65. The same may be said in the matter of Roman Law (see Vol. II, pp. 96 et seq.) and of coinage (see Vol. II, pp. 616 et seq.).
66. That is, “it is impossible to part a cube into two cubes, a biquadrate into two biquadrates, and generally any power above the square into two powers having the same exponent.” Fermat claimed to possess a proof of the proposition, but this has not been preserved, and no general proof has hitherto been obtained.—Tr.
67. Thus Bishop Berkeley’s Discourse addressed to an infidel mathematician (1735) shrewdly asked whether the mathematician were in a position to criticize the divine for proceeding on the basis of faith.—Tr.
68. From the savage conjuror with his naming-magic to the modern scientist who subjects things by attaching technical labels to them, the form has in no wise changed. See Vol. II, pp. 116 et seq., 322 et seq.
69. See Vol. II, pp. 137 et seq.
70. A beginning is now being made with the application of non-Euclidean geometries to astronomy. The hypothesis of curved space, closed but without limits, filled by the system of fixed stars on a radius of about 470,000,000 earth-distances, would lead to the hypothesis of a counter-image of the sun which to us appears as a star of medium brilliancy. (See translator’s footnote, p. 332.)
71. That only one parallel to a given straight line is possible through a given point—a proposition that is incapable of proof.
72. It is impossible to say, with certainty, how much of the Indian mathematics that we possess is old, i.e., before Buddha.
73. The technical difference (in German usage) between Grenz and Grenzwert is in most cases ignored in this translation as it is only the underlying conception of “number” common to both that concerns us. Grenz is the “limit” strictly speaking, i.e., the number a to which the terms a1₁, a2₂, a₃ ... of a particular series approximate more and more closely, till nearer to a than any assignable number whatever. The Grenzwert of a function, on the other hand, is the “limit” of the value which the function takes for a given value a of the variable x. These methods of reasoning and their derivatives enable solutions to be obtained for series such as (1⁄m¹,) (1⁄m²,) (1⁄m³,) ... (1⁄mx) or functions such as
where x is infinite or indefinite.—Tr.
74. “Function, rightly understood, is existence considered as an activity” (Goethe). Cf. Vol. II, p. 618, for functional money.
75. Built for August II, in 1711, as barbican or fore-building for a projected palace.—Tr.
76. From the standpoint of the theory of “aggregates” (or “sets of points”), a well-ordered set of points, irrespective of the dimension figure, is called a corpus; and thus an aggregate of n - 1 dimensions is considered, relatively to one of n dimensions, as a surface. Thus the limit (wall, edge) of an “aggregate” represents an aggregate of lower “potentiality.”
78. “Anti-historical,” the expression which we apply to a decidedly systematic valuation, is to be carefully distinguished from “ahistorical.” The beginning of the IV Book (53) of Schopenhauer’s Welt als Wille und Vorstellung affords a good illustration of the man who thinks anti-historically, that is, deliberately for theoretical reasons suppresses and rejects the historical in himself—something that is actually there. The ahistoric Greek nature, on the contrary, neither possesses nor understands it.
79. “There are prime phenomena which in their godlike simplicity we must not disturb or infringe.”
80. The date of Napoleon’s defeat, and the liberation of Germany, on the field of Leipzig.—Tr.
81. See Vol. II, pp. 25 et seq., 327 et seq.
82.
From the final stanza of Faust II (Anster’s translation).—Tr.
83. This phrase, derived by analogy from the centre of gravity of mechanics, is offered as a translation of “mithin in einim Zeitpunkte ger nicht zusammengefasst werden können.”—Tr.
84. Cf. Vol. II, p. 33 et seq.
85. Not the dissecting morphology of the Darwinian’s pragmatic zoology with its hunt for causal connexions, but the seeing and overseeing morphology of Goethe.
86. See Vol. II, pp. 41 et seq.
87. See Vol. II, pp. 227 et seq.
88. See Vol. II, pp. 116 et seq. What constitutes the downfall is not, e.g., the catastrophe of the Great Migrations, which like the annihilation of the Maya Culture by the Spaniards (see Vol. II, p. 51 et seq.) was a coincidence without any deep necessity, but the inward undoing that began from the time of Hadrian, as in China from the Eastern Han dynasty (25-220).
89. St. Bernward was Bishop of Hildesheim from 993 to 1022, and himself architect and metal-worker. Three other churches besides the cathedral survive in the city from his time or that of his immediate successors, and Hildesheim of all North German cities is richest in monuments of the Romanesque.—Tr.
90. By “Saxony,” a German historian means not the present-day state of Saxony (which was a small and comparatively late accretion), but the whole region of the Weser and the lower Elbe, with Westphalia and Holstein.—Tr.
91. Vases from the cemetery adjoining the Dipylon Gate of Athens, the most representative relics that we possess of the Doric or primitive age of the Hellenic Culture (about 900 to 600 B.C.).—Tr.
92. See Vol. II, pp. 381 et seq.
93. In English the word “cast” will evidently satisfy the sense better on occasion. The word “stil” will therefore not necessarily be always rendered “style.”—Tr.
94. See Vol. II, pp. 109 et seq.
95. See Vol. II, pp. 36 et seq.
96. I will only mention here the distances apart of the three Punic Wars, and the series—likewise comprehensible only as rhythmic—Spanish Succession War, Silesian wars, Napoleonic Wars, Bismarck’s wars, and the World War (cf. Vol. II, p. 488). Connected with this is the spiritual relation of grandfather and grandson, a relation which produces in the mind of primitive peoples the conviction that the soul of the grandfather returns in the grandson, and has originated the widespread custom of giving the grandson the grandfather’s name, which by its mystic spell binds his soul afresh to the corporeal world.
97. The word is used in the sense in which biology employs it, viz., to describe the process by which the embryo traverses all the phases which its species has undergone.—Tr.
98. The first draft of Faust I, discovered only comparatively recently.—Tr.
99. See Ency. Brit., XIth Ed., articles Owen, Sir Richard; Morphology and Zoology (p. 1029).—Tr.
100. It is not superfluous to add that there is nothing of the causal kind in these pure phenomena of “Living Nature.” Materialism, in order to get a system for the pedestrian reasoner, has had to adulterate the picture of them with fitness-causes. But Goethe—who anticipated just about as much of Darwinism as there will be left of it in fifty years from Darwin—absolutely excluded the causality-principle. And the very fact that the Darwinians quite failed to notice its absence is a clear indication that Goethe’s “Living Nature” belongs to actual life, "cause"-less and "aim"-less; for the idea of the prime-phenomenon does not involve causal assumptions of any sort unless it has been misunderstood in advance in a mechanistic sense.
101. Reigned 246-210 B.C. He styled himself “first universal emperor” and intended a position for himself and his successors akin to that of “Divus” in Rome. For a brief account of his energetic and comprehensive work see Ency. Brit., XI Ed., article China, p. 194.—Tr.
102. The sensuous life and the intellectual life too are Time; it is only sensuous experience and intellectual experience, the “world,” that is spatial nature. (As to the nearer affinity of the Feminine to Time, see Vol. II, pp. 403 et seq.)
103. The expression “space of time” (Zeitraum) which is common to many languages, is evidence of our inability to represent direction otherwise than by extension.
104. I.e., the translated Bible.—Tr.
105. See Vol. II, pp. 19 et seq.
107. See Vol. II, p. 137.
108. The nearest English equivalent is perhaps the word “fear.” “Fearful” would correspond exactly but for the fact that in the second sense the word is objective instead of subjective. The word “shy” itself bears the second meaning in such trivial words as gun-shy, work-shy.—Tr.
109. The Relativity theory, a working hypothesis which is on the way to overthrowing Newton’s mechanics—which means at bottom his view of the problem of motion—admits cases in which the words “earlier” and “later” may be inverted. The mathematical foundation of this theory by Minkowski uses imaginary time units for measurement.
110. The dimensions are x, y, z (in respect of space) and t (in respect of time), and all four appear to be regarded as perfectly equivalent in transformations. [The English reader may be referred to A. Einstein, “Theory of Relativity,” Ch. XI and appendices I, II.—Tr.]
111. Si nemo ex me quaerat, scio; si quaerenti explicari velim, nescio. (Conf. XI, 14.)
112. Save in elementary mathematics. (It may be remarked that most philosophers since Schopenhauer have approached these questionsquestions with the prepossessions of elementary mathematics.)
113. The “inverse circular functions” of English text-books.—Tr.
114. The Newtonian form of the differential calculus was distinct from the Leibnizian, which is now in general use. Without going into unnecessary detail, the characteristic of Newton’s method was that it was meant not for the calculation of quadratures and tangents (which had occupied his predecessors), nor as an organ of functional theory as such (as the differential calculus became much later), but quite definitely as a method of dealing with rate of change in pure mechanics, with the “flowing” or “fluxion” of a dependent variable under the influence of a variable which for Newton was the “fluent,” and which we call the argument of a function.—Tr.
115. See Vol. II, pp. 13, 19.
116. See Vol. II, p. 16.
117. The original reads: “(So ist jede Art von Verstehen ... nur dadurch möglich ...) dass ein Begriffspaar von innerem Gegensatz gewissermassen durch Auseinandertreten erst Wirklichkeit erhält.”—Tr.
118. At this point the German text repeats the paragraph which in this edition begins at “But inquiry” (p. 121) and ends at the close of section I (p. 121).—Tr.
119. See Vol. II, pp. 137, 159.
120. Here the author presumably means history in the ordinary acceptation of the word.—Tr.
121. Œd. Rex., 642. κακῶς εἴληφα τοὐμὸν σῶμα σὺνκακῶς εἴληφα τοὐμὸν σῶμα σὺν τέχνῃ κακῇ. (Cf. Rudolf Hirsch, Die Person (1914), p. 9.)
123. Choëphoræ, 710. ἐπὶ ναυάρχῳ σώματι ... τῷ βασιλείῳ.
124. Phidias, and through him his patron Pericles, were attacked for alleged introduction of portraits upon the shield of Athene Parthenos. In Western religious art, on the contrary, portraiture was, as everyone knows, a habitual practice. Every Madonna, for instance, is more or less of a portrait.
With this may be compared again the growing resistance of Byzantine art, as it matured, to portraiture in sacred surroundings, evidenced for instance in the history of the nimbus or halo—which was removed from the insignia of the Prince to become the badge of the Saint—in the legend of the miraculous effacement of Justinian’s pompous inscription on Hagia Sophia, and in the banishment of the human patron from the celestial part of the church to the earthly.—Tr.
125. Who was criticized as “no god-maker but a man-maker” and as one who spoilt the beauty of his work by aiming at likeness.
Cresilas, the sculptor from whom the only existing portrait of Pericles is derived, was a little earlier; in him, however, the “ideal” was still the supreme aim.—Tr.
126. The writers immediately succeeding Aristophanes.—Tr.
127. See Vol. II, pp. 360 et seq.
128. Diels, Antike Technik (1920), p. 159.
129. About 400 B.C. savants began to construct crude sun-dials in Africa and Ionia, and from Plato’s time still more primitive clepsydræ came into use; but in both forms, the Greek clock was a mere imitation of the far superior models of the older East, and it had not the slightest connexion with the Greek life-feeling. See Diels, op. cit., pp. 160 et seq.
130. Horace’s monumentum ære perennius (Odes III, 30) may seem to conflict with this: but let the reader reconsider the whole of that ode in the light of the present argument, and turn also to Leuconoe and her “Babylonian” impieties (Odes I, 11) inter alia, and he will probably agree that so far as Horace is concerned, the argument is supported rather than impugned.—Tr.
131. Ordered, for us, by the Christian chronology and the ancient-mediæval-modern scheme. It was on those foundations that, from early Gothic times, the images of religion and of art have been built up in which a large part of Western humanity continues to live. To predicate the same of Plato or Phidias is quite impossible, whereas the Renaissance artists could and did project a classical past, which indeed they permitted to dominate their judgments completely.
132. See pp. 9. et seq.
133. The Indian history of our books is a Western reconstruction from texts and monuments. See the chapter on epigraphy in the “Indian Gazetteer,” Vol. II.—Tr.
134. See Vol. II, pp. 482, 521 et seq.
135. There is one famous episode in Greek history that may be thought to contradict this—the race against time of the galley sent to Mitylene to countermand the order of massacre (Thucydides, III, 49). But we observe that Thucydides gives twenty times the space to the debates at Athens that he gives to the drama of the galley-rowers pulling night and day to save life. And we are told that it was the Mitylenean ambassadors who spared no expense to make it worth the rowers’ while to win, whereupon “there arose such a zeal of rowing that....” The final comment is, strictly construing Thucydides’s own words: “Such was the magnitude of the danger that Mitylene passed by” (παρὰ τοσοῦτον μὲν ἡτοσοῦτον μὲν ἡ Μυτιλήνη ἦλθε κινδύνου), a phrase which recalls forcibly what has just been said regarding the “situation-drama.”—Tr.
136. Besides the clock, the bell itself is a Western “symbol.” The passing-bell tolled for St. Hilda of Whitby in 680, and a century before that time bells had come into general use in Gaul both for monasteries and for parish churches. On the contrary, it was not till 865 that Constantinople possessed bells, and these were presented in that year by Venice. The presence of a belfry in a Byzantine church is accounted a proof of “Western influence”: the East used and still largely uses mere gongs and rattles for religious purposes. (British Museum “Handbook of Early Christian Antiquities)”.Antiquities)”.—Tr.
137. May we be permitted to guess that the Babylonian sun-dial and the Egyptian water-clock came into being “simultaneously,” that is, on the threshold of the third millennium before Christ? The history of clocks is inwardly inseparable from that of the calendar; it is therefore to be assumed that the Chinese and the Mexican Cultures also, with their deep sense of history, very early devised and used methods of time-measurement.
(The Mexican Culture developed the most intricate of all known systems of indicating year and day. See British Museum “Handbook of May on Antiquities.”Antiquities.”—Tr.)
138. Let the reader try to imagine what a Greek would feel when suddenly made acquainted with this custom of ours.
139. The Chinese ancestor-worship honoured genealogical order with strict ceremonies. And whereas here ancestor-worship by degrees came to be the centre of all piety, in the Classical world it was driven entirely into the background by the cults of present gods; in Roman times it hardly existed at all.
(Note the elaborate precautions taken in the Athenian “Anthesteria” to keep the anonymous mass of ghosts at bay. This feast was anything but an All Souls’ Day of re-communion with the departed spirits.—Tr.)
140. With obvious reference to the resurrection of the flesh (ἐκ νεκρῶν). But the meaning of the term “resurrection” has undergone, from about 1000 A.D., a profound—though hardly noticed—change. More and more it has tended to become identified with “immortality.” But in the resurrection from the dead, the implication is that time begins again to repeat in space, whereas in “immortality” it is time that overcomes space.
141. For English readers, the most conspicuous case of historic doubt is the Shakespeare-Bacon matter. But even here, it is only the work of Shakespeare that is in question, not his existence and personality, for which we have perfectly definite evidence.—Tr.
142. Originally a philosophical and scientific lecture-temple founded in honour of Aristotle, and later the great University of Alexandria, bore the title Μουσεῖον. Both Aristotle and the University amassed collections but they were collections of (a) books, (b) natural history specimens, living or taken from life. In the West, the collection of memorials of the past as such dates from the earliest days of the Renaissance.—Tr.
143. The connotation of “care” is almost the same as that of “Sorge,” but the German word includes also a certain specific, ad hoc apprehension, that in English is expressed by “concern” or “fear.”—Tr.
144. The Lingayats are one of the chief sects of the Saivas (that is, of the branch of Hinduism which devotes itself to Shiva) and Paewati worshippers belong to another branch, having the generic name of Saktas, who worship the “active female principle” in the persons of Shiva’s consorts, of whom Paewati is one. Vaishnavism—the Vishnu branch of Indian religion—also contains an erotic element in that form which conceives Vishnu as Krishna. But in Krishna worship the erotic is rather less precise and more amorous in character.
See “Imperial Gazetteer of India,” Vol. I, pp. 421 et seq., and Ency. Brit., XI Edition, article Hinduism.—Tr.
145. British Museum.—Tr.
146. Dresden.—Tr.
147. See Vol. II, p. 316.
148. In connexion with this very important link in the Author’s argument, attention may be drawn to a famous wall-painting of very early date in the Catacomb of St. Priscilla. In this, Mary is definitely and unmistakably the Stillende Mutter. But she is, equally unmistakably, different in soul and style from her “Early-Christian-Byzantine” successor the Theotokos. Now, it is well known that the art of the catacombs, at any rate in its beginnings, is simply the art of contemporary Rome, and that this “Roman” art had its home in Alexandria. See Woermann’s Geschichte der Kunst, III, 14-15, and British Museum “Guide to Early Christian Art,” 72-74, 86. Woermann speaks of this Madonna as the prototype of our grave, tenderly-solicitous Mother-Madonnas. Dr. Spengler would probably prefer to regard her as the last Isis. In any case it is significant that the symbol disappears: in the very same catacomb is a Theotokos of perhaps a century later date.—Tr.
149. Vol. II, pp. 403 et seq.
150. See, further, the last two sections of Vol. II (Der Staat and Wirtschaftsleben).—Tr.
151. Sesenheim is the home of Friederike, and a student’s holiday took him thither: Weimar, of course, is the centre from which all the activity of his long life was to radiate.—Tr.
152. Vermeintlich. The allusion is presumably to the fact that Copernicus, adhering to the hypothesis of circular orbits, was obliged to retain some elements of Ptolemy’s geocentric machinery of epicycles, so that Copernicus’s sun was not placed at the true centre of any planetary orbit.—Tr.
153. Sprüche in Reimen.
154. See Vol. II, pp. 294 et seq., 359 et seq.
155. The path from Calvin to Darwin is easily seen in English philosophy.
156. This is one of the eternal points of dispute in Western art-theory. The Classical, ahistorical, Euclidean soul has no “evolution”; the Western, on the contrary, extends itself in evolving like the convergent function that it is. The one is, the other becomes. And thus all Classical tragedy assumes the constancy of the personality, and all Western its variability, which essentially constitutes a “character” in our sense, viz., a picture of being that consists in continuous qualitative movement and an endless wealth of relationships. In Sophocles the grand gesture ennobles the suffering, in Shakespeare the grand idea (Gesinnung) ennobles the doing. As our æsthetic took its examples from both Cultures, it was bound to go wrong in the very enunciation of its problem.
157. “The older one becomes, the more one is persuaded that His Sacred Majesty Chance does three-quarters of the work of this miserable Universe.” (Frederick the Great to Voltaire.) So, necessarily, must the genuine rationalist conceive it.
158. See Vol. II, pp. 20 et seq.
159. The incident which is said to have precipitated the French war on Algiers (1827).—Tr.
160. Act. II, Scene VII.—Tr.
161. In the general upheaval of 1848 a German national parliament was assembled at Frankfurt, of a strongly democratic colour, and it chose Frederick William IV of Prussia as hereditary emperor. Frederick William, however, refused to “pick up a crown out of the gutter.” For the history of this momentous episode, the English reader may be referred to the Cambridge Modern History or to the article Germany (History) in the Ency. Brit., XI Edition.—Tr.
162. It is the fact that a whole group of these Cultures is available for our study that makes possible the “comparative” method used in the present work. See Vol. II, pp. 42 et seq.
163. Derived from μείρομαι, to receive as one’s portion, to have allotted to one, or, colloquially, to “come in for” or “step into.”—Tr.
164. The expedition of the Ten Thousand into Persia is no exception. The Ten Thousand indeed formed an ambulatory Polis, and its adventures are truly Classical. It was confronted with a series of “situations.”—Tr.
165. Helios is only a poetical figure; he had neither temples nor cult. Even less was Selene a moon-goddess.
166. The original is somewhat obscure. It reads: "Welche Form die Wahrscheinlichkeit für sich hat, ist bereits eine Frage des historischen—und also des tragischen—Stils."—Tr.
167. The words of Canning at the beginning of the XIXth century may be recalled. “South America free! and if possible English!” The expansion idea has never been expressed in greater purity than this.
168. The Western Culture of maturity was through-and-through a French outgrowth of the Spanish, beginning with Louis XIV. But even by Louis XVI’s time the English park had defeated the French, sensibility had ousted wit, London costume and manners had overcome Versailles, and Hogarth, Chippendale and Wedgwood had prevailed over Watteau, Boulle and Sèvres.
169. The allusion is to the voyage of Linois’s small squadron to Pondichéry in 1803, its confrontation by another small British squadron there, and the counter-order which led Linois to retire to Mauritius.—Tr.
170. Hardenberg’s reorganization of Prussia was thoroughlythoroughly English in spirit, and as such incurred the severe censure of the old Prussian Von der Marwitz. Scharnhorst’s army reforms too, as a breakaway from the professional army system of the eighteenth-century cabinet-wars, are a sort of “return to nature” in the Rousseau-Revolutionary sense.
171. Where in 295 B.C. the Romans decisively defeated the last great Samnite effort to resist their hegemony over Italy.—Tr.
172. Which, inasmuch as it has been detached from time, is able to employ mathematical symbols. These rigid figures signify for us a destiny of yore. But their meaning is other than mathematical. Past is not a cause, nor Fate a formula, and to anyone who handles them, as the historical materialist handles them, mathematically, the past event as such, as an actuality that has lived once and only once, is invisible.
173. That is, not merely conclusions of peaces or deathdays of persons, but the Renaissance style, the Polis, the Mexican Culture and so forth—are dates or data, facts that have been, even when we possess no representation of them.