/0 a0 00 000 000 0000 0000 Z?

Zr I 2345 6789 10 II 19 20 21 30 40 50 60 100 200 300 up, and in one way only, number from I to 2k-I the parts I, 2, 4,.. 2k-1.

x81 1 - x81 _ - x27 x40. - x' With the old Semitic alphabet of 22 letters this system broke down at n= 400, and the higher hundreds had to be got by juxtaposition; but when the Hebrew square character got the distinct final forms 1, o, 1, + l, these served for the hundreds from Soo to coo. The Greeks with their larger alphabet required but three supplemental signs, which they got by keeping for this purpose two old Phoenician letters which were not used in writing (*F* or g= i = 6, and y = p = 90), and by adding sampi-m for 900.1 Among the Greeks the first certain use of this system seems to be on coins of Ptolemy II. The first trace of it on Semitic ground is on Jewish coins of the Hasmoneans. It is the foundation of gematria as we find it in Jewish book and in the apocalyptic number of the beast ("DP 1113= 666). But we do not know how old gematria is; the name is borrowed from the Greek.

The most familiar case of the use of letters as numerals is the Roman system. Here C is the initial of centum and M of mille; but instead of these signs we find older forms, consisting of a circle divided vertically for z000 and horizontally, 0, or in the cognate Etruscan system divided into quadrants, *G,* for too. From the sign for loco, still sometimes roughly shown in print as dIo, comes D, the half of the symbol for half the number; and the older forms of L, viz. or .L, suggest that this also was once half of the hundred symbol. So V (Etruscan A) is half of X, which itself is not a true Roman letter. The system, therefore, is hardly alphabetic in origin, though the idea has been thrown out that the signs for io, 50, and zoo were originally the Greek X, 'Y, 4), which were not used in writing Latin.2 When high numbers had to be expressed systems such as we have described became very cumbrous, and in alphabetic systems it became inevitable to introduce a principle of periodicity by which, for example, the signs for I, 2, *3,* &c., might be used with a difference to express the same number of thousands. Language itself suggested this principle, and so we find in Hebrew N or in Greek ,a = z000. So further 1 3111v, or simply 20,000 (2 myriads). If now the larger were always written to the left of the smaller elements of a number the diacritic mark could be dispensed with in such a case as f wXa (instead of ,/3coXa) = 2831, for here it was plain that (3 = 2000, not = 2, since otherwise it would not have preceded *w =* Soo. We have here the germ of the very important notion that the value of a symbol may be periodic and defined by its position. The same idea had appeared much earlier among the Babylonians, who reckoned by powers of 60, calling 60 a *soss* and 60 sixties a *sar.* On the tablets of Senkerah a list of squares and cubes is given on this principle, and here the square of 59 is written 58r - that is, 58 X 60+1; and the cube of 30 is 7.30 - that is, 7 sar+30 soss = 7 X 60 2 -30 X 60. Here again we have value by position;. but, as there is no zero, it is left to the judgment of the reader to know which power of 60 is meant in each case. The sexagesimal system, long specially associated with astronomy, has left a trace in our division of the hour and of the circle, but as language goes by powers of 10 it is practically very inconvenient for most purposes of reckoning. The Greek mathematicians used a sort of decimal system; thus Archimedes was able to solve his problem of stating a number greater than that of the grains of sand which would fill the sphere of the fixed stars by dividing numbers into octades, the unit of the second octade being 10 8 and of the third io' 6. So too Apollonius of Perga teaches multiplication by regarding 7 as the 7rvOµnv or 70, 700 and so forth. One must then find successively the product of the several pythmens of the multiplier and the multiplicand, noticing in each case what are tens, what hundreds, and so on, and adding the results. The want of a sign for zero made it impossible mechanically to distinguish the tens, hundreds, &c., as we now do.

1 The Arabs, who quite changed the order of the alphabet and extended it to twenty-eight letters, kept the original values of the old letters (putting for ? and w for vi), while the hundreds from 500 to woo were expressed by the new letters in order from v to t. In the time of Caliph Walid (A.D. 705-715) the Arabs had as yet no signs of numeration.

2 See further Fabretti, *Palaographische Studien. * Very early, however, a,mechanical contrivance, the abacus, had been introduced for keeping numbers of different denominations apart. This was a table with compartments or columns for counters, each column representing a different value to be given to a counter placed on it. This might be used either for concrete arithmetic - say with columns for pence, shillings and pounds; or for abstract reckoning - say with the Babylonian sexagesimal system. An old Greek abacus found at Salamis has columns which, taken from right to left, give a counter the value of a, 10, zoo, 1000 drachms, and finally of i talent (6000 drachms) respectively. An abacus on the decimal system might be ruled on paper or on a board strewed with fine sand, and was then a first step to the decimal system. Two important steps, however, were still lacking: the first was to use instead of counters distinctive marks (ciphers) for the digits from one to nine; the second and more important was to get a sign for zero, so that the columns might be dispensed with, and the denomination of each cipher seen at once by counting the number of digits following it. These two steps taken, we have at once the modern so-called Arabic numerals and the possibility of modern arithmetic; but the invention of the ciphers and zero came but slowly, and their history is a most obscure problem.

What is quite certain is that our present decimal system, in its complete form, with the zero which enables us to do without the ruled columns of the abacus, is of Indian origin. From the Indians it passed to the Arabians, probably along with the astronomical tables brought to Bagdad by an Indian ambassador in 773 A.D. At all events the system was explained in Arabic in the early part of the 9th century by the famous Abu Ja`far Mohammed b. Musa al-Khwarizmi (Hovarezmi), and from that time continued to spread, though at first slowly, through the Arabian world.

In Europe the complete system with the zero was derived from the Arabs in the 12th century, and the arithmetic based on this system was known by the name of algoritmus, algorithm, algorism. This barbarous word is nothing more than a transcription of Al-Khwarizmi, as was conjectured by Reinaud, and has become plain since the publication of a unique Cambridge MS. containing a Latin translation - perhaps by Adelhard of Bath - of the lost arithmetical treatise of the Arabian mathematician.3 The arithmetical methods of Khwarizmi were simplified by later Eastern writers, and these simpler methods were introduced to Europe by Leonardo of Pisa in the West and Maximus Planudes in the East. The term zero appears to come from the Arabic sifr through the form zephyro used by Leonardo.

Thus far modern inquirers are agreed. The disputed points are (I) the origin and age of the Indian system, and (2) whether or not a less developed Indian system, without the zero but with the nine other ciphers used on an abacus, entered Europe before the rise of Islam, and prepared the way for a complete decimal notation.

Boetius 9 **GJ ** 1. The use of numerals in India can be followed back to the Nana Ghat inscriptions, supposed to date from the early part of the 3rd century B.C. These are signs for units, tens and hundreds, as 3 Published by Boncompagni in *Trattati d'aritmetica* (Rome, 1857).

4 From Sir E. C. Bayley's paper in *J.R.A.S.* (1882).

5 From Burnell's *South Indian Palaeography* (1874).

6 Of the 10th century. (From Burnell, *op. cit.) * 7 Of the 10th century; from a MS. written at Shiraz. (From Woepcke, *Memoire sur la propagation des chiffres indiens.) * 8 From a MS. at Paris. (From Woepcke, *op. cit.) * 9 Erlangen (Altdorf) MS. (From Woepcke, *op. cit.) * TABLE II.

1234 Nana Ghat (Indian) 4 = Cave Inscriptions (Indian)5 Devanagari 6. .

Eastern Arabic 7 .

Ghobar **s. ** in the other old systems we have dealt with. Like the Indian alphabet, they are probably derived from abroad, but, as in the case of the alphabet, their origin is obscure. The forms of the later Indian numerals for the nine digits appear to be clearly derived from the earlier system. In table II. the first two lines give forms earlier than the introduction of the system of position, while the Devanagari in the third line was used with a zero and position value. The " cave " numerals were employed during the first centuries of the Christian era. The earliest known example of a date written on the modern system is of A.D. 738, while the old system is found in use as late as the early part of the 7th century (Bayley). On the other hand, there is some evidence that a system of value by position was known to Sanskrit writers on arithmetic in the 6th Christian century. These writers, however, do not use ciphers, but symbolical words and letters, so that it is not quite clear whether they refer to a system which had a zero, or to a system worked on an abacus, where the zero is represented by a blank column. There is no proof as yet for the use of any system of position in India before the 6th century, and nothing beyond conjecture can be offered as to its origin.

2. In Europe, before the introduction of the algorithm of full Indo-Arabic system with the zero, we find a transition system in which calculations were made on the decimal system with an abacus, but instead of unit counters there were placed in the columns ciphers, with values from one to nine, and of forms that are at bottom the Indian forms and agree most nearly with the numerals used by the Arabs of Africa and Spain. For among the Arabs themselves there were varieties in the forms of the Indian numeral, and in particular an eastern and a western type. The latter is called ghobar (dust), a name which seems to connect it with the use of a sand-spread tablet for calculation. The abacus with ciphers instead of counters was used at Rheims about 970-980 by Gerbert, who afterwards was pope under the title of Sylvester II., and it became well known in the tith century. Where did Gerbert learn the use of the abacus with ciphers ? There is no direct evidence as to this, for the story in William of Malmesbury, that he stole it from an Arab in Spain, is generally given up as fabulous. On the other hand, no evidence is offered for an earlier use of the abacus with ciphers, except a passage describing the system in the *Geometria* ascribed to Boetius. If this book is genuine the Indian numerals were known in Europe and applied to the abacus in the 5th century, and Gerbert only revived the long-forgotten system. On this view we have to explain how Boetius got the ciphers. The *Geometria* ascribes the system to the " Pythagorici " - i.e. the Neo-Pythagoreans - and it has been thought possible that the Indian forms for the numerals reached Alexandria, along with the cruder form of value by position involved in the use of the abacus without a zero, before direct communication between Europe and India ceased, which it did about the 4th century A.D. It is then further conjectured by Woepcke that the ghobar numerals of the western Arabs were by them borrowed from the system of Boetius before the full Indian method with the zero reached them; and thus the resemblance between these forms and those in MSS. of Boetius, which are essentially the same as in other MSS. of the ttth century, would be explained. This view, however, presents great difficulties, of which the total disappearance of all trace of the system between Boetius and Gerbert is only one. We have no proof that the Indians ever used such an abacus, or that they had value by position at so early a date as is required, and the ghobar numerals are too similar to those of the eastern Arabs to make it very credible that the two systems had been separated for centuries. The genuineness of the *Geometria* is maintained by Moritz Cantor, but it has been attacked on other grounds than that of the passage about the abacus; and on the whole it is still an open question whether the abacus with ciphers is not the outcome of an early imperfect knowledge of the Arabic system, Gerbert or some other having got the signs and a general idea of value by position without having an explanation of the zero.

See M. Cantor, *Geschichte der Mathematik,* vol. i. (Leipzig, 1880); also M. Chasles, papers in the *Comptes rendus* (1843); G. Friedlein, *Die Zahlzeichen and das elementare Rechnen der Griechen and Romer,* &c. (1869); F. Woepcke, *Sur l'introduction de l'arithmetique indien en occident* (Rome, 1859), and *Mimoire sur la propagation des chiffres indiens* (Paris, 1863). For the palaeography of the Indian numerals see Burnell, *Elements of S. Indian Palaeography* (1874); and Sir E. C. Bayley in *J.R.A.S.* (1882, 1883). For Boetius compare Friedlein's edition of his arithmetic and geometry (Leipzig, 1867), and Weissenborn in *Zeitsch. Math. Phys.* xxiv. Other references to the copious literature will be found in Cantor and Friedlein, who also discuss the subject of the notation for fractions, which cannot be entered on here. For systems passed over here, see Pihan, *Exposé des signes de numeration usites chez les peuples orientaux* (Paris, 1860). (W. R. S.)

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