Werk:Aufdeckung missender Geheimnisse über Magnetismus/Kha und andere Wörter, die 'Null' bedeuten

Kha and other words denoting ‘zero’, the metaphysics of counterspace, and the ancient wheel analogy of space & counterspace

• Kha und andere Wörter, die 'Null' bedeuten, die Metaphysik des Gegenraums und die alte Radanalogie von Raum und Gegenraum

Adapted from the works of Dr. A.K. Coomaraswamy, of which his son, Dr. R.P. Coomaraswamy, has granted me full rights to all his fathers’ works in 2006.

Below: All cubic space is the resultant of magnetic divergence, as necessitated without exceptions, 0 = X-X. Zero does not exist, rather as both meant and implied unmanifest inertia in counterspace; i.e. the Ether. The ‘square’ piece lower left is the exact same piece on the right, only from a different angle.

Kha, cf. Greek Xaos, is generally "cavity"; and in the Rg Veda, particularly "the hole in the nave of a wheel through which the axle runs" (Monier Williams). A. N. Singh has shown conclusively that in Indian mathematical usage, current during the earlier centuries of the Christian era, kha means "zero";Suryadeva, commenting on Aryabhata, says that "the khas refer to voids (khaini sunya upa lakptani) …thus khadvinake means the eighteen places denoted by zeros." Among other words denoting zero are sunya, akasa, vyoma, antariksa, nabha, ananta, and purna. We are immediately struck by the fact that the words sunya, "void," and purna, "plenum," should have a common reference; the implication being that all numbers are virtually or potentially present in that which is without number; expressing this as an equation, 0= X - X, it is apparent that zero is to number as possibility is to actuality. Again, employment of the term ananta with the same reference implies an identification of zero with infinity; the beginning of all series being thus the same as their end. This last idea, we may observe, is met with already in the earlier metaphysical literature, for example RV IV.I.II, where Agni is described as "hiding both his ends (guhamano anta) "; AB 111.43, "the Agnistoma is like a chariot wheel, endless (ananta)"; JUB 1.35, "the Year is endless (ananta), its two ends (anta) are Winter and Spring . . . so is the endless chant (anantam saman)." These citations suggest that it may be possible to account for the later mathematicians selection of technical terms by reference to an earlier usage of the same or like terms in a purely metaphysical context.

Our intention being to demonstrate the native connection of the mathematical terms kha (zero, counterspace), etc., with the same terms as employed in purely metaphysical contexts, it will be necessary to prepare the diagram of a circle or cosmic wheel (cakra,mandala) and to point out the significance of the relationships of the parts of such a diagram according to universal tradition and more particularly in accordance with the formulation of the Rg Veda. Take a piece of blank paper of any dimensions, mark a point anywhere upon it, and With this point as center draw two concentric circles of any radii, but one much less than the other; draw any radius from the center to the outer circumference. With exception of the center, which as a point is necessarily without dimension, note that every part of our diagram is merely representative; that is, the number of circles may be indefinitely increased, and the number of radii likewise, each circle thus filled up becoming at last a plane continuum, the extended ground of any given world or state of being; for our purpose we are considering only two such worlds mythologically speaking, Heaven and Earth, or psychologically, the worlds of subject and object as forming together the world or cosmos, typical of any particularized world which may be thought of as partial within it. Finally, our diagram may be thought of either as consisting of two concentric circles with their common radii and one common center, or as the diagram of a wheel, with its felly, nave, spokes, and axle point. Nabha, from nabh, to "expand," etc., as also in nabhi, "navel" and "nave." A secondary sense of nabh is "to destroy” (in counterspace, voidance). Actually to represent all possible states of being, the Wheel would have to be conceived in the manner of a gyroscope, revolving simultaneously in an indefinite number of planes, though still with a motionless center: just as the Cross must be thought of from this point of view asconstituted of three arms, mutually at right angles, intersecting at the one common point which is also the center of the sphere in which the Cross stands. Actually, however, this would be to introduce a needless complication, and in fact the symbol as employed is essentially an ordinary chariot wheel, just as also in common usage the two armed cross stands for a cross extended in three directions. Although, then, the Wheel, as the "round of the world " and "earth plain," strictly speaking corresponds only to a given ensemble of conditions it represents analogically the indefinite totality of all possible conditions, the entirety of samsara. As thus representing the Universe in its entirety, the Wheel symbol remains in use unchanged from Rg Veda. The content of the wheel symbolism is extraordinarily rich, and can only be outlined here.. Its dimensions are indefinite, its radius the variable distance between an undimensioned (amatra) point and an immeasurable (asankhya) circumference; there in the `middle space" (antariksa, akasa), between the "I" and the "not I," essence and nature, lie procession and recession (pravrtti, nivrtti), there are good and evil (dharmadharmau), joy and sorrow (sukha, duhkha), light and shade (chayatapa), birth end death, all local movement and affection; and that motion and passibility are greater the greater the distance from the centre. Beyond the felly lies only the inexistence of the irrational, an impossibility of existence, as of square circles or the horns' of a hare; within the nave, the non existence of the supra rational. The cycle of ego consciousness implies an outward movement from the nave to the ever-receding felly, and a return from the however distant felly to the unchanging center. A progressive enlightenment (krama mukti) can then be expressed 'as a gradual contraction of the radius, bringing the circumference ever closer to the center, until that which seemed to enclose the point is seen to lie contained within it, knowledge being thus con centrated into a single form, which is the form of very different things. That is Nirvana, unitary being, "with residual existential elements," and by a vanishment of the point becomes also epistrophe, synthesis, without residuum of existence.

Below: The magneto-dielectric archetype, seen face on.

Now in the first place, as a geometrical symbol, that is to say with respect to measure or numeration, our diagram represents the logical relationships of the concepts naught or zero, inconnumerable unity, and indefinite multiplicity; the blank (sunnya) surface having no numerical significance; the central point (Indu, bindu) being an inconnumerable unity (inconnumerable, advaita, because there cannot be conceived a second center); and either circumference an endless (ananta) series of points, which may be thought of as numbers; the totality (sarvam) of the numbered, that is to say individual, points representing the sum of a mathematically infinite series extending from one to "infinity," and conceivable as plus or minus according to the direction of procedure. The whole area (sarira) delimited corresponds to place (desa), a revolution of the circles about their center corresponds to time (kala). It will be observed further that any radius connects analogous or corresponding points or numbers on the two circumferences; if, now, we suppose the radius of one or both circles indefinitely reduced, which brings us to the central point as limiting concept (that is also "as it was in the beginning"), it is evident that even this point can be thought of only as a plenum of all the numbers represented on either circumference. On the other hand, this point, at the same time that it represents an inconnumerable unity and, as we have seen, a plenum, must also be thought of as representing, that is, as the symbol of, zero, for two reasons: (1) inasmuch as the concept to which it refers is by definition without place and without dimensions, and therefore nonexistent, and (2) the mathematically infinite series, thought of as both plus and minus according to direction, cancel out where all directions meet in common focus.

So far as I know, Indian literature does not provide a specifically geometrical exegesis exactly corresponding to what is given in the preceding paragraph. What we do find in the metaphysical and religious traditions is a corresponding usage of the symbol of the Wheel (primarily the solar chariot, or a wheel thereof), and it is in this connection that we first meet with some of the most significant of those terms which are later on employed by the mathematicians. In RV 1.155.6 and 1.164.2, II, 13, 14, 48; AV x.8.4 7; KB xx.i; JUB 1.35; BU 1.5.15; Svet. Up. 1.4; PS 6, and like texts, the year as an everlasting sequence is thought of as an unwasting wheel of life, a revolving wheel of the Angels, in which all things have their being and are manifested in succession; "none of its spokes is last in order" (RV v.85.5). The parts of the wheel are named as follows: ani, the axle point within the nave (note that the axle causes revolution, but does not itself revolve); kha, nabhi, the nave (usually as space within the hub, occasionally as the hub itself); ara, spoke, connecting hub and felly; nemi, pavi, the felly. It should be observed that nabhi, from nabh, to expand, is also "navel"; similarly in anthropomorphic formulation, "navel" corresponds to "space" (MU vi.6); in the Rg Veda, the cosmos is constantly thought of as "expanded" (pin) from this chthonic center. We considered above mainly the case in which the cosmic wheel is thought of as singleperhaps more often the chariot of the Sun is thought of as running on twin wheels connected by a common axle tree (aksa), and this involves a consideration of the world from two distinct but inseparable points of view. As the Sure shines equally for angels and for men (leg Veda, I, 50, 5, etc.), so of the twin wheels of his chariot one touches Heaven, the other Earth (Rg Veda, I, 30, 19, and. X, 85, 18); and their common axle tree is identified with the axis of the universe that holds apart (vitaram, visvak) Heaven and Earth (Rg Veda., V, Z9, 4, and X, 89, 4). Or again, when the chariot of the Sun is thought of as three wheeled (tricakra), Rg Veda, X, 85, two of the wheels are identified, as aforesaid with Heaven and Earth ("one looks down upon the several worlds, the other ordains the seasons and is born again," cf. I, 164, 44 and 32), and these "proceed by magic”, mayaya caranti; but the third is hidden (guha’ = guhayam nihitam, sc. "in the heart "), and only the adepts (addhatayah) are Comprehensors (viduh) thereof. This third wheel evidently corresponds to the "secret name”. These doctrines of three wheels, three lights, etc., are tantamount to the trikaya doctrine. The axle tree of the twin wheels (which axle must be thought of analogically also as penetrating the third wheel) is the primary source of moving power, or Brahma (as rooted incidentally in leg Veda, I,166,.9): not itself revolving (important), it is the unmoved mover in relation to the wheels. But to complete our understanding of the pratika it must be realized that the revolution of the wheel requires the operation of an opposing force operative at the felly, where in actual experience contact with the ground supplies a fulcrum. In other words, revolution depends on the interaction of conjoint principles, which may be galled Heaven and Earth, Purusa and Prakrti, sattva and tames, I and not I, subject and object, etc. This is recognized in several passages in which the infixation of the axle, or the movement of the wheels, is effected by the deity by means of his abilities (sacibhih, Rg Veda, 1, 30, 15, and X, 89, 4), powers (8`aktibhih, X, 88, 10), or magic (mayaya, X, 85, 18), saci, Sakti, and maya being synonymous feminine designations of his "means whereby," the "ground" of manifestation, cooperating with his "essence," who is Sacipati, Mayin, etc. The axle tree is also the axis of the universe, as most clearly stated in Rg Veda, X, 89, 4, yo akseneva cakriya sacibhih visvak tastambha prthivim uta dyam, "by the axle of his wheeled car indeed, by his abilities, he pillars apart Heaven and Earth”.

In China we shall be concerned with only two stone objects, or perforates; these objects of jade are symbols of Heaven and Earth. Of these two perforates, the Tsung, or Earth symbol is internally tubular and externally cubical (figure 15), while the pi, or Heaven symbol is a perforated circular disk or ringstone (figure 16). It is interesting that these Tsung are regularly thought of “cart wheels” or “wheel hubs”. But the “ancient jade chariot” no more implies an actual chariot of jade used than does the Vedic chariot of light or biblical chariot of fire refer to vehicles that might be unearthed by archeologists. These are purely symbolic representation of space and counterspace, earth and the unseen, counterspace, conventionally ‘Heaven’.

Below: Symbols of Heaven and Earth. Of these two perforates, the Tsung, or Earth symbol is internally tubular and externally cubical (figure 15), while the pi, or Heaven symbol is a perforated circular disk or ringstone (figure 16).

Certain passages indicating the metaphysical significance of the terms ani, kha, and nabhi in the Rg Veda may now be cited. It should be premised that we find here in connection with the constant use of the wheel symbol, and absence of a purely geometrical formulation, the term ani employed to express ideas later on referred to by the words indu or bindu. Vedic ani, being the axle point within the nave of the wheel, and on which the wheel revolves, corresponds exactly to Dante's "il pumadello stelo al cui la prima rota va dintorno" (Paradiso XIII.I I I2) . The metaphysical significance of the ani is fully brought out in RV 1.35.6, anim na rathyam amrta adhi tasthuh, "as on the axle point of the chariot wheel are actually existent the undying [Angels or intellectual principles]," which also supplies the answer to the well known problem, "How many Angels can stand on the point of a needle?" More often the nave of thewheel, rather than the axle point specifically, is treated as its center; nor need this confuse us if we reflect that just as under limiting conditions (indefinite reduction of the radius, or when the central point has been identified but the circle not yet drawn) the center represents the circle, so under similar conditions (metaphysically, in principio) the axle point implies the nave or even the whole wheel the point without dimension, and a principial space not yet expanded (or as the Rg Veda would express it, "closed") being the same in reference. The nave then, kha or nabhi, of the world wheel is regarded as the receptacle and fountain of all order, formative ideas, and goods: for example, 11.28.5, rahydmato varuna kham rtasya, "may we, O Varuna, win thy nave of Law"; viii.4I.6, where in Trita Aptya "all oracles (kavyd) are set as is the nave within the wheel (cakre nabhir iva) "; IV.28, where Indra opens the closed or hidden naves or rocks (apihita . . . khani in verse I, apihitani asna in verse 5) and thus releases the Seven Rivers of Life. In v.32.I, where

Indra breaks open the Fountain of Life (utsam), this is again an emptying out of the hollows (khani), whereby the fettered floods are released. The Rivers, of course, represent ensembles of possibility (hence they are often spoken of as "maternal") with respect to a like number of "worlds," or planes of being, as in 1.22.16, prthivya sapta dhamabhih. Our terms kha, asna, etc., are necessarily employed in the plural when the "creation" is envisaged with respect to the cosmos not as a single "world," but as composed of two, three, or seven originally unmanifested but now to be conceptually distinguished "worlds"; the solar chariot having one, two, three, or seven wheels, accordingly. It is perhaps because the chariot of the Year is more often than not thought of as two wheeled (Heaven and Earth, Counterspace and Space), and therefore provided with two analogous axle points, that ani was not later employed as a verbal symbol of "one."

According to an alternative formulation, all things are thought of as ante principium shut up within, and in principio as proceedingfrom, a common ground, rock, or mountain (budhna, adri, parvata, etc.): this ground, thought of as resting island like within the undifferentiated sea of universal possibility (x.89.4, where the waters pour sagarasya budhnat), is merely another aspect of our axle point (,dni), regarded as the primary assumption toward which the whole potentiality of existence is focused by the primary acts of intellection and will. This means that a priori undimensioned space (kha, akasa, etc.) underlies and is the mother of the point, rather than that the latter has an independent origin; and this accords with the logical order of thought, which proceeds from potentiality to actuality, nonbeing to being. This ground or point is, in fact, the "rock of ages" (asmany anante, 1.130.3; adrim . . . acyutam, v1.i7.5).

Here ante principium Agni lies occulted (guha santam, 1.141.3, etc.) as Ahi Budhnya, "in the ground of space, concealing both his ends" (budhne rajaso . . . guhamdno ants, iv.i.ii, where it may be noted that guhamano anta is tantamount to ananta, literally "end less," "in-finite," "eternal"), hence he is called "chthonic" (nabhir agni prthivya, 1.59.2, etc.), and is born in this ground (jayata prathamah . . . budhne, iv.i.ii) and stands erect, Janus like, at the parting of the ways (ayor ha skambha . . . patham visarge, x.5.6) ; hence he gets his chthonic steeds and other treasures (asvabudhna, x.8.3; budhnya vasuni, v11.6.7) . It is only when this rock is cleft that the hidden trine are freed, the waters flow (1.62.3, where Brhaspati bhinad adrim and vidadgah; v.4i.i2, srnvanty dpah . . . adreh). This is, moreover, a center without place, and hence when the Waters have come forth (that is, when the cosmos has come to be) one asks, as in x.iii.8, "where is their beginning (agram), where their ground (budnah), where now, ye Waters, your innermost center (madhyam . . . antah).

Thus metaphysically, in the symbolism of the Wheel, of counterspace, the surface blank (sunya) in the initial nonbeing (asat) of any formulation (samkalpa) represents the truly infinite (aditi) and maternal possibility of being; the axle point or nave, exemplary being (visvam ekam, RV 111.54.8 = integral omnipresence); the actual construction, a mentally accomplished partition of being into existences; each spoke, the integration of an individual as nama-rupa (psycho physical), that is, as archetypal inwardly and phenomenal outwardly; the felly, the principle of multiplicity (visamatva). Or, employing a more theological terminology: the

undetermined surface represents the Godhead (aditi, parabrahman, tamas, apah) ; the axle point or immovable rock, God (aditya, aparabrahman, isvara, jyoti) ; the circle of the nave, Heaven (svarga);any point on the circumference of the nave, an intellectual principle (nama, deva) ; the felly, Earth with its analogous (anurupa) phenomena (visva rupani); the construction of the wheel, the sacrificial act of creation (karma, srsti), its abstraction, the act of dissolution (laya). Furthermore, the course (gati) of any individual upon the pathway of a spoke is in the beginning centrifugal (pravrtta, bhava, samsarati, paticcasamuppada) and then again centripetal (nivrtta, nisbandhu, epistrophe, samadhi, synthesis etc.), until the center (madhya) is found; and when the center of individual being coincides with the center of the wheel, he is emancipated (mukta), the extension of the wheel no longer involving him in local motion, at the same time that its entire circuit now becomes for him one picture (jagaccitra) seen in simultaneity, who as "round about seer," paridrastr, now "overlooks everything," visvam . . . abhicaste, 1.164.44.

In order to understand the use of terms for "space" (kha, ukasa, antariksa, sunya, etc.), rather counterspace, as verbal symbols of zero (which represents privation of number, and is yet a matrix of number in the sense 0 = X-X), it must be realized that akasa, etc., represent primarily a concept not of physical space, but of a purely principal space without dimension, though the matrix of dimension." For example, "all these beings arise out of the space (akasad samapadyanta) and return into the space (akasam pratyastam yanti). For the space is older than they, prior to them, and is their last resort (parayanam)," CU I.9.I; "space is the name of the permissive cause of individual integration (akaso vai nama namarupayor nirvahita)," CU VIII.I4; and just as India "opens the closed spaces (apihitj khani)," RV IV.2H.I, so the Self "awakens this rational [cosmos] from chat space (akasat esa khalu idam cetamatram dobhayati)," MU VI.I7, in other words, ex nihilo fit. Furthermore, the locus of this "space" is "within you": "what i's the intrinsic aspect of expansion is the supernal fiery energy in the vacance of the inner man (tat svarupam nabhasah khe antarbhutasya yat param tejah)," MU VII.II; and this same "space in the heart" (antarhrdaya akasa) is the locus (dyatana, vesma, nada, kosla, etc.) where are deposited in secret (guha nihitam) all that is ours already or may be ours on any plane (loka) of experience (CU VIII.I.I 3). At the same time, in BU v.i, this "ancient space" (kha) is identified with the Absolute and with the Spirit (kham brahma, kham puranam, vayuram kham iti), and this Absolute is at the same time a plenum or pleroma (purna) such that "when plenum is taken from plenum, plenum yet remains.”

Here we get precisely that equivalence of kha and puma, void and plenum, which was remarked upon as noteworthy in the verbalnotation of the mathematicians. The thought, moreover, is almost literally repeated when Bhaskara in the Bijaganita defines the term ananta thus: ayam ananto rasih khahara ity ucyate. Asmin vikarah khahare na rasavapi pravistesvapi nihsrtesu bahusvapi syal layasrstikale 'nante 'cyute bhutaganesu yadvat, that is, "This fraction of which the denominator is zero, is called an infinite quantity. In this quantity consisting of that which has cipher for its divisor, there is no alteration, though many be added or subtracted; just as there is no alteration in the Infinite Immovable (anante acyute) at the time of the emanation or resolution of worlds, though hosts of beings are emanated or withdrawn." In Rg Veda, X, 85, 12, "the chariot is in the mode of Intellect (anomanasmayam), the Breath of Life (vyana) was the axle (aksa) fastened there." It will be understood that the axle point (ani) that penetrates the hollow (kha) in the nave (na’bhi) is central in each wheel; so in Rg Veda, I, 35, 6, the Undying Angels (se. the Several Angels, visve devah, Adityas) are said to depend upon Savitr (the Supernal Sun as prime mover) "as on the chariot's axle point (ani)," and in Aitareya A.ranyaka, II, 7, the Self is compared to the "twin axle points (ani) "'of the Veda. We have thus dwelt at some length on the Vedic implications of the wheel or wheels, because it is important to realize the wider content and consequent power of this symbol, which was so extensively employed in metaphysics, though with a more restricted application.

It may be observed further that while in the Rg Veda we "do not find the use of names of things to denote numbers, we do find instances of numbers demoting things." In vii.103.1, for example, the number "twelve" denotes the "year"; in X.71.3, "seven" stands for "rivers of life" or "states of being." It is thus merely a converse usage of words when the mathematicians make use of the names of things to denote numbers; to take the most obvious examples, it is just what should be expected, when we find that r is expressed by such words as adi, indu, abja, prthvi; 2 by such as yama, asvina; 3 by such as agni, vaisvanara, haranetra, bhuvana; 4 by veda, dis, yuga, samudra, etc.; 5 by prana; 6 by rtu; and so forth. It is not to be understood, of curse, that the number words are all of Vedic origin; many suggest rather an Epic vocabulary, e.g., pandava for 5, while others, such as netra for 2, have an obvious and secular source. In certain cases an ambiguity arises, for example, loka as representing either 3 or 14, dis as representing 4 or 10, but this can be readily understood; in the last mentioned case, for example, the quarters have been thought of in one and the same cosmology as either four, or if we count up eight quarters and half quarters, adding the zenith and nadir, as ten. Taken in its entirety as cited by Singly the numeral vocabulary can hardly antedate the beginning of the Christian era (we find that 10 is represented, among other words, by avatara; and 6 by raga).

If we attempt to account for the forms of the ideograms of numbers in a similar fashion, we shall be on much less certain ground. A few suggestions may nevertheless be made. For example, a picture writing of the nation "axle point" could only have been a "point," and of the concept "nave" could only have been a "round O," and both of these signs are employed at the present day to indicate "zero." The upright line that represents "one" may be regarded as a pictogram of the axis that penetrates the naves of the dual wheels, and thus at once unites and separates Heaven and Earth. The Devandgari and Arabic signs for "three" correspond to the trident (trisula), which is known to have been from very ancient times a symbol of Agni or Siva. A priori it might be expected that a sign for "four" should be cruciform, following the notion of extension in the directions of the four airs; and in fact we find in Saka script that "four" is represented by a sign X, and that the Devandgari may well be thought of as a cursive form derived from a like prototype. ´Even if there be sufficient foundation for such suggestions, it is hardly likely that a detailed interpretation of ideograms of numbers above four could now be deduced. We can only say that the foregoing suggestions as to the nature of numerical ideograms rather support than counter the views of those who seek to derive the origins of symbolism, script, and speech from the concept of the circuit of the year.

It is, however, beyond question that many of the verbal symbols the case of kha for "zero" or counterspace is conspicuous; used by Indian mathematicians had an earlier currency, that is to say before a development of mathematical science as such, in a more universal, metaphysical context. That a scientific terminology should thus have been formulated on the basis of a metaphysical terminology, and by no means without a full consciousness of what was being done (as the citation from Bhaskara clearly shows), is not only in accordance with all that we know of the natural course of Indian thought, which takes the universal for granted and proceeds to the particular, but also admirably illustrates what from a traditionally orthodox point of view would be regarded as constituting a natural and right relationship of any special science to the metaphysical background of all sciences. One is reminded of words in the Encyclical of Pope Leo XIII, dated 1879, on the "Restoration of Christian Philosophy": "Hence, also, the physical sciences, which now are held in so much repute, and everywhere draw to themselves a singular admiration, because of the wonderful discoveries made in them, would not only take no harm from a restoration of the philosophy of the ancients, but would derive great protection from it. For the fruitful exercise and increase of these sciences it is not enough that we consider facts and contemplate Nature. When the facts are well known we must rise higher, and give our thoughts with great care to understanding the nature of corporeal things, as well as to the investigation of the laws which they obey, and of the principles from which spring their order, their unity in variety, and their common likeness in diversity. It is marvelous what power and light and help are given to these investigations by Scholastic philosophy, if it be wisely used…there is no contradiction, truly so called, between the certain and proved conclusions of recent physics, and the philosophical principles of the Schools." These words by no, means represent a merely Christian apologetic, but rather enunciate u generally valid procedure, in which the theory of the universal acts at the same time with suggestive force and normatively with respect to more specific applications. We may reflect, on the one hand, that the decimal system, with which the concept "zero" is inseparably connected, was developed by Indian scholars who were very surely, as their own words prove, deeply versed in and dependent upon an older and traditional metaphysical interpretation of the meaning of the world; and on the other, that had it not been for its boasted and long maintained independence of traditional metaphysics (in which the principles, if not the facts, of relativity are explicit), modern scientific thought might have reached much sooner than has actually been the case a scientifically valid formulation and proof of such characteristic notions as those of an expanding universe and the finity of physical space. What has been outlined above with respect to the special science of mathematics represents a principle no less valid in the case of the arts, as could easily be demonstrated at very great length. For example, what is implied by the statement in ABVI.27, that "it isin imitation of the angelic works of art that any work of art such as a garment or chariot is made here, is actually to be seen in the hieratic arts of every traditional culture, and in the characteristic motifs of the surviving folk arts everywhere. Or in the case of literature: epic (Volsunga Saga, Beowulf, the Cuchullain and Arthurian cycles, Mahdbhdrata, Buddhacarita, etc.) and fairy tale (notably, for example, jack and the Beanstalk) repeat with infinitely varied local coloring the one story of jatavida, Genesis. The whole point of view can, indeed, be recognized in the Indian classification of traditional literature, in which the treatises (sastras) on auxiliary science such as grammar, astronomy, law, medicine, architecture, etc., are classed as Vedanga, "limbs or powers of the Veda," or as Upaveda", accessory with respect to the Veda".