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LEFT INDEPENDENCE 133 INDEX NUMBERS common with those of that city. Pop. (1910) 9,859; (1920) 11,686^ INDEPENDENCE, DECLARATION OF. See Declaration of Independ- ence. INDEPENDENCIA, also called Fray Bentos, a town in Uruguay, the capital of the Department of Rio Negro. It is on the Uruguay river. It has handsome public buildings and has an important port. The surrounding country is de- voted to stock-raising. The city has important meat-packing establishments. Pop., about 10,000. INDEPENDENT CATHOLIC CHURCH IN THE UNITED STATES, THE (POLISH), an independent church founded in Chicago by the Rev. Anthony Kozlowski, and consisting of Polish im- migrants who broke away from Catholic Church discipline. The church since its establishment has entered into relations with old Catholic congTegations in Eu- rope and the Protestant Episcopal Church in the United States, without however merging with either. The founder was made bishop at an Old Cath- olic conference in Europe. The church reports a growth in membership, and has schools, hospitals, orphanages, and homes for the aged. In 1915 40 churches, 35 ministers, and 18,500 members were re- ported. INDETERMINATE, in mathematics, having an indefiriite number of values or solutions. Indeterminate analysis is a branch of algebra in which there are always given a greater number of un- known quantities than there are inde- pendent equations, by which means the number of solutions is indefinite. INDETERMINATE SENTENCE, a .sentence of a person convicted of crime which is not a definite, but an indefinite length of time. Under the preceding practice of courts all prisoners were sen- tenced for a definite time, but in places where the indeterminate sentence is in use the judge simply gives the sentence as the law establishes it, e. g., not less than five nor more than twelve years. This method allows the judge to free the prisoner if he has demonstrated the gen- uineness of his reform and to hold him to the maximum sentence if he has not. This method has been approved by all the eminent penologists who maintain that reform of the prisoner is the aim of penology. INDEX, a compilation of figured or numbered entries for purposes of refer- ence. In mathematics, the index of a radical is a number written over the radical sign to denote the degree of the root to be extracted. An index is gen- erally a whole number greater than 2. When the square root is indicated, the index is generally omitted, being under- stood. Congregation of the Index is a congre- gation founded by Pius V. in 1571. It consists chiefly of cardinals, nominated by the Roman Pontiff, with other mem- bers, who are called consultors. The secretary is always a Dominican. Its duty is to examine and correct, or pro- hibit the reading of, books which it deems heretical, or which contain matter dan- gerous to faith or morals. INDEX, CEPHALIC, CRANIAL, PELVIC, terms used to designate the proportions of parts of the body or of a skeleton, employed especially by anthro- pologists in determining racial groups. The most commonly used index is the ratio of the width of the head to the length, known as the cephalic index when applied to living beings, and cranial index when applied to the skull. The length of the skull is taken just above the eyes where the forehead protrudes most, while the width is taken at any part of the head having the greatest breadth. The metric system is used in all measure- ments. Tables of index numbers have been compiled so that indices may be determined without actually working out the accepted formula, — Width x 100 =1. Length Heads may be classified also accord- ing to indices other than those of length and breadth of the skull: — facial, nasal, dental, and pelvic being often used. The cephalic index does not always give ac- curate information, because it is taken only on the upper part of the head, and is not a sufficient guide to the shape of the head as a whole. Heads having an entirely different cross sectional outline would fall into the same class by this method of measurement, so it can be taken as an exact guide only when a large number of skulls of the same race are observed to give the same result. INDEX NUMBERS are used by eco- nomic statisticians to measure tba amount of change that takes place in the value of money, as shown in its rela- tive purchasing power. It is obvious that money itself could not be used as the measure of its own value, nor could any one commodity which would be pur- chased with a fixed sum of money be used, since single commodities fluctuate in value even more than money. How- ever, the average price fluctuations of a number of different commodities shor