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COLUMBIA — COMBINATORIAL

Five railways enter it, namely, the Atlantic Coast Line, the Southern, the South Carolina and Georgia, the Florida Central and Peninsula, and the Columbia, Newberry, and Laurens. It is the seat of South Carolina College, which in 1898 had twelve professors and 188 students. Population (1880), 10,036; (1890), 15353; (1900), 21,108. Columbia, capital of Maury county, Tennessee, U.S.A., situated on Duck river, in the central part of the state, at an altitude of 646 feet. It has two railways, the Louisville and Nashville and the Nashville, Chattanooga, and St Louis. Population (1880), 3400; (1890), 5370; (1900), 6052. Columbia, District of“. See Washington. Co I u m b i a U n i ve rs i t y, in the city of New York, U.S.A., includes both a college and a university in the strict sense of the word as used in the United States. It comprises the faculties of law, medicine, philosophy, political science, pure science, and applied science. It is the successor of the corporation known as “The Governors of King’s College, in the Province of New York,” founded in 1754 by royal charter. In the educational system is also included Barnard College for Women, a separate corporation founded in 1889, and a teachers’ college, also a separate corporation. In 1897 the university moved from the centre of the city northwards to Morningside Heights, which overlook the Hudson from an altitude of 150 feet. The Medical School (the College of Physicians and Surgeons) remains in its old location opposite Roosevelt Hospital. The entire plant of the university represents a cost of about $9,500,000. In the year ending 30th June 1900 the expenditures for educational purposes were $942,460, leaving a deficiency of $17,328, which was met by a special guarantee fund. In 1901 there were registered in the college for men (Columbia) 475 students, and in the college for women (Barnard) 293 students, making a total of 768 undergraduates. The total of non - professional graduate students was 412. The scientific schools contained 539 students, the law school 422 students, the medical school 775 students, the teachers’ college 498 students—making a total of 2234 professional students. The total number of students in the university was thus 3830 (including 417 summer session students). In addition to these there were 29 auditors and 679 members of extension courses, making a grand total of 4538. The number of teachers of all grades for the same year was 375. The library,which numbers about 300,000 volumes, is thoroughly modern, and is selected with special reference to scholarly uses. The university is growing in all departments. (See also Universities and Education.) (s. l*.) Columbus, capital of Muscogee county, Georgia, U.S.A., situated on the western boundary of the state, at an altitude of 260 feet, on Chattahoochee river, which is navigable to this point. Just above the city the river crosses the fall line, producing falls and rapids which furnish excellent water-power. This has been turned to account in extensive cotton manufactures. Three railways enter the city, the Southern, the Central of Georgia, and the Georgia and Alabama. Population (1880), 10,123; (1890), 17,303; (1900), 17,614. ColumbUS, capital of Bartholomew county, Indiana, U.S.A., situated on the east fork of White river, a little south of the centre of the state, at an altitude of 629 feet. It is at the intersection of lines of the Pittsburg, Cincinnati, Chicago, and St Louis and the Cleveland, Cincinnati, Chicago, and St Louis Railways. The centre of population of the United States was in 1900 very near this place. Population (1880), 4813; (1890), 6719; (1900), 8130.

ANALYSIS

ColumbUS, capital of Lowndes county, Mississippi, U.S.A., at the intersection of the Southern and the Mobile and Ohio Railways, on the Tombigbee river. It contains large cotton mills. Population (1890), 4559; (1900, with limits enlarged), 6484, including 3366 negroes. ColumbUS, capital of Franklin county, Ohio, U.S.A., and of the state of Ohio. The site was purposely selected in 1813 near the centre of the state, in 39° 57' N. lat. and 82° 59' W. long., at an altitude of 743 feet, the elevation of the lines at the Union Station. It is a railway centre of the first importance. Fourteen different lines of railway, belonging to eight companies, enter the fine new Union Station in the heart of the city, thence radiating in all directions. The manufactures employed in 1890 a capital of $16,178,703, with 13,421 hands and an output of $22,887,586. The principal manufacture was that of carriages and waggons, valued at $3,199,287. Foundry and machine-shop products were second, with a value of $2,139,185. Then followed the manufacture of steam cars, the product of which was valued at $1,670,078. The Ohio State University, situated here, had in 1898 a faculty of 95 professors, and was attended by 1150 students, one-fifth of whom were women. Its property was valued at $2,600,000, and its income at $292,000. It has schools of law, medicine, dentistry, pharmacy, and veterinary surgery. Capital University, a Lutheran institution, also here, had in 1898 a faculty of 10 teachers and 113 students. The death-rate in 1899 was but 10‘83 per thousand; this is less than half the average of American cities, and little more than half that of the Union. The assessed valuation of property, real and personal, was, in 1899, $64,344,990. The income from all sources was $2,612,301, the expenditure $2,570,038, and the net debt $6,059,146. The tax rate per $1000 was $27.50. Population (1880), 51,647 ; (1890), 88,150; (1900), 125,560; death-rate (1900), 15-8. Com bacon um, or Kumbakonam, a city of British India, in the Tanjore district of Madras, in the delta of the Kaveri; with a railway station on the South Indian Railway, 194 miles from Madras. In 1881 it had a population of 50,098, and in 1891 of 54,307, of whom nearly one-fifth were Brahmans. In 1901 the population was 59,688, showing an increase of 10 per cent. The municipal income in 1897-98 was Rs.80,480. It contains a Government college, two high schools, four printingpresses, and a reading-room. Combmatoraal Analysis.—The Combina torial Analysis, as it was understood up to the end of the 18th century, was of limited scope and restricted application. P. Nicholson, in his Essays on the H^°ncal Combinatorial Analysis, published in 1818, auction. states that “the Combinatorial Analysis is a branch of mathematics which teaches us to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that we may be certain that we have not missed any collection or arrangement of these things that has not been enumerated.” Writers on the subject seemed to recognize fully that it was in need of cultivation, that it was of much service in facilitating algebraical operations of all kinds, and that it was the fundamental method of investigation in the theory of Probabilities. Some idea of its scope may be gathered from a statement of the parts of algebra to which it was commonly applied, viz., the expansion of a multinomial, the product of two or more multinomials, the quotient of one multinomial by another, the reversion and conversion of series, the theory of indeterminate equations, &c. Some of the elementary theorems and various particular problems appear in the works of the