ARITHMET´ICAL, pertaining to arithmetic or its operations.--_Arithmetical mean_, the middle term of three quantities in arithmetical progression, or half the sum of any two proposed numbers; thus 11 is the arithmetical mean to 8 and 14.--_Arithmetical progression_, a series of numbers increasing or decreasing by a common difference, as 1, 3, 5, 7, &c.--_Arithmetical signs_, certain symbols used in arithmetic, and indicating processes or facts. The common signs used in arithmetic are the following: [plus sign] signifying that the numbers between which it is placed are to be added; [minus sign] - that the second is to be subtracted from the first; [times sign] that the one is to be multiplied by the other; [divide sign] that the former is to be divided by the latter; [equals sign] signifies that the one number is equal to the other; [proportional signs - colons] are the signs placed between the members of a proportional series, as 4 : 6 :: 8 : 12. A small figure placed on the right hand of another at the top signifies the corresponding power of the number beside which it is placed, as 5^2, 4^3, meaning the square of 5 and the cube of 4. [cube root] placed before or over a number signifies the square root of that number; with a figure it signifies the root of a higher power, as [cube root], which means cube root. A period placed to the left of a series of figures indicates that they are decimal fractions.
A´RIUS, the originator of the Arian heresy. See _Arians_.
ARIZO´NA, a former territory of the United States, admitted into the union as a sovereign State on 14th Feb., 1912, is bounded south by Mexico, west by California and Nevada (the River Colorado forming the greater part of the boundary), north by Utah, and east by New Mexico; area, 113,956 sq. miles. The surface is generally mountainous, but many fertile and well-watered valleys lie between the ridges. Part of the surface consists of deserts often entirely destitute of vegetation. The territory belongs to the basin of the Colorado, which passes through a portion of it, besides forming the boundary; while the Gila and Little Colorado, tributaries of the Colorado, traverse it from east to west. The canyons of the Colorado form a wonderful feature, the river flowing for hundreds of miles in a deep rocky channel with walls rising perpendicularly to the height of 1500 to 6000 feet. In some parts timber is plentiful. The rainfall is small, and irrigation has been employed for agricultural purposes. Large tracts of elevated land have been found excellently adapted as pastures for sheep and cattle. The territory is rich in copper, gold, silver, lead, and other minerals, and mining is largely carried on, with much copper smelting and refining. The capital is Phoenix. Pop. 29,053. The Southern and the Santa Fé Pacific Railways traverse it. Pop. (1920), 333,903 (of which 171,468 are white), exclusive of Apaches and other Indians on reservations (area, 29,017 sq. miles; pop. (1920), 42,400).