Kha and other words denoting
"Zero"
in Connection with the Indian metaphysics of space
by Ananda K. Coomaraswamy
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";(1) 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.(2) 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, o=x-x, it is apparent that zero is to
number as possibility is to actuality. ,.gain,
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, 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.
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;(3) 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.(4) 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" (pi n) from this chthonic center.
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.(5) 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 the wheel, 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 princi pio) 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.(6) 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.
According to an alternative formulation, all things
are thought of as ante principium shut up within, and in principio
as proceeding from, 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 princi pium 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 Water s 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) ?(7)
Thus metaphysically, in the symbolism of the Wheel,
the surface blank (sunya) in the initial nonbeing (asat) of any formulation (samkalpa)
represents the truly infinite (adin) 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,
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,(8) 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) and then again centripetal (nivrtta),
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)(9) seen
in simultaneity, who as "round about seer," paridrastr,
n1ow "overlooks everything," visvam . . . abhicaste, 1.164.44.
In order to understand the use
of terms for "space" (kha, ukasa, antarik~a, sunya, etc.)(10) as verbal symbols of zero (which
represents privation of number, and is yet a matrix of number in the sense
o=x-x),(11) 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;(13) 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 Brahman and with the Spirit
(kham brahma, kham puranam, vayuram
kham iti), and this Brahman
is at the same time a plenum or pleroma (purna) such chat "when plenum is taken from plenum,
plenum yet remains (14)
Here we get precisely that equivalence of kha
and puma, void and plenum, which was remarked upon as noteworthy in the verbal
notation of the mathematicians. The thought, moreover, is almost literally
repeated when Bhaskara in the Bijaganita(15) 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)(16)
at the time of the emanation or resolution of worlds, though hosts of beings
are emanated or withdrawn."
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."(17) 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 wheel's, 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 airts (dis); 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"
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
(18) 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), (19) 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 Is
in imitation of the angelic works of art that any work of art such as a garment
or chariot is. made here, (20) 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.(21) 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,(22)
medicine, architecture, etc., are classed as Vedanga,
"limbs or powers of the Veda," or as Upaveda",
accessory with respect to the Veda".
________________________________________________________________________________________
NOTES
1. Journal of the United .Provinces
Historical Society, VII, 44 45, 62.
2 It may as well be pointed out here that although "the decimal
notation must have been in existence and in common use among the mathematicians
long before the idea of applying the place value principle to a system of word
names could have been conceived" (ibid., p. 61 ), and although a decimal
scale has actually been found at Mohenjo Daro (E.l.H. Mackay,
"Further Excavations at Mohenjodaro,"
journal of the Royal Society of .Arts, LXXXII, 1934, 222) it is by no means the
intention of the present article to present an argument for a Rg Vedic knowledge of either the decimal system or the
concept of "zero" as such. Our purpose is merely to exhibit the
metaphysical and ontological implications of the terms which were later on
actually used by Aryabhata and Bhaskara,
etc., to designate "zero," "one," and some higher numbers.
3. The familiar principle "as above, so below" is illustrated
here.
4. The notion of exemplarism is expressed
here, with respect to number or mathematical individuality.
5. Indu a occurs in
the Rg Veda as "drop" in connection with
Soma: in AV vii. r o9.6 as "point on a die";
and grammatically as the designation of Anusvara. PB
vi.9.19-20 is of interest: indava
Iva hi pitarah, mana iva, i.e., "the
Patriarchs are as it were drops (indu in pl.), as it
were the intellectual principle." In RV vI.44.22, Indu
is evidently Soma; in vii.54.2, Vastospati.
6. 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 I.22J6, 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),
and therefore provided with two analogous axle points, that ani
was not later employed as a verbal symbol of "one."
7. Madhya is "middle" in all senses, and also algebraically
"mean." For the metaphysical values, cf. RV madhye
samudre, and utsasya madhye = sindhunam upodaye, as the place of Agni or Varuna, and in CU iii.ii.i, ekata madhye sthane,
"single in the midmost station."
8. For the construction of the wheel, cf. RV vIII.77.3, akhidat khe aran
iva khedaya,
and the discussion in Coomaraswamy, "Angel and
Titan: An Essay in Vedic Ontology," 1935.
9. Sankaracarya, Svatmanirupana
95.
10. Sunya does not appear in RV, though sunam occurs in the sense of "privation."
11. Observe that the dual series of plus and minus numbers represents
"pairs of opposites," dvandvau.
12. C. A. Scharbrau, "Transzendenter Raum der Ewigkeit ist der
Akasa vor allem auch da, wo er als Ausgangspunkt, als Schopfungsgrund and als
Ziel, als A and 0 der Welt angcschaut wird," Die Idee der Schopfung in der
vedischen Literatur (Stuttgart, tc~32), p. 56; "size which has no size,
though the principle of size," Moister Eckhart, Evans ed., i.114.
13. Nabha, from nabh,
to "expand," etc., as also in nabhi,
"navel" and "nave." A secondary sense of nabh is "to destroy."
14. This text occurs in almost the same form in AV x.8.29.
15.
16. Cf. asmany anante
and adrim acyutam cited
above, with the meaning "rock of Ages."
17. Singh, p. 56, (as cited in n. I).
18. "The place system of the Babylonians . . . fell on fertile soil
only among the Hindus . . . . .Algebra, which is distinctly Hindu . . . uses
the principle of local value" (M. J. Babb, in JAOS, LI, 1931, 52). That
the "Arabic" numerals are ultimately of Indian origin is now
generally admitted; what their adoption meant for the development of European
science need not be emphasized.
19. Aryabhata, Aryabhatiya
iv.9, "As a man in a boat going forward sees a stationary object moving
backward, just so at Lanka a man sees the stationary asterisms moving
backward."
20. See Coonmaraswamy, The Transformation o f
Nature in Art, 1934, p. 8 and n. 8.
21. Cf. Ernest Siecke, Die Liebesgeschichte des Himmcls (Strassburg,
1892) ; and Alfred Jcretnlas, ..Band bueh der altoriewalisc hen Geiste shultur
(Berlin, 19,29) p. x: ie Metischheitsbildung ist ein einheitliches Ganzes, and
in den verschiedenen Kulturen fndet man die Dialekte der cinen Geis
tessprache."
22. Even the "Machiavellian" Arthasastra
(i.3) proceeds from the principle svadharnnah svargaya anantyaya ca, tasya atikrame
lokah sankarad acchidyeta, "vocation leads to heaven and aeviternity; in case of a digression from this norm, the
world is brought to ruin by confusion."