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)Google Scholar, and although a decimal scale has actually been found at Mohenjodaro (Mackay, , ‘Further Excavations at Mohenjodaro,’ Journ. Roy. Soc. Arts, No. 4233, 1934, p. 222Google Scholar), it is by no means the intention of the present article to present an argument for a Ṛg Vedic knowledge of either the decimal system or the concept “zero” as such. Our purpose is merely to exhibit the metaphysical and ontological implications of the terms which were later on actually used by Āryabhaṭa and Bhaskara, etc., to designate “zero”, “one”, and some higher numbers.

¹ The familiar principle “as above, so below” is illustrated here.

² The notion of exemplarism is expressed here, with respect to number or mathematical individuality.

¹ Indu occurs in the Ṛg Veda as “drop” in connection with Soma; in Atharva Veda, vii, 109, 6Google Scholar, as “point on a die”; and grammatically as the designation of Anusvāra. Pañcaviṁśa Brāhmaṅa, vi, 9, 19-20 is of interest; indava iva hi pitaraḥ mana iva, i.e. “the Patriarchs are as it were drops (indu in pi.), as it were the intellectual principle”. In Ṛg Veda, vi, 44, 22, Indu is evidently Soma; in vii, 54, 2, Vāstospati.

¹ 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, 22, 16 pṛthivyā sapta dhāmabhiḥ. Our terms kha, aśna, 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 āṅi was not later employed as a verbal symbol of “one”.

¹ Madhya is “middle” in all senses, and also algebraically “mean”. For the metaphysical values, cf. in the Ṛg Veda madhye samudre and utsasya madhye = sindhūnam upodnye as the place of Agni or Varuṅa and in Chāndogya Up., iii, 11, 1 ekata madhye sthāe “single in the midmost station“.

² For the construction of the wheel, cf. Ṛg Veda, viii, 77, 3, akhidat khe arāii iva khedayā and discussion in my Angel and Titan, an essay in Vedic ontology, to appear in the JAOS. 1934.

³ Śaṅkarâcārya, Svâtmanirūpaṅa, 95.

⁴ Śūnya does not appear in the Ṛg Veda, though śūnam occurs in the sense of “privation”.

¹ Observe that the dual series of plus and minus numbers represents “pairs of opposites”, dvandvau.

² “Transzendenter Raum der Ewigkeit ist der Ākāśa vor allem auch da, wo er als Ausgangspunkt, als Schöpfungsgrund und als Ziel, als A und O der Welt angeschaut wird.” Scharbrau, , Die Idee der Schôpfung in der vedischen Literatur, 1932, p. 56; “size which has no size, though the principle of size,” Eekhart, i, 114.Google Scholar

³ Nabha, from root nabh “to expand”, etc., as also in nābhi “navel” and “nave”. A secondary sense of nabh is “to destroy”.

⁴ This text occurs in almost the same form in Atharvz Veda, x, 8, 29.Google Scholar

¹ Cf. aśmany anante and adrim acyulam cited above with the meaning “rock of Agea”.

¹ “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., vol. 51, p. 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.

² Cf. Ārvabhaṇa, Āryabkaṇīya, iv, 9, “As a man in a boat going forward sees a stationary object moving backward, just so at Laṅkā a man sees the stationary sterisms moving backward.”

³ See my Transformation of Nature in Art, 1934, p. 8 and note 8.Google Scholar

⁴ Cf. Siecke, , Die Liebesgeschichte des Himmeh, Strassburg, 1892Google Scholar; and Jeremias, “Die Menschheitsbildung ist ein einheitliches Ganzes, und in den verschiedenen Kulturen findet man die Dialekte der einen Geistessprache,” Altorientalische Geisteskultur, ed. 2, p..Google Scholar

¹ Even the “Machiavellian” Artha-śāstra (i, 3) proceeds from the principle svadharmaḥ svargāya ānantyāya ca, tasya atikrame lolcaḥ sankarād ucchidyeta “vocation leads to heaven and eternity; in case of a digression from this norm, the world is brought to ruin by confusion”.

² Guénon, René, Orient et Occident, Paris, 1930 (extracts from ch. ii).Google Scholar