CHAPTER II

THE MEANING OF NUMBERS

It is necessary to begin by drawing attention to certain basic terms which, as used in this work, carry strict and in some cases novel connotations. Though the metaphysical content of these terms would gradually become evident in following the course of the reasoning, nevertheless, the exact significance to be attached to them ought to be made clear beyond misunderstanding from the very outset.

The popular distinction—current also in philosophy—between “being” and “becoming” seems to miss the essential point in the contrast it is meant to express. An endless becoming—“action,” “actuality”—will always be thought of also as a condition (as it is, for example, in physical notions such as uniform velocity and the condition of motion, and in the basic hypothesis of the kinetic theory of gases) and therefore ranked in the category of “being.” On the other hand, out of the results that we do in fact obtain by and in consciousness, we may, with Goethe, distinguish as final elements “becoming” and “the become” (_Das Werden_, _das Gewordne_). In all cases, though the atom of human-ness may lie beyond the grasp of our powers of abstract conception, the very clear and definite _feeling_ of this contrast—fundamental and diffused throughout consciousness—is the most elemental something that we reach. It necessarily follows therefore that “the become” is always founded on a “becoming” and not the other way round.

I distinguish further, by the words “proper” and “alien” (_das Eigne_, _das Fremde_), those two basic facts of consciousness which for all men in the waking (not in the dreaming) state are established with an immediate inward certainty, without the necessity or possibility of more precise definition. The element called “alien” is always related in some way to the basic fact expressed by the word “perception,” i.e., the outer world, the life of sensation. Great thinkers have bent all their powers of image-forming to the task of expressing this relation, more and more rigorously, by the aid of half-intuitive dichotomies such as “phenomena and things-in-themselves,” “world-as-will and world-as-idea,” “ego and non-ego,” although human powers of exact knowing are surely inadequate for the task.

Similarly, the element “proper” is involved with the basic fact known as feeling, i.e., the inner life, in some intimate and invariable way that equally defies analysis by the methods of abstract thought.

I distinguish, again, “soul” and “world.” _The existence of this opposition is identical with the fact of purely human waking consciousness (Wachsein)._ There are degrees of clearness and sharpness in the opposition and therefore grades of the consciousness, of the spirituality, of life. These grades range from the feeling-knowledge that, unalert yet sometimes suffused through and through by an inward light, is characteristic of the primitive and of the child (and also of those moments of religious and artistic inspiration that occur ever less and less often as a Culture grows older) right to the extremity of waking and reasoning sharpness that we find, for instance, in the thought of Kant and Napoleon, for whom soul and world have become subject and object. This elementary structure of consciousness, as a fact of immediate inner knowledge, is not susceptible of conceptual subdivision. Nor, indeed, are the two factors distinguishable at all except verbally and more or less artificially, since they are always associated, always intertwined, and present themselves as a unit, a totality. The epistemological starting-point of the born idealist and the born realist alike, the assumption that soul is to world (or world to soul, as the case may be) as foundation is to building, as primary to derivative, as “cause” to “effect,” has no basis whatever in the pure fact of consciousness, and when a philosophic system lays stress on the one or the other, it only thereby informs us as to the personality of the philosopher, a fact of purely biographical significance.

Thus, by regarding waking-consciousness structurally as a tension of contraries, and applying to it the notions of “becoming” and “the thing- become,” we find for the word _Life_ a perfectly definite meaning that is closely allied to that of “becoming.” We may describe becomings and the things-become as the form in which respectively the facts and the results of life exist in the waking consciousness. To man in the waking state his proper life, progressive and constantly self-fulfilling, is presented through the element of Becoming in his consciousness—_this fact we call “the present”_—and it possesses that mysterious property of _Direction_ which in all the higher languages men have sought to impound and—vainly—to rationalize by means of the enigmatic word _time_. It follows necessarily from the above that there is a fundamental connexion between _the become_ (the _hard-set_) and _Death_.

If, now, we designate the Soul—that is, the Soul as it is felt, not as it is reasonably pictured—as the _possible_ and the World on the other hand as the _actual_ (the meaning of these expressions is unmistakable to man’s inner sense), we see life as _the form in which the actualizing of the possible is accomplished_. With respect to the property of Direction, the possible is called the _Future_ and the actualized the _Past_. The actualizing itself, the centre-of-gravity and the centre-of- meaning of life, we call the _Present_. “Soul” is the still-to-be- accomplished, “World” the accomplished, “life” the accomplishing. In this way we are enabled to assign to expressions like moment, duration, development, life-content, vocation, scope, aim, fullness and emptiness of life, the definite meanings which we shall need for all that follows and especially for the understanding of historical phenomena.

Lastly, the words _History_ and _Nature_ are here employed, as the reader will have observed already, in a quite definite and hitherto unusual sense. These words comprise _possible_ modes of understanding, of comprehending the totality of knowledge—becoming as well as things- become, life as well as things-lived—as a homogeneous, spiritualized, well-ordered _world-picture_ fashioned out of an indivisible mass- impression in this way or in that according as the becoming or the become, direction (“time”) or extension (“space”) is the dominant factor. And it is not a question of one factor being alternative to the other. The possibilities that we have of possessing an “outer world” that reflects and attests our proper existence are infinitely numerous and exceedingly heterogeneous, and the purely organic and the purely mechanical world-view (in the precise literal sense of that familiar term[42]) are only the extreme members of the series. Primitive man (so far as we can imagine his waking-consciousness) and the child (as we can remember) cannot fully see or grasp these possibilities. One condition of this higher world-consciousness is the possession of _language_, meaning thereby not mere human utterance but a culture-language, and such is non-existent for primitive man and existent but not accessible in the case of the child. In other words, neither possesses any clear and distinct notion of the world. They have an inkling but no real knowledge of history and nature, being too intimately incorporated with the ensemble of these. _They have no Culture._

And therewith that important word is given a positive meaning of the highest significance which henceforward will be assumed in using it. In the same way as we have elected to distinguish the Soul as the possible and the World as the actual, we can now differentiate between _possible_ and _actual_ culture, i.e., culture as _an idea in the_ (general or individual) _existence_ and culture as the _body_ of that idea, as the total of its visible, tangible and comprehensible expressions—acts and opinions, religion and state, arts and sciences, peoples and cities, economic and social forms, speech, laws, customs, characters, facial lines and costumes. _Higher history_, intimately related to life and to becoming, _is the actualizing of possible Culture_.[43]

We must not omit to add that these basic determinations of meaning are largely incommunicable by specification, definition or proof, and in their deeper import must be reached by feeling, experience and intuition. There is a distinction, rarely appreciated as it should be, between experience as lived and experience as learned (zwischen Erleben und Erkennen), between the immediate certainty given by the various kinds of intuition—such as illumination, inspiration, artistic flair, experience of life, the power of “sizing men up” (Goethe’s “exact percipient fancy”)—and the product of rational procedure and technical experiment.

The first are imparted by means of analogy, picture, symbol, the second by formula, law, scheme. The become is experienced by learning—indeed, as we shall see, the having-become is for the human mind identical with the completed act of cognition. A becoming, on the other hand, can only be experienced by living, felt with a deep wordless understanding. It is on this that what we call “knowledge of men” is based; in fact the understanding of history implies a superlative knowledge of men. The eye which can see into the depths of an alien soul—owes nothing to the cognition-methods investigated in the “Critique of Pure Reason,” yet the purer the historical picture is, the less accessible it becomes to any other eye. The mechanism of a pure nature-picture, such as the world of Newton and Kant, is cognized, grasped, dissected in laws and equations and finally reduced to system: the organism of a pure history-picture, like the world of Plotinus, Dante and Giordano Bruno, is intuitively seen, inwardly experienced, grasped as a form or symbol and finally rendered in poetical and artistic conceptions. Goethe’s “living nature” is a _historical_ world-picture.[44]

-----

Footnote 39:

The work referred to is embodied in Vol. II (pp. 521 et seq., 562 et seq., 631 et seq.).

Footnote 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.

Footnote 41:

At the end of the volume.

Footnote 42:

Weltanschauung im wörtlichen Sinne; Anschauung der Welt.

Footnote 43:

The case of mankind in the historyless state is discussed in Vol. II, pp. 58 et seq.

Footnote 44:

With, moreover, a “biological horizon.” See Vol. II, p. 34.

-----

II

In order to exemplify the way in which a soul seeks to actualize itself in the picture of its outer world—to show, that is, in how far Culture in the “become” state can express or portray an idea of human existence— I have chosen _number_, the primary element on which all mathematics rests. I have done so because mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form; it is, lastly, a metaphysic of the highest rank, as Plato and above all Leibniz show us. Every philosophy has hitherto grown up in conjunction with a mathematic _belonging_ to it. Number is the symbol of causal necessity. Like the conception of God, it contains the ultimate meaning of the world-as- nature. The existence of numbers may therefore be called a mystery, and the religious thought of every Culture has felt their impress.[45]

Just as all becoming possesses the original property of _direction_ (irreversibility), all things-become possess the property of _extension_. But these two words seem unsatisfactory in that only an artificial distinction can be made between them. The real secret of all things-become, which are _ipso facto_ things extended (spatially and materially), is embodied in mathematical number as contrasted with chronological number. Mathematical number contains in its very essence the notion of a _mechanical demarcation_, number being in that respect akin to _word_, which, in the very fact of its comprising and denoting, fences off world-impressions. The deepest depths, it is true, are here both incomprehensible and inexpressible. But the actual number with which the mathematician works, the figure, formula, sign, diagram, in short the _number-sign which he thinks, speaks or writes exactly_, is (like the exactly-used word) from the first a symbol of these depths, something imaginable, communicable, comprehensible to the inner and the outer eye, which can be accepted as representing the demarcation. The origin of numbers resembles that of the myth. Primitive man elevates indefinable nature-impressions (the “alien,” in our terminology) into deities, _numina_, at the same time capturing and impounding them by a _name_ which limits them. So also numbers are something that marks off and captures nature-impressions, and it is by means of names and numbers that the human understanding obtains power over the world. In the last analysis, the number-language of a mathematic and the grammar of a tongue are structurally alike. Logic is always a kind of mathematic and vice versa. Consequently, in all acts of the intellect germane to mathematical number—measuring, counting, drawing, weighing, arranging and dividing[46]—men strive to delimit the extended in words as well, i.e., to set it forth in the form of proofs, conclusions, theorems and systems; and it is only through acts of this kind (which may be more or less unintentioned) that waking man begins to be able to use numbers, normatively, to specify objects and properties, relations and differentiæ, unities and pluralities—briefly, that structure of the world-picture which he feels as necessary and unshakable, calls “Nature” and “cognizes.” _Nature is the numerable_, while History, on the other hand, is the aggregate of that which has no relation to mathematics— hence the mathematical certainty of the laws of Nature, the astounding rightness of Galileo’s saying that Nature is “written in mathematical language,” and the fact, emphasized by Kant, that exact natural science reaches just as far as the possibilities of applied mathematics allow it to reach. In number, then, as the _sign of completed demarcation_, lies the _essence_ of everything actual, which is cognized, is delimited, and has become all at once—as Pythagoras and certain others have been able to see with complete inward certitude by a mighty and truly religious intuition. Nevertheless, mathematics—meaning thereby the capacity to think practically in figures—must not be confused with the far narrower scientific mathematics, that is, the _theory_ of numbers as developed in lecture and treatise. The mathematical vision and thought that a Culture possesses within itself is as inadequately represented by its written mathematic as its philosophical vision and thought by its philosophical treatises. Number springs from a source that has also quite other outlets. Thus at the beginning of every Culture we find an archaic style, which might fairly have been called geometrical in other cases as well as the Early Hellenic. There is a common factor which is expressly mathematical in this early Classical style of the 10th Century B.C., in the temple style of the Egyptian Fourth Dynasty with its absolutism of straight line and right angle, in the Early Christian sarcophagus- relief, and in Romanesque construction and ornament. Here every line, every deliberately non-imitative figure of man and beast, reveals a mystic number-thought in direct connexion with the mystery of death (the hard-set).

Gothic cathedrals and Doric temples are _mathematics in stone_. Doubtless Pythagoras was the first in the Classical Culture to conceive number scientifically as the principle of a world-order of comprehensible things—as _standard_ and as _magnitude_—but even before him it had found expression, as a noble arraying of sensuous-material units, in the strict canon of the statue and the Doric order of columns. The great arts are, one and all, modes of interpretation by means of limits based on number (consider, for example, the problem of space- representation in oil painting). A high mathematical endowment may, without any mathematical science whatsoever, come to fruition and full self-knowledge in _technical_ spheres.

In the presence of so powerful a number-sense as that evidenced, even in the Old Kingdom,[47] in the dimensioning of pyramid temples and in the technique of building, water-control and public administration (not to mention the calendar), no one surely would maintain that the valueless arithmetic of Ahmes belonging to the New Empire represents the level of Egyptian mathematics. The Australian natives, who rank intellectually as thorough primitives, possess a mathematical instinct (or, what comes to the same thing, a power of thinking in numbers which is not yet communicable by signs or words) that as regards the interpretation of pure space is far superior to that of the Greeks. Their discovery of the boomerang can only be attributed to their having a sure feeling for numbers of a class that we should refer to the higher geometry. _Accordingly_—we shall justify the adverb later—they possess an extraordinarily complicated ceremonial and, for expressing degrees of affinity, such fine shades of language as not even the higher Cultures themselves can show.

There is analogy, again, between the Euclidean mathematic and the absence, in the Greek of the mature Periclean age, of any feeling either for ceremonial public life or for loneliness, while the Baroque, differing sharply from the Classical, presents us with a mathematic of spatial analysis, a court of Versailles and a state system resting on dynastic relations.

It is the style of a Soul that comes out in the world of numbers, and the world of numbers includes something more than the science thereof.

-----

Footnote 45:

See Vol. II, pp. 327 et seq.

Footnote 46:

Also “thinking in money.” See Vol. II, pp. 603 et seq.

Footnote 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._

-----

III

From this there follows a fact of decisive importance which has hitherto been hidden from the mathematicians themselves.

_There is not, and cannot be, number as such._ There are several number- worlds as there are several Cultures. We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number—each type fundamentally peculiar and unique, an expression of a specific world-feeling, a symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one. For indubitably the inner structure of the Euclidean geometry is something quite different from that of the Cartesian, the analysis of Archimedes is something other than the analysis of Gauss, and not merely in matters of form, intuition and method but above all in essence, in the intrinsic and obligatory meaning of number which they respectively develop and set forth. This number, the horizon within which it has been able to make phenomena self- explanatory, and therefore the whole of the “nature” or world-extended that is confined in the given limits and amenable to its particular sort of mathematic, are not common to all mankind, but specific in each case to one definite sort of mankind.

The style of any mathematic which comes into being, then, depends wholly on the Culture in which it is rooted, the sort of mankind it is that ponders it. The soul can bring its inherent possibilities to scientific development, can manage them practically, can attain the highest levels in its treatment of them—but is quite impotent to alter them. The idea of the Euclidean geometry is actualized in the earliest forms of Classical ornament, and that of the Infinitesimal Calculus in the earliest forms of Gothic architecture, centuries before the first learned mathematicians of the respective Cultures were born.

A deep inward experience, the genuine _awakening of the ego_, which turns the child into the higher man and initiates him into community of his Culture, marks the beginning of number-sense as it does that of language-sense. It is only after this that objects come to exist for the waking consciousness as things limitable and distinguishable as to number and kind; only after this that properties, concepts, causal necessity, system in the world-around, _a form of the world_, and _world laws_ (for that which is set and settled is _ipso facto_ bounded, hardened, number-governed) are susceptible of exact definition. And therewith comes too a sudden, almost metaphysical, feeling of anxiety and awe regarding the deeper meaning of measuring and counting, drawing and form.

Now, Kant has classified the sum of human knowledge according to syntheses _a priori_ (necessary and universally valid) and _a posteriori_ (experiential and variable from case to case) and in the former class has included mathematical knowledge. Thereby, doubtless, he was enabled to reduce a strong inward feeling to abstract form. But, quite apart from the fact (amply evidenced in modern mathematics and mechanics) that there is no such sharp distinction between the two as is originally and unconditionally implied in the principle, the _a priori_ itself, though certainly one of the most inspired conceptions of philosophy, is a notion that seems to involve enormous difficulties. With it Kant postulates—without attempting to prove what is quite incapable of proof—both _unalterableness of form_ in all intellectual

## activity and _identity of form for all men_ in the same. And, in

consequence, a factor of incalculable importance is—thanks to the intellectual prepossessions of his period, not to mention his own—simply ignored. This factor is the _varying degree_ of this alleged “universal validity.” There are doubtless certain characters of very wide-ranging validity which are (seemingly at any rate) independent of the Culture and century to which the cognizing individual may belong, but along with these there is a quite particular necessity of form which underlies all his thought as axiomatic and to which he is subject by virtue of belonging to his own Culture and no other. Here, then, we have two very different kinds of _a priori_ thought-content, and the definition of a frontier between them, or even the demonstration that such exists, is a problem that lies beyond all possibilities of knowing and will never be solved. So far, no one has dared to assume that the supposed constant structure of the intellect is an illusion and that the history spread out before us contains more than one _style of knowing_. But we must not forget that unanimity about things that have not yet become problems may just as well imply universal error as universal truth. True, there has always been a certain sense of doubt and obscurity—so much so, that the correct guess might have been made from that non-agreement of the philosophers which every glance at the history of philosophy shows us. But that this non-agreement is not due to imperfections of the human intellect or present gaps in a perfectible knowledge, in a word, is not due to defect, but to destiny and historical necessity—this is a _discovery_. Conclusions on the deep and final things are to be reached not by predicating constants but by studying differentiæ and developing the _organic logic_ of differences. The _comparative morphology of knowledge forms_ is a domain which Western thought has still to attack.

IV

If mathematics were a mere science like astronomy or mineralogy, it would be possible to define their object. This man is not and never has been able to do. We West-Europeans may put our own scientific notion of number to perform the same tasks as those with which the mathematicians of Athens and Baghdad busied themselves, but the fact remains that the theme, the intention and the methods of the like-named science in Athens and in Baghdad were quite different from those of our own. _There is no mathematic but only mathematics._ What we call “the history of mathematics”—implying merely the progressive actualizing of a single invariable ideal—is in fact, below the deceptive surface of history, a complex of self-contained and independent developments, an ever-repeated process of bringing to birth new form-worlds and appropriating, transforming and sloughing alien form-worlds, a purely organic story of blossoming, ripening, wilting and dying within the set period. The student must not let himself be deceived. The mathematic of the Classical soul sprouted almost out of nothingness, the historically- constituted Western soul, already possessing the Classical science (not inwardly, but outwardly as a thing learnt), had to win its own by apparently altering and perfecting, but in reality destroying the essentially alien Euclidean system. In the first case, the agent was Pythagoras, in the second Descartes. In both cases the act is, at bottom, the same.

The relationship between the form-language of a mathematic and that of the cognate major arts,[48] is in this way put beyond doubt. The temperament of the thinker and that of the artist differ widely indeed, but the expression-methods of the waking consciousness are inwardly the same for each. The sense of form of the sculptor, the painter, the composer is essentially mathematical in its nature. The same inspired ordering of an infinite world which manifested itself in the geometrical analysis and projective geometry of the 17th Century, could vivify, energize, and suffuse contemporary music with the harmony that it developed out of the art of thoroughbass, (which is the geometry of the sound-world) and contemporary painting with the principle of perspective (the felt geometry of the space-world that only the West knows). This inspired ordering is that which Goethe called “_The Idea, of which the form is immediately apprehended in the domain of intuition_, whereas pure science does not apprehend but observes and dissects.” The Mathematic goes beyond observation and dissection, and in its highest moments finds the way by vision, not abstraction. To Goethe again we owe the profound saying: “the mathematician is only complete in so far as he feels within himself the _beauty_ of the true.” Here we feel how nearly the secret of number is related to the secret of artistic creation. And so the born mathematician takes his place by the side of the great masters of the fugue, the chisel and the brush; he and they alike strive, and must strive, to actualize the grand order of all things by clothing it in symbol and so to communicate it to the plain fellow-man who hears that order within himself but cannot effectively possess it; the domain of number, like the domains of tone, line and colour, becomes an image of the world-form. For this reason the word “creative” means more in the mathematical sphere than it does in the pure sciences— Newton, Gauss, and Riemann were artist-natures, and we know with what suddenness their great conceptions came upon them.[49] “A mathematician,” said old Weierstrass “who is not at the same time a bit of a poet will never be a full mathematician.”

The mathematic, then, is an art. As such it has its styles and style- periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes from epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) side-glance at contemporary mathematics. In the very deep relation between changes of musical theory and the analysis of the infinite, the details have never yet been investigated, although æsthetics might have learned a great deal more from these than from all so-called “psychology.” Still more revealing would be a history of musical instruments written, not (as it always is) from the technical standpoint of tone-production, but as a study of the deep spiritual bases of the tone-colours and tone-effects aimed at. For it was the wish, intensified to the point of a longing, to fill a spatial infinity with sound which produced—in contrast to the Classical lyre and reed (lyra, kithara; aulos, syrinx) and the Arabian lute—the two great families of keyboard instruments (organ, pianoforte, etc.) and bow instruments, and that as early as the Gothic time. The development of both these families belongs spiritually (and possibly also in point of technical origin) to the Celtic-Germanic North lying between Ireland, the Weser and the Seine. The organ and clavichord belong certainly to England, the bow instruments reached their definite forms in Upper Italy between 1480 and 1530, while it was principally in Germany that the organ was developed into the _space-commanding_ giant that we know, an instrument the like of which does not exist in all musical history. The free organ-playing of Bach and his time was nothing if it was not analysis—analysis of a strange and vast tone-world. And, similarly, it is in conformity with the Western number-thinking, and in opposition to the Classical, that our string and wind instruments have been developed not singly but in great groups (strings, woodwind, brass), ordered within themselves according to the compass of the four human voices; the history of the modern orchestra, with all its discoveries of new and modification of old instruments, is in reality the self-contained history of one tone-world—a world, moreover, that is quite capable of being expressed in the forms of the higher analysis.

V

When, about 540 B.C., the circle of the Pythagoreans arrived at the idea that _number is the essence of all things_, it was not “a step in the development of mathematics” that was made, but a wholly new mathematic that was born. Long heralded by metaphysical problem-posings and artistic form-tendencies, now it came forth from the depths of the Classical soul as a formulated theory, a mathematic born in one act at one great historical moment—just as the mathematic of the Egyptians had been, and the algebra-astronomy of the Babylonian Culture with its ecliptic co-ordinate system—and new—for these older mathematics had long been extinguished and the Egyptian was never written down. Fulfilled by the 2nd century A.D., the Classical mathematic vanished in its turn (for though it seemingly exists even to-day, it is only as a convenience of notation that it does so), and gave place to the Arabian. From what we know of the Alexandrian mathematic, it is a necessary presumption that there was a great movement within the Middle East, of which the centre of gravity must have lain in the Persian-Babylonian schools (such as Edessa, Gundisapora and Ctesiphon) and of which only details found their way into the regions of Classical speech. In spite of their Greek names, the Alexandrian mathematicians—Zenodorus who dealt with figures of equal perimeter, Serenus who worked on the properties of a harmonic pencil in space, Hypsicles who introduced the Chaldean circle-division, Diophantus above all—were all without doubt Aramæans, and their works only a small part of a literature which was written principally in Syriac. This mathematic found its completion in the investigations of the Arabian- Islamic thinkers, and after these there was again a long interval. And then a perfectly new mathematic was born, the Western, _our own_, which in our infatuation we regard as “Mathematics,” as the culmination and the implicit purpose of two thousand years’ evolution, though in reality its centuries are (strictly) numbered and to-day almost spent.

The most valuable thing in the Classical mathematic is its proposition that number is the essence of all things _perceptible to the senses_. Defining number as a measure, it contains the whole world-feeling of a soul passionately devoted to the “here” and the “now.” Measurement in this sense means the measurement of something near and corporeal. Consider the content of the Classical art-work, say the free-standing statue of a naked man; here every essential and important element of Being, its whole rhythm, is exhaustively rendered by surfaces, dimensions and the sensuous relations of the parts. The Pythagorean notion of the harmony of numbers, although it was probably deduced from music—a music, be it noted, that knew not polyphony or harmony, and formed its instruments to render single plump, almost fleshy, tones— seems to be the very mould for a sculpture that has this ideal. The worked stone is only a something in so far as it has considered limits and measured form; what it _is_ is what it _has become_ under the sculptor’s chisel. Apart from this it is a _chaos_, something not yet actualized, in fact for the time being a null. The same feeling transferred to the grander stage produces, as an opposite to the state of chaos, that of _cosmos_, which for the Classical soul implies a cleared-up situation of the external world, a harmonic order which includes each separate thing as a well-defined, comprehensible and present entity. The sum of such things constitutes neither more nor less than the whole world, and the interspaces between them, which for us are filled with the impressive symbol of the Universe of Space, are for them the nonent (τὸ μὴ ὅν).

Extension means, for Classical mankind body, and for us space, and it is as a function of space that, to us, things “appear.” And, looking backward from this standpoint, we may perhaps see into the deepest concept of the Classical metaphysics, Anaximander’s ἄπειρον—a word that is quite untranslatable into any Western tongue. It is that which possesses no “number” in the Pythagorean sense of the word, no measurable dimensions or definable limits, and therefore no being; the measureless, the negation of form, the statue not yet carved out of the block; the ἀρχὴ optically boundless and formless, which only becomes a something (namely, the world) after being split up by the senses. It is the underlying form _a priori_ of Classical cognition, bodiliness as such, which is replaced exactly in the Kantian world-picture by that Space out of which Kant maintained that all things could be “thought forth.”

We can now understand what it is that divides one mathematic from another, and in particular the Classical from the Western. The whole world-feeling of the matured Classical world led it to see mathematics only as the theory of relations of magnitude, dimension and form between bodies. When, from out of this feeling, Pythagoras evolved and expressed the decisive formula, number had come, for him, to be an _optical_ symbol—not a measure of form generally, an abstract relation, but a frontier-post of the domain of the Become, or rather of that part of it which the senses were able to split up and pass under review. By the whole Classical world without exception numbers are conceived as units of measure, as magnitude, lengths, or surfaces, and for it no other sort of extension is imaginable. The whole Classical mathematic is at bottom _Stereometry_ (solid geometry). To Euclid, who rounded off its system in the third century, the triangle is of deep necessity the bounding surface of a body, never a system of three intersecting straight lines or a group of three points in three-dimensional space. He defines a line as “length without breadth” (μῆκος ἀπλατές). In our mouths such a definition would be pitiful—in the Classical mathematic it was brilliant.

The Western number, too, is not, as Kant and even Helmholtz thought, something proceeding out of Time as an _a priori_ form of conception, but is something specifically spatial, in that it is an order (or ordering) of like units. Actual time (as we shall see more and more clearly in the sequel) has not the slightest relation with mathematical things. Numbers belong exclusively to the domain of extension. But there are precisely as many possibilities—and therefore necessities—of ordered presentation of the extended as there are Cultures. Classical number is a thought-process dealing not with spatial relations but with visibly limitable and tangible units, and it follows naturally and necessarily that the Classical knows only the “natural” (positive and whole) numbers, which on the contrary play in our Western mathematics a quite undistinguished part in the midst of complex, hypercomplex, non- Archimedean and other number-systems.

On this account, the idea of irrational numbers—the unending decimal fractions of our notation—was unrealizable within the Greek spirit. Euclid says—and he ought to have been better understood—that incommensurable lines are “_not related to one another like numbers_.” In fact, it is the idea of irrational number that, once achieved, separates the notion of number from that of magnitude, for the magnitude of such a number (π, for example) can never be defined or exactly represented by any straight line. Moreover, it follows from this that in considering the relation, say, between diagonal and side in a square the Greek would be brought up suddenly against a quite other sort of number, which was fundamentally alien to the Classical soul, and was consequently feared as a secret of its proper existence too dangerous to be unveiled. There is a singular and significant late-Greek legend, according to which the man who first published the hidden mystery of the irrational perished by shipwreck, “for the unspeakable and the formless must be left hidden for ever.”[50]

The fear that underlies this legend is the selfsame notion that prevented even the ripest Greeks from extending their tiny city-states so as to organize the country-side politically, from laying out their streets to end in prospects and their alleys to give vistas, that made them recoil time and again from the Babylonian astronomy with its penetration of endless starry space,[51] and refuse to venture out of the Mediterranean along sea-paths long before dared by the Phœnicians and the Egyptians. It is the deep metaphysical fear that the sense- comprehensible and present in which the Classical existence had entrenched itself would collapse and precipitate its cosmos (largely created and sustained by art) into unknown primitive abysses. And to understand this fear is to understand the final significance of Classical number—that is, _measure in contrast to the immeasurable_—and to grasp the high ethical significance of its limitation. Goethe too, as a nature-student, felt it—hence his almost terrified aversion to mathematics, which as we can now see was really an involuntary reaction against the _non-Classical_ mathematic, the Infinitesimal Calculus which underlay the natural philosophy of his time.

Religious feeling in Classical man focused itself ever more and more intensely upon physically present, _localized_ cults which alone expressed a college of Euclidean deities. Abstractions, _dogmas_ floating homeless in the space of thought, were ever alien to it. A cult of this kind has as much in common with a Roman Catholic dogma as the statue has with the cathedral organ. There is no doubt that something of cult was comprised in the Euclidean mathematic—consider, for instance, the secret doctrines of the Pythagoreans and the Theorems of regular polyhedrons with their esoteric significance in the circle of Plato. Just so, there is a deep relation between Descartes’ analysis of the infinite and contemporary dogmatic theology as it progressed from the final decisions of the Reformation and the Counter-Reformation to entirely desensualized deism. Descartes and Pascal were mathematicians and Jansenists, Leibniz a mathematician and pietist. Voltaire, Lagrange and D’Alembert were contemporaries. Now, the Classical soul felt the principle of the irrational, which overturned the statuesquely-ordered array of whole numbers and the complete and self-sufficing world-order for which these stood, as an impiety against the Divine itself. In Plato’s “Timæus” this feeling is unmistakable. For the transformation of a series of discrete numbers into a continuum challenged not merely the Classical notion of number but the Classical world-idea itself, and so it is understandable that even _negative_ numbers, which to us offer no conceptual difficulty, were impossible in the Classical mathematic, let alone _zero as a number_, that refined creation of a wonderful abstractive power which, for the Indian soul that conceived it as base for a positional numeration, was nothing more nor less than the key to the meaning of existence. _Negative magnitudes_ have no existence. The expression -2×-3=+6 is neither something perceivable nor a representation of magnitude. The series of magnitudes ends with +1, and in graphic representation of negative numbers

( + 3 + 2 + 1 0 - 1 - 2 - 3 ) — ･ — ･ — ･ ･ — ･ — ･ — ･

we have suddenly, from zero onwards, _positive_ symbols of something negative; they _mean_ something, but they no longer _are_. But the fulfilment of this act did not lie within the direction of Classical number-thinking.

Every product of the waking consciousness of the Classical world, then, is elevated to the rank of actuality by way of sculptural definition. That which cannot be drawn is not “number.” Archytas and Eudoxus use the terms surface- and volume-numbers to mean what we call second and third powers, and it is easy to understand that the notion of higher integral powers did not exist for them, for a fourth power would predicate at once, for the mind based on the plastic feeling, an extension in four dimensions, and four _material_ dimensions into the bargain, “which is absurd.” Expressions like ε^{ix} which we constantly use, or even the fractional index (e.g., 5^½) which is employed in the Western mathematics as early as Oresme (14th Century), would have been to them utter nonsense. Euclid calls the factors of a product its sides πλευραί and fractions (finite of course) were treated as whole-number relationships between two lines. Clearly, out of this no conception of zero as a number could possibly come, for from the point of view of a draughtsman it is meaningless. We, having minds differently constituted, must not argue from our habits to theirs and treat their mathematic as a “first stage” in the development of “Mathematics.” Within and for the purposes of the world that Classical man evolved for himself, the Classical mathematic was a complete thing—it is merely not so _for us_. Babylonian and Indian mathematics had long contained, as essential elements of _their_ number-worlds, things which the Classical number- feeling regarded as nonsense—and not from ignorance either, since many a Greek thinker was acquainted with them. It must be repeated, “Mathematics” is an illusion. A mathematical, and, generally, a scientific way of thinking is right, convincing, a “necessity of thought,” when it completely expresses the life-feeling proper to it. Otherwise it is either impossible, futile and senseless, or else, as we in the arrogance of our historical soul like to say, “primitive.” The modern mathematic, though “true” only for the Western spirit, is undeniably a master-work of that spirit; and yet to Plato it would have seemed a ridiculous and painful aberration from the path leading to the “true”—to wit, the Classical—mathematic. And so with ourselves. Plainly, we have almost no notion of the multitude of great ideas belonging to other Cultures that we have suffered to lapse because _our_ thought with its limitations has not permitted us to assimilate them, or (which comes to the same thing) has led us to reject them as false, superfluous, and nonsensical.

VI

The Greek mathematic, as a science of perceivable magnitudes, deliberately confines itself to facts of the comprehensibly present, and limits its researches and their validity to the near and the small. As compared with this impeccable consistency, the position of the Western mathematic is seen to be, practically, somewhat illogical, though it is only since the discovery of Non-Euclidean Geometry that the fact has been really recognized. Numbers are images of the perfectly desensualized understanding, of pure thought, and contain their abstract validity within themselves.[52] Their exact application to the actuality of conscious experience is therefore a problem in itself—a problem which is always being posed anew and never solved—and the congruence of mathematical system with empirical observation is at present anything but self-evident. Although the lay idea—as found in Schopenhauer—is that mathematics rest upon the direct evidences of the senses, Euclidean geometry, superficially identical though it is with the popular geometry of all ages, is only in agreement with the phenomenal world approximately and within very narrow limits—in fact, the limits of a drawing-board. Extend these limits, and what becomes, for instance, of Euclidean parallels? They meet at the line of the horizon—a simple fact upon which all our art-perspective is grounded.

Now, it is unpardonable that Kant, a Western thinker, should have evaded the mathematic of distance, and appealed to a set of figure-examples that their mere pettiness excludes from treatment by the specifically Western infinitesimal methods. But Euclid, as a thinker of the Classical age, was entirely consistent with its spirit when he refrained from proving the phenomenal truth of his axioms by referring to, say, the triangle formed by an observer and two infinitely distant fixed stars. For these can neither be drawn nor “intuitively apprehended” and his feeling was precisely the feeling which shrank from the irrationals, which did not dare to give nothingness a value as zero (i.e., a number) and even in the contemplation of cosmic relations shut its eyes to the Infinite and held to its symbol of Proportion.

Aristarchus of Samos, who in 288-277 belonged to a circle of astronomers at Alexandria that doubtless had relations with Chaldaeo-Persian schools, projected the elements of a heliocentric world-system.[53] Rediscovered by Copernicus, it was to shake the metaphysical passions of the West to their foundations—witness Giordano Bruno[54]—to become the fulfilment of mighty premonitions, and to justify that Faustian, Gothic world-feeling which had already professed its faith in infinity through the forms of its cathedrals. But the world of Aristarchus received his work with entire indifference and in a brief space of time it was forgotten—designedly, we may surmise. His few followers were nearly all natives of Asia Minor, his most prominent supporter Seleucus (about 150) being from the Persian Seleucia on Tigris. In fact, the Aristarchian system had no spiritual appeal to the Classical Culture and might indeed have become dangerous to it. And yet it was differentiated from the Copernican (a point always missed) by something which made it perfectly conformable to the Classical world-feeling, viz., the assumption that the cosmos is _contained_ in a materially finite and optically appreciable _hollow sphere_, in the middle of which the planetary system, arranged as such on Copernican lines, moved. In the Classical astronomy, the earth and the heavenly bodies are consistently regarded as entities of two different kinds, however variously their movements in detail might be interpreted. Equally, the opposite idea that the earth is _only a star among stars_[55] is not inconsistent in itself with either the Ptolemaic or the Copernican systems and in fact was pioneered by Nicolaus Cusanus and Leonardo da Vinci. But by this device of a celestial sphere the principle of infinity which would have endangered the sensuous-Classical notion of bounds was smothered. One would have supposed that the infinity-conception was inevitably implied by the system of Aristarchus—long before his time, the Babylonian thinkers had reached it. But no such thought emerges. On the contrary, in the famous treatise on the grains of sand[56] Archimedes proves that the filling of this stereometric body (for that is what Aristarchus’s Cosmos is, after all) with atoms of sand leads to very high, but _not_ to infinite, figure-results. This proposition, quoted though it may be, time and again, as being a first step towards the Integral Calculus, amounts to a denial (implicit indeed in the very title) of everything that we mean by the word analysis. Whereas in our physics, the constantly-surging hypotheses of a material (i.e., directly cognizable) æther, break themselves one after the other against our refusal to acknowledge material limitations of any kind, Eudoxus, Apollonius and Archimedes, certainly the keenest and boldest of the Classical mathematicians, completely worked out, in the main with rule and compass, a _purely optical_ analysis of things-become on the basis of sculptural-Classical bounds. They used deeply-thought-out (and for us hardly understandable) methods of integration, but these possess only a superficial resemblance even to Leibniz’s definite-integral method. They employed geometrical loci and co-ordinates, but these are always specified lengths and units of measurement and never, as in Fermat and above all in Descartes, unspecified spatial relations, values of points in terms of their positions in space. With these methods also should be classed the exhaustion-method of Archimedes,[57] given by him in his recently discovered letter to Eratosthenes on such subjects as the quadrature of the parabola section by means of inscribed rectangles (instead of through similar polygons). But the very subtlety and extreme complication of his methods, which are grounded in certain of Plato’s geometrical ideas, make us realize, in spite of superficial analogies, what an enormous difference separates him from Pascal. Apart altogether from the idea of Riemann’s integral, what sharper contrast could there be to these ideas than the so-called quadratures of to-day? The name itself is now no more than an unfortunate survival, the “surface” is indicated by a bounding function, and the _drawing_ as such, has vanished. Nowhere else did the two mathematical minds approach each other more closely than in this instance, and nowhere is it more evident that the gulf between the two souls thus expressing themselves is impassable.

In the cubic style of their early architecture the Egyptians, so to say, concealed pure numbers, fearful of stumbling upon their secret, and for the Hellenes too they were the key to the meaning of the become, the stiffened, the mortal. The stone statue and the scientific system deny life. Mathematical number, the formal principle of an extension-world of which the phenomenal existence is only the derivative and servant of waking human consciousness, bears the hall-mark of causal necessity and so is linked with _death_ as chronological number is with becoming, with _life_, with the necessity of destiny. This connexion of strict mathematical form with the _end_ of organic being, with the phenomenon of its organic remainder the corpse, we shall see more and more clearly to be the origin of all great art. We have already noticed the development of early ornament on funerary equipments and receptacles. _Numbers are symbols of the mortal._ Stiff forms are the negation of life, formulas and laws spread rigidity over the face of nature, numbers make dead—and the “Mothers” of Faust II sit enthroned, majestic and withdrawn, in

“The realms of Image unconfined. ... Formation, transformation, Eternal play of the eternal mind With semblances of all things in creation For ever and for ever sweeping round.”[58]

Goethe draws very near to Plato in this divination of one of the final secrets. For his unapproachable Mothers are Plato’s Ideas—the possibilities of a spirituality, the unborn forms to be realized as

## active and purposed Culture, as art, thought, polity and religion, in a

world ordered and determined by that spirituality. And so the number- thought and the world-idea of a Culture are related, and by this relation, the former is elevated above mere knowledge and experience and becomes a view of the universe, there being consequently as many mathematics—as many number-worlds—as there are higher Cultures. Only so can we understand, as something _necessary_, the fact that the greatest mathematical thinkers, the creative artists of the realm of numbers, have been brought to the decisive mathematical discoveries of their several Cultures by a deep religious intuition.

Classical, Apollinian number we must regard as the creation of Pythagoras—_who founded a religion_. It was an instinct that guided Nicolaus Cusanus, the great Bishop of Brixen (about 1450), from the idea of the unendingness of God in nature to the elements of the Infinitesimal Calculus. Leibniz himself, who two centuries later definitely settled the methods and notation of the Calculus, was led by purely metaphysical speculations about the divine principle and its relation to infinite extent to conceive and develop the notion of an _analysis situs_—probably the most inspired of all interpretations of pure and emancipated space—the possibilities of which were to be developed later by Grassmann in his _Ausdehnungslehre_ and above all by Riemann, their real creator, in his symbolism of two-sided planes representative of the nature of equations. And Kepler and Newton, strictly religious natures both, were and remained convinced, like Plato, that it was precisely through the medium of number that they had been able to apprehend intuitively the essence of the divine world- order.

VII

The Classical arithmetic, we are always told, was first liberated from its sense-bondage, widened and extended by Diophantus, who did not indeed create algebra (the science of undefined magnitudes) but brought it to expression within the framework of the Classical mathematic that we know—and so suddenly that we have to assume that there was a pre- existent stock of ideas which he worked out. But this amounts, not to an enrichment of, but a complete victory over, the Classical world-feeling, and the mere fact should have sufficed in itself to show that, inwardly, Diophantus does not belong to the Classical Culture at all. What is

## active in him is a new number-feeling, or let us say a new limit-feeling

with respect to the actual and become, and no longer that Hellenic feeling of sensuously-present limits which had produced the Euclidean geometry, the nude statue and the coin. Details of the formation of this new mathematic we do not know—Diophantus stands so completely by himself in the history of so-called late-Classical mathematics that an Indian influence has been presumed. But here also the influence it must really have been that of those early-Arabian schools whose studies (apart from the dogmatic) have hitherto been so imperfectly investigated. In Diophantus, unconscious though he may be of his own essential antagonism to the Classical foundations on which he attempted to build, there emerges from under the surface of Euclidean _intention_ the new limit- _feeling_ which I designate the “Magian.” He did not widen the idea of number as magnitude, but (unwittingly) eliminated it. No Greek could have stated anything about an _undefined_ number _a_ or an _undenominated_ number 3—which are neither magnitudes nor lines—whereas the new limit-feeling sensibly expressed by numbers of this sort at least underlay, if it did not constitute, Diophantine treatment; and the letter-notation which we employ to clothe our own (again transvalued) algebra was first introduced by Vieta in 1591, an unmistakable, if unintended, protest against the classicizing tendency of Renaissance mathematics.

Diophantus lived about 250 A.D., _that is, in the third century of that Arabian Culture_ whose organic history, till now smothered under the surface-forms of the Roman Empire and the “Middle Ages,”[59] comprises everything that happened after the beginning of our era in the region that was later to be Islam’s. It was precisely in the time of Diophantus that the last shadow of the Attic statuary art paled before the new space-sense of cupola, mosaic and sarcophagus-relief that we have in the Early-Christian-Syrian style. In that time there was once more _archaic_ art and strictly geometrical ornament; and at that time too Diocletian completed the transformation of the now merely sham Empire into a Caliphate. The four centuries that separate Euclid and Diophantus, separate also Plato and Plotinus—the last and conclusive thinker, the Kant, of a fulfilled Culture and the first schoolman, the Duns Scotus, of a Culture just awakened.

It is here that we are made aware for the first time of the existence of those higher individualities whose coming, growth and decay constitute the _real substance of history_ underlying the myriad colours and changes of the surface. The Classical spirituality, which reached its final phase in the cold intelligence of the Romans and of which the whole Classical Culture with all its works, thoughts, deeds and ruins forms the “body,” had been born about 1100 B.C. in the country about the Ægean Sea. The Arabian Culture, which, under cover of the Classical Civilization, had been germinating in the East since Augustus, came wholly out of the region between Armenia and Southern Arabia, Alexandria and Ctesiphon, and we have to consider as expressions of this new soul almost the whole “late-Classical” art of the Empire, all the young ardent religions of the East—Mandæanism, Manichæism, Christianity, Neo- Platonism, and in Rome itself, as well as the Imperial Fora, that Pantheon which is the _first of all mosques_.

That Alexandria and Antioch still wrote in Greek and imagined that they were thinking in Greek is a fact of no more importance than the facts that Latin was the scientific language of the West right up to the time of Kant and that Charlemagne “renewed” the Roman Empire.

In Diophantus, number has ceased to be the measure and essence of _plastic things_. In the Ravennate mosaics man has ceased to be a _body_. Unnoticed, Greek designations have lost their original connotations. We have left the realm of Attic καλοκάγαθία the Stoic ἀταραξία and γαλήνη. Diophantus does not yet know zero and negative numbers, it is true, but he has _ceased_ to know Pythagorean numbers. And this Arabian indeterminateness of number is, in its turn, something quite different from the controlled variability of the later Western mathematics, the variability of the _function_.

The Magian mathematic—we can see the outline, though we are ignorant of the details—advanced through Diophantus (who is obviously not a starting-point) boldly and logically to a culmination in the Abbassid period (9th century) that we can appreciate in Al-Khwarizmi and Alsidzshi. And as Euclidean geometry is to Attic statuary (the same expression-form in a different medium) and the analysis of space to polyphonic music, so this algebra is to the Magian art with its mosaic, its arabesque (which the Sassanid Empire and later Byzantium produced with an ever-increasing profusion and luxury of tangible-intangible organic motives) and its Constantinian high-relief in which uncertain deep-darks divide the freely-handled figures of the foreground. As algebra is to Classical arithmetic and Western analysis, so is the cupola-church to the Doric temple and the Gothic cathedral. It is not as though Diophantus were one of the great mathematicians. On the contrary, much of what we have been accustomed to associate with his name is not his work alone. His accidental importance lies in the fact that, so far as our knowledge goes, he was the first mathematician in whom the new number-feeling is unmistakably present. In comparison with the masters who _conclude_ the development of a mathematic—with Apollonius and Archimedes, with Gauss, Cauchy, Riemann—Diophantus has, in his form- language especially, something _primitive_. This something, which till now we have been pleased to refer to “late-Classical” decadence, we shall presently learn to understand and value, just as we are revising our ideas as to the despised “late-Classical” art and beginning to see in it the tentative expression of the nascent Early Arabian Culture. Similarly archaic, primitive, and groping was the mathematic of Nicolas Oresme, Bishop of Lisieux (1323-1382),[60] who was the first Western who used co-ordinates so to say elastically[61] and, more important still, to employ fractional powers—both of which presuppose a number-feeling, obscure it may be but quite unmistakable, which is completely non- Classical and _also_ non-Arabic. But if, further, we think of Diophantus together with the early-Christian sarcophagi of the Roman collections, and of Oresme together with the Gothic wall-statuary of the German cathedrals, we see that the mathematicians as well as the artists have something in common, which is, that they stand in their respective Cultures at _the same_ (viz., the primitive) level of abstract understanding. In the world and age of Diophantus the stereometric sense of bounds, which had long ago reached in Archimedes the last stages of refinement and elegance proper to the megalopolitan intelligence, had passed away. Throughout that world men were unclear, longing, mystic, and no longer bright and free in the Attic way; they were men rooted in the earth of a young country-side, not megalopolitans like Euclid and D’Alembert.[62] They no longer understood the deep and complicated forms of the Classical thought, and their own were confused and new, far as yet from urban clarity and tidiness. Their Culture was in the _Gothic_ condition, as all Cultures have been in their youth—as even the Classical was in the early Doric period which is known to us now only by its Dipylon pottery. Only in Baghdad and in the 9th and 10th Centuries were the young ideas of the age of Diophantus carried through to completion by ripe masters of the calibre of Plato and Gauss.

-----

Footnote 48:

As also those of law and of money. See Vol. II, pp. 68 et seq., pp. 616 et seq.

Footnote 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 soudaineté 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._

Footnote 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._

Footnote 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._

Footnote 52:

See Vol. II, pp. 11 et seq.

Footnote 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.

Footnote 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._

Footnote 55:

F. Strunz, _Gesch. d. Naturwiss. im Mittelalter_ (1910), p. 90.

Footnote 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._

Footnote 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).

Footnote 58:

Dr. Anster’s translation.—_Tr._

Footnote 59:

See Vol. II,