^{page 336 note 1} Since this was written I have become aware that Professor Sayce is about to publish yet earlier examples of ancient Indian writing, but they will not, I believe, include any examples of numerals.

^{page 336 note 2} The results of the examination of the Náná Ghát inscriptions by the Bombay Archæological Survey have not yet reached England. The numerals of the Náná Ghát period are therefore taken from the facsimiles given by Bhagwán Lál Indraji in the Indian Antiquary for 1877, vol. vi. pp. 44–47.

^{page 337 note 1} A priori, numerals are likely, at least in all ancient systems, to be of later date than written expressions of ideas, for they seem to have been originally little else than “shorthand” modes of writing numbers. Of course, however, when an alphabet is borrowed from an external source, as the Indian alphabet in its initial form probably was, the alphabet so borrowed may have already had a system of numeration attached to it, which was imported with it. There is, however, one fact which might be held to indicate that the Indian alphabet originally possessed no numeral system. The old Pali writings of Ceylon, that is, the books of the Buddhist religion written in the sacred language of Buddhism, do not employ any numerical signs. This religion and its language were introduced from India into Ceylon apparently during the fourth, or late in the fifth century B.C. In these books the numbers are either expressed in words, or by a certain arrangement of written syllables. It does not, however, necessarily follow that the Indians were at that time altogether ignorant of numerical notation by separate signs; only that in all probability such a method had not come into use in sacred writings, or in MSS. of any kind. Some such system indeed very probably existed in India even before 400 B.C., though perhaps in a more or less imperfect state, for, as will be seen presently, there is some ground for believing that it received improvements by successive borrowings down to the middle of the second century B.C.—[M. Rodet and Professor Rask, quoted by Pihan, Signes de Numeration, pp. 142–43; also Cantor, Mathematische Beitrage, pp. 58–59.]

^{page 337 note 2} The references attached to this Plate indicate the authority on which each figure is adduced; for the most part it will be seen that these are given from original photographs or coins, and only when that is not possible from trustworthy facsimiles.

^{page 339 note 1} Dr. Bühler's memorandum was attached in the first instance to a private letter, and was originally intended only for my personal use, and not for publication.

^{page 344 note 1} ḍa is the northern equivalent for ḷa in the language also.

^{page 345 note 1} Since seeing the tables given in Pl. II. Dr. Bühler informs me he is convinced that the “Aksharapalli” is of an origin extraneous to India, though he still finds it difficult to believe that its signs are borrowed from four or five different sources.—E. C. B.

^{page 345 note 2} “Zeitschrift der Deutschen Morgenländischen gesellschaft” for 1877, vol. xxxi. p. 598.

^{page 346 note 1} It is by no means intended to intimate any dissent from Dr. Bühler's views on this part of the subject; on the contrary, they seem at least a priori reasonable. It may perhaps be a question how far the examples given by Dr. Bühler go to prove the derivation of the Maurya forms of writing from the Andhra,— they seem at least equally consistent with the supposition that both may have been derived from some earlier common original, to which perhaps, in its square and more archaic forms, the Maurya may preserve a closer resemblance than the Andhra does. The more rounded forms of the latter may perhaps be due to the nature of the substance written upon—which to some extent is even still locally employed for writing, viz. the palm-leaf. On the other hand, Dr. Bühler's arguments may perhaps be held to make it at least probable that the early modifications of the letters which he traces to Brahman influence actually grew up for the most part in the Andhra alphabet, and that they were adopted thence by the Mauryas. Indeed, this is a priori probable, inasmuch as the Andhra court seems to have flourished for some time previous to the consolidation of the Maurya power, and would thus have furnished a centre of civilization and learning, where Brahmans would be more likely to find extensive patronage, than elsewhere in Central India at that day. Indeed, even afterwards, the atmosphere of the Maurya Court was possibly not (especially during its later years) altogether favourable to the development of Brahmanical ideas. Otherwise the divergence between the Andhra and the Maurya alphabetical types hardly seems greater than local circumstances would usually produce in India, within a moderate period of time.

^{page 347 note 1} It may be observed that the influence of the aksharas and the Brahmanical manipulation of the numerals would hardly begin to take effect till the numerals were employed for manuscript purposes,—or perhaps for use in sacred MSS.; and, as will be seen from what has been said in a previous note (p. 337), of Professor Rask's remarks on the ancient Cingalese numerals, this use did not probably begin till the fourth or fifth century B.C.

^{page 348 note 1} See p. 370.

^{page 349 note 1} See General Cunningham's paper, J.A.S.B. vol. xxiii. for 1854, p. 703, note. It will be observed further on, that I have not entirely adopted the details of General Cunningham's identifications; indeed, with the fuller knowledge of the Bactrian letters which we now possess, and which is so largely due to General Cunningham's own labours, that writer would not probably himself now maintain them all, or indeed the theory founded on the facts as then understood by him.

^{page 350 note 1} Cf. Gesenius, , Monumnt. Vet. pp. 80–88Google Scholar; Pihan, , Signes de Numeration, p. 165Google Scholar; and De Luynes, , Numismatique des Satrapies et de la Phénicie, pp. 112–114.Google Scholar

^{page 350 note 2} See Pihan, , Signes de Numeration, p. 164.Google Scholar

^{page 350 note 3} Pihan, , Signes de Numeration, p. 26.Google Scholar

^{page 351 note 1} The old form, however, occurs on one of Kumára Gupta's inscriptions at Garhwa (of 129 Gupta), Cunningham, Arch. Survey, vol. x. p. 7, pl. iv.

^{page 351 note 2} Though in this case also the sign is differentiated to express two hundred.

^{page 352 note 1} Indeed of hardly of any except the five.

^{page 353 note 1} It need hardly be said that the use of the “abacus” is still common in every village bázár in India, and has been universal apparently from time immemorial.

^{page 353 note 2} The term “akshara” (from the negative “a” and “kshar”), signifying “indestructible,” “incorruptible,” seems to be a term invented after the introduction of writing, or at least of numeral signs, as indicating the superiority in respect of durability and accuracy of the phonetic signs.

^{page 354 note 1} M. Woepcke (Sur l'Introduction de l'Arithmetique Indienne eu Occident, p. 68) quotes from Sibth ul Máridíní, who died in 1527–28 A.D., another mnemonic arrangement of letters according to the Abjad system, in groups according to the powers of each unit, thus 1, 10, 100, 1000; and 2, 20, 200, 2000, etc.

^{page 355 note 1} See Thompson's Hindi Dictionary; also Fallon's Hindustani Dictionary, in voce “Chhápa.” In the latter, a quotation of a Hindi verse will be found, in which the word occurs in this sense.

^{page 357 note 1} The presumed date of the “Rhind” Papyrus, but the first use of these symbols was probably older still.

^{page 358 note 1} This fact may perhaps explain the following quotation from Sibth ul Máridíní (Woepcke, Sur l'Introduction de l'Arithmetique Indienne en Occídent, p. 67): “Sachez que les ordres elementaires des nombres sont au nombre de trois: unités, dixaines, et centaines, dont chacun comprend neuf nœuds.”

^{page 358 note 2} The term “latest” is used with the knowledge that an apparently new symbol for a hundred (which Gen. Cunningham supposed to be a Bactrian letter) occurs among the Indo-Scythian and Gupta numerals, but this seems rather a cursive modification of the ‘s’ shaped, or second oldest, form of the symbol (of which it has been suggested that it came from the Bactro-Phœnician form), the ‘crook’ on the left side only being omitted.

^{page 359 note 1} This has already been suggested by Pandit Bhagwán Lál (Indian Antiquary, vol. vi. for 1877, p. 46).

^{page 359 note 2} I am indebted for these signs to the kindness of Mr. Pinches, of the British Museum.

^{page 359 note 3} The Himyaritic ‘50’ is also of the same form as the Akkadian ‘60,’ and is also augmented by signs for ‘10’ (Ind. Ant. vol. iv. p. 27Google Scholar).

^{page 360 note 1} The Indian sign for eighty might perhaps be taken from the Akkadian sign for sixty placed between two Akkadian signs for “ten,” thus . Cf. the Assyrian hieratic numerals as given by Menant, New Assyrian Grammar (1882).

^{page 360 note 2} It is a corollary of this conclusion that at the time when these indents were made on their alphabet, the Bactrians possessed no regular alphabetical system of notation. It has been suggested that certain letters occurring on the coins of later Bactrian kings, e.g. Hippostratus, Azas and Azilisas, express numbers and dates. If so, the idea, or even the system, must have been obtained from the Greeks, and this is rather rendered probable by the fact that these signs often seem differentiated by the vowel i, which was used by the Greeks to express ‘ten.’ If these figures represent numbers at all, therefore, they are probably low numbers, and if dates, regnal dates only.

^{page 360 note 3} Of course they were not altogether debarred from expressing the intermediate numbers, for they could have used the Phœnician and Bactrian mode by which the highest of these were expressed by groups compounded of the signs for 20 and for 10, and in which 20+10 stood for 30; 20+20 for 40, etc. This Phœnician method of counting by twenty and tens together, must apparently have been the origin of the Modern French “soixante-dix,” “quatre-vingt,” etc., coming down from the usages of the early Phœnician colonists of Marseilles and other seaports. It is curious that this awkward and antiquated method should have superseded the far more convenient and expressive “septante,” “octante,” and “nonnante.”

^{page 361 note 1} Specially that in J. A. vol. i. series 6.

^{page 361 note 2} Maspero, , “Histoire Ancienne des Peuples de l'Orient,” pp. 146–148, 168–170Google Scholar; Lenormant, F., “Manuel d'Histoire Ancienne de l'Orient,” vol. ii. pp. 240–244.Google Scholar

^{page 361 note 3} Cf. also Heeren (Asiatic Nations), vol. i. Chapters on Babylonian and Phœnician commerce.

^{page 361 note 4} See 1 Kings xxii. 45.

^{page 362 note 1} Journal Asiatique, series 6, vol. i. p. 354.Google Scholar

^{page 362 note 2} Numismatique des Satrapies et de la langue de la Phénicie, pp. 112, 114.

^{page 364 note 1} Reinaud, in Journal Asiatique for 1863, vol. i. series vi. p. 97; see also rest of memoir, pp. 93 to 234. If we compare this Indian traffic to that with Europe before the employment of steam navigation, and deduct from the latter the demand for military and civil organization, the result will show that the purely commercial intercourse of modern times was not very greatly in advance of that of the last century B.C.

^{page 366 note 1} Dr. Leitner has recently collected a number of forms of numerals used in Kashmir by shawl-weavers and others, which exactly answer the description here given.—“Linguistic Fragments,” Sec. I. Lahore, 1882.

^{page 367 note 1} For a general description of the phonetic equivalents used in Sanskrit and cognate languages, see Nouveau Journal Asiatique, vol. xvi. pp. 1–42Google Scholar (Jacquet); series 6, vol. i. pp. 284–90 (Woepcke); and series 7, vol. xvii. p. 47–130 (Rodet).

^{page 368 note 1} These facts are given on the authority mainly of my friend Mr. E. S. Poole; of the British Museum.

^{page 368 note 2} SirWilkinson, Gardner's Ancient Egyptians, vol. ii. p. 367, ed. 1878.Google Scholar

^{page 369 note 1} For example, in the Sanskrit (Native) Colleges at Kishnagur in Bengal—and the following amusing note, borrowed from Dr. Burnell's South Indian Palæography (p. 65, 1st ed.), illustrates this state of things from the early Arab point of view. Albiruni (Reinaud, Memoire, p. 234) gives a remarkable instance of the Indian tendency this way: “Les livres des Indiens sont rédigés en vers, les indigènes croient par là, les rendre plus aisés à retenir dans la mémoire, ils ne recourent pas aux livres qu'à la dernière extrémité. On les voit même s'attacher a apprendre des vers dont ils ignorent tout a fait le sens. J'ai reconnu à mes depens l'inconvénient de cet usage. J'avais fait pour les indigènes des extraits du traité d'Euclide et de l'Almageste; j'avais composé nu traité de l'Astrolabe afin de les initier aux méthodes Arabes, mais aussitôt ils mirent ces morceaux en ‘slokas,’ de manière qu'il était devenu peu facile de s'y reconnaître.” Dr. Burnell adds, “I have myself seen the Penal code put into Tamil verse.”

^{page 369 note 2} It does not follow that these were always merely syllables—they were in many cases doubtless, as they even now are in Sanskrit, words bearing other significations—see the papers on the subject already cited in a previous note at p. 33.

^{page 369 note 3} Perhaps in the Greek form of the name of this instrument some trace exists of the use of the ‘aksharas.’ Admitting that it was probably in its origin, the Semitic term for the material on which the signs were traced, ‘fine dust’ or ‘abak,’ yet it was an occasional practice of the Greeks to adapt foreign terms and even names, so as to bear a signification in their own tongue. Ἄβαξ, the Greek form of the term, is identical with an adjective ἄβαξ, given by Eustathius as the base of the word βάκησαν, which is found in the Odyssey—in the sense of being ‘unconscious’ or ‘helplessly ignorant,’ ‘like infants’; the word occurs in a speech of Helen to Menelaus, who, speaking of the visit of Ulysses to Troy, says:

Τῷ ἴκελος κατέδυ Τρώων пόλιν, οἱ δ᾽ αβακήσαν

пάντες.—Odyssey, δ. 249.

“Like unto this (sc. a beggar) he entered the city of the Trojans and they (other people) were unaware” (or ‘like babies’).

The derivation given is from the verb βαζω ‘speak,’ with the negative ‘alpha,’ that is, ‘speechless (like a baby),’ quâ ‘in-fans’; other words with the same derivation, such as άβακης (adjective), βακέως (adverb), βακί-ζομαι, βακήμων, all with the same general sense, are also quoted in Liddell and Scott's dictionary (see ἄβαξ). Ἄβαξ would thus in Greek mean ‘speechless,’ ‘wordless,’ or ‘non-phonetic’—surely a very appropriate term for a ‘silent’ mode of calculation which superseded the phonetic ‘aksharas.’ The common Sanskrit term for the instrument seems to be páṭhi, which signifies ‘a board’ or ‘calculating board’; but the exact derivation is not given with certainty in any dictionary which it has been possible to consult. The Hindi word seems also to signify primarily ‘a board,’ though it may have reference to lines or divisions. But this derivation is not quite clear. In Russia (where the introduction of the instrument is attributed to the Mongols) its name signifies a ‘counting’ or ‘computing’ board. These later etymologies, however, do not throw much light on the original character of the instrument.

^{page 370 note 1} It was the recognition of the old mode of differentiation as employed in these which led to an examination of the principle of that method, which was the commencement of the inquiry on which this paper is founded.

^{page 371 note 1} The evidence (from coins) on which this statement is based will be the subject of a separate paper in the Numismatic Chronicle, in connection with the era to which the dates belong, which these figures are used to denote.

^{page 371 note 2} There can be little doubt as will be explained in the Numismatic Chronicle that the Arabs obtained their numerals from Kabal.

^{page 373 note 1} The division between the earlier and later Kshatrapahs is taken, somewhat arbitrarily perhaps, at the close of the reign of Rudra Séna, son of Vira Dama, whose dates extend to 198.

^{page 375 note 1} The Valabhi dates extend over a period of about 240 years. These dates give three nearly equal periods of about 80 years, say from 206 to 290 (Valabhi), 290 to 365, and 365 to 447; the first period terminating with the reign of Siladitya I., and the third commencing with that of Siladitya III.