Nor was it possible, as has sometimes been suggested, for the scientist or the artisan of the day merely by purging his mind of the prejudices and distorting theories of the schools and looking on the naked face of nature to make a beginning of "modern science." Such an attempt would have left him destitute of any tools whatsoever for the investigation of nature, and lacking any intelligible "language" in which to co-ordinate his discoveries. For as Jowett wrote, pointing the "positive value" for later ages of the "general notions" provided by Greek philosophy, these are always "necessary to the apprehension of particular facts....Before men can observe the world they must be able to conceive it." (16) The most primitive act of classification by the mind is already an implicit interpretation in terms of an order, which cannot be considered as "given" in that which is classified (17). Though it be allowed that we make contact with the world through "experience," "experience," as a crude datum, is incapable of providing the methods and the controlling scheme for the analysis of itself, by which process alone it is rendered intelligible; indeed some such process, together with the implicit claim that it has been validly and successfully applied is the inescapable if unconscious reference, whenever the term "Reality" is employed with any degree of significance. For "the fact," as soon as it becomes anything above the purely qualitative subrelational intuition, is already theoretically orientated and known only in regard to some context which implicitly determines it. The relations and characteristics which compound its nature become known only as it is ascribed a place in an hierarchical set of concepts and categories - each one of which is in some respects an abbreviation of the total system - in which "experience" is made to participate. "The fact as such reflects, therefore, all the systematic problems of construction and interpretation. And in appealing to facts, we unconsciously appeal, whether we admit it or not, to the systems which are responsible for their formulation, for whatever we know of them we know only in terms of the system we have presupposed." (18) Indeed much of the power, the high survival value of aspects of the "new science" of the Renaissance, results from its development of novel methods of assessing, or rather constructing facts, its establishment of a revolutionary criterion of what should be considered "typical" or of fundamental importance in natural process. But it is to be noticed that although it arrived at this from general assumptions, and with bold confidence, by enormous "extrapolation," was prepared to assert the universal validity of its theories and launch into speculations on how the total workings of the cosmos might be represented on such patterns, nevertheless, the actual range of "experience" that it could immediately deal with and satisfactorily interpret, was at that time considerably more restricted than that which other natural philosophies were able to take into account. It was much more narrowly selective and, many thought, in a highly artificial way, in the data it was able to admit as significant, than were for instance an empirically minded "Naturalist" school or an Aristotelianism with its constant appeal to common sense, which might therefore claim to take their stand on much more solid and extensive "factual" foundations.
The present relevance and importance of this may be more clearly shown by a brief notice of one of the ideal, but self-confessed aims of scientific theory: that is its constant endeavour to reduce itself as far as possible to the form of a purely deductive system. It attempts to render experience rational by interpreting it in terms of laws which are held to possess, or which arise directly from principles possessing, a quality of "necessity." But this "necessity" is a product of the definitions and axioms which form the groundwork of the mental approach, and cannot, logically, be held in any way to be a quality of the sequence of sense impressions which the former are intended to describe approximately, and to account for satisfactorily. The methods which scientific procedure employs to establish "necessary" - and therefore - intelligible relations, reveal this procedure clearly as "an active endeavour to create something, and not a passive endeavour to discover something." (19) The underlying aim emerges strikingly from such classical instances of scientific activity as Fourier's treatment of heat - dominated by the attempt to determine a minimum number of simple "facts" from the scientific expression of which all related phenomena would mathematically be discoverable; or Lagrange's Mechanique Analytique which claimed to do for mechanics what Descartes had effected for geometry, to dispense with all figures, and constructions and exhibit the subject as a branch of analysis; or the same constancy of endeavour exhibited by Maxwell's total achievement, controlled by the explicitly confessed principle that a fact could only be held to be "explained" when it was reduced by analysis to a form in which it could be deduced from the fundamental laws of mechanics (20). From this standpoint, the ultimate problem for contemporary science appears as Sir Edmund Whittaker expressed it as "What are the fundamental data or postulates from which the complete set of laws of the material universe can be deduced by pure mathematics?" (21); while an indication of how far scientists may believe that theory has progressed in this direction is given by Eddington's assertion, in connection with the "Four Constants of Nature," that "ideally they are all that need to be measured, and when he has supplied us with these numbers, the experimental physicist might retire and leave all the rest of physics to the mathematician." (22)
This tendency to metamorphosize itself into a formally explicit deductive system is probably a feature of all organised and coherent thought. What distinguishes differing systems more particularly are the underlying criteria of intelligibility which inform such systems with the particular characteristics on the grounds of which they are accepted as psychologically satisfactory. Differing but equally ultimate criteria of this kind are represented by magical anthropomorphism, teleology, mechanical causation (as illustrated by Kelvin's declaration that he could only "understand" something when he could build a working mechanical model to reproduce it "right the way through") and, one of the more recent entrants to this class, the Calculus of probabilities. Much of the importance for subsequent scientific development of late Renaissance thinkers lies in their adoption of such criteria, which however archaic in source, were still a revolutionary departure from those of the majority of their more immediate predecessors; in their provision of a philosophical framework which rendered these theoretically acceptable, and which was to allow of a gradual redefinition of scientific method and a reassessment of the foundations of man's knowledge of the external world. The assimilation of the Elements of Euclid - then first disseminated in its complete form - and recently discovered treatises of Archimedes (both of which apparently offered mathematical systems made up of a series of pure deductions which also corresponded, exactly as far as was verifiable, with the presentations of sensible intuition), to a psychology and metaphysic drawn from platonist writings was to prove crucial (23). The methods they suggested commenced a rapid invasion of numerous other fields of thought hitherto considered as quite foreign to mathematics. Thus Cassirer writes: "Mathematics had been an element in culture before the Renaissance; but in the Renaissance with thinkers like Leonardo or Galileo it became a new cultural force. It is the intensity with which this new force fills the whole intellectual life of the time and transforms it from within that we should regard as what is significantly new." (24a) At the same time it may be noted that despite some brilliant individual discoveries, the services of the sixteenth century to pure mathematics lay less in any original work it produced than in vindicating its study and firmly re-establishing its importance - effecting indeed in large measure an entire revaluation of it. Thus, although it has been sometimes pointed out that it was mathematicians of the seventeenth century who first make any considerable advance on the ancient knowledge, that Cavalieri, Barrow and Wallis took up the study of geometry "precisely at the point where the Academy had left off," (24b) yet nevertheless it was an earlier generation that had, by its own faith and enthusiasm, resurrected the ancient achievement, had given it wide currency, and first adumbrated the character and importance of the role it was to play in future thought.
The triumphant, almost unhesitating career, of the methods whose seeds first found generally favourable ground in the thought of the Renaissance is perhaps the outstanding feature of "the modern period." So impressive has it been that it has sometimes been urged that the entire history of thought and science should be interpreted and assessed in relation to the history of mathematics (25), and a plausible defence can be made for Kelvin's statement, that only when a thing can be measured and a number ascribed to it, can it be said that anything may properly be known about it (26). Be the general validity of such views what it may, they undoubtedly reflect the nature of the premisses, which now seem unchallenged in their claim to provide the necessary Archimedean point from which alone Science can effectively survey, and attempt to operate on, the world. "Tout progres important dans la connaissance de la nature," it has been claimed, "etait marque par la conquete d'une decimale exacte de plus," (27) and Poincare, holding firmly that in themselves mathematics were no more than "un language commode," insisted none the less that they must not be considered as a veil set between the mind and nature but were the sole instruments permitting to the mind knowledge of "l'harmonie interne du monde, qui est la seule vertable realite objective." (28) On such standards it is clear that "the discovery of the true relevance of the mathematical relations disclosed in presentational immediacy was the first step in the intellectual conquest of nature. Accurate science was then born," (29) and it is hoped that the following study, by an examination of the thought of a typical 16th century "scientist," may make apparent some important aspects of the contribution of that period to such a beginning.
It is a period somewhat earlier than that perhaps usually given detailed consideration from such a point of view, and still occasionally stigmatised as largely "pre-scientific," yet in the late Renaissance may be observed a growing clearness and exactness in the determination of the direction in which later science was to move. Once begun, its progress is marked by a fine inevitability differentiating it strongly from other historically developing intellectual movements. Thus, to take a striking example, it is difficult to over-rate the importance of Newton's advance in under-propping the purely kinematic description of the solar system, which was all previous exact science had been able to provide, with an equally exact, related dynamic "explanation." Nevertheless Newton's expression of the law of gravitation can be seen as the solution of a purely mathematical problem, having for its data Kepler's expression of the laws of planetary motion, taken in conjunction with a transformation of Kepler's vague suggestion of a central controlling force into the exact question, what must the nature or magnitude of the action of such a force be, to permit such motions, given the respective masses of the bodies concerned? For two centuries after Newton, advances and discoveries in the study of the mechanism of nature are characterized by their conformity with the original Newtonian schema (30). When a change apparently began with the gradual intrusion into scientific thought of such features as the "stresses in the dielectric" by which Maxwell sought to explain electrostatic attractions but which he confessed himself unable to account for by mechanical considerations, or the necessary introduction in a rapidly increasing number of physical problems of considerations relative to the character of a "field," with its accompanying abandonment of attempts to seek "explanation" in terms of the nature and interactions of the component "entities," these marked, not the failure of the general methods of mathematical analysis which had produced Newtonian thought, but only the outgrowing of the concrete image, the intuitively attractive analogy of "mechanical causation," which had understandably been suggested by the application of these methods in the past, but with which they had no intrinsic connection. The psychological value of such pictures in fostering procedural development has frequently been high, and their interaction with it of utmost consequence, especially in the early stages of any science, but whatever the mental need of them, they become progressively less important, their survival more and more misleading, in a science controlled by a method in which they have no natural place; it is doubtful whether the atom can ever be thought of again under any other guise than that of a set of differential equations or be fitted with any more "understandable" description which it will not immediately repulse. Brief interludes when departments of science have appeared to hesitate in this path, have in their outcome only served to confirm the inevitability of the general pattern. Comte in 1828 haled the rise of an apparently independent and self sufficient chemistry - and biology - as prophetic signs that the tyranny of mathematics over scientific thought, and the metaphysical prejudice that certitude was only to be found in mathematics was approaching its end - or at least discovering its limited validity (31). But in the event, chemistry was more and more driven to seek an explanation for the fundamental operations it sought to account for from a mathematicised physic, and already by the end of the century Karl Pearson was extensively applying mathematical weapons to attack biological questions (32), a procedure thoroughly vindicated by subsequent developments, especially in Mendelian genetics. Today we occupy a world in which the failure of an otherwise adequate principle to account for a residue of 43 seconds of arc per century in the perihelion of Mercury can lead (in conjunction with a few other problems involving data of comparable magnitude) to a radical alteration in our total cosmology; and this in a period when as Hermann Weyl has authoritatively pronounced: "In spite of, or because of our deepened critical insight we are today less sure than at any previous time of the ultimate foundations on which mathematics rests." (33) It is as well, especially when considering the history of science, not to overlook, because of more obvious fundamental differences, such generic similarities as between, for example, the Pythagoreans' postulation of "antichthon," to satisfy the demands of the mathematic pattern they held as underlying the cosmos (34), the modern prediction, in order to bring observed planetary motions into conformity with the laws of Newton, of the existence of Neptune, which was shortly after verified, and the subsequent no less confident prediction of the size and position of the planet Vulcan on precisely similar grounds, a body which had later to be recognised as entirely mythical.
It is in the centuries immediately preceding the age of Newton that we may observe the gradual formation and dissemination of several of the fundamental beliefs - or if one will, conventional assumptions - that have so permeated and coloured later scientific thought, and have provided so many of the categories and relationships through which man has claimed to "understand" - if he did not experience - "reality." Dee, and many of his contemporaries, advocated an approach to nature the rich potentialities of which were only gradually to be revealed, but at the same time, he, as did also somewhat later thinkers in the same tradition such as Kepler or Galileo, largely derived this approach from prior beliefs about the general order of creation, and defended it by a metaphysical evaluation of it. It is necessary to emphasize this, since perhaps no later than the seventeenth century this process had been consciously reversed (35), and thus reversed - a general picture of nature being now "constructed" from the results of the application of some selected method to particular phenomena - was to receive general acceptance as the correct procedural sequence of thought. But when, as it would seem in such cases, metaphysics are dispensed with, it is usually to be found, as Burtt has pointed out, that what has occurred is merely that, unobserved, some "method" has been surreptitiously elevated to the status of a metaphysic (36); (while the assumption that a given method may be universally applied with unvarying success is already rich in implications about the Real Nature of Things, which follow from the asserted consonancy between things and the particular method). If Newton was able to believe that he "made no hypotheses," it was only because some of what had once been the "metaphysical fantasies" of earlier men, who had insisted on the necessity of replacing a humanly experienced qualitative interpretation of nature by an abstract quantitative one, had then become current and unquestioned coin: an instrument had been transmitted to him, which by then could be seen to produce sufficiently remarkable practical results to secure him from any feelings of a need for a priori certainty as to its validity, or for enquiry into whether it was possessed of a pragmatically respectable provenance (37). None the less, especially when the high reputation of the empiricism of Francis Bacon in this period is remembered, the question remains, as Pierre Boutroux phrases it (while investigating the underlying premise of Newton's thought, "Cet acte de foi dans la valeur des mathematiques") "Comment se fait - il qu'une vue aussi theorique ait ete si facilement acceptee par les heritiers directs d'une ecole de savants qui pretendait repousset systematiquement tous les principes poses a priori?" (38)
These two apparently opposed procedures - the one largely deriving its methods and approach from an acknowledged metaphysic or defending them by reference to it, the other confidently assuming a method and declaring it advances without hypotheses, are however, in reality, closely related, they merge into one another and historically it is usually difficult to distinguish, except very approximately, between them as present in various stages of the total development of a science. Any absolute distinction between them or clear cut discrimination between periods on such a basis is probably only acceptable in the interests of temporary convenience - generally these procedures are co-present, though in constantly varying proportions as regards influence and effectivity. Periods may be more readily distinguished by the change in the consciously acknowledged importance of each. In that form of thought, which from this standpoint usually precedes and contributes to the formation of the special character of the other type of procedure, that is later more generally accepted as correct, some unified Wholeness, an already typified cosmology, which is believed to be, at least in the general principles, already in the possession of the mind, controls and conditions the view taken of the parts, endowing them with their nature and significance by dominating and directing the detailed study of particular phenomena; in the second "modern" procedure, certain initial assumptions are present which define only the type of approach to be made to the parts, the method of approach then reveals aspects of the parts, the particulars it discriminates, which are accepted as primary, and generates in the end an idea of the whole, mediately, through discovered "laws." (These to a strict empiricism may appear as only the statistical grouping of results, but even so it is a grouping under previously chosen headings representing themselves the original determinative assumptions, or analogies, and frequently embodying ideas which impart to such laws a certain formal necessity.) It is clear however that both procedures must ultimately rely on premisses which, unless it is claimed these are to be admitted only in virtue of their material consequences and these in turn regarded as arising only by lucky and unanalysable chance from them - a view powerless to provide or suggest any guide or direction for further investigations, evidently stand in need of further justification and make demand for a foundation which could inevitably only be of a metaphysical type. Though whether in any case such foundation may be adequately supplied has, it would seem, nothing to do with the possible value which may result from their acceptance for thought in general or for scientific theory or practice (39); just as, conversely, though it is possible to pass an historical judgment made in the light of its subsequent developments on the justification, or the limited validity of any particular system arising from either of these procedures, this will usually be based on reasons, and certainly made from data bearing little relation to those which determined its original acceptance. Thus, whatever the particular merits of any system judged from this or any other standpoint, it would seem that, in the final analysis, at the source of even the most rigorously exact science is to be found only the Augustinian affirmation "Credimus ut Cognoscamus."
The close dependence of practice upon theory which is here in question is perhaps best illustrated by the nature of the "experimental method," in the early history of which Dee's Preface holds a not unimportant place. It draws its value, and it is this that distinguishes it from an uncritical contemplation or accumulation of experiences, from the fact that each experiment is envisaged within a coherent framework of theory, might indeed be regarded as merely a testing of the premisses which suggest it. The continual adjustments that result are what redeems the apparently circular course - the "hermeneutic circle" - of theory and observation from viciousness. Most hypotheses it will be found have died at the hands of their own offspring, that is have been eventually unable to account conveniently for data provided by experimentation which they have themselves conceived. But at all times the significance of the experiment lies in its relation to the theory which has designed it, and it is frequently a complicated artificial construct. A pure empiricism not only could never have lent any assistance to the development of this procedure, but belief in the very doubtful existence of such a form of thought is itself rather to be explained as a historical by-product of the successful application of the experimental method, and of misunderstandings as to its nature. For "at all times is rigorous reasoning, the essence of mathematics, the necessary condition of accurate and fruitful observation." (41) Moreover it is the body of theory, which has been thus organised by reason, that provides a systematic language of relations into which the experimental results may be interpreted and hence co-ordinated and connected. Indeed only through the medium of some such "language," however primitive in terms or structure, however unconsciously employed, can anything of the external world be known or even observed at all; while some particular syntax will provide the fundamental character of any body of thought, from the simplest magical anthropomorphism upwards, that can, in the largest sense, be described as science. Hence it is clear that the type of experimental question put, by itself providing the terms of the answer, conditions the nature of the observations and the type of result, for the form of its expression makes a preliminary delimitation of an area of acceptable "meaningfulness" in which the answer is confined (42). Thus Priestley's largely qualitative observation of his own experiments on air were barren of any "profitable" results, in-so-far as he himself was only able to make use of them to elaborate an outworn phlogiston hypothesis, while those same experiments repeated by Lavoisier to whom they had been described, and subjected to his searching but purely quantitative analysis, led directly to new and important chemical discoveries. Aristotelians in the Renaissance also claimed to "experiment" widely and constantly appealed to "experience" in opposing the theories of the Neo-Platonic Mathematicists. However, it was the devising of means for investigating experience in mathematical terms that was undertaken by various Renaissance theoreticians, that marks the beginning of specifically modern experimental methods. Yet significantly many of the "experiments" figuring, for instance, in the work of Stevin or Galileo, are of a purely "ideal" nature, that is, the conditions of the problem had only to be expressed in the new terms for the result to become immediately evident, without any necessity for an attempt at physical reproduction. Nevertheless, in their logical rigour, and conscious application of carefully designed, clearly understood theoretical tools they are "experiments" in a far truer sense than the practices of the thoroughly empirical and qualitative Averroist medicine which proved one of the chief founts of opposition to the new theories, and to which the teaching of Francis Bacon, with his relegation of mathematics to the role of an occasionally useful, practical auxiliary, is fundamentally far more akin than it is to the intellectual tradition that lies at the central core of scientific development.
Now "the history of science, in so far as it is a history of scientific progress, consists not so much in the progressive accumulation of facts as in the progressive clarification of problems." (43) But the phases in the natural development of a science are not at all reflected in the general formal structure it assumes at a later well-advanced stage, and to investigate its past merely with reference to this standard, exhibiting earlier stages as being in themselves merely primitive imitations of, and partial approximations to, this form, is but high-level teleological fantasy. Poincare's dictum on mathematics may be applied equally to all well-developed, clearly organised systems: "en devenant rigoureuse, la Science mathematique prend un caractere artificiel...; elle oublie ses origines historiques, on voit comment les questions peuvent se resoudre, on ne voit plus comment et pourquoi elles se posent." (44) Moreover basic assumptions, and ways of viewing problems, transmitted by the late Renaissance to later physical science, have been so hallowed by success in action that the original, less pragmatically reputable, motives for their formulation and adoption have been apt to be overlooked or to be dismissed as unworthy of serious analysis. It is perhaps only necessary to instance here Kepler's determined and influential defence of the, empirically unsupported, heliocentric system and his persistent rejection of the more plausible Tychonic compromise, which largely resulted from his ideas of the innate mystical dignity of the sun, and his erroneous identifications of it with the Pythagorean Central Fire; beliefs which suggested his "valuable error" of defining the planetary orbits with reference to the position of the sun, and not as Copernicus had done with reference to the centre of the earth's eccentric orbit. He was upheld through the eighteen years of incessant calculation, unaided by logarithms, leading to his discovery of the elliptical form of the orbit of Mars, and in his insistence on the tremendous importance of the utmost accuracy of observation, by the conviction that thus would be revealed the exact correspondence (that he had first suggested in Mysterium Cosmographicum, 1596) between the relative proportions of the planetary orbits, and the spheres inscribed in and circumscribing the, to him metaphysically significant, five regular solids (45). Or again, Galileo, despite the air of modernity pervading much of his work, which has won him the appellation of "the grandfather" of modern physical science, indicated quite clearly that it was only the continued "illogical" objections of opponents, that compelled him to a supererogatory search for practical demonstrations in support of a cosmology, of which he was convinced on far other grounds (46).
It is not merely that in innumerable individual cases, "misconceptions," "false analogies," "metaphysical notions" have been the immediate source of particular discoveries, subsequently verified predictions, significant methodological developments (47), but that it is these which have generally contributed the necessary total framework for the mind without which thought could hardly function directedly at all. There is no necessary proportion between the "psychological value" of a hypothesis, which may well be immense, and its "value" as assessed in terms of the extent of verifiable fact it embodies (48), for "the utility of a belief and its validity are independent variables, and erroneous hypotheses are often avenues to Truth." (49) Thus it has been quite properly pointed out that, despite Descartes' rejection of the Harveian circulation of the blood, upon theoretical considerations, and although "Cartesian physiology....is exceedingly poor in direct matter of fact observations...., the framework of the whole is of the most portentous nature and constitutes, indeed, one of the greatest landmarks in the evolution of biological science, and of natural science generally." (50)
Hence it is not merely insufficient but positively misleading to regard any past scheme of thought merely as "un melange de verite et d'erreurs, de resultats positifs fournis par l'experience et de donnees imaginaires enfantees par la speculation; elements disparates dont l'etrange association s'est trouvee favourisee par le demi-jour dans lequel ont forcement vecu et se sont developpes ces corps de doctrine."(51) It deserves instead examination as a total system, of which the active principles may be exposed, and the gradual unfolding of the potentialities it includes exhibited as supplying the logic, and the organic connections of successive scientific discoveries and general advance. Achievements, or branches of study, lying outside, apparently unrelated to, such general systems, have an appearance of isolation, even randomness, setting them apart from the main current of intellectual progress. This is inevitably the case with the work of even the foremost sixteenth century naturalists - Rondelet, Salviani, Gesner, Belon, Aldrovandi (52). It explains also why, after some centuries of professed empiricism, Renaissance medicine had made no significant advance beyond Galen, but developed rapidly when it entered into the sphere of influence of the new metaphysic and set about utilising in its own field the theoretical weapons this had provided for other branches of science. Thus "die neure Physiologie (of the Galilean era) beginnt mit der neueren Mechanik. Die Betrachtung der Naturerscheinungen ruckt vom Objekte zu den Funktionen, von der Statik zur Dynamik."(53) One of the immediate, more famous, fruits that may be claimed here was the discovery of the circulation of the blood, for "it is significant that the question Harvey put was a mechanical one and not a chemical one,"(54) and one in which he was guided to a solution by a quantitative evaluation of the data.(55)
It would seem then illegitimate, for even the History of "Science" as it must be written for any period, to attempt to establish any supposedly essential, or rigidly applied convential, distinctions between experimentally verified conclusions, and the underlying metaphysic, the exploded or abandoned conceptions of nature from which these sprung, and even those other conclusions born of the same parentage which died contributing no issue to a later assemblage of "recognised facts."(56) The error would lie in not making allowance for the distortions produced by our almost spontaneously assumed "Ptolomaic" attitude towards the past; which looking back from the standpoint of a present synthesis, ends by accepting this as a constant pole permitting an evaluation of precedent development according to its degree of apparent conformity with a rectilinear progress in this direction. Though true in certain respects this does not take account of the fact, that the teleological character of mental activity engenders in various periods a diversity of conscious, and conditioning aims which are likely to bear little resemblance to that more unified and permanent end which is gradually emergent from, or rather increasingly defined by, the development of the methodology running through, though differingly viewed within, these past systems. Accepted uncritically, such "Ptolomaicism" results in a confusion of overt, superficial, frequently chance similarities of form with real anticipation, and a failure to detect the genuinely homologous or properly comparable (57). In contrast, it may be confidently claimed that despite the thoroughly remote, or "pre-scientific" appearance presented by much of Dee's work, or many of the subjects which engaged his attention, here may nevertheless be detected in an active form, the germs - "seminal reasons" would have been his own phrase - of much subsequent theory and scientific practice; and discernible as present in a manner in which they are not to be found in the work of the majority of his predecessors, of contemporary "Aristotelian scientists," or adherents of an empirical but largely magical "naturalism" such as Porta or Cardan, or even in that of such propagandists of utilitarianism as the more "modern" Francis Bacon. However there was perhaps very little in which thinkers such as Dee could claim entire originality. The works of Duhem have ably demonstrated that in almost every respect their views have a long genealogy and had made frequent previous appearances in the history of thought. Nevertheless such appearances as they had made had been sporadic, or partial, represented merely by individual thinkers, or confined in practice, within the narrow bounds of the formal instruction of one or other of the schools - the University of paris did much to preserve and transmit various opinions on physical questions, differing from the traditional Aristotelian ones, which later became of critical importance in the general interpretation of nature - but it was only in the sixteenth century that these views became generally active, and a significant feature in the "climate of opinion." A testimony to this is the close connection they then rapidly established with the activities of the artisan and technical classes of the time.
The historical development of bodies of thought, more especially in scientific fields, presents, in some respects, a not unimportant resemblance to systems of geometry. That is, they may both be viewed as the unfolding of very often elaborate consequences, implicit in an original set of axioms and postulates. These give at first sight the misleading impression of having to do with primitive entities of which they serve to describe certain primary interrelations, and the natures of which might be expected to be known by previous definition. But the attempt so to know them by initial exhaustive description of a kind independent of the operational context in which they are later to figure, is ultimately only pandering to irrelevant demands of intuition. The entities exist only relationally; their "definition" is indirect and retrospective, a result of the total system, which they seemed to assist in generating. Examples are Euclid's attempts at a definition of "a straight line" which by its unsatisfactory uninformative character lured on generations of commentators to vain endeavours to better it, and his failure to relate satisfactorily his particular concept of "parallelism" to such simple previous definitions; the reason for his failure being in both cases that the nature of the entities Euclid tries intuitively to represent emerges only from the totality of operations in which they appear as signs, which totality produces and serves to define a certain type of abstract "space" on which the operations themselves seem to be dependent, but the nature of which is itself nothing more than the sum of the conditions permitting such operations to be performed - in this case a "space" infinite, homogeneous, allowing the transposition, or change in magnitude, of figures without distortion. But the "meanings" of these conditions are in turn only to be found in the operations. However the historical appearance of such an interpretation, especially in so far as it is maintained as being widely applicable, is a very late-occurring phenomenon, indicative of the attainment of a high level of self consciously critical sophistication. A "modern" can write "It is no wonder that the contemporary physicist finds operational conventions lurking within his definitions of physical facts"; (such ultimates as "mass," "energy," etc., on analysis cannot be shown to denote anything but various theoretical or laboratory "operations") "physical theory has never been anything else than this elaborate and ingenious code of regulations defining how anyone may undertake the imitation and analysis of experimental procedures...It is not a definition of the processes which constitute Nature."(58) At the same time it is obviously not at all necessary to adopt such a theory in order to contribute valuably to progress in fields of which this theory offers quite fundamental reinterpretation; especially is it an outlook very foreign to those Renaissance philosophers in whose work one may detect the first tentative formulation and application, of various axioms of approach and the presence of certain presuppositions that were to play so large a part in the growth of science. They relied on metaphysical doctrines, or the presence of "clear and distinct ideas" to provide foundations for their axiomatic beliefs, to render intelligible the elements these seemed to have reference to and resolve such questions as the "nature of number," and so to justifying their methods of analysis. The philosophy which performed this momentous function for the Renaissance, as the type of Neoplatonism to which Dee subscribed, and if any attempt at a neatly concise summary finds itself continually baffled by its occasionally confusing lack of apparent order, the extensive and indiscriminate nature of its contents, the diversity of materials which it found itself able unperturbedly to find room for, the relatively wide range of standpoints it could adopt on varying questions yet it is this very breadth and flexibility that provides, in some measure, a clue to its strength and contributed largely to its historical value. It must be remembered in considering this rich confusion that although it drew much inspiration from, and founded itself in part upon mathematics, this is not one of those well-organised, logically developed philosophical systems - Spinoza's, or Leibniz's for example, which seem inevitably to arise in those times when some special branch of mathematics attaining a successful, well-developed maturity, in that form provides - in a very thorough and profoundly significant sense - a "model," but is a body of thought in which a new, almost rudimentary mathematics was able to find a congenial place, where nourished and protected, the vigorous development it was capable of, when so assisted, and for which the time was ripe, could occur unimpeded.