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Art. I.—On the Genealogy of Modern Numerals. Part II. Simplification of the Ancient Indian Numeration

Published online by Cambridge University Press:  15 March 2011

E. Clive Bayley
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Extract

The second part of this paper will be occupied by an attempt to show how the ancient Indian system of numeral signs, described in Part L, was simplified. In other words, it will be attempted to show how this old system became the parent of that now used in India, which employs only nine units and a zero,—indeed of that system as used not in India alone, but now almost universally both in eastern and western countries

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Original Communications
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Journal of the Royal Asiatic Society , Volume 15 , Issue 1 , January 1883 , pp. 1 - 72
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Copyright © The Royal Asiatic Society 1883

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page 3 note 1 It is not necessary to explain here the methods by which the still higher numbers used for mathematical calculations were expressed. It is sufficient for the present inquiry to take note of the early, and it may be said universal employment of the decimal arrangement. Nor is it necessary to dwell on the much wider question of the causes which led to its adoption. It is possible that there was a stage in the very early history of civilization, when mankind were more restricted in their power of numeration, as is the case to this day, with some of the savage races on the Andamanese Islands, who cannot count beyond three,—indeed indications may, perhaps, still be traced that such a condition once existed among the most highly civilized nations, and that even when this was exceeded they continued to count by groups of threes, —still it is certain that the extension of this power must have been one of the earliest steps in the progress of civilization. The system of numbering by decimal stages or “rests” has been very generally supposed to have been suggested, at any rate, by the use of the human hand as an instrument to assist the process of reckoning numbers. Indeed, it is quite possible that the structure of the human hand suggested not only the decimal, but the earlier supposed methods of counting by ‘triads,’ or ‘threes,’ the quinary, the quaternary, and the duodecimal modes of numeration. The first being suggested by the ten fingers and thumbs of the joined hands, the second by the ‘three’ joints, the third by the four fingers, the fourth by the fingers and thumb of one hand (the Akkadian name for ‘five’ is synonymous with that for ‘hand’; Pinches, Proc. Soc. Bibl. Arch., June, 1882), and the duodecimal by the multiplication of the ‘three’ joints by the four fingers. It is singular, too, that the Babylonian sexagesimal unit of ‘sixty,’ or šuš, will result from the further multiplication of twelve by five (perhaps better of 2×2×3×5; see Pinches, Proc. Soc. Bibl. Arch., June, 1882, p. 116), and the still further multiplication of this result by ‘ten,’ gives the Babylonian ‘ner,’ or ‘six hundred.’ At any rate, there are many curious facts which seem to indicate at least, this origin for the decimal system, and which also show the universal use of the human hand as a ‘reckoning board.’ It will suffice to mention a few of these only here. In Egypt, for example, in the hieroglyphic signs, the human hand and its portions were employed to signify measures of length. The cubit was divided into ‘diti,’ of which twenty-eight went to the royal, and twenty-one to the common cubit. One, two, or three ‘diti,’ were indicated by one, two, or three fingers respectively; four ‘diti’ by the human hand displaying four open fingers; five ‘diti’ by a similar figure with the thumb also displayed; six ‘diti’ by a closed fist; and eight by a reduplication of the sign for four ‘diti.’ Again, in general numeration, the finger with the top joint bent designated ten thousand, and there are perhaps in the hieroglyphics other, though less palpable reminiscences, of the human hand. Another curious piece of evidence is suggested by a notice published by MrFleet, J. in the Indian Antiquary for 1875, vol. iv. p. 85Google Scholar. Mr. Fleet mentions Professor Hunfalvy's remarks at the Oriental Congress of the preceding year to the effect, that in a very considerable number of languages of the Turanian stock, the ‘ring’ finger is always termed ‘the finger without a name.’ Mr. Fleet illustrates this by quoting a curions anecdote recorded by a Sanskrit author with reference to the poet Kalidása and his eight contemporaries of literary fame at the Court of Kanouj, who were termed its “Nine Gems,” and by it proves that a similar custom had existed for so long a period in India, that even at that date (the seventh century A.D.) its origin had been forgotten; for in reckoning these nine gems on the fingers, the writer says “Kalidása” was always reckoned first (on the little finger of the left hand), but no one was counted on the next (or ring finger), because none of his contemporaries could be reckoned as even second to him; and, adds this author, hence was assigned at last some reason for calling that finger ‘anámika,’ or ‘without a name.’ Mr. Fleet, it is true, goes on to suggest that this may not be the true signification, and that the term might mean in Sanskrit ‘unbent,’ in allusion to the difficulty of bending that finger, but in the face of the Turanian parallel, this explanation can hardly stand. Thisancient custom, however, may easily be accounted for by referring back to the origin of decimal notation on the hand. If the ten fingers and thumbs suggested the origin of the decimal notation, it is nevertheless evident that in using the hand as an instrument for reckoning, one finger would be superfluous, nine symbols only being required, as the tenth became the first of the new and next highest stage of the decimal series. One finger, therefore, would necessarily be ‘skipped,’ or laid aside. It is not perhaps easy to suggest any reason why the ring finger should have been specially chosen for omission, but it would be only natural that the omission should be by common custom of one selected finger, and if, as Mr. Fleet suggests, the process of counting commenced in ancient as it still does in Modern India, with the little finger of the left hand, then it would be natural that the calculator should wish to put his calculation right as soon as possible, and should therefore omit the finger next after the initial one, which would be of course the ring finger. These facts may suffice to illustrate the antiquity of counting on the fingers. Its wide general diffusion need hardly be pointed out. The Chinese, to this day, have a mode of counting up to 99,999 on the fingers of one hand alone, which will be seen illustrated on Fig. I. Plate I. The nine units are reckoned on the joints, commencing along the outside of the little finger; then counting four, five, and six on the joints at the back of the finger; seven, eight, and nine on the joints along the inside of the finger; the next finger is similarly used to represent the tens; the next the hundreds; the next the thousands; and the thumb for the tens of thousands. In England the venerable Bede describes another system, which he states to be great antiquity, while the practice of concealed bargaining by pressure of the fingers has been used from time immemorial and is still used among the nations of the East. In India (where the hands are concealed under a cloth), Tavernier (Voyages, Part II. pp. 326–7, ed. 1712) describes this mode of settling prices. Halhed says that (in Bengal) the practice is limited to counting up to fifteen; this may be an error, but even this would enable bargains to be made in pies, annas, rupees, and mohurs, and a limit of fifteen mohurs or 240 rupees would suffice for the requirements of most Bengal markets. In Barbary and Arabia the hands are manipulated under cover ot the long sleeves of the burnous. Enough has, however, been said to indicate the probability of the derivation of the decimal system from the structure of the human hand, and to show that, at any rate, it is apparently the most primitive and simple and most widely spread of all extant methods of numeration. On the other hand, however, it is not to be forgotten that other suggestions have been made as to the origin of this and especially of the quaternary, quinary, and duodecimal methods of notation, which are in themselves not improbable, particularly those derived from astronomy and the natural divisions of time. Indeed, as regards the sign for ‘five’ employed in Egyptian hieroglyphics, such an origin is expressly assigned bythe Egyptian priest Horapollo, and may be taken as correct. “l 'Aστέρα γράϕοντες δηλοσι”… τν πντε ἄριθμον, πειδ πλθονς ντος εν οὐρανῷ πντε μνοι ξ αὐτν κινομενοι τν το κσμον οἰκονομαν κτελοσιν (Horapollo Hierog. liber i. c. 13, apud Cantor, M.B. p. 18 and note p. 17); that is to say, the idea of the five pointed stars was taken from the ‘five’ planets, then alone known to Egyptian observers. No doubt, too, the Egyptians used both quinary and quaternary methods of notation, for eight stars were used to represent ‘forty,’ and a single star with two to make seven. So the Egyptians early used quaternary multiples of the ‘hen’ or unit of capacity in their scale of measures of capacity (see Rossi, Grammatica Copto-Geroglyfica, p. 89 note and p. 97).

While thus referring back to the oldest pyramids for evidence as to the origin of decimal notation, it may not be out of place to remark that if the theory adopted by this paper be correct, all the signs of the Indian numerals may also be referred back directly or indirectly to the same source. This is even the case with the unit signs, which it has been proposed to derive from the Bactrian alphabet, for since Prinsep assigned these characters to some form of the Phoenician alphabet, this point has never been questioned seriously by subsequent writers, and has been, indeed, supported by several of high authority (see Thomas, Num. Chron. N.S. vol. iii. p. 229, and Prinsep's Essays, vol. ii. pp. 144–162; also Cunningham, Successors of Alexander in the East, pp. 30–44), though they were modified to meet the requirements of an Aryan language, and perhaps also (as Dr. Bühler has suggested) of a Brahmanical liturgy. Again, the Vicomte de Rougé and M. Lenormant (Introduction à une memoire sur la propagation de l'alphabet Phæcenicien, Paris, 1866, pp. 108–9) have made it almost certain that the Phoenician characters came, through the hieratic, from the Egyptian hieroglyphics.

page 7 note 1 As will be presently explained, in some rare instances the Indians arranged numbers perpendicularly one above the other—as, in fact, they did letters also; in either case, however, the first letters and the highest numbers occupied the uppermost positions; the fact does not, however, affect the general argument as respects the ordinary arrangement of the Indian numeral signs.

page 8 note 1 εἰς τν δ ριθμν μθησιν κα γεωμετρας ενγειν αὐτν πειρτο επ' ἄβακος τς κστου ποδεξεις ποιομενος.—Jamblichus De Vitˇ Pyth. cap. v. § 22.

page 9 note 1 εὔρημα δ' αὐτς ϕασν ῖναι Bαβυλωνίων κα Πυθαγόρου πρώτυ εἰς Έλλῄνας λθειν. (jamblichus Comment, ad Nicomach: Arith., the second word ought apparently to be δ'αυτς.) See also Isidore Hispaliensis (Bishop of Seville) Origines, liber iii. c. 2. Numeri disciplinam primum apud Graecos Pythagoracis autumnant conscripsisse et deinde a Nicomacho diffusius essedispositam quam apud Latinos Appuleius deinde Boethius transtulisse (for the quotations in this note see Cantor, pp. 369 and 391). Porphyry, in his Life of Pythagoras, credits the Phoenicians with the invention, or at least perfection of arithmetic, while assigning that of geometry to the Egyptians, and of astronomy to the Chaldeans, Гεομετρας μἓν γρ κ παλαιν χρνων πεμεληθναι Aιγυπτιυς τ δ περ ριθμος τε κα λογισμοὺς Φονικας Xαλδαους δ τ περ τν οὐρανν θεορματα (De Vit. Pythag. 56, ed. Krissler, p. 12). But the point is not of importance for the present argument; if the Babylonians or Chaldeans were far advanced in astronomy, they could hardly have made much progress without some considerable use of arithmetic, and Pythagoras, who is reported to have been carried as a prisoner into Babylon by Cambyses, and who spent a long captivity there, may well have learnt his arithmetic and the use of the abacus in that country.

page 9 note 2 The idea may have arisen from some such practice as still obtains in many a village school in India, where the smallest boys are made to lie upon the ground and scrawl letters and figures in the dust or sand of the floor (sometimes on the ground outside) with a bit of stick till they acquire some familiarity with the shape of these; they are then promoted to the use of a writing-board. Of this more will be said when treating in Part III. of the Gobar numerals.

page 10 note 1 Aβκες (or abaci) was also the term employed in the language of ‘decorative art,’ to signify the rectangular parallelograms or ‘pannels’ used in painting the walls of rooms.

page 11 note 1 Όντως γρ εἴσιν οὗτοι παραπλσισι ταῖς πι τν βακιον Ψϕοις 'Eκεῖνα τε γρ κατ τν το Ψηϕσοντος βολησιν ἄρτι ϰαλϰον κα παραυτκα τλαντα ἴσϰουσιν; οΊ τε περ τς αὐλς κατ τ τυ βασιλως νεα μακριοι, κα παρ πδας λεεινο γγνονται.—Polybius, v. 26, 13 (Cantor, p. 390).

page 14 note 1 The Chinese abacus, the lines of which are used horizontally, has also a similar perpendicular dividing line. The Chinese methods of using this instrument, however, are peculiar, and it is not possible to discuss them here at length.

page 16 note 1 It was possibly by drawing lines between each series of numbers that the ‘tableau à colonnes’ was eventually transformed into the ‘exchequer table’ or ‘chequers.’ This last is described by an English mediaeval writer, Richard Fitznigel, as consisting of a space covered by a black cloth with white lines on it, drawn both transversely and perpendicularly about a palm apart, on which calculations were made by means of counters. The calculations in the extreme column to the right advancing by ‘twelves’ (for ‘pennies,’ as in the case of the columns for the subdivisions of the ‘as’ on the Roman abacus), the others by ‘tens.’ On this cloth the calculations of payments into the Royal Treasury were made, and the term ‘chequers’ is supposed to be derived from the mediæval term for a chess-board, or ‘scaccum,’ to which the tableau à colonnes in this shape bore a strong resemblance. See Edin. Review for 1811, vol. xviii. art. vii. p. 207.

page 17 note 1 See M. Woepcke in Journal Asiatique, torn. i. series 6, pp. 247–248.

page 18 note 1 See Woepcke on the authority of the Tárikh ul Huḳamá, Journal Asiatique as above, and also pp. 472–480.

page 18 note 2 Part II. of vol. ii. 3rd series, pp. 138–146.

page 19 note 1 Tárikh ul Huḳamá. See Woepcke's Traité sur l'Introduction de l'Arithmétique Indienne en Occident, p. 19.

page 20 note 1 That the Indians not only had a knowledge of algebra at a remote period, but made great progress in the employment of it, is doubtless true; but the Greeks also knew it at a very early date. (Diophantus can hardly have been its first originator among the Greeks, and have advanced per saltum to a stage beyond even the Indian algebra.) And though it is quite possible that, through the intercourse between the two nations, one may have borrowed from the other algebra and similar inventions, yet there is nothing to prove that it was indigenous with either, or may not even have been borrowed by both from some common source. (Cf. Reinaud, , Memoire sur l'Inde, p. 303Google Scholar.)

page 21 note 1 Representations of these figures will be found on Plate I. Fig. 6. They will be seen to be for the most part derived from an Arabic model, though one set given by Cantor, from a MS. of Planudes, clearly comes direct from an Indian source. The chain of descent of these figures, and of the Böethian apices will, however, be more fully treated in Part III.

page 22 note 1 This principle was probably known to the Indians long before. See remark by Dr. Bühler in Part I., but this particular application of it is new.

page 22 note 2 It is evident that possessing signs both for the units and for thirty, for ten and for twenty, i.e. the intermediate places between twenty-five and thirty would be expressed by the use of these.

page 23 note 1 The force of the argument, as will be seen later on, rests mainly on the use of these terms. The actual employment of this mode of notation might have been suggested by a knowledge of the Greek ‘octads,’ as hinted by Reinaud, , Memoire sur l'Inde, p. 303Google Scholar.

page 23 note 2 These words are of course ‘aksharas’ or ‘phonetic numerals.’

page 23 note 3 =3000 =700 =50.

page 24 note 1 It is, no doubt, possible that similar evidence may be discovered as to the knowledge of still earlier writers; but it is enough for the purpose of this inquiry that the case goes back even as far as the first half of the sixth century A.D.

page 25 note 1 “The method of the Pythagorean abacus as we find it descrihed in Boethius' Geometry, is almost identical with the positive value of the Indian system, but that method, long unfruitful with the Greeks and Romans, first obtained general extension in the middle ages, especially after the zero sign had superseded the vacant space” (Murray's, Kosmos ed. vol. ii. p. 164)Google Scholar. “Even the existence of the cipher or character for ‘0’ is not a necessity for the simple positive value, as the scholium of Neophytus shows” (Murray's, Kosmos ed. vol. ii. p. lxxxi)Google Scholar. “What a revolution would have been effected in the more rapid development of mathematical knowledge.… if the Brahman Sphines, called by the Greeks Calanos, or. … the Brahman Bargosa had been able to communicate the knowledge of the Indian system of numbers to the Greeks” (Murray's, Kosmos ed. vol. ii. p. 164)Google Scholar.

page 25 note 2 “Au moment que j'allais conclure et attribuer à A'ryabháta l'usage de notre système decimal ecrit, un scrupule m'est-venu: les calculs qu'il enseigne a faire peuvent s'effectuer conformément à son règle sur un abaque; le nom que les Indiens ses successeurs comme lui donnait au zero, a du étre inventé à une epoque ou Ton faisait usage d'un abaque sur lequel le zero n'est marqué que par une place vide. A'ryabháta effectuait il ses calculs sur l'abaque, et, ⃜ se contentait et de transcrire les resultes a l'aide d'un systéme de chiffres decimaux mixtes …? Voila un point capital que je suis contraint de laisser sans solution, attendant que des documents nouveaux viennent nous fournir des éclaircissementa qui nous manquent.”—Journal Asiatique, series vii. vol. xvi. p. 443.

page 26 note 1 This remark refers to the later forms of the Sanskrit zero the ‘0’ and the —As to this, more will be said immediately.

page 27 note 1 These are dated in 441 and 447, which I have given in the Numismatic Chronicle reasons for believing to be in an era dating from 189 or 190 A.D.

page 27 note 2 This grant, which is yet unpublished, is in the possession of Dr. Bühler, who kindly furnished me with a facsimile. It is one by Jaika Rashtrakúta of Bharuj and is dated in 794 ‘Vikramaya.’ It was found at Okamandel.

page 28 note 1 Found at Rádhanptúr in 1873–4. See Indian Antiquary, vol. vi. for 1877, p. 59Google Scholar.

page 28 note 2 Professor Jacóbi has kindly favoured me with other similar examples from Jain books.

page 29 note 1 Dr. Kielhorn gives facts which seem to bear out this statement, in the succeeding pages of his report, to which it is only necessary to refer in this place.

page 29 note 2 It may be remarked that Dr. Bühler has more than once drawn attention to a similar fact—disclosed by recently discovered inscriptions—viz. that the early Indians certainly employed two modes of writing contemporaneously—one stiff and formal for official purposes, the other cursive for general use.

page 30 note 1 L'emploi est formellement prescrit dans un traité d'Arithmétique, probablement assez ancien, qui fait partie dumanuscrit 169 fonds persans de la Bibliothéque nationale. L'auteur (Mahmûd ben Mohammed ‘Qiwám ul Qázy, de Valisthán, surnomme Mahmúd de Herat), ne manque pas de dire a chaque operation: “Tariq é amal ienan ast, ke Jadúli rasm huttand, ke adad é sutûr e tûlî é ú matasavi e adad é mafaradüt é án adad shavad ké,” “la manière de faire cette operation est celle-ci: on trace un tableau dont le nombre des lignes (colonnes, bandes) en longitude (cette à dire comprises entre deux meridians d'une carte) soit egal au nombre des places du nombre que.” Cet auteur n'efface pas les chiffres a modifier il ecrit le nouveau chifire “dar zîr é dignr ba ad az Khat i ké án ra Khat-i-é máhy khwánand” au-dessous de l'autre après une ligne que Ton appelle ‘lineaoccultans.’ Cette dernière expression, empruntée a la grammaire syriaque, doit elle faire croire à une origine syriaque de ‘jadûl’ de notre auteur, (may not ‘jadúli’ rather mean a form for a ‘magical table,’ such as used for incantations, and amulets, from the old Persian ‘jádú’ ‘magic,’ or ‘witchcraft’).

page 31 note 1 I omit here a note by M. Rodet, which I hope to reproduce when the subject of the “Gobar” ciphers comes under consideration.

page 32 note 1 Though, as Prince Buoncampagni shows, he had been anticipated by a writer in the thirteenth century.

page 32 note 2 See Woepcke, M., Journal Asiatique, series vi. vol. i. p. 518Google Scholar. M. Woepcke considers that it came probably through the school of Toledo, where Adelard of Bath studied in 1130, Robert of Beading in 1140, William Shelly in 1145, Daniel Morley in 1180 (all Englishmen), and Gerard of Cremona about the same time. M. Woepcke quotes Wallis, De Algebra, tract, hist, et praet. Operum Math. vol. ii. p. 1216.

page 35 note 1 This may be explained as below, the thick letters in the figure above expressing the ultimate product. The results are written without carrying the tens, etc., but these are set down (mentally) as follows: thus—

A. 2326× 4 = 8284, which write, carrying 1020.

B. 2326×10 = 23260 (of which write only 200000 before 8284) and carry nothing.

C. is A+B = 2144(4, which write, omitting the last four, which is already entered, and carry 10100.

D. 2326×200 = 464200, carrying 1000.

Add C. 2144(4 (N.B.—This is a mental operation not shown at all.)

Result=E: 234)8564(4, carrying 4200.

Write down above C., with the 8, however, in the line with C, and the 4 in a line with A., in which also the first ‘4’ will be included. Now commence to add the sums carried; 1st, the 1020, from A., which makes the 5000 and the ‘40’ in line E. 6000 and 60 respectively. Write the 6 of the 60 in line E., and the 6 of the 6000 in line F., and add the 10,100 from C. This will make the 80,000 into 90,000, and the 600 of E. 7000. Write the 9 in line E. and 7 in line F; then add the 1000 from D., which makes the 6000 in line F. into 7000; write the 7 in line G. which completes the operation.

page 35 note 2 The figure ‘5’ is substituted for the ‘2,’ given at p. 23 of M. Woepcke's Traité, whence the example is taken, and which is clearly a typographical error.

page 37 note 1 See Journal Asiatique, series vi. vol. i. p. 497, where a quotation is given.

page 40 note 1 I do not here speak of the Cingalese ancient numerals, still used for some purposes, and which present even a still closer resemblance to the ancient Indian modes of numeration, and are therefore shown in PI. II. Table I.

page 42 note 1 See Wilkinson's, Ancient Egyptians, vol. ii. p. 493, edition 1878Google Scholar; also Pihan, Signes de Numeration.

page 45 note 1 The oldest actual example of the Indian ‘zero,’ with which I am at present acquainted, occurs on a coin in my own cabinet, of the Hindu Kabul series, which seems to read 707 (Gupta according to my view, and equal to 897 A.D.). Unfortunately the coin is in poor preservation, and the precise shape of the sign is hardly certain. It seems to be a kind of irregularly formed dot. See Numismatic Chronicle, vol. ii. N.S. for 1882, p. 111, pi. i. fig. 7.

page 47 note 1 For facility of reference, it may be well to set out the Greek system of alphabetical numerals as employed by their later arithmeticians, bearing in mind that it was not quite identical with the Hebrew or Arabic alphabetic methods. In the Greek system, after the first five letters, which were used to express the first five units, a special sign the ‘epistemon’ or ‘’ was inserted to represent six. The alphabetical order was then resumed till ‘iota’s represented 10; from this point the power of the letters rose by tens, κ representing 20, λ 30, and so on until ninety was reached, which was expressed also by a special sign, the ‘koppa’ or ς; then the ρ represented 100, from which the power of the letters rose by hundreds, thus σ = 200, ϕ = 500, ϰ = 600, Ψ = 700, but the nine hundred had also its own special sign , or , termed ‘Sampi.’ But the thousand introduced a new mode of marking, the power of a thousand being given to the nine first units by inserting an iota beneath them, thus The tens of thousands were expressed by the letter M (or Mu) for Mυρ, similarly subjoined to the unit letters, thus and . Of the modes of expressing yet higher numbers, whether by octads, or tetrads, or otherwise, mention will be made in the text, and special signs were also used to mark certain fractions. The mode of writing fractions, however, does not bear on the subject immediately under discussion.

page 48 note 1 The passage in which Sir J. Leslie gives his views as to the origin of this sign is omitted, as the explanation already adopted from M. Woepcke seems, for the reasons he gives, preferable.

page 51 note 1 It is hardly necessary here to refer to the supposed discovery announced by Niebuhr (as having been established to the satisfaction both of Playfair and of himself) of the Arabic numeral signs and of the zero (the decimal zero) used according to the true value of position, in a Greek MS. (a palimpsest in the Vatican Library), which is supposed to be of the seventh century. Supposing even the fact as stated to have been correctly ascertained, still so far as the figures themselves and the value of position are concerned, these still might well, looking at the date of the MS., have had an Indian origin, although the discovery would have militated against the comparatively late date which has been assigned to the zero. Professor Spezi has, however, demonstrated by a careful reexamination of the MS. itself, that Niebuhr's decipherment was clearly erroneous, and that in fact the supposed numerals, so far as they are numerals at all, are the ordinary Greek alphabetical numerals. Cantor, pp. 386–388 and note p. 248.

page 56 note 1 The MS. of Altdorf is that of Boethius, in which this passage was first discovered in full.

page 56 note 2 This refers to Prinsep's theory that the Indian numeral signs were in reality the initial letters of their written equivalents, a theory which has long since been abandoned, and which has been dealt with virtually in the discussion as to Aksharas in Part I.

page 58 note 1 The passage had been printed as early as 1499, and again in two or three later editions, but in a corrupt and unintelligible condition.

page 59 note 1 See Pl. IV. Table III., where sets of these “Apices” are given from various sources.

page 61 note 1 I am again indebted for these (in the form now generally accepted) to the kindness of Mr. Pinches of the British Museum.

page 62 note 1 ‘š’ is the Hebrew ‘sin,’ and ‘s’ the Hebrew Sameth or Semcath ḫ is the Arabic (rarely ) and the Hebrew (kheth) and on the principle adopted in the Hebrew and Arabic columns, may be read as “kh.”

It is to be remembered that the language of the ancient Egyptians was, like the Assyrian, of the Semitic stock, and some of the Assyrian terms for the numerals show strong resemblance to the Egyptian; thus the Assyrian ‘sana’ two, ‘sissu’ six, and ‘samnu’ eight, are palpably the same as their equivalents in Egyptian ‘sen,’ ‘sas,’ and ‘sesennu.’ Even the Assyrian ‘arba’ four, ‘sab'a’ seven, and ‘tisa’ nine, may be perhaps severally identified with Egyptian ‘aft,’ or ‘avt,’ ‘sefech’ or ‘sevech,’ ‘sechef,’ or ‘sechev,’ and ‘peset’ or ‘psit’; there exists indeed further evidence of this connection, but important as the subject is, it is not possible to pursue it further here. It might be thought that the Neo-Pythagoreans, so closely connected by their founder and by their long settlement at Alexandria with Egypt, may possibly have got these terms direct from the Egyptians, but the ‘arba,’ ‘quimas,’ ‘temenias,’ and still more notably the ‘zenis,’ so obviously come through the medium of Arabic or Hebrew (in all probability the latter), which themselves descend from the ancient Assyrian, that there seems no room for such an hypothesis.

page 64 note 1 It may be objected that these words might hare come not directly from the Tamil, but from some older Dravidian form lingering more to the West. But the words for ‘two’ and ‘three,’ in what are deemed the older Drayidian tongues, such as the Biluch, differ almost wholly from “Andras” and Ormis. The Malayalam approaches rather more closely, but the Tamil affords the nearest analogues.

page 65 note 1 The limit may be a survival of the primeval plan of counting by groups, but this question cannot be discussed now.

page 65 note 2 That this trinal mode of grouping is a point of some importance may be seen from Professor Leslie's words already quoted on p. 50.