Logarithms, the greatest arithmetical achievement of the Seventeenth Century, were introduced by Napier (q.v.) in 1614. Many changes for the better have also been made in the operations for finding the square root and in the algorisms of applied arithmetic. As to improvements that may be expected to be introduced in the future, the 'Italian' method of division seems destined to yield to the so-called 'Austrian' algorisms, which is represented by the following example: Austrian. 10.48 31416)329200 32.92 3.1416 3142 150 126 24 24 Further, proportion, or the old 'rule of three,' will probably be replaced by the equation as such, and the same may be expected for unitary analysis.
Regarding the teaching of arithmetic, the student of pedagogy will look for information to the writers of the Nineteenth Century. The following may be recommended as valuable works on the history and pedagogy of the subject: Unger, Die Methodik der praktischen Arithmetik in historischer Entwickelung (Leipzig, 1888); Kehr, Geschichte der Methodik (Gotha, 1888, Vol. III.); W. Rein, A. Pickel, and E. Scheller, Theorie und Praxis des Volksschulunterrichts nach Herbartischen Grundsützen (Leipzig, 1898); J. A. McLellan and J. Dewey, Psychology of Number (New York, 1895); D. E. Smith, Teaching of Elementary Mathematics (New York, 1900). Notable among higher arithmetics are: Tannery, Leçons d'arithmétique théorique et pratique (Paris, 1894); and W. W. Beman and D. E. Smith's Higher Arithmetic (Boston, 1897).
ARITHMETICAL PROGRES'SION.See Series.
ARITHMETIC AND GE'OMET'RIC SIGNS. Arbitrary symbols used to indicate: (1) the nature of a magnitude, as + α, a positive quantity, and — α, a negative quantity; (2) operations to be performed upon magnitudes, as a . b, i.e., b multiplied by a; (3) relations between magnitudes, as a > b, i.e., a is greater than b. The following are a few signs in common use:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, are the Hindu numerals.
I, II, III. IV, V, VI, etc., are Roman numerals.
+ (plus); — (minus); ± (plus or minus); X or · (times or multiplied by); ÷, / or the fraction bar (divided by); = (equal to); a ≡ b (a is identical with b); a ≡ b (mod m) (a is congruent to b, modulus m); a > b (a is greater than b); a < b (a is less than b); a b (a is similar to b); a b (a varies as b): ∞ (infinity); a b (a approaches b as a limit); (therefore); (since); |-a| (absolute value of a; A (angle A); (perpendicular to); (congruent to, i.e., similar and equal): (root of); π (ratio of the circumference of a circle to its diameter); e (base of the hyperbolic logarithms).
ARITHMETIC COMPLEMENT. The difference between a number and the next larger number representing a power of 10. Thus, the arithmetic complement of 7 is 10—7, or 3; the arithmetic complement of 85 is 100—85, or 15; that of 125 is 1000—125, or 875. The arithmetic complement is much used in numerical calculations where differences are to be found. Since a—b = a + (10n—b)—10n, it follows that instead of subtracting a number its arithmetic complement may be added, the corresponding power of 10 being then deducted. Thus, 456—273 = 456 + 727—1000 = 183. The advantage consists in this, that since the arithmetic complement of a number is easily found by subtracting each digit from 9, except the unit's (which is taken from 10), it is often easier to add the arithmetic complement than to subtract the number. In working with logarithms arithmetic complements are often used instead of their numbers. under the name of co-logarithms. See Logarithm.
ARITHMETIC MEAN.See Mean.
ARITHMETIC TRI'ANGLE. See Pascal.
ARITHMOGRAPH. See Calculating Machine.
ARI THORGILSSON, ä´rë tôr'gêl-sõn (1067-1148). The father of Icelandic literature. His Islendingabók, the first literary work of the island, was finished between 1134 and 1138, and is a concise account of the history of Iceland from its settlement, about 870. till 1120. See Icelandic Literature.
ARI´US (c.256-336). The father of Arianism, the doctrine that Christ was not of the same essence as God the Father, but was a creature, though the first and highest of creation. He was born in Libya, the North African province to the west of Egypt, about 250. He went to Alexandria and there was made deacon and presbyter, and was the highly esteemed pastor of a church called, from its shape, the Baucalis (the Greek name of a kind of vase). In 318 he denied the statement which Alexander of Alexandria made upon the Trinity: viz., that there was only a single essence. This he declared was Sabellian. Defining his own position, he affirmed that if the Son were truly a son, there must have been a time when he was not. For this statement he was applauded by many, but Alexander called a council of a hundred Egyptian and Libyan bishops, which condemned Arius and his allies and deposed them (321). The fight had now begun. Arius had numerous supporters, chief of whom was Eusebius, Bishop of Nicomedia. Alexander also rallied a large contingent. He wrote numerous letters (two of which are still extant), exhorting the bishops not to receive the heretic. Notwithstanding this active canvass by Alexander, Eusebius of Nicomedia absolved Arius, who had retired to Palestine and then to Nicomedia, from the Alexandrian's condemnation, and had Arius's position approved by a synod held in 323, probably in Nicomedia. Arius wrote The Banquet, a work in prose and verse, of which fragments remain. It sets forth his view of the person of Christ and put it in a form so that it could be sung to popular tunes. This is said to have aided his cause greatly. The strife attracted the attention of the Emperor Constantine, as it was troubling the peace of the Church and disuniting it. Con-
