708 ARITHMETIC on arithmetic employing the Arabian or Indian figures, and the decimal system, is undoubtedly that of Avicenna, the Arabian physician, who lived in Bokhara about A. D. 1000 ; it was found in manuscript in the library at Cairo, E<m>t and contains, besides the rules for addi- tion, subtraction, multiplication, and division, many peculiar properties of numbers. (For a translation of a portion of this remarkable manuscript by Marcel, see De Montfevrier, Dic- tionnaire des sciences mathematiques, vol. i., p. 141 et seq.) It was not till the beginning of the 13th century that the science of arithmetic be- gan to be diffused in Europe. One of the earli- est writers on the subject was John Halifax, better known as Sacro-Bosco, who in the 13th century composed an arithmetic in Latin rhymes, in which the shapes of the figures are nearly identical with those of the present day. The monk Planudes, who flourished in the early part of the 14th century, wrote a book entitled " Indian Arithmetic, or the Manner of Reckoning after the Indian Style," of which several manuscripts still exist. Contemporary with him was Jordanus of Namur, author of the Algorithmic Demonstratus, and also of a treatise on arithmetic which Jacques Faber published with commentaries immediately after the invention of printing. A great develop- ment of the science now took place. In the 16th century Clavius and Stifelius (Stiefel) in Germany and Digges in England were conspic- uous for their services to this science, and the Arabian or Indian figures came into use among the learned ; but it was not till the 17th century that arithmetic began to be a regular branch of common education. The value of our system of arithmetical notation, as is well known, consists in the adoption of a scale and of a system by which the place of the figure in the order in which it appears causes its value to increase in multiples of that scale. The universally adopted scale is the decimal, probably derived from the number of fingers of the human hand, but other scales might have been adopted as well ; and the advantages which some persons suppose might have been derived from the adoption of a dif- ferent scale, as the duodecimal or twelve, the tonal or sixteen, &c., are more apparent than real. A smaller scale would, however, have sim- plified arithmetical operations, as was forcibly demonstrated by Leibnitz, who showed how with the smallest possible scale, the binary, and the consequent use of only two figures, 1 and 0, operations were so much simplified that there might be even a saving of time in redu- cing a decimal expression into a binary one, performing the operation, and restoring it back again into the decimal system. Th regular series of numbers, one, two, three, foW, five, six, seven, eight, nine, &c., is expressed in the binary system thus: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, &c. ; in the ternary system, in which three is adopted as the basis, it is 1, 2, 10, 11, 12, 20, 21, 22, 100, &c. When arith- ARIUS metic goes beyond the practical calculations by numbers, and treats of the properties of num- bers in general, it enters the field of algebra. The properties of numbers are of two kinds : some are general and inherent in the numbers themselves, while others depend on the deci- mal system adopted. Thus the law that the sum of two numbers multiplied by their dif- ference is equal to the difference of their squares is a general property ; while the fact that if the sum of the figures is divisible by 9, the whole number is divisible by 9, is a property depending on the adoption of the decimal sys- tem ; if we had adopted the duodecimal system, 11 would have that property. Besides ordi- nary arithmetic, we may distinguish a palpable arithmetic performed by the sense of feeling by the blind ; an instrumental arithmetic, where the solutions are obtained by peculiarly con- trived instruments ; a tabular arithmetic, where problems are solved by means of tables com- puted for the purpose, &c. Pestalozzi, the great German pedagogue, applied his method to instruction in arithmetic with the most eminent success. It was introduced into the United States by Warren Colburn of Massa- chusetts, by the publication of treatises on this subject which have largely influenced the authors of. arithmetical text books, a great vari- ety of excellent practical works having since been published, to which we refer for further information in regard to the practical details of this science. For many curious facts on the properties of numbers, see Gauss, Disquisitions Arithmetical, or Legendre, Theorie des nonibres. ARIL'S, the founder of Arianism, according to some a Libyan, according to others a native of Alexandria, died in 336. He joined the Me- letians in Alexandria, but left them, and in 306 was ordained a deacon by Bishop Peter of Alexandria. He afterward returned to the Meletians and was excommunicated, but was readmitted to the church by Achillas, successor of Peter, and ordained priest. After the death of Achillas, Arius came near being elected bishop of Alexandria ; but Alexander was pre- ferred to him. According to the Arian histo- rian Philostorgius, Arius himself brought about the election of Alexander. It is reported that for several years Alexander held Arius in high esteem, and that the most perfect agreement existed between them. The great controversy with which their names are connected began when Alexander made an address to his clergy in which he spoke of the Trinity as consisting of a single essence. Arius exclaimed against this, af- firmed the distinct personality of the Father and the Son, and accused Alexander of Sabellianism. Alexander demanded from Arius a recantation ; but the latter not only refused this, but sent a written confession of faith to several bishops, requesting, in case they agreed with him, their intercession with Alexander in his behalf. A number of prominent bishops responded favor- ably ; among them were Eusebius of Caesarea, the church historian, and Eusebius of Nico-
