INCOMMENSURABLE 132 INDEPENDENCE in general details to the Federal tax, in 1918 and the year following. This tax was to provide for increased expenses. INCOMMENSURABLE, a state of deprivation or non-existence (as regards two kindred quantities) of a common unit, or, more strictly, a magnitude which cannot be measured by another, taken as unity. Two quantities of the same kind are incommensurable with respect to each other, when they have not a common unit — that is, when there is no quantity of the same kind so small that it is con- tained in both an exact number of times. Thus the diagonal and side of a square are incommensurable, for it has been shown that if we denote the side of the square by 1, the diagonal will be denoted by the square root of 2; but the square root of 2 is incommensurable with 1, be- cause the square root of an imperfect square cannot be expressed in exact terms of 1. INCREMENT, in mathematics, a quantity, generally variable, added to the independent variable in a variable ex- pression. The function also undergoes a corresponding change, which is called an increment or decrement, according as the function is increasing or decreasing. When the increment or decrement is in- finitely small, it is called a differential. INCREMENT, UNEARNED, the in- crease in the rent of land due to the growth of industrial undertakings and of towns, and the general progress of so- ciety. INCUBATION, in pathology, the ma- turation of morbific matter introduced into the system. The word has refer- ence, also, to the act of sleeping for oracular dreams. Artificial incubation is the hatching of eggs by means of heat artificially ap- plied. The Egyptians have long done so successfully by means of heated ovens. Hatching eggs by incubation has become a lucrative business in many sec- tions of the United States. Period of incubation, in pathology, this is the interval that elapses between the introduction of morbific matter into the body, and the commencement of the disease thence resulting. INCUBATOR, a piece of apparatus in which a constant, predetermined tem- perature may be maintained. Its prin- cipal scientific use is for the growth of bacterial culture. It is frequently a double box of copper or other metal in- sulated with miner wool or felt. The space between the walls is usually filled with water, making it in effect a water bath. The source of heat is generally z Bunsen burner, although in some cases electric coils have been used. An ad- justable thermostat which operates on the gas flow, is used to control the tem- perature. The temperature desired, and the type of work for which the apparatus is to be used, largely determine the do- sign, but it is usual to provide a heavily insulated glass door, in order that the condition of the contents may be observed from time to time. See Incubation. INCUS, one of the small bones of the ear. It stands between the malleus and the stapes, and is connected with them by articular surfaces. INDEMNITY, an act designed to re- lieve the government or any of its of- ficers from penalties when they have been compelled by exceptional circumstances to omit the performance of some duty, or to violate or even to suspend some law. INDENTED, in heraldry, a term sig- nifying notched like the teeth of a saw, but smaller than dancette. Applied to one of the lines of partition; ordinaries are also thus borne. INDENTURE, a deed entered into be- tween two or more parties, and so called because duplicates of every deed between two or more parties were once written on one skin, which was cut in half, with a jagged or indented edge; so that they were seen to belong to one another. INDEPENDENCE, a city of Kansas, the county seat of Montgomery co. It is on the Verdigris River, and on the Atchison, Topeka and Santa Fe and the Missouri Pacific railroads. It is the center of an important gas and oil field region. Its industries include cotton mills, planing mills, fiour mills, a glass factory, cement plants, machine shops, etc. It has hospitals, a public library, an excellent park system, a courthouse, high schools, etc. Pop. (1910) 10,480; (1920) 11,920. INDEPENDENCE, a city and county- seat of Jackson co.. Mo.; on the Chicago and Alton, the Kansas City Southern, and the Missouri Pacific railroads; 4 miles S. of the Missouri river, and 10 miles E. of Kansas City. It is the seat of St. Mary's Academy. The city has public high school, public library, Wash- ington and Fairmount Parks, electric light plants, a National and other banks, and many important industrial estab- lishments. Independence was occupied by the Mormons in 1831-1838, and during this period was a noted rendezvous for their emigrants. Its nearness to Kansas City makes many of its business Interests
