BULLETIN
OF THE
SCHOOL OF ORIENTAL STUDIES
LONDON INSTITUTION
PAPERS CONTRIBUTED
Kha and other Words denoting “ Zero” in Connection
with the Metaphysics of Space
By Ananpa K. Coomaraswamy
HA, cf. Greek xaos, is generally “cavity” ; and in the Ry Veda,
particularly, “the hole in the nave of a wheel through which
the axle runs” (Monier-Williams). In Journ. U.P. Hist. Soc., vii,
44-5 and 62, Mr. A. N. Singh shows conclusively that in Indian mathe-
matical usage, current during the earlier centuries of the Christian
era, kha means “‘ zero” ; Siiryadeva, commenting on Aryabhata, says
“the khas refer to voids (khani sinya upa laksitani) . . . thus khad-
vinake means the eighteen places denoted by zeros”. Amongst other
words denoting zero are éiinya, akaéa, vyoma, antariksa, nabha, ananta,
and piirna.1 We are immediately struck by the fact that the words
Sanya“ void”, and piirna “ 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 — 2, it is apparent that zero is to number as possibility
1 It may as well be pointed out here that although “The decimal notation must
have been in existence and in common use amongst the mathematicians long before
the idea of applying the place-value principle to a system of word names could have
been conceived ” (Singh, loc, cit., p. 61), and although a decimal scale has actually
been found at Mohenjodaro (Mackay, “Further Excavations at Mohenjodaro,”
Journ, Roy. Soc. Arts, No. 4233, 1934, p. 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 “zero” as such. Our purpose is merely to exhibit the meta-
physical and ontological implications of the terms which were later on actually used
by Aryabhata and Bhaskara, ote., to designate “zero”, “one”, and some higher
numbers.
Vou. VII, PART 3. 32488 ANANDA K. COOMARASWAMY
to actuality. Again, employment of the term anania 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 Rg Veda, iv, 1, 11, where Agni is described as “ hiding both
his ends” (guhamdno anta); Aitareya Br., iii, 43, “the Agnistoma is
like a chariot-wheel, endless” (ananta); Jaiminiya Up. Br., i, 35,
“ the Year is endless (ananta), its two ends (ant@) are Winter and Spring
. . 80 is the endless chant” (anantarn saman). These citations suggest
that it may be possible to account for the later mathematicians’ selec-
tion of technical terms by reference to an earlier usage of the sume
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 centre draw two concentric circles of any radius, but one
much less than the other; draw any radius from the centre to the
outer circumference. With exception of the centre, which as 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 centre, 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 (siya) surface having
no numerical significance; the central point (indw, bindu) being anKHA AND OTHER WORDS DENOTING “ZERO” 489
inconnumerable unity (inconnumerable, advaita, because there cannot
be conceived a second centre); 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 (deéa), a revolution of the circles about their
centre corresponds to time (Kila). 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 only be thought of 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 just 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 non-existent, 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 specific
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, or
awheel of the, solar chariot), 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 Rg Veda i, 155, 6; i, 164,
2, 11, 13, 14, and 48; Atharva Veda, x, 8, 4-7; Kausitaki Br., xx, 1;
Jaiminiya Up. Br., i, 35; Brhadéranyaka Up., i, 5, 15 ; Svetdsvatara
Up., i, 4; Prasna Up., vi, 5-6; and like texts, the Year as an ever-
lasting 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”, Rg
Veda, v, 85, 5. The parts of the wheel are named as follows: dni,
1 The familiar principle “ as above, so below ” is illustrated here.
® The notion of exemplariam is expressed here, with respect to number or mathe-
matical individuality.