The American Cyclopædia (1879)/Arithmetic
ARITHMETIC (Gr. ἀριθμητική, from ἀριθμεῖν, to count), the science of the properties and relations of numbers when expressed with figures or relations of figures. The accepted opinion is that we have derived this science from the Greeks, who obtained it from the Phœnicians; but if we consider that the Chaldeans, one of the oldest nations, have given us the knowledge of certain astronomical cycles or periods, of which the determination required an advanced knowledge of arithmetic, it is evident that its origin is of much earlier date. The Hebrews and Greeks used the first nine letters of their alphabet for the numbers 1 to 9; the next nine letters for 10, 20, &c., to 90; and the others for hundreds; while for thousands they recommenced the alphabet and added to each letter a mark or iota. The Romans followed a similar system, of which our Roman numerals are a specimen. But arithmetic did not reach its more modern state of progress until the introduction of the Arabic figures now used by all civilized nations. The Arabs admit that they obtained these figures from Hindostan in the 10th century. They call them Indian figures, and arithmetic the Indian science. Boëthius, in his work De Geometria, informs us that the disciples of Pythagoras used in their calculations nine peculiar figures, while others used the letters of the alphabet; and it is probable that this philosopher, who had travelled considerably, had obtained this knowledge in Hindostan, and communicating it as a secret to his disciples, caused it to remain sterile in their hands. The Greeks in the ordinary way of writing expressed the fractions thus: while β, γ, δ, &c., stood for 2, 3, 4, &c., β', γ', δ', represented ½, ⅓, ¼, &c. The oldest text book 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, Egypt and contains, besides the rules for addition, subtraction, multiplication, and division, many peculiar properties of numbers. (For a translation of a portion of this remarkable manuscript by Marcel, see De Montfévrier, Dictionnaire des sciences mathématiques, vol. i., p. 141 et seq.) It was not till the beginning of the 13th century that the science of arithmetic began to be diffused in Europe. One of the earliest 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 Algorithmus Demonstratus, and also of a treatise on arithmetic which Jacques Faber published with commentaries immediately after the invention of printing. A great development of the science now took place. In the 16th century Clavius and Stifelius (Stiefel) in Germany and Digges in England were conspicuous 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 different scale, as the duodecimal or twelve, the tonal or sixteen, &c., are more apparent than real. A smaller scale would, however, have simplified 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 reducing a decimal expression into a binary one, performing the operation, and restoring it back again into the decimal system. The regular series of numbers, one, two, three, four, 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 arithmetic goes beyond the practical calculations by numbers, and treats of the properties of numbers 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 decimal system adopted. Thus the law that the sum of two numbers multiplied by their difference 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 system; if we had adopted the duodecimal system, 11 would have that property. — Besides ordinary 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 contrived instruments; a tabular arithmetic, where problems are solved by means of tables computed 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 Massachusetts, by the publication of treatises on this subject which have largely influenced the authors of arithmetical text books, a great variety 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 Arithmeticæ, or Legendre, Théorie des nombres.