Above all, this is manifested in the bizarre hypotheses of atomic disintegration which elucidate the phenomena of radioactivity, and according to which uranium atoms that have kept their essence unaltered, in spite of all external influences, for millions of years and then suddenly without assignable cause explode, scattering their smallest particles over space with velocities of thousands of kilometres per second. Only a few individuals in an aggregate of radioactive atoms are struck by Destiny thus, the neighbours being entirely unaffected. Here too, then, is a picture of history and not “Nature,” and although statistical methods here also prove to be necessary, one might almost say that in them mathematical number has been replaced by chronological.[527]
With ideas like these, the mythopoetic force of the Faustian soul is returning to its origins. It was at the outset of the Gothic, just at the time when the first mechanical clocks were being built, that the myth of the world’s end, Ragnarök, the Twilight of the Gods, arose. It may be that, like all the reputedly old-German myths Ragnarök (whether in the Völuspa form or as the Christian Muspilli) was modelled more or less on Classical and particularly Christian-Apocalyptic motives. Nevertheless, it is the expression and symbol of the Faustian and of no other soul. The Olympian college is historyless, it knows no becoming, no epochal moments, no aim. But the passionate thrust into distance is Faustian. Force, Will, has an aim, and where there is an aim there is for the inquiring eye an end. That which the perspective of oil-painting expressed by means of the vanishing point, the Baroque park by its point de vue, and analysis by the nth term of an infinite series—the conclusion, that is, of a willed directedness—assumes here the form of the concept. The Faust of the Second Part is dying, for he has reached his goal. What the myth of Götterdämmerung signified of old, the irreligious form of it, the theory of Entropy, signifies to-day—world’s end as completion of an inwardly necessary evolution.
XV
It remains now to sketch the last stage of Western science. From our standpoint of to-day, the gently-sloping route of decline is clearly visible.
This too, the power of looking ahead to inevitable Destiny, is part of the historical capacity that is the peculiar endowment of the Faustian. The Classical died, as we shall die, but it died unknowing. It believed in an eternal Being and to the last it lived its days with frank satisfaction, each day spent as a gift of the gods. But we know our history. Before us there stands a last spiritual crisis that will involve all Europe and America. What its course will be, Late Hellenism tells us. The tyranny of the Reason—of which we are not conscious, for we are ourselves its apex—is in every Culture an epoch between man and old-man, and no more. Its most distinct expression is the cult of exact sciences, of dialectic, of demonstration, of causality. Of old the Ionic, and in our case the Baroque were its rising limb, and now the question is what form will the down-curve assume?
In this very century, I prophesy, the century of scientific-critical Alexandrianism, of the great harvests, of the final formulations, a new element of inwardness will arise to overthrow the will-to-victory of science. Exact science must presently fall upon its own keen sword. First, in the 18th Century, its methods were tried out, then, in the 19th, its powers, and now its historical rôle is critically reviewed. But from Skepsis there is a path to “second religiousness,” which is the sequel and not the preface of the Culture. Men dispense with proof, desire only to believe and not to dissect.
The individual renounces by laying aside books. The Culture renounces by ceasing to manifest itself in high scientific intellects. But science exists only in the living thought of great savant-generations, and books are nothing if they are not living and effective in men worthy of them. Scientific results are merely items of an intellectual tradition. It constitutes the death of a science that no one any longer regards it as an event, and an orgy of two centuries of exact scientific-ness brings satiety. Not the individual, the soul of the Culture itself has had enough, and it expresses this by putting into the field of the day ever smaller, narrower and more unfruitful investigators. The great century of the Classical science was the third, after the death of Aristotle; when Archimedes died and the Romans came, it was already almost at its end. Our great century has been the 19th. Savants of the calibre of Gauss and Humboldt and Helmholtz were already no more by 1900. In physics as in chemistry, in biology as in mathematics, the great masters are dead, and we are now experiencing the decrescendo of brilliant gleaners who arrange, collect and finish-off like the Alexandrian scholars of the Roman age. Everything that does not belong to the matter-of-fact side of life—to politics, technics or economics—exhibits the common symptom. After Lysippus no great sculptor, no artist as man-of-destiny, appears, and after the Impressionists no painter, and after Wagner no musician. The age of Cæsarism needed neither art nor philosophy. To Eratosthenes and Archimedes, true creators, succeed Posidonius and Pliny, collectors of taste, and finally Ptolemy and Galen, mere copyists. And, just as oil-painting and instrumental music ran through their possibilities in a few centuries, so also dynamics, which began to bud about 1600, is to-day in the grip of decay.
But before the curtain falls, there is one more task for the historical Faustian spirit, a task not yet specified, hitherto not even imagined as possible. There has still to be written a morphology of the exact sciences, which shall discover how all laws, concepts and theories inwardly hang together as forms and what they have meant as such in the life-course of the Faustian Culture. The re-treatment of theoretical physics, of chemistry, of mathematics as a sum of symbols—this will be the definitive conquest of the mechanical world-aspect by an intuitive, once more religious, world-outlook, a last master-effort of physiognomic to break down even systematic and to absorb it, as expression and symbol, into its own domain. One day we shall no longer ask, as the 19th Century asked, what are the valid laws underlying chemical affinity or diamagnetism—rather, we shall be amazed indeed that minds of the first order could ever have been completely preoccupied by questions such as these. We shall inquire whence came these forms that were prescribed for the Faustian spirit, why they had to come to our kind of humanity particularly and exclusively, and what deep meaning there is in the fact that the numbers that we have won became phenomenal in just this picture-like disguise. And, be it said, we have to-day hardly yet an inkling of how much in our reputedly objective values and experiences is only disguise, only image and expression.
The separate sciences—epistemology, physics, chemistry, mathematics, astronomy—are approaching one another with acceleration, converging towards a complete identity of results. The issue will be a fusion of the form-worlds, which will present on the one hand a system of numbers, functional in nature and reduced to a few ground-formulæ, and on the other a small group of theories, denominators to those numerators, which in the end will be seen to be myths of the springtime under modern veils, reducible therefore—and at once of necessity reduced—to picturable and physiognomically significant characters that are the fundamentals. This convergence has not yet been observed, for the reason that since Kant—indeed, since Leibniz—there has been no philosopher who commanded the problems of all the exact sciences.
Even a century ago, physics and chemistry were foreign to one another, but to-day they cannot be handled separately—witness spectrum analysis, radioactivity, radiation of heat. Fifty years ago the essence of chemistry could still be described almost without mathematics, and to-day the chemical elements are in course of volatilizing themselves into the mathematical constants of variable relation-complexes, and with the sense-comprehensibility of the elements goes the last trace of magnitude as the term is Classically and plastically understood. Physiology is becoming a chapter of organic chemistry and is making use of the methods of the Infinitesimal Calculus. The branch of the older physics—distinguished, according to the bodily senses concerned in each, as acoustics, optics and heat—have melted into a dynamic of matter and a dynamic of the æther, and these again can no longer keep their frontiers mathematically clear. The last discussions of epistemology are now uniting with those of higher analysis and theoretical physics to occupy an almost inaccessible domain, the domain to which, for example, the theory of Relativity belongs or ought to belong. The sign-language in which the emanation-theory of radioactivity expresses itself is completely de-sensualized.
Chemistry, once concerned with defining as sharply as possible the qualities of elements, such as valency, weight, affinity and reactivity, is setting to work to get rid of these sensible traits. The elements are held to differ in character according to their derivation from this or that compound. They are represented to be complexes of different units which indeed behave (“actually”) as units of a higher order and are not practically separable but show deep differences in point of radioactivity. Through the emanation of radiant energy degradation is always going on, so that we can speak of the lifetime of an element, in formal contradiction with the original concept of the element and the spirit of modern chemistry as created by Lavoisier. All these tendencies are bringing the ideas of chemistry very close to the theory of Entropy, with its suggestive opposition of causality and destiny, Nature and History. And they indicate the paths that our science is pursuing—on the one hand, towards the discovery that its logical and numerical results are identical with the structure of the reason itself, and, on the other, towards the revelation that the whole theory which clothes these numbers merely represents the symbolic expression of Faustian life.
And here, as our study draws to its conclusion, we must mention the truly Faustian theory of “aggregates,” one of the weightiest in all this form-world of our science. In sharpest antithesis to the older mathematic, it deals, not with singular quantities but with the aggregates constituted by all quantities [or objects] having this or that specified morphological similarity—for instance all square numbers or all differential equations of a given type. Such an aggregate it conceives as a new unit, a new number of higher order, and subjecting it to criteria of new and hitherto quite unsuspected kinds such as “potency,” “order,” “equivalence,” “countableness,” and devising laws and operative methods for it in respect of these criteria. Thus is being actualized a last extension of the function-theory.[528] Little by little this absorbed the whole of our mathematic, and now it is dealing with variables by the principles of the Theory of Groups in respect of the character of the function and by those of the Theory of Aggregates in respect of the values of the variables. Mathematical philosophy is well aware that these ultimate meditations on the nature of number are fusing with those upon pure logic, and an algebra of logic is talked of. The study of geometrical axioms has become a chapter of epistemology.
The aim to which all this is striving, and which in particular every Nature-researcher feels in himself as an impulse, is the achievement of a pure numerical transcendence, the complete and inclusive conquest of the visibly apparent and its replacement by a language of imagery unintelligible to the layman and impossible of sensuous realization—but a language that the great Faustian symbol of Infinite space endows with the dignity of inward necessity. The deep scepticism of these final judgments links the soul anew to the forms of early Gothic religiousness. The inorganic, known and dissected world-around, the World as Nature and System, has deepened itself until it is a pure sphere of functional numbers. But, as we have seen, number is one of the most primary symbols in every Culture; and consequently the way to pure number is the return of the waking consciousness to its own secret, the revelation of its own formal necessity. The goal reached, the vast and ever more meaningless and threadbare fabric woven around natural science falls apart. It was, after all, nothing but the inner structure of the “Reason,” the grammar by which it believed it could overcome the Visible and extract therefrom the True. But what appears under the fabric is once again the earliest and deepest, the Myth, the immediate Becoming, Life itself. The less anthropomorphic science believes itself to be, the more anthropomorphic it is. One by one it gets rid of the separate human traits in the Nature-picture, only to find at the end that the supposed pure Nature which it holds in its hand is—humanity itself, pure and complete. Out of the Gothic soul grew up, till it overshadowed the religious world-picture, the spirit of the City, the alter ego of irreligious Nature-science. But now, in the sunset of the scientific epoch and the rise of victorious Skepsis, the clouds dissolve and the quiet landscape of the morning reappears in all distinctness.
The final issue to which the Faustian wisdom tends—though it is only in the highest moments that it has seen it—is the dissolution of all knowledge into a vast system of morphological relationships. Dynamics and Analysis are in respect of meaning, form-language and substance, identical with Romanesque ornament, Gothic cathedrals, Christian-German dogma and the dynastic state. One and the same world-feeling speaks in all of them. They were born with, and they aged with, the Faustian Culture, and they present that Culture in the world of day and space as a historical drama. The uniting of the several scientific aspects into one will bear all the marks of the great art of counterpoint. An infinitesimal music of the boundless world-space—that is the deep unresting longing of this soul, as the orderly statuesque and Euclidean Cosmos was the satisfaction of the Classical. That—formulated by a logical necessity of Faustian reason as a dynamic-imperative causality, then developed into a dictatorial, hard-working, world-transforming science—is the grand legacy of the Faustian soul to the souls of Cultures yet to be, a bequest of immensely transcendent forms that the heirs will possibly ignore. And then, weary after its striving, the Western science returns to its spiritual home.
1. Kant’s error, an error of very wide bearing which has not even yet been overcome, was first of all in bringing the outer and inner Man into relation with the ideas of space and time by pure scheme, though the meanings of these are numerous and, above all, not unalterable; and secondly in allying arithmetic with the one and geometry with the other in an utterly mistaken way. It is not between arithmetic and geometry—we must here anticipate a little—but between chronological and mathematical number that there is fundamental opposition. Arithmetic and geometry are both spatial mathematics and in their higher regions they are no longer separable. Time-reckoning, of which the plain man is capable of a perfectly clear understanding through his senses, answers the question “When,” not “What” or “How Many.”
2. One cannot but be sensible how little depth and power of abstraction has been associated with the treatment of, say, the Renaissance or the Great Migrations, as compared with what is obviously required for the theory of functions and theoretical optics. Judged by the standards of the physicist and the mathematician, the historian becomes careless as soon as he has assembled and ordered his material and passes on to interpretation.
3. In the original, these fundamental antitheses are expressed simply by means of werden and sein. Exact renderings are therefore impossible in English.—Tr.
4. The attempts of the Greeks to frame something like a calendar or a chronology after the Egyptian fashion, besides being very belated indeed, were of extreme naïveté. The Olympiad reckoning is not an era in the sense of, say, the Christian chronology, and is, moreover, a late and purely literary expedient, without popular currency. The people, in fact, had no general need of a numeration wherewith to date the experiences of their grandfathers and great-grandfathers, though a few learned persons might be interested in the calendar question. We are not here concerned with the soundness or unsoundness of a calendar, but with its currency, with the question of whether men regulated their lives by it or not; but, incidentally, even the list of Olympian victors before 500 is quite as much of an invention as the lists of earlier Athenian archons or Roman consuls. Of the colonizations, we possess not one single authentic date (E. Meyer. Gesch. d. Alt. II, 442. Beloch. Griech. Gesch. I, 2, 219) “in Greece before the fifth century, no one ever thought of noting or reporting historical events.” (Beloch. I, 1, 125). We possess an inscription which sets forth a treaty between Elis and Heraea which “was to be valid for a hundred years from this year.” What “this year” was, is however not indicated. After a few years no one would have known how long the treaty had still to run. Evidently this was a point that no one had taken into account at the time—indeed, the very “men of the moment” who drew up the document, probably themselves soon forgot. Such was the childlike, fairy-story character of the Classical presentation of history that any ordered dating of the events of, say, the Trojan War (which occupies in their series the same position as the Crusades in ours) would have been felt as a sheer solecism.
Equally backward was the geographical science of the Classical world as compared with that of the Egyptians and the Babylonians. E. Meyer (Gesch. d. Alt. II, 102) shows how the Greeks’ knowledge of the form of Africa degenerated from Herodotus (who followed Persian authorities) to Aristotle. The same is true of the Romans as the heirs of the Carthaginians; they first repeated the information of their alien forerunners and then slowly forgot it.
5. Contrast with this the fact, symbolically of the highest importance and unparalleledunparalleled in art-history, that the Hellenes, though they had before their eyes the works of the Mycenæan Age and their land was only too rich in stone, deliberately reverted to wood; hence the absence of architectural remains of the period 1200-600. The Egyptian plant-column was from the outset of stone, whereas the Doric column was wooden, a clear indication of the intense antipathy of the Classical soul towards duration.
6. Is there any Hellenic city that ever carried out one single comprehensive work that tells of care for future generations? The road and water systems which research has assigned to the Mycenæan—i.e., the pre-Classical—age fell into disrepair and oblivion from the birth of the Classical peoples—that is, from the Homeric period. It is a remarkably curious fact, proved beyond doubt by the lack of epigraphic remains, that the Classical alphabet did not come into use till after 900, and even then only to a limited extent and for the most pressing economic needs. Whereas in the Egyptian, the Babylonian, the Mexican and the Chinese Cultures the formation of a script begins in the very twilight of dawn, whereas the Germans made themselves a Runic alphabet and presently developed that respect for writing as such which led to the successive refinements of ornamental calligraphy, the Classical primitives were entirely ignorant of the numerous alphabets that were current in the South and the East. We possess numerous inscriptions of Hittite Asia Minor and of Crete, but not one of Homeric Greece. (See Vol. II, pp. 180 et seq.)
7. From Homer to the tragedies of Seneca, a full thousand years, the same handful of myth-figures (Thyestes, Clytæmnestra, Heracles and the like) appear time after time without alteration, whereas in the poetry of the West, Faustian Man figures, first as Parzeval or Tristan, then (modified always into harmony with the epoch) as Hamlet, Don Quixote, Don Juan, and eventually Faust or Werther, and now as the hero of the modern world-city romance, but is always presented in the atmosphere and under the conditions of a particular century.
8. It was about 1000 A.D. and therefore contemporaneously with the beginning of the Romanesque style and the Crusades—the first symptoms of a new Soul—that Abbot Gerbert (Pope Sylvester II), the friend of the Emperor Otto III, invented the mechanism of the chiming wheel-clock. In Germany too, the first tower-clocks made their appearance, about 1200, and the pocket watch somewhat later. Observe the significant association of time measurement with the edifices of religion.
9. Newton’s choice of the name “fluxions” for his calculus was meant to imply a standpoint towards certain metaphysical notions as to the nature of time. In Greek mathematics time figures not at all.
10. Here the historian is gravely influenced by preconceptions derived from geography, which assumes a Continent of Europe, and feels himself compelled to draw an ideal frontier corresponding to the physical frontier between “Europe” and “Asia.” The word “Europe” ought to be struck out of history. There is historically no “European” type, and it is sheer delusion to speak of the Hellenes as “European Antiquity” (were Homer and Heraclitus and Pythagoras, then, Asiatics?) and to enlarge upon their “mission” as such. These phrases express no realities but merely a sketchy interpretation of the map. It is thanks to this word “Europe” alone, and the complex of ideas resulting from it, that our historical consciousness has come to link Russia with the West in an utterly baseless unity—a mere abstraction derived from the reading of books—that has led to immense real consequences. In the shape of Peter the Great, this word has falsified the historical tendencies of a primitive human mass for two centuries, whereas the Russian instinct has very truly and fundamentally divided “Europe” from “Mother Russia” with the hostility that we can see embodied in Tolstoi, Aksakov or Dostoyevski. “East” and “West” are notions that contain real history, whereas “Europe” is an empty sound. Everything great that the Classical world created, it created in pure denial of the existence of any continental barrier between Rome and Cyprus, Byzantium and Alexandria. Everything that we imply by the term European Culture came into existence between the Vistula and the Adriatic and the Guadalquivir and, even if we were to agree that Greece, the Greece of Pericles, lay in Europe, the Greece of to-day certainly does not.
11. See Vol. II, pp. 31, 175.
12. Windelband, Gesch. d. Phil. (1903), pp. 275 ff.
13. In the New Testament the polar idea tends to appear in the dialectics of the Apostle Paul, while the periodic is represented by the Apocalypse.
14. As we can see from the expression, at once desperate and ridiculous, “newest time” (neueste Zeit).
15. K. Burdach, Reformation, Renaissance, Humanismus, 1918, pp. 48 et seq. (English readers may be referred to the article Joachim of Floris by Professor Alphandery in the Encyclopædia Britannica, XI ed., Tr.)
16. The expression “antique”—meant of course in the dualistic sense—is found as early as the Isagoge of Porphyry (c. 300 A.D.).
17. “Mankind? It is an abstraction. There are, always have been, and always will be, men and only men.” (Goethe to Luden.)
18. “Middle Ages” connotes the history of the space-time region in which Latin was the language of the Church and the learned. The mighty course of Eastern Christianity, which, long before Boniface, spread over Turkestan into China and through Sabæa into Abyssinia, was entirely excluded from this “world-history.”
19. See Vol. II, p. 362, foot-note. To the true Russian the basic proposition of Darwinism is as devoid of meaning as that of Copernicus is to a true Arab.
20. This is conclusively proved by the selection that determined survival, which was governed not by mere chance but very definitely by a deliberate tendency. The Atticism of the Augustan Age, tired, sterile, pedantic, back-looking, conceived the hall-mark “classical” and allowed only a very small group of Greek works up to Plato to bear it. The rest, including the whole wealth of Hellenistic literature, was rejected and has been almost entirely lost. It is this pedagogue’s anthology that has survived (almost in its entirety) and so fixed the imaginary picture of “Classical Antiquity” alike for the Renaissance Florentine and for Winckelmann, Hölderlin, and even Nietzsche.
[In this English translation, it should be mentioned, the word “Classical” has almost universally been employed to translate the German antike, as, in the translator’s judgment, no literal equivalent of the German word would convey the specific meaning attached to antike throughout the work, “antique,” “ancient” and the like words having for us a much more general connotation.—Tr.]
21. As will be seen later, the words zivilisierte and Zivilisation possess in this work a special meaning.—Tr.
22. English not possessing the adjective-forming freedom of German, we are compelled to coin a word for the rendering of grossstädtisch, an adjective not only frequent but of emphatic significance in the author’s argument.—Tr.
23. See Vol. II, pp. 117 et seq.
24. One cannot fail to notice this in the development of Strindberg and especially in that of Ibsen, who was never quite at home in the civilized atmosphere of his problems. The motives of “Brand” and “Rosmersholm” are a wonderful mixture of innate provincialism and a theoretically-acquired megalopolitan outlook. Nora is the very type of the provincial derailed by reading.
25. Who forbade the cult of the town’s hero Adrastos and the reading of the Homeric poems, with the object of cutting the Doric nobility from its spiritual roots (c. 560 B.C.).
26. A profound word which obtains its significance as soon as the barbarian becomes a culture-man and loses it again as soon as the civilization-man takes up the motto “Ubi bene, ibi patria.”
27. Hence it was that the first to succumb to Christianity were the Romans who could not afford to be Stoics. See Vol. II, pp. 607 et seq.
28. In Rome and Byzantium, lodging-houses of six to ten stories (with street-widths of ten feet at most!) were built without any sort of official supervision, and frequently collapsed with all their inmates. A great part of the cives Romani, for whom panem et circenses constituted all existence, possessed no more than a high-priced sleeping-berth in one of the swarming ant-hills called insulæ. (Pohlmann, Aus Altertum und Gegenwart, 1911, pp. 199 ff.)
29. See Vol. II, 577.
30. German gymnastics, from the intensely provincial and natural forms imparted to it by Jahn, has since 1813 been carried by a very rapid development into the sport category. The difference between a Berlin athletic ground on a big day and a Roman circus was even by 1914 very slight.
31. See Vol. II, 529.
32. The conquest of Gaul by Cæsar was frankly a colonial, i.e., a one-sided, war; and the fact that it is the highest achievement in the later military history of Rome only shows that the well of real achievement was rapidly drying up.
33. The modern Germans are a conspicuous example of a people that has become expansive without knowing it or willing it. They were already in that state while they still believed themselves to be the people of Goethe. Even Bismarck, the founder of the new age, never had the slightest idea of it, and believed himself to have reached the conclusion of a political process (cf. Vol. II, 529).
34. This is probably the meaning of Napoleon’s significant words to Goethe: “What have we to-day to do with destiny? Policy is destiny.”
35. Corresponding to the 300-50 B.C. phase of the Classical world.
36. Which in the end gave its name to the Empire (Tsin = China).
37. See Vol. II, 521-539.
38. See Vol. II, 373 ff.
39. The work referred to is embodied in Vol. II (pp. 521 et seq., 562 et seq., 631 et seq.).
40. The philosophy of this book I owe to the philosophy of Goethe, which is practically unknown to-day, and also (but in a far less degree) to that of Nietzsche. The position of Goethe in West-European metaphysics is still not understood in the least; when philosophy is being discussed he is not even named. For unfortunately he did not set down his doctrines in a rigid system, and so the systematic philosophy has overlooked him. Nevertheless he was a philosopher. His place vis-à-vis Kant is the same as that of Plato—who similarly eludes the would-be-systematizer—vis-à-vis Aristotle. Plato and Goethe stand for the philosophy of Becoming, Aristotle and Kant the philosophy of Being. Here we have intuition opposed to analysis. Something that it is practically impossible to convey by the methods of reason is found in individual sayings and poems of Goethe, e.g., in the Orphische Urworte, and stanzas like “Wenn im Unendlichen” and “Sagt es Niemand,” which must be regarded as the expression of a perfectly definite metaphysical doctrine. I would not have one single word changed in this: "The Godhead is effective in the living and not in the dead, in the becoming and the changing, not in the become and the set-fast; and therefore, similarly, the reason (Vernunft) is concerned only to strive towards the divine through the becoming and the living, and the understanding (Verstand) only to make use of the become and the set-fast" (to Eckermann). This sentence comprises my entire philosophy.
41. At the end of the volume.
42. Weltanschauung im wörtlichen Sinne; Anschauung der Welt.
43. The case of mankind in the historyless state is discussed in Vol. II, pp. 58 et seq.
44. With, moreover, a “biological horizon.” See Vol. II, p. 34.
45. See Vol. II, pp. 327 et seq.
46. Also “thinking in money.” See Vol. II, pp. 603 et seq.
47. Dynasties I-VIII, or, effectively, I-VI. The Pyramid period coincides with Dynasties IV-VI. Cheops, Chephren and Mycerinus belong to the IV dynasty, under which also great water-control works were carried out between Abydos and the Fayum.—Tr.
48. As also those of law and of money. See Vol. II, pp. 68 et seq., pp. 616 et seq.
49. Poincaré in his Science et Méthode (Ch. III), searchingly analyses the “becoming” of one of his own mathematical discoveries. Each decisive stage in it bears “les mêmes caractères de brièveté, de soudainetésoudaineté et de certitude absolue” and in most cases this “certitude” was such that he merely registered the discovery and put off its working-out to any convenient season.—Tr.
50. One may be permitted to add that according to legend, both Hippasus who took to himself public credit for the discovery of a sphere of twelve pentagons, viz., the regular dodecahedron (regarded by the Pythagoreans as the quintessence—or æther—of a world of real tetrahedrons, octahedrons, icosahedrons and cubes), and Archytas the eighth successor of the Founder are reputed to have been drowned at sea. The pentagon from which this dodecahedron is derived, itself involves incommensurable numbers. The “pentagram” was the recognition badge of Pythagoreans and the ἄλογον (incommensurable) their special secret. It would be noted, too, that Pythagoreanism was popular till its initiates were found to be dealing in these alarming and subversive doctrines, and then they were suppressed and lynched—a persecution which suggests more than one deep analogy with certain heresy-suppressions of Western history. The English student may be referred to G. J. Allman, Greek Geometry from Thales to Euclid (Cambridge, 1889), and to his articles “Pythagoras,” “Philolaus” and “Archytas” in the Ency. Brit., XI Edition.—Tr.
51. Horace’s words (Odes I xi): “Tu ne quæsieris, scire nefas, quem mihi quem tibi finem di dederint, Leuconoë, nec Babylonios temptaris numeros ... carpe diem, quam minimum credula postero.”—Tr.
52. See Vol. II, pp. 11 et seq.
53. In the only writing of his that survives, indeed, Aristarchus maintains the geocentric view; it may be presumed therefore that it was only temporarily that he let himself be captivated by a hypothesis of the Chaldaean learning.
54. Giordano Bruno (born 1548, burned for heresy 1600). His whole life might be expressed as a crusade on behalf of God and the Copernican universe against a degenerated orthodoxy and an Aristotelian world-idea long coagulated in death.—Tr.
55. F. Strunz, Gesch. d. Naturwiss. im Mittelalter (1910), p. 90.
56. In the “Psammites,” or “Arenarius,” Archimedes framed a numerical notation which was to be capable of expressing the number of grains of sand in a sphere of the size of our universe.—Tr.
57. This, for which the ground had been prepared by Eudoxus, was employed for calculating the volume of pyramids and cones: “the means whereby the Greeks were able to evade the forbidden notion of infinity” (Heiberg, Naturwiss. u. Math. i. Klass. Alter. [1912], p. 27).
58. Dr. Anster’s translation.—Tr.
59. See Vol. II, Chapter III.
60. Oresme was, equally, prelate, church reformer, scholar, scientist and economist—the very type of the philosopher-leader.—Tr.
61. Oresme in his Latitudines Formarum used ordinate and abscissa, not indeed to specify numerically, but certainly to describe, change, i.e., fundamentally, to express functions.—Tr.
62. Alexandria ceased to be a world-city in the second century A.D. and became a collection of houses left over from the Classical civilization which harboured a primitive population of quite different spiritual constitution. See Vol. II, pp. 122 et seq.
63. Born 1601, died 1665. See Ency. Brit., XI Ed., article Fermat, and references therein.—Tr.
64. Similarly, coinage and double-entry book-keeping play analogous parts in the money-thinking of the Classical and the Western Cultures respectively. See Vol. II, pp. 610 et seq.