526 AKITHMETIC and the system is thus traced back to the 7th, and in all probability to the 5th century of the Christian era. Even then it was evidently no novelty, but is alluded to as hold ing an established position; and the Hindu writers nowhere lay claim to the invention of it, but constantly assign to it a supernatural origin. The method was known to the Arabians in the 9th century; and in the course of the 10th it seems to have come into general use among them, espe cially in their astronomical writings and tables. It was probably in the following century that the Arabs intro duced the notation into Spain; but in regard to this we have no explicit information, and different accounts are given of the earliest instances of the use of the system in Europe. On the one hand, it is alleged that the figures first occur in a translation of Ptolemy, of the date 1136, while others maintain that they were introduced (about 1252) by means of the celebrated astronomical tables pub lished by and named from Alphonso the Wise. That their use was known in Italy at the commencement of the 13th century, appears to be satisfactorily established, for there is no good reason to doubt the genuineness of the MS. writings of Leonardo of Pisa, copies of which have been found bearing the dates 1202 and 1220. Numerous other instances are given of the early use of the nine figures and the cipher, especially by astronomers, and in calendars. The great superiority of this to earlier modes of numerical ex pression became gradually apparent, and in course of time it came into almost universal use among civilised nations. For a time there was, not unnaturally, considerable in exactness or confusion in the employment of the notation. In early writings such combinations are found, for example, as X2 for 12, 301 for 31, &c. In the latter case the law of local value is lost sight of, and the characters 30 are used as equivalent to thirty, irrespective of their position. 3. Calculation or computation by means of numerical characters is what is ordinarily regarded as the distinctive province of arithmetic, and the worth of a system of nota tion is to be estimated by the facilities it affords for the operations of reckoning. The methods in common use will be detailed, and the principles on which they depend briefly expounded, in subsequent sections of this article. Computation of a comparatively rude kind was often carried on in ancient times, and is practised still in some countries, by what are called palpable methods, as, for instance, by means of counters, or by balls strung on rods or running in grooves. Of this the abacus of the Romans (used too, but not so generally, among the Greeks), the swan-pan, still in constant use among the Chinese, &c., are examples. (See ABACUS, vol. i. p. 4.) The methods that preceded the adoption of the Arabic numerals were all comparatively unwieldy, and very simple processes involved great labour. The notation of the Romans, in particular, could adapt itself so ill to arithmetical operations, that nearly all their calculations had to be made by the abacus. One of the best and most manageable of the ancient systems is the Greek, though that too is very clumsy, as may be seen from an elementary example. Let it be required to muliply 862 by 523, i.e., w fi by <$>Ky. The product of o> and </> is 400,000, i.e., yn M ; of and <, 30,000, or y M ; of /3 < and <f>, 1000, or <a.. Similarly the multipli- _ ^JLV cation of w by K gives 16,000, or a x/ $-; of p-x.yx- by K, 0.0-; of (3 by K, p; of w by y, ftv; of by y, /DTT; and of /3 by y, $-. These may be arranged in some such order as in the margin, and summed. The result is H K ewKr , or 450,856. The notation employed here extends to hundreds of millions. A scheme proposed by Archi medes, the most distinguished mathematician of anti ,r acr/x, w K 5- quity, in his work entitled SI a/i/uT^s, Arenarius, goes far beyond this. Taking the limit of the ordinary system, viz., a myriad of myriads, or 100,000,000, as the basis of a new scale, he treats of numbers formed by the continued pro duct of that amount repeated eight times, extending to sixty-four places in our notation. These he divided into what we should call periods, of eight figures each, to which he gave the name of octades. He thus appears to have to some extent anticipated the modern method of grouping, though, from the want of knowledge of the principle of local value, the practical advance he made was unimportant. Apollonius of Perga, who flourished a little after the time of Archimedes, took the myriad as the basis of his system, and repeated the sign Mu for each product of a myriad; thus A8Mu.Mu.Mu indicates what we would write as 34,000,000,000,000. But his most important contribution to arithmetical science was his grouping the tens in mul tiplication, so as to connect large numbers, as far as pos sible, with those we represent by the nine digits. He thus endeavoured, and with some measure of success, to remedy the defect of the general system arising from the want of apparent connection of such characters as X and T, for example, or TT and to, with what he called their bases, y and T). It is chiefly in the commentaries of Eutocius on the works of Archimedes and Apollonius that the examples of the ancient Greek arithmetic which we possess are preserved. The operations of the Greek arithmetic involving frac tions were necessarily very complicated. A simpler system was introduced (by Ptolemy, according to his commentator Theon, but it appears to have existed before his time) in what are known as sexagesimals, which are precisely analo gous to the duodecimals by which we sometimes calculate areas. The division of the circle into 360 degrees arose, no doubt, from 360 approximating the number of days in the year, and containing a large number of divisors. The radius of the circle, or side of the inscribed hexagon, subtended at the centre 60 of these degrees, and from this the degree was again divided into 60 parts called minutes, the minute into 60 seconds, &c., just as the foot is divided into 12 inches, the inch into 12 seconds, &c., in duo decimals. The sexagesimal system, though applied in the first instance to circular, was also employed in linear, measurements. As an example, the square of A 8 ve, i.e., of 37 4 55 , will be found to be /rroe 8 18 t /ce, i.e., 1375 4 14" 10 " 25"". The numerals beyond 60 or being dis pensed with in this system, the next letter, o, was used to denote zero; and it has been conjectured that this may have been the origin of the form of our cipher. The most important step in the progress of modern arith metic was the introduction of decimal fractions, and the extension of the Arabic notation to the expression of them. The first writer who advocated and exemplified the use of decimals was Simon Stevin de Bruges, better known as Stevinus, in a paper (La Disme) published about 1585; but he employed an awkward notation, and it was only after a considerable period that they assumed the form entirely consistent with the notation of integers with which we are familiar. From a form which Lord Napier employs in his liabdologla (1617), the introduction of the decimal point (as a comma) has been ascribed to him, but apparently without sufficient evidence. The following writers on arithmetic may be named, in addition to those already mentioned: Diophantus, who flourished in or about the 4th century; Maximus Planudes, who died about 1350; Lucas Pacioli (de Burgo, or di Borgo), whose Summa de Arithmetica (1494) was the first work on algebra printed, and one of the earliest on arith metic ; Bishop Tonstall, whose De Arte Supputandi (1522)
was the first work on the subject printed in English;