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 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 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 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 ghobār (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 11th 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 Boëtius. 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 Boëtius 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 It is then further conjectured by Woepcke that the ghobār numerals of the western Arabs were by them borrowed from the system of Boëtius before the full Indian method with the zero reached them; and thus the resemblance between these forms and those in MSS. of Boëtius, which are essentially the same as in other MSS. of the 11th century, would be explained. This view, however, presents great difficulties, of which the total disappearance of all trace of the system between Boëtius 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 ghobār 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 und das elementare Rechnen der Griechen und Römer, &c. (1869); F. Woepcke, Sur l’introduction de l’arithmétique indien en occident (Rome, 1859), and Mémoire 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 Boëtius compare Friedlein’s edition of his arithmetic and geometry (Leipzig, 1857), 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 numération usités chez les peuples orientaux (Paris, 1860).

NUMERIANUS, MARCUS AURELIUS, son of the Roman emperor Carus. On the death of his father, whom he accompanied on his expedition against the Persians, he was proclaimed emperor (December, 283). He resolved to abandon the campaign, and died mysteriously on his way back to Europe, eight months afterwards. Arrius Aper, praefect of the praetorian guards, his father-in-law, who was suspected of having murdered him, was slain by Diocletian, whom the soldiers had already proclaimed his successor. Numerianus is represented as having been a man of considerable literary attainments, and of remarkably amiable character.

NUMIDIA, the name given in ancient times to a tract of country in the north of Africa, extending along the Mediterranean from the confines of Mauretania to those of the Roman province to Africa. When the Romans first came into collision with Carthage in the 3rd century, the name was applied to the whole country from the river Mulucha (now the Muluya), about 100 m. W. 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 name Numidia is used by Polybius and all 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 being the river Ampsaga, which enters the sea to the west of the promontory called Tretum, now known as the Seven Capes. At the time of the second Punic War the eastern tribe was governed by Massinissa, who took the side of the Romans in the contest, while Syphax his rival, king of the Massaesyli, supported the Carthaginians. At the end of the war the victorious Romans confiscated the dominions of Syphax, and gave them to Massinissa, whose sway extended from the frontier of Mauretania to the boundary of the Carthaginian territory, and also south and east as far as the Cyrenaica (Appian, Punica, 106), so that the Numidian kingdom entirely surrounded Carthage except towards the sea, Massinissa. who reached a great age, retained the whole of these dominions till his death in 148 and was succeeded in them by his son Micipsa, who died in 118. For the war with Rome which followed the death of Micipsa see.

After the death of Jugurtha as a captive at Rome in 106, the western part of his dominions was added to those of Bocchus, king of Mauretania, while the remainder (excluding perhaps the territory towards Cyrene) continued to be governed by native princes until the civil war between Caesar and Pompey, in which Juba I., then king of Numidia, who had espoused the cause of the Pompeians, was defeated by Caesar, and put an end to his own life (46 ). Numidia, in the more restricted sense which it had now acquired, became for a short time a Roman province under the title of Africa Nova, but in the settlement of affairs after the battle of Actium it was restored to Juba II. (son of Juba I.), who had acquired the favour of Augustus. Soon afterwards, in 25, Juba was transferred to the throne of Mauretania, including the whole western portion of the ancient Numidian monarchy as far as the river Ampsaga, while the eastern part was added to the province of Africa, i.e. that part which had been called Africa Nova before it was given to Juba. It retained the official title, though it may also have been known as Numidia; together with Africa Vetus it was governed by a proconsul, and was the only senatorial province in which a legion was permanently stationed, under the orders of the senatorial governor. In 37 the emperor Gaius put an end to this arrangement by sending a legatus of his own to take over the command of the legion, thus separating the military from the civil administration, and practically separating Numidia or Africa Nova from Africa Vetus, though the two were still united in name (Tac. Hist. 4. 48). Under Septimus Severus ( 193–211) Numidia was separated from Africa Vetus, and governed by an imperial procurator (procurator per Numidiam); finally, under the new organization of the empire by Diocletian, Numidia became one of the seven provinces of the diocese of Africa, being known as Numidia Cirtensis, and after Constantine as N. Constantina, corresponding closely in extent to the modern French province of Constantine. During all this period it reached a high degree of civilization, and was studded with numerous towns, the importance of which is attested by inscriptions (see vol. viii. of the Corpus inscriptionum), and by the massive remains of public buildings. The invasion of the Vandals in 428 reduced it to a condition of gradual decay; and the invasion of