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ARITHMETICAL COMPLEMENT  so continually presented to Parliament or Congress. Finally, universal arithmetic is a name sometimes applied to algebra. The chief subjects generally treated under the science or art of arithmetic are: (1) numeration and notation; (2) addition; (3) subtraction; (4) multiplication; (5) division; (6) reduction; (7) compound addition; (8) compound subtraction; (9) compound multiplication; (10) compound division; (11) simple proportion (rule of three); (12) compound proportion; (13) vulgar fractions; (14) decimal fractions; (15) duodecimals; (16) involution; (17) evolution; (18) ratios, proportions, and progressions; (19) fellowship or partnership; (20) simple interest; (21) compound interest, and (22) position. Of these, the most important are the simple processes of addition, subtraction, multiplication and division, the judicious use of which, singly or in combination, will solve the most complex arithmetical questions.  ARITHMETICAL COMPLEMENT, that which a number wants to make it reach the next highest decimal denomination. Thus the arithmetical complement of 4 is 6, for 4 + 6 are = 10, and that of 642 is 358, because 642 + 358 are = 1,000. The arithmetical complement of a logarithm is what it wants to make it reach 10.  ARITHMETICAL MEAN. (1) The number, whether it be an integer or a fraction, which is exactly intermediate between two others. Thus, 5 is the arithmetical mean between 2 and 8; for 2 + 3 are = 5,and 5 + 3 are = 8. To find such a mean add the numbers together and divide their sum by 2; thus, 2 + 8 = 10, and 10 ÷ 2 = 5. (2) Any one of several numbers in an arithmetical ratio interposed between two other numbers. Thus, if 6, 9 and 12 be interposed between 3 and 15, any one of them may be called an arithmetical mean between these two numbers.  ARITHMETICAL PROGRESSION, a series of numbers increasing or diminishing uniformly by the same number. If they increase, the arithmetical progression is said to be ascending, and if they decrease, descending. Thus the series 3, 6, 9, 12, 15 is an ascending arithmetical progression, mounting up by the continued addition of 3; and the series 8, 6, 4, 2 is a descending one, falling regularly by 2.  ARITHMETICAL PROPORTION, the relation existing between four numbers, of which the first is as much greater or less than the second as the third is of

the fourth; the equality of two differences or arithmetical ratios. In such cases the sum of the extremes is=that of the means.  ARI THORGILSSON , the father of Icelandic literature (1067-1148). He was the first Icelander to use his mother tongue as a literary medium in writing his “Islendingabok,” a concise history of Iceland from its settlement (about 870) until 1120. This work was finished between 1134 and 1138.  ARIUS, the founder of Arianism. See .  ARIZONA, a State of the Western Division of the North American Union, bounded by Nevada, Utah, New Mexico, California and the Mexican State of Sonora; gross area, 113,870 square miles; organized Feb. 14, 1863; pop., (1900) 122,931; (1910) 204,354; (1920) 334,162. Capital, Phœnix.

Topography.—The surface in general is a series of plateaus, ranging in altitude from 80 to 7,500 feet above sea-level. It is traversed by the Northside, San Francisco, Black, Black Mesa, Gila, Dragon, Santa Ana, Zuni, Santa Catarina, Mogollon and Penaleno Mountains, with peaks stretching to an extreme height of 12,572 feet (Humphrey Peak). The watercourses are the Colorado river and its tributaries, the Little Colorado, Gila, Zuni, San Juan, and several smaller streams. The principal rivers pass through cañons that are among the greatest wonders of the world. Dotting the plains are enormous mesas or tablelands, some with perpendicular sides more than 1,000 feet high. No part of the world has so rich a field of archæological and ethnological investigation as Arizona. Long-buried dwellings and cities, with other ruins of an exceedingly ancient people, are being continually disclosed.

Geology.—The weird cañons are remarkable in that they exhibit all the geological formations of North America. Geologists declare that the Colorado river, in its whole course, has cut through strata representing a thickness of 25,000 feet, and exposed the gradations from the quarternary alluvial deposits through volcanic alterations to the primary azoic rocks. The Grand Cañon of the Colorado alone shows upper carboniferous limestone, cross-stratified sandstone, red calcareous sandstone with gypsum, lower carboniferous limestone, shales, grits, Potsdam sandstone and granite and other formations.

Mineral Production.—Arizona is one of the most important of the mineral producing States. Its most important<section end="Arizona" />