaction between them and the ease with which practitioners could pass from one to the next. Inquiries of that sort can
2. On this point, in addition to the material below, see Kuhn, “Scientific Growth: Reflections on Ben-David’s ‘Scientific Role’,” Minerva, X (1972), 166-178.
gradually provide a map of the complex structure of the scientific enterprise of a selected period, and some such map is prerequisite to an examination of the complex effects of metascientific factors, whether intellectual or social, on the development of the sciences. But a structural map alone is not sufficient. To the extent that the effects to be examined vary from field to field, the historian who aims to understand them will also have to examine at least representative parts of the sometimes recondite technical activities within the field or fields that concern him. Whether in history or sociology of science, the list of topics that can usefully be studied without attention to the content of the relevant sciences is extremely short.
Historical research of the sort just demanded has barely begun. My conviction that its pursuit will be fruitful derives not from new work, my own or someone else’s, but from repeated attempts as a teacher to synthesize the apparently incompatible products of the two non-communicating traditions just described.  Inevitably, all results of that synthesis are tentative and partial, regularly straining and sometimes overstepping the limits of existing scholarship. Nevertheless, schematic presentation of one set of those results may serve both to illustrate what I have had in mind when speaking of the changing natural divisions between the sciences and also to suggest the gains which might be achieved by closer attention to them. One consequence of a more developed version of the position to be examined below could be a fundamental reformulation of an already overlong debate about the
3. These problems of synthesis go back to the very
beginning of my career, at which time they took two forms which initially seemed
entirely distinct. The first,
sketched in note 2,
above, was how to make socioeconomic concerns relevant to narratives
about the development of scientific ideas. The second, highlighted by the appearance
of Herbert Butterfield’s admirable and influential Origins of Modern Science
origins of modern science. Another would be the isolation of an important novelty which, during the nineteenth century, helped to produce the discipline of modern physics.
My main theme may be introduced by a question. Among the large number of topics now included in the physical sciences, which ones were already in antiquity foci for the continuing activity of specialists? The list is extremely short. Astronomy is its oldest and most developed component; during the Hellenistic period, as research in that field advanced to a previously unprecedented level, it was joined by an additional pair, geometrical optics and statics, including hydrostatics. These three subjects - astronomy, statics, and optics - are the only parts of physical science which, during antiquity, became the objects of research traditions characterized by vocabularies and techniques inaccessible to laymen and thus by bodies of literature directed exclusively to practitioners. Even today Archimedes’ Floating Bodies and Ptolemy’s Almagest can be read only by those with developed technical expertise. Other subjects which, like heat and electricity, later came to be included in the physical sciences remained throughout antiquity simply interesting classes of phenomena, subjects for passing mention or for philosophic speculation and debate. (Electrical effects, in particular, were parceled out among several such classes.) Being restricted to initiates does not, of course, guarantee scientific advance, but the three fields just mentioned did advance in ways that required the esoteric knowledge and technique responsible for their isolation. If, furthermore, the accumulation of concrete and apparently permanent problem solutions is a measure of scientific progress, these fields are the only parts of what were to become the physical sciences in which unequivocal progress was made during antiquity.
At that time, however, the three were not practiced alone but were instead intimately associated with two others - mathematics and harmonics — no longer ordinarily regarded as physical sciences. Of this
4. Henry Guerlac first urged on me the necessity of
including music theory in the cluster of classical sciences. That I should initially have omitted a
field no longer conceived as science indicates how easy it is to miss the force
of the methodological precept offered in my opening pages. Harmonics was not, however, quite the
field we would now call music theory. Instead, it was a mathematical science
which attributed numerical proportions to the numerous intervals of various
Greek scales or modes. Since there
were seven of these, each available in three genera and in fifteen tonoi
or keys, the discipline [was complex,
specification of some intervals requiring four and five digit numbers. Since only the simplest intervals were
empirically accessible as the ratios of the lengths of vibrating strings,
harmonics was also a highly abstract subject. Its relation to musical practice was at
best indirect, and it
Historically, harmonics dates from the fifth century B.C. and was highly
developed by the time of Plato and Aristotle.
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pair, mathematics was even older and more developed than astronomy. Dominated, from the fifth century B.C., by geometry, it was conceived as the science of real physical quantities, especially spatial, and it did much to determine the character of the four others which clustered around it. Astronomy and harmonics dealt, respectively, with positions and ratios, and they were thus literally mathematical. Statics and geometric optics drew concepts, diagrams, and technical vocabulary from geometry, and they shared with it also a generally logical deductive structure common to both presentation and research. Not surprisingly, under these circumstances, men like Euclid, Archimedes, and Ptolemy, who contributed to one of these subjects, almost always made significant contributions to others as well. More than developmental level thus made the five a natural cluster, setting them apart from other highly evolved ancient specialties such as anatomy and physiology. Practiced by a single group and participating in a shared mathematical tradition, astronomy, harmonics, mathematics, optics, and statics are therefore grouped together here as the classical physical sciences or, more simply, as the classical sciences.  Indeed, even listing them as distinct topics is to some extent anachronistic. Evidence to be encountered below will suggest that, from some significant points of view, they might better be described as a single field, mathematics.
To the unity of the classical sciences one other shared characteristic was also prerequisite, and it will play an especially important role in the balance of this paper. Though all five fields, including ancient mathe-
5. The abbreviation “classical sciences” is a possible source of confusion, for anatomy and physiology were also highly developed sciences in classical antiquity, and they share a few, but by no means all, of the developmental characteristics here attributed to the classical physical sciences. These bio-medical sciences were, however, parts of second classical cluster, practiced by a distinct group of people, most of them closely associated with medicine and medical institutions. Because of these and other differences, the two clusters may not be treated together, and I restrict myself here to the physical sciences, partly on grounds of competence and partly to avoid excessive complexity. See, how ever, notes 6 and 9 below.
matics, were empirical rather than a priori, their considerable ancient development required little refined observation and even less experiment. For a person schooled to find geometry in nature, a few relatively accessible and mostly qualitative observations of shadows, mirrors, levers, and the motions of stars and planets provided an empirical basis sufficient for the elaboration of often powerful theories. Apparent exceptions to this broad generalization (systematic astronomical observation in antiquity as well as experiments and observations on refraction and prismatic colors then and in the Middle Ages) will only reinforce its central point when examined in the next section. Although the classical sciences (including, in important respects, mathematics) were empirical, the data their development required were of a sort which everyday observation, sometimes modestly refined and systematized, could provide.  That is among the reasons why this cluster of fields could advance so rapidly under circumstances that did not significantly promote the evolution of a second natural group, the one to which my title refers as the products of an experimental tradition,
Before examining that second cluster, consider briefly the way in which the first developed after its origin in antiquity. All five of the classical sciences were actively pursued in Islam from the ninth century, often at a level of technical proficiency comparable to that of antiquity. Optics advanced notably, and the focus of mathematics was in some places shifted by the intrusion of algebraic techniques and concerns not ordinarily valued within the dominantly geometric Hellenistic tradition. In the Latin West, from the thirteenth century, further technical elaboration of these generally mathematical fields was subordinated to a dominantly philosophical-theological tradition, important novelty being restricted primarily to optics and statics. Significant portions of the corpus of ancient and Islamic mathematics and astronomy were, however, preserved and occasionally studied for their own sake until
6. Elaborate or refined data generally become available only when their collection fulfills some perceived social function. That anatomy and physiology, which require such data, were highly developed in antiquity must be a consequence of their apparent relevance to medicine. That even that relevance was often hotly disputed (by the Empirics!) should help to account for the relative paucity, except in Aristotle and Theophrastus, of ancient data applicable to the more general taxonomic, comparative, and developmental concerns basic to the life sciences from the sixteenth century. Of the classical physical sciences, only astronomy required data of apparent social use (calendars and, from the second century B.C., horoscopy). If the others had depended upon the availability of refined data, they would probably have advanced no further than study of topics like heat.
they became again the objects of continuing erudite European research during the Renaissance.  The cluster of mathematical sciences then reconstituted closely resembled its Hellenistic progenitor. As these fields were practiced during the sixteenth century, however, research on a sixth topic was increasingly associated with them. Partly as a result of fourteenth-century scholastic analysis, the subject of local motion was separated from the traditional philosophic problem of general qualitative change, becoming a subject of study in its own right. Already highly developed within the ancient and medieval philosophical tradition, the problem of motion was a product of everyday observation, formulated in generally mathematical terms. It therefore fitted naturally into the cluster of mathematical sciences with which its development was thereafter firmly associated.
Thus enlarged, the classical sciences continued from the
Renaissance onward to constitute a closely knit set. Copernicus specified the audience
competent to judge his astronomical classic with the words, “Mathematics is
written for mathematicians.” Galileo, Kepler, Descartes, and
7. This paragraph has considerably benefited from discussions with John Murdoch, who emphasizes the historiographic problems encountered if the classical sciences are conceived as continuing research traditions in the Latin middle ages. On this topic see his “Philosophy and the Enterprise of Science in the Later Middle Ages,” in Y. Elkana (ed.), The Interaction between Science and Philosophy (New York, 1974), 51-74.
One last remark about the classical sciences will prepare the way for consideration of the movement that promoted new experimental methods. All but harmonics  were radically reconstructed during the sixteenth and seventeenth centuries, and in physical science such transformations occurred nowhere else.  Mathematics made the transition from geometry and “the art of the cross” to algebra, analytic geometry, and calculus; astronomy acquired non-circular orbits based on the newly central sun; the study of motion was transformed by new fully quantitative laws; and optics gained a new theory of vision, the first acceptable solution to the classical problem of refraction, and a drastically altered theory of colors. Statics, conceived as the theory of machines, is an apparent exception. But as hydrostatics, the theory of fluids, it was extended during the seventeenth century to pneumatics, the “sea of air,” and it can therefore be included in the list of reconstructed fields. These conceptual transformations of the classical sciences are the events through which the physical sciences participated in a more general revolution of Western thought. If, therefore, one thinks
8. Although harmonics was not transformed, its status
declined greatly from the late fifteenth to the early eighteenth century. More and more it was relegated to the first
section of treatises directed primarily to more practical subjects: composition,
temperament, and instrument construction. As these subjects became more and more
central to even quite theoretical treatises, music was increasingly divorced
from the classical sciences. But
the separation came late and was never complete. Kepler, Mersenne, and Descartes, all
wrote on harmonics; Galileo, Huyghens, and
9. They did, of course, occur in the classical life sciences, anatomy and physiology. Also, these were the only parts of the bio-medical sciences transformed during the Scientific Revolution. But the life sciences had always depended on refined observation and occasionally on experiment as well; they had drawn their authority from ancient sources (e.g., Galen) sometimes distinct from those important to the physical sciences; and their development was intimately involved with that of the medical profession and corresponding institutions. It follows that the factors to be discussed when accounting either for the conceptual transformation or for the newly enlarged range of the life sciences in the sixteenth and seventeenth centuries are by no means always the same as those most relevant to the corresponding changes in the physical sciences. Nevertheless, recurrent conversations with my colleague Gerald Geison reinforce my longstanding impression that they too can fruitfully be examined from a viewpoint like the one developed here. For that purpose the distinction between experimental and mathematical traditions would be of little use, but a division between the medical and non-medical life sciences might be critical.
of the Scientific Revolution as a revolution of ideas, it is the changes in these traditional, quasi-mathematical fields which one must seek to understand. Although other vitally important things also happened to the sciences during the sixteenth and seventeenth centuries (the Scientific Revolution was not merely a revolution in thought), they prove to be of a different and to some extent independent sort.
Turning now to the emergence of another cluster of research fields, I again begin with a question, this time with one about which there is much confusion and disagreement in the standard historical literature. What, if anything, was new about the experimental movement of the seventeenth century? Some historians have maintained that the very idea of basing science upon information acquired through the senses was novel. Aristotle, according to this view, believed that scientific conclusions could be deduced from axiomatic first principles; not until the end of the Renaissance did men escape his authority sufficiently to study nature rather than books. These residues of seventeenth-century rhetoric are, however, absurd. Aristotle’s methodological writings contain many passages which are just as insistent upon the need for close observation as the writings of Francis Bacon. Randall and Crombie have isolated and studied an important medieval methodological tradition which, from the thirteenth century into the early seventeenth, elaborated rules for drawing sound conclusions from observation and experiment.  Descartes’ Regulae and Bacon’s New Organon owe much to that tradition. An empirical philosophy of science was no novelty at the time of the Scientific Revolution.
Other historians point out that, whatever people may have believed about the need for observations and experiments, they made them far more frequently in the seventeenth century than they had before. That generalization is doubtless correct, but it misses the essential qualitative differences between the older forms of experiment and the new. The participants in the new experimental movement, often called Baconian after its principal publicist, did not simply expand and elaborate the empirical elements present in the tradition of classical physical science. Instead they created a different sort of empirical science, one that for a time existed side by side with, rather than
10. A. C. Crombie,
Robert Grosseteste and the Origins of
Experimental Science, 1100-1700
supplanting, its predecessor. A brief characterization of the occasional role played in the classical sciences by experiment and systematic observation will help to isolate the qualitative differences which distinguish the older form of empirical practice from its seventeenth-century rival.
Within the ancient and medieval tradition, many experiments prove on examination to have been “thought experiments,” the construction in mind of potential experimental situations the outcome of which could safely be foretold from previous everyday experience. Others were performed, especially in optics, but it is often extremely difficult for the historian to decide whether a particular experiment discovered in the literature was mental or real. Sometimes the results reported are not what they would be now; on other occasions the apparatus required could not have been produced with existing materials and techniques. Real problems of historical decision result, and they also haunt students of Galileo. Surely he did experiments, but he is even more noteworthy as the man who brought the medieval thought-experimental tradition to its highest form. Unfortunately, it is not always clear in which guise he appears. 
Finally, those experiments which clearly were performed seem invariably to have had one of two objects. Some were intended to demonstrate a conclusion known in advance by other means. Roger Bacon writes that, though one can in principle deduce the ability of flame to burn flesh, it is more conclusive, given the mind’s propensity for error, to place one’s hand in the fire. Other actual experiments, some of them consequential, were intended to provide concrete answers to questions posed by existing theory. Ptolemy’s experiment on the refraction of light at the boundary between air and water is an important example. Others are the medieval optical experiments that generated colors by passing sunlight through globes filled with water. When Descartes and
11. For a useful and easily accessible example of medieval experimentation, see Canto II of Dante’s Paradiso. Passages located through the index-entry “experiment, role of in Galileo’s work” in Ernan McMullin (ed.), Galileo, Man of Science (New York, 1965), will indicate how complex and controversial Galileo’s relation to the medieval tradition remains.
times and positions were needed to prepare ephemerides or to compute parameters called for by existing theory.
Contrast this empirical mode with the one for which Bacon became the most effective proponent. When its practitioners, men like Gilbert, Boyle, and Hooke, performed experiments, they seldom aimed to demonstrate what was already known or to determine a detail required for the extension of existing theory. Rather they wished to see how nature would behave under previously unobserved, often previously non-existent, circumstances. Their typical products were the vast natural or experimental histories in which were amassed the miscellaneous data that many of them thought prerequisite to the construction of scientific theory. Closely examined, these histories often prove less random in choice and arrangement of experiments than their authors supposed. From 1650 at the latest, the men who produced them were usually guided by one or another form of the atomic or corpuscular philosophy. Their preference was thus for experiments likely to reveal the shape, arrangement, and motion of corpuscles; the analogies which underlie their juxtaposition of particular research reports often reveal the same set of metaphysical commitments.  But the gap between metaphysical theory on the one hand and particular experiments on the other was initially vast. The corpuscularism which underlies much seventeenth-century experimentation seldom demanded the performance or suggested the detailed outcome of any individual experiment. Under these circumstances, experiment was highly valued and theory often decried. The interaction which did occur between them was usually unconscious.
That attitude towards the role and status of experiment is only the first of the novelties which distinguish the new experimental movement from the old. A second is the major emphasis given to experiments which Bacon himself described as “twisting the lion’s tail.” These were the experiments which constrained nature, exhibiting it under conditions which it could never have attained without the forceful intervention of man. The men who placed grain, fish, mice, and various chemicals seriatim in the artificial vacuum of a barometer or an air pump exhibit just this aspect of the new tradition.
Reference to the barometer and air pump highlights a third novelty of the Baconian movement, perhaps the most striking of all. Before
12. An extended example is provided by Kuhn, ~Robert Boyle and Structural Chemistry in the Seventeenth Century,” Isis, XLIII (1952), 12-36.
1590 the instrumental armory of the physical sciences consisted solely of devices for astronomical observation. The next hundred years witnessed the rapid introduction and exploitation of telescopes, microscopes, thermometers, barometers, air pumps, electric charge detectors, and numerous other new experimental devices. The same period was characterized by the rapid adoption by students of nature of an arsenal of chemical apparatus previously to be found only in the workshops of practical craftsmen and the retreats of alchemical adepts. In less than a century physical science became instrumental.
These marked changes were accompanied by several others, one of which merits special mention. The Baconian experimentalists scorned thought experiments and insisted upon both accurate and circumstantial reporting. Among the results of their insistence were sometimes amusing confrontations with the older experimental tradition. Robert Boyle, for example, pilloried Pascal for a book on hydrostatics in which, though the principles were found to be unexceptionable, the copious experimental illustrations had clearly been mentally manufactured to fit. Monsieur Pascal does not tell us, Boyle complained, how a man is to sit at the bottom of a twenty-foot tub of water with a cupping glass held to his leg. Nor does he say where one is to find the superhuman craftsman able to construct the refined instruments upon which some of his other experiments depend.  Reading the literature of the tradition within which Boyle stands, the historian has no difficulty telling which experiments were performed. Boyle himself often names witnesses, sometimes supplying their patents of nobility.
Granting the qualitative novelty of the Baconian
movement, how did its existence affect the development of science? To the conceptual transformations of the
classical sciences, the contributions of Baconianism were very small. Some experiments did play a role, but
they all have deep roots in the older tradition. The prism which
13. “Hydrostatical Paradoxes, Made out by New Experiments” in
A. Millar (ed.), The Works of the Honourable Robert Boyle
string or of a body falling through the center of the earth and then back towards it. The barometer was first conceived and analyzed as a hydrostatic device, a water-filled pumpshaft without leaks designed to realize the thought experiment with which Galileo had “demonstrated” the limits to nature’s abhorrence of a vacuum.  Only after an extended vacuum had been produced and the variation of column height with weather and altitude had been demonstrated did the barometer and its child the air pump join the cabinet of Baconian instruments.
Equally to the point, although the experiments just
mentioned were of consequence, there are few like them, and all owe their
special effectiveness to the closeness with which they could confront the
evolving theories of classical science which had called them forth. The outcome of Torricelli’s barometer
experiment and of Galileo’s with the inclined plane had been largely foreseen.
Under those circumstances, numerous historians, Koyré included, have described the Baconian movement as a fraud, of no consequence to the development of science. That evaluation is, however, like the one it sometimes stridently opposed, a product of seeing the sciences as one. If Baconianism contributed little to the development of the classical
14. For the medieval
prelude to Galileo’s approach to the
pendulum, see Marshall Clagett, The Science of Mechanics in the Middle Ages
15. Alexandre Koyré, Etudes Galiléennes
sciences, it did give rise to a large number of new scientific fields, often with their roots in prior crafts. The study of magnetism, which derived its early data from prior experience with the mariner’s compass, is a case in point. Electricity was spawned by efforts to find the relation of the magnet’s attraction for iron to that of rubbed amber for chaff. Both these fields, furthermore, were dependent for their subsequent development upon the elaboration of new, more powerful, and more refined instruments. They are typical new Baconian sciences. Very nearly the same generalization applies to the study of heat. Long a topic for speculation within the philosophical and medical traditions, it was transformed into a subject for systematic investigation by the invention of the thermometer. Chemistry presents a case of a different and far more complex sort. Many of its main instruments, reagents, and techniques had been developed long before the Scientific Revolution. But until the late sixteenth century they were primarily the property of craftsmen, pharmacists, and alchemists. Only after a reevaluation of the crafts and of manipulative techniques were they regularly deployed in the experimental search for natural knowledge.
Since these fields and others like them were new foci for scientific activity in the seventeenth century, it is not surprising that their pursuit at first produced few transformations more striking than the repeated discovery of previously unknown experimental effects. If the possession of a body of consistent theory capable of producing refined predictions is the mark of a developed scientific field, the Baconian sciences remained underdeveloped throughout the seventeenth and much of the eighteenth centuries. Both their research literature and their patterns of growth were less like those of the contemporary classical sciences than like those discoverable in a number of the social sciences today. By the middle of the eighteenth century, however, experiment in these fields had become more systematic, increasingly clustering about selected sets of phenomena thought to be especially revealing. In chemistry, the study of displacement reactions and of saturation were among the newly prominent topics; in electricity, the study of conduction and of the Leyden jar; in thermometry and heat, the study of the temperature of mixtures. Simultaneously, corpuscular and other concepts were increasingly adapted to these particular areas of experimental research, the notions of chemical affinity or of electric fluids and their atmospheres providing particularly well-known examples.
The theories in which concepts like these functioned remained for some time predominantly qualitative and often correspondingly vague,
but they could nonetheless be confronted by individual experiments with a precision unknown in the Baconian sciences when the eighteenth century began. Furthermore, as the refinements which permitted such confrontations continued into the last third of the century and became increasingly the center of the corresponding fields, the Baconian sciences rapidly achieved a state very like that of the classical sciences in antiquity. Electricity and magnetism became developed sciences with the work of Aepinus, Cavendish, and Coulomb; heat with that of Black, Wilcke, and Lavoisier; chemistry more gradually and equivocally, but not later than the time of Lavoisier’s Chemical Revolution. At the beginning of the following century the qualitatively novel optical discoveries of the seventeenth century were for the first time assimilated to the older science of optics. With the occurrence of events like these, Baconian science had at last come of age, vindicating the faith, though not always the methodology, of its seventeenth-century founders.
How, during the almost two centuries of maturation, did the cluster of Baconian sciences relate to the cluster here called “classical”? The question has been far too little studied, but the answer, I think, will prove to be: not a great deal and then often with considerable difficulty - intellectual, institutional, and sometimes political. Into the nineteenth century the two clusters, classical and Baconian, remained distinct. Crudely put, the classical sciences were grouped together as “mathematics”; the Baconian were generally viewed as “experimental philosophy” or, in France, as “physique expérimentale”; chemistry, with its continuing ties to pharmacy, medicine, and the various crafts, was in part a member of the latter group, and in part a congeries of more practical specialties. 
This separation between the classical and Baconian sciences can be traced from the origin of the latter. Bacon himself was distrustful, not only of mathematics, but of the entire quasi-deductive structure of classical science. Those critics who ridicule him for failing to recognize the best science of his day have missed the point. He did not reject Copernicanism because he preferred the Ptolemaic system. Rather, he rejected both because he thought that no system so complex, abstract,
16. For an early stage in the
development of chemistry as a subject of intellectual concern, see Marie Boas,
Robert Boyle and Seventeenth-Century
and mathematical could contribute to either the understanding or the control of nature. His followers in the experimental tradition, though they accepted Copernican cosmology, seldom even attempted to acquire the mathematical skill and sophistication required to understand or pursue the classical sciences. That situation endured through the eighteenth century: Franklin, Black, and Nollet display it as clearly as Boyle and Hooke.
Its converse is far more equivocal. Whatever the causes of the Baconian movement, they impinged also on the previously established classical sciences. New instruments entered those fields, too, especially astronomy. Standards for reporting and evaluating data changed as well. By the last decade of the seventeenth century confrontations like that between Boyle and Pascal are no longer imaginable. But, as previously indicated, the effect of these developments was a gradual refinement rather than a substantial change in the nature of the classical sciences. Astronomy had been instrumental and optics experimental before; the relative merits of quantitative telescopic and naked eye observation were in doubt throughout the seventeenth century; excepting the pendulum, the instruments of mechanics were predominantly tools for pedagogic demonstration rather than for research. Under these circumstances, though the ideological gap between Baconian and classical science narrowed, it by no means disappeared. Through the eighteenth century, the main practitioners of the established mathematical sciences performed few experiments and made fewer substantive contributions to the development of the new experimental fields.
change. It isolates a figure who could and did make epochal contributions to the classical sciences but not, except through instrumental design, to the Baconian.
Educated during the years when British Baconianism was
at its height,
With the partial exception, however, of his continental contemporaries, Huyghens and Mariotte,
18. Boyle, Works, II, 42-43.
institutions and career lines, at least into the nineteenth century. Although much additional research in this area is needed, the following remarks will suggest the gross pattern which research is likely to refine. At least at the elementary level, the classical sciences had established themselves in the standard curriculum of the medieval university. During the seventeenth and eighteenth centuries the number of chairs devoted to them increased. The men who held them, together with those appointed to positions in the newly founded national scientific academies of
The example of the
sorts of talent for which the mechanics section had been designed, found no place in the Academy until, in 1816 at age 69, his name was inscribed on its rolls by royal ordinance.
What these isolated cases suggest is indicated also by the Academy’s formal organization. A section for physique expérimentale was not introduced until 1785, and it was then grouped in the mathematical division (with geometry, astronomy, and mechanics) rather than in the division for the more manipulative sciences physique (anatomy, chemistry and metallurgy, botany and agriculture, and natural history and mineralogy). After 1815, when the new section’s name was changed to physique générale, the experimentalists among its members were for some time very few. Looking at the eighteenth century as a whole, the contributions of academicians to the Baconian physical sciences were minor compared with those of doctors, pharmacists, industrialists, instrument makers, itinerant lecturers, and men of independent means. Again the exception is
Return now briefly from the end of the eighteenth century to the middle of the seventeenth. The Baconian sciences were then in gestation, the classical being radically transformed. Together with concomitant changes in the life sciences, these two sets of events constitute what has come to be called the Scientific Revolution. Although no part of this essay purports to explain its extraordinarily complex causes, it is worth noting how different the question of causes becomes when the developments to be explained are subdivided.
That only the classical sciences were transformed during the Scientific Revolution is not surprising. Other fields of physical science scarcely existed until late in the period. To the extent that they did, furthermore, they lacked any significant body of unified technical doctrine to reconstruct. Conversely, one set of reasons for the transformation of the classical sciences lies within their own previous lines of development. Although historians differ greatly about the weight to be attached to them, few now doubt that some medieval reformulations of ancient doctrine, Islamic or Latin, were of major significance to figures like Copernicus, Galileo, and Kepler. No similar scholastic roots for the Baconian sciences are visible to me, despite the claims sometimes made for the methodological tradition that descends from Grosseteste.
Many of the other factors now frequently invoked to explain the Scientific Revolution did contribute to the evolution of both classical and Baconian sciences, but often in different ways and to different degrees. The effects of new intellectual ingredients - initially Hermetic and then corpuscular – mechanical - in the environment where early modern science was practiced provide a first example of such differences. Within the classical sciences, Hermetic movements sometimes promoted the status of mathematics, encouraged attempts to find mathematical regularities in nature, and occasionally licensed the simple mathematical forms thus discovered as formal causes, the terminus of the scientific causal chain.  Both Galileo and Kepler provide examples of this increasingly ontological role of mathematics, and the latter displays a second, more occult, Hermetic influence as well. From Kepler and Gilbert to
After the first third of the seventeenth century, when Hermetic mysticism was increasingly rejected, its place, still in the classical sciences, was rapidly taken by one or another form of corpuscular philosophy derived from ancient atomism. Forces of attraction and repulsion between either gross or microscopic bodies were no longer favored, a source of much opposition to
19. The increased value ascribed to mathematics, as tool or as ontology, by many early-modern scientists has been recognized for almost half a century and was for many years described as a response to Renaissance neo-Platonism. Changing the label to “Hermeticism” does not improve the explanation of this aspect of scientific thought (though it has assisted in the recognition of other important novelties), and the change illustrates a decisive limitation of recent scholarship, one which I have not known how to avoid here. As currently used, “Hermeticism” refers to a variety of presumably related movements: neo-Platonism, Cabalism, Rosicrucianism, and what you will. They badly need to be distinguished: temporally, geographically, intellectually, and ideologically.
The same new intellectual milieux affected the Baconian sciences, but often for other reasons and in different ways. Doubtless, the Hermetic emphasis on occult sympathies helps to account for the growing interest in magnetism and electricity after 1550; similar influences promoted the status of chemistry from the time of Paracelsus to that of van Helmont. But current research increasingly suggests that the major contribution of Hermeticism to the Baconian sciences and perhaps to the entire Scientific Revolution was the Faustian figure of the magus, concerned to manipulate and control nature, often with the aid of ingenious contrivances, instruments, and machines. Recognizing Francis Bacon as a transition figure between the magus Paracelsus and the experimental philosopher Robert Boyle has done more than anything else in recent years to transform historical understanding of the manner in which the new experimental sciences were born. 
For these Baconian fields, unlike their classical contemporaries, the effects of the transition to corpuscularism were equivocal, a first reason why Hermeticism endured longer in subjects like chemistry and magnetism than in, say, astronomy and mechanics. To declare that sugar is sweet because its round particles soothe the tongue is not obviously an advance on attributing to it a saccharine potency. Eighteenth-century experience was to demonstrate that the development of Baconian sciences often required guidance from concepts like affinity and phlogiston, not categorically unlike the natural sympathies and antipathies of the Hermetic movement. But corpuscularism did separate the experimental sciences from magic, thus promoting needed independence. Even more important, it provided a rationale for experiment, as no form of Aristotelianism or Platonism could have done. While the tradition governing scientific explanation demanded the specification of formal causes or essences, only data provided by the natural course of events could be relevant to it. To experiment or to constrain nature was to do it violence, thus hiding the role of the “natures” or forms which made things what they were. In a corpuscular universe, on the other hand, experimentation had an obvious relevance to the sciences. It could not change and might specially illuminate the mechanical conditions and laws from which natural phenomena followed. That was the lesson Bacon attached repeatedly to the fable of Cupid in chains.
A new intellectual milieu was not, of course, the sole cause of the
20. Frances A. Yates, “The Hermetic Tradition in Renaissance
Science,” in C. S. Singleton (ed.), Science and History in the Renaissance
Scientific Revolution, and the other factors most often
invoked in its explanation also gain cogency when examined separately in
classical and Baconian fields. During the Renaissance the medieval
university’s monopoly on learning was gradually broken. New sources of wealth, new ways of life,
and new values combined to promote the status of a group formerly classified as
artisans and craftsmen. The
invention of printing and the recovery of additional ancient sources gave its
members access to a scientific and technological heritage previously available,
if at all, only within the clerical university setting. One result, epitomized in the careers of
Brunelleschi and Leonardo, was the emergence from craft guilds during the
fifteenth and sixteenth centuries of the artist-engineers whose expertise
included painting, sculpture, architecture, fortification, water supply, and the
design of engines of war and construction. Supported by an increasingly elaborate
system of patronage, these men were at once employees and increasingly also
ornaments of Renaissance courts and later sometimes of the city governments of
The importance of this new group to the Scientific Revolution is indisputable. Galileo, in numerous respects, and Simon Stevin, in all, are among its products. What requires emphasis, however, is that the sources its members used and the fields which they primarily influenced belong to the cluster here called classical. Whether as artists (perspective) or as engineers (construction and water supply), they mainly exploited works on mathematics, statics, and optics. Astronomy, too, occasionally entered their purview, though to a lesser extent. One of Vitruvius’ concerns had been the design of precise sundials; the Renaissance artist-engineers sometimes extended it to the design of other astronomical instruments as well.
21. P. Rossi (trans. Salvator Attanasio), Philosophy, Technology, and the Arts in the Early Modern Era (New York, 1970). Rossi and earlier students of the subject do not, however, discuss the possible importance of distinguishing between the crafts practiced by the artist-engineers and those later introduced to the learned world by figures like Vanoccio Biringuccio and Agricola. For some aspects of that distinction, introduced below, I am much indebted to conversation with my colleague, Michael S. Mahoney.
Although only here and there seminal, the concern of the artist-engineers with these classical fields was a significant factor in their reconstruction. It is probably the source of Brahe’s new instruments and certainly of Galileo’s concern with the strength of materials and the limited power of water pumps, the latter leading directly to Torricelli’s barometer. Plausibly, but more controversially, engineering concerns, promoted especially by gunnery, helped to separate the problem of local motion from the larger philosophical problem of change, simultaneously making numbers rather than geometric proportions relevant to its further pursuit. These and related subjects are the ones that led to the inclusion of a section for arts mécaniques in the French Academy and that caused that section to be grouped with the sections for geometry and astronomy. That it thereafter provided no natural home for the Baconian sciences finds its counterpart in the concerns of the Renaissance artist-engineers, which did not include the non-mechanical, non-mathematical aspects of such crafts as dyeing, weaving, glass-making, and navigation. These were, however, precisely the crafts that played so large a role in the genesis of the new experimental sciences. Bacon’s programmatic statements called for natural histories of them all, and some of those histories of non-mechanical crafts were written.
Because the possible utility of even an analytic separation between the mechanical and non-mechanical crafts has not previously been suggested, what follows must be even more tentative than what precedes. As subjects for learned concern, however, the latter appear to have arrived later than the former. Presumably promoted at the start by Paracelsan attitudes, their establishment is demonstrated in such works as Biringuccio’s Pyrotechnia, Agricola’s De re metallica, Robert Norman’s Newe Attractive, and Bernard Palissy’s Discours, the earliest published in 1540. The status previously achieved by the mechanical arts doubtless helps to explain the appearance of books like these, but the movement which produced them is nevertheless distinct. Few practitioners of the non-mechanical crafts were supported by patronage or succeeded before the late seventeenth century in escaping the confines of craft guilds. None could appeal to a significant classical literary tradition, a fact which probably made the pseudo-classical Hermetic literature and the figure of the magus more important to them than to their contemporaries in the mathematical-mechanical fields.  Except in chemistry,
22. Although neither deals quite directly with this point, two recent articles suggest the way in which, first, Hermeticism and, then, corpuscularism could figure in seventeenth century battles for intellectual-social status: P. M. Rattansi, “The Helmontian-[Galenist Controversy in Restoration England,” Ambix, XII (1964), 1-23; T. M. Brown, “The College of Physicians and the Acceptance of latromechanism in England, 1665- 1695,” Bulletin of the History of Medicine, XLIV (1970), 12-30.]
HHC: [bracketed] displayed on page 25 of original.
among pharmacists and doctors, actual practice was seldom combined with learned discourse about it. Doctors do, however, figure disproportionately large among those who wrote learned works not only on chemistry but also on the other non-mechanical crafts which provided data required for the development of the Baconian sciences. Agricola and Gilbert are only the earliest examples.
These differences between the two traditions rooted in
prior crafts may help to explain still another. Although the Renaissance artist-engineers
were socially useful, knew it, and sometimes based their claims upon it, the
utilitarian elements in their writings are far less persistent and strident than
those in the writings of men who drew upon the non-mechanical crafts. Remember how little Leonardo cared
whether or not the mechanical contrivances he invented could actually be built;
or compare the writings of Galileo, Pascal, Descartes, and
Excepting chemistry, which had found a variegated
institutional base by the end of the seventeenth century, the Baconian and
classical sciences flourished in different national settings from at least 1700. Practitioners of both can be
found in most European countries, but the center for the Baconian sciences was
23. Information relevant to this point is scattered
throughout Pierre Brunet, Les physi cien Hollandais et la méthode
striking. Such investigation may also show that the national differences just sketched emerged only after the mid-seventeenth century, becoming slowly more striking during the generations that followed. Are not the differences between the eighteenth-century activities of the
Among the numerous competing explanations of the Scientific Revolution, only one provides a clue to this pattern of geographical divergences. It is the so-called Merton thesis, a redevelopment for the sciences of explanations for the emergence of capitalism provided earlier by Weber, Troeltsch, and Tawney.  After their initial evangelical proselytizing phases, it is claimed, settled Puritan or protestant communities provided an “ethos” or “ethic” especially congenial to the development of science. Among its primary components were a strong utilitarian strain, a high valuation of work, including manual and manipulative work, and a distrust of system which encouraged each man to be his own interpreter first of Scripture and then of nature. Leaving aside, as others may not, the difficulties of identifying such an ethos and of determining whether it may be ascribed to all protestant or only to certain Puritan sects, the main drawbacks of this viewpoint have always been that it attempts to explain too much. If Bacon, Boyle, and Hooke seem to fit the Merton thesis, Galileo, Descartes, and Huyghens do not. It is in any case far from clear that post-evangelical Puritan or protestant communities existed anywhere until the Scientific Revolution had been underway for some time. Not surprisingly the Merton thesis has been controversial.
Its appeal is, however, vastly larger if it is applied not to the Scientific Revolution as a whole, but rather to the movement which advanced the Baconian sciences. That movement’s initial impetus towards power over nature through manipulative and instrumental techniques was doubtless supported by Hermeticism. But the corpuscular philosophies which in the sciences increasingly replaced Hermeticism from the 1630s carried no similar values, and Baconianism continued to flourish. That it did so especially in non-Catholic countries suggests that it may yet be worth discovering what, with respect to the
24. R. K. Merton, Science, Technology and Society in
sciences, a “Puritan” and an “ethos” are. Two isolated bits of biographical information may make that problem especially intriguing. Denis Papin, who built Boyle’s second air pump and invented the pressure cooker, was a Huguenot driven from
My final topic must be presented as an epilogue, a tentative sketch of a position to be developed and modified by further research. But, having traced the generally separate development of the classical and Baconian sciences into the late eighteenth century, I must at least ask what happened next. Anyone acquainted with the contemporary scientific scene will recognize that the physical sciences no longer fit the pattern sketched above, a fact which has made that pattern itself difficult to see. When and how did the change occur? What was its nature?
Part of the answer is that the physical sciences during the nineteenth century participated in the rapid growth and transformation experienced by all learned professions. Older fields like medicine and law gained new institutional forms, more rigid and with intellectual standards more exclusive than any they had known before. In the sciences, from the late eighteenth century, the number of journals and societies rapidly increased, and many of them, unlike the traditional national academies and their publications, were restricted to individual scientific fields. Longstanding disciplines like mathematics and astronomy became for the first time professions with their own institutional forms.  Similar phenomena occurred only slightly more slowly in the newer Baconian fields, and one result was a loosening of ties which had previously bound them together. Chemistry, in particular, had by mid-century at the latest become a separate intellectual profession, still with ties to industry and to other experimental fields but with an identity now distinct from either. Partly for these institutional reasons and partly because of the effect on chemical research, first, of
elsewhere in the physical sciences. As this occurred, topics like heat and electricity were increasingly barred from chemistry and left to experimental philosophy or to a new field, physics, that was increasingly taking its place.
A second important source of change during the nineteenth century was a gradual shift in the perceived identity of mathematics. Until perhaps the middle of the century such topics as celestial mechanics, hydrodynamics, elasticity, and the vibrations of continuous and discontinuous media were at the center of professional mathematical research. Seventy-five years later, they had become “applied mathematics,” a concern separate from and usually of lower status than the more abstract questions of “pure mathematics” which had become central to the discipline. Though courses in topics like celestial mechanics or even electromagnetic theory were sometimes still taught by members of mathematics faculties, they had become service courses, their subjects no longer on the frontier of mathematical thought.  The resulting separation between research in mathematics and in the physical sciences urgently needs more study, both for itself and for its effect on the development of the latter. That is doubly the case because it occurred in different ways and at different rates in different countries, a factor in the development of the additional national differences to be discussed below.
A third variety of change, especially relevant to the topics considered in this essay, was the remarkably rapid and full mathematization of a number of Baconian fields during the first quarter of the nineteenth century. Among the topics which now constitute the subject matter of physics, only mechanics and hydrodynamics had demanded advanced mathematical skills before 1800. Elsewhere the elements of geometry, trigonometry, and algebra were entirely sufficient. Twenty years later, the work of Laplace, Fourier, and Sadi Carnot had made higher mathematics essential to the study of heat; Poisson and Ampere had done the same for electricity and magnetism; and Jean Fresnel, with his immediate followers, had had a similar effect on the field of optics. Only as their new mathematical theories were accepted as
26. Relevant recollections about the relation of mathematics
and mathematical physics in
models did a profession with an identity like that of modern physics become one of the sciences. Its emergence demanded a lowering of the barriers, both conceptual and institutional, that had previously separated classical and Baconian fields.
Why those barriers were lowered when and as they were is a problem demanding much additional research. But a major part of the answer will doubtless lie in the internal development of the relevant fields during the eighteenth century. The qualitative theories so rapidly mathematized after 1800 had come into existence only during and after the I780s. Fourier’s theory demanded the concept of specific heat and the consequent systematic separation of notions of heat and temperature. The contributions of Laplace and Carnot to thermal theory required in addition the recognition at the end of the century of adiabatic heating. Poisson’s pioneering mathematization of static electrical and magnetic theory was made possible by the prior work of Coulomb, most of which appeared only in the I790s.  Ampere’s mathematization of the interaction between electric currents was supplied almost simultaneously with his discovery of the effects that his theory treated. Especially for the mathematization of electrical and thermal theory, recent developments in mathematical technique also played a role. Except perhaps in optics, the papers which between 1800 and 1825 made previously experimental fields fully mathematical could not have been written two decades before the burst of mathematization began.
Internal development, primarily of Baconian fields, will not, however, explain the manner in which mathematics was introduced after 1800. As the names of the authors of the new theories will already have suggested, the first mathematizers were uniformly French. Excepting in some initially little known papers by George Green and Gauss, nothing of the same sort occurred elsewhere before the I840s, when the British and Germans began belatedly to adopt and adapt the example set by the French a generation before. Probably institutional and individual factors will prove primarily responsible for that early French leadership. Beginning very slowly in the 1760s, with the appointments of Nollet and then of Monge to teach physique expérimentale at the Ecole du genie at Mézières, Baconian subjects increasingly penetrated the
27. Aspects of the problem of mathematizing physics are
considered in Kuhn, “The Function of Measurement in Modem Physical Science,”
education of French military engineers.  That movement culminated in the establishment during the 1790s of the Ecole polytechnique, a new sort of educational institution at which students were exposed not only to the classical subjects relevant to the arts mécanique but also to chemistry, the study of heat, and other related subjects. It can be no accident that all of those who produced mathematical theories of previously experimental fields were either teachers or students at the Ecole polytechnique. Also of great importance to the direction taken by their work was the magistral leadership of
For reasons that are currently both obscure and
controversial, the practice of the new mathematical physics declined rapidly in
What had begun in
28. Relevant information will be found in René Taton,
“L’dcole royale du genie de Mézières,” in R. Taton (ed.), Enseignement et
djffusion des sciences en
29. R. Fox, “The Rise and Fall of Laplacian Physics,” Historical Studies in the Physical Sciences, IV (1976), 89-136; R. H. Silliman, “Fresnel and the Emergence of Physics as a Discipline,” ibid., 137-162.
30. Relevant information as well as guidance to the still sparse literature on this topic will be found in R. Fox, “Scientific Enterprise and the Patronage of Research in France 1800-70,” Minerva, XI (1973), 442-473; H. W. Paul, “La science française de la seconde partie du XIXe siècle vue par les auteurs anglais et américains,” Revue d’histoire des sciences, XXVII (1974), 147-163. Note, however, that both are concerned primarily with the alleged decline in French science as a whole, an effect surely less pronounced and perhaps quite distinct from the decline of French physics. Conversations with Fox have reinforced my convictions and helped me to organize my remarks on these points.
the other. Part of Germany’s quite special success - attested by the preponderant role of Germans in the twentieth-century conceptual transformations of physics - must be due to the rapid growth and consequent plasticity of German educational institutions during the years when men like Neumann, Weber, Helmholtz, and Kirchhoff were creating a new discipline in which both experimentalists and mathematical theorists would be associated as practitioners of physics. 
During the first decades of this century that German model increasingly spread to the rest of the world. As it did so, the longstanding division between the mathematical and the experimental physical sciences was more and more obscured and may even seem to have disappeared. But, from another viewpoint, it is perhaps more accurately described as having been displaced - rom a position between separate fields to the interior of physics itself, a location from which it continues to provide a source of both individual and professional tensions. It is only, I suggest, because physical theory is now everywhere mathematical that theoretical and experimental physics appear as enterprises so different that almost no one can hope to achieve eminence in both. No such dichotomy between experiment and theory has characterized fields like chemistry or biology in which theory is less intrinsically mathematical. Perhaps, therefore, the cleavage between mathematical and experimental science still remains, rooted in the nature of the human mind. [32
31. Russel McCormmach, “Editor’s Foreword,” Historical Studies in the Physical Sciences, III (1971), ix-xxiv.
32. Other frequently remarked but still little investigated phenomena also hint at a psychological basis for this cleavage. Many mathematicians and theoretical physicists have been passionately interested in and involved with music, some having had great difficulty choosing between a scientific and a musical career. No comparably widespread involvement is visible in the experimental sciences including experimental physics (nor I think, in other disciplines without an apparent relationship to music). But music, or part of it, was once a member of the cluster of mathematical sciences, never of the experimental. Also likely to be revealing is further study of a subtle distinction often remarked by physicists: that between a “mathematical” and a “theoretical” physicist. Both use much mathematics, often on the same problems. But the first tends to take the physics problem as conceptually fixed and to develop powerful mathematical techniques for application to it; the second thinks more physically, adapting the conception of his problem to the often more limited mathematical tools at his disposal. Lewis Pyenson, to whom I am indebted for helpful comments on my earliest draft, is developing interesting ideas on the evolution of the distinction.