Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/685

N U M N U M 627 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 2 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 738 A.D., while the old system is found in use as late as the early part of the 7th century (Bay ley). 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 or 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 Kheims about 970-980 by Gerbert, who afterwards was pope under the title of Sylvester II., and it became well known in the llth 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 " Pytha- gorici " i.e., the Neo-Pythagoreans and it has been thought pos sible 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 be tween 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 llth century, would be explained. This view, however, presents great difficulties, of which the total dis appearance 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 still ably maintained by 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 See Cantor, Geschichte der Mathenwtik, vol. i., Leipsic, 1880, as the most recent feneral account of the subject ; also Chasles, papers in the Comptes Bendus, 1843 ; rit-dlein, Die Zahlzeichen und das elementare Rechnen der Griechen und Bomer, <tc., 1869 ; Woepcke, Sur I introduction de I arithmetiqne Indien en Occident, Rome, 1859, and Memoire sur la projiagation des chijfres Indiens, Paris, 1863. For the palaeography of the Indian numerals see Burnell, Elements ofS. Indian Paleography, 1874 ; and Sir B. C. Bayley in J. R. A. S., 1882, 1883. For Boetius compare Friedlein s edition of his arithmetic and geometry, Leipsic, 1807, and AVeissenborn in Zeitsch. Math. Phijs., 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, Expose des signes de numeration usites chez ks peuplcs orientaux, Paris, 1860. (W. R. S.) NUMERIANUS, M. ATJRELIUS, Roman emperor, accom panied his father, the emperor Carus, on the Persian expedition beyond the Tigris, and along with his absent brother, Carinus, was proclaimed emperor on the death of the former (December 283). Having resolved to abandon the campaign, he was returning towards Europe when he mysteriously died before Chalcedon was reached, eight months afterwards. Arrius Aper, prefect of the pretorians, his father-in-law, suspected of having murdered him, was hastily stabbed by Diocletian, his successor. Numerianus is represented as having been a man of considerable literary attainments, as well as of singular gentleness, amiability, and purity. NUMIDIA was the name given to a large tract of country in the north of Africa, extending along the Medi terranean Sea from the confines of Mauretania to those of the Roman province of Africa. The term was, however, employed in very different senses, and within very different limits, at different periods of time. When Carthage was at the height of its power, and the Romans first came into contact with the nations of northern Africa, the name of Numidia was applied to the whole country from the river Mulucha (now called the Muliiya), about 100 miles west of Oran, to the frontier of the Carthaginian territory, which nearly coincided with the modern regency of Tunis. It is in this sense that the term is employed by Polybius, and all succeeding historians down to the close of the Roman republic. The Numidians, as thus defined, were divided into two great tribes, the Massyli on the east, and the Massaesyli on the west, the limit between the two territories being the river Ampsaga, which enters the sea to the west of the remarkable promontory called Tretum, now known as the Seven Capes. At the time of the Second Punic War the eastern tribe was under the government of Masinissa, who took part with the Romans in the contest, while his rival Syphax, king of the Massaesylians, supported the cause of the Carthaginians. In consequence of this, after the close of the war, Syphax s dominions were forfeited, and united with those of Masi nissa, who now ruled the whole Numidian people from the frontier of Mauretania to the boundary of the Cartha ginian territory. That monarch, who attained to a great age, retained the whole of these extensive dominions till his death in 148 B.C., as was the case also with his son and successor Micipsa ; but after the death of the latter in 118, the ambition of his nephew JUGURTHA (q.v.) in volved him in a war with Rome, which ended in his defeat and death in 106. Numidia was not, however, incorporated with the Roman empire until a later period. After the death of Jugurtha the western portion of his dominions was added to those of Bocchus, king of Mauretania, while the remainder con tinued to be governed by native princes until the civil war between Caesar and Pompey, in which Juba, then king of Numidia, having espoused the cause of the latter, and supported Scipio and Cato in Africa, was defeated by Caesar, and put an end to his own life (46 -3.C.). Numidia, in the more restricted sense which it had now acquired, became for a short time a Roman province, but in the settle ment of affairs after the battle of Actium (30) it was restored to Juba II., son of the preceding monarch, who had acquired the favour of Augustus. A few years later, however, Juba was transferred to the throne of Mauretania, including the whole western portion of the ancient Numi dian monarchy as far as the river Ampsaga, while the Roman province of Numidia, which was now definitely constituted, comprised only the tract between that river and the Tusca, which formed the western limit of the Roman province of Africa. But though thus restricted in extent so as nearly to correspond with the modern French province of Constantine, while the kingdom of Numidia in the wider sense had included the whole of Algeria the Roman province of Numidia attained a high degree of pros perity and civilization, and was studded with numerous towns, the importance of which is attested by inscriptions still extant, as well as by the massive remains of their ancient monuments and works of public utility. This period of prosperity continued to be favoured by unbroken