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his public duties and position. His character in other respects was always of stainless integrity.

Bibliography.—A collected edition of Calhoun’s Works (6 vols., New York, 1853–1855) has been edited by Richard K. Crallé. The most important speeches and papers are:—The South Carolina Exposition (1828); Speech on the Force Bill (1833); Reply to Webster (1833); Speech on the Reception of Abolitionist Petitions (1836), and on the Veto Power (1842); a Disquisition on Government, and a Discourse on the Constitution and Government of the United States (1849–1850)—the last two, written a short time before his death, defend with great ability the rights of a minority under a government such as that of the United States. Calhoun’s Correspondence, edited by J. Franklin Jameson, has been published by the American Historical Association (see Report for 1899, vol. ii.). The biography of Calhoun by Dr Hermann von Holst in the “American Statesmen Series” (Boston, 1882) is a condensed study of the political questions of Calhoun’s time. Gustavus M. Pinckney’s Life of John C. Calhoun (Charleston, 1903) gives a sympathetic Southern view. Gaillard Hunt’s John C. Calhoun (Philadelphia, 1908) is a valuable work.  (H. A. M. S.) 

CALI, an inland town of the department of Cauca, Colombia, South America, about 180 m. S.W. of Bogotá and 50 m. S.E. of the port of Buenaventura, on the Rio Cali, a small branch of the Cauca. Pop. (1906 estimate) 16,000. Cali stands 3327 ft. above sea-level on the western side of the Cauca valley, one of the healthiest regions of Colombia. The land-locked character of this region greatly restricts the city’s trade and development; but it is considered the most important town in the department. It has a bridge across the Cali, and a number of religious and public edifices. A railway from Buenaventura will give Cali and the valley behind it, with which it is connected by over 200 m. of river navigation, a good outlet on the Pacific coast. Coal deposits exist in the immediate vicinity of the town.

CALIBRATION, a term primarily signifying the determination of the “calibre” or bore of a gun. The word calibre was introduced through the French from the Italian calibro, together with other terms of gunnery and warfare, about the 16th century. The origin of the Italian equivalent appears to be uncertain. It will readily be understood that the calibre of a gun requires accurate adjustment to the standard size, and further, that the bore must be straight and of uniform diameter throughout. The term was subsequently applied to the accurate measurement and testing of the bore of any kind of tube, especially those of thermometers.

In modern scientific language, by a natural process of transition, the term “calibration” has come to denote the accurate comparison of any measuring instrument with a standard, and more particularly the determination of the errors of its scale. It is seldom possible in the process of manufacture to make an instrument so perfect that no error can be discovered by the most delicate tests, and it would rarely be worth while to attempt to do so even if it were possible. The cost of manufacture would in many cases be greatly increased without adding materially to the utility of the apparatus. The scientific method, in all cases which admit of the subsequent determination and correction of errors, is to economize time and labour in production by taking pains in the subsequent verification or calibration. This process of calibration is particularly important in laboratory research, where the observer has frequently to make his own apparatus, and cannot afford the time or outlay required to make special tools for fine work, but is already provided with apparatus and methods of accurate testing. For non-scientific purposes it is generally possible to construct instruments to measure with sufficient precision without further correction. The present article will therefore be restricted to the scientific use and application of methods of accurate testing.

General Methods and Principles.—The process of calibration of any measuring instrument is frequently divisible into two parts, which differ greatly in importance in different cases, and of which one or the other may often be omitted. (1) The determination of the value of the unit to which the measurements are referred by comparison with a standard unit of the same kind. This is often described as the Standardization of the instrument, or the determination of the Reduction factor. (2) The verification of the accuracy of the subdivision of the scale of the instrument. This may be termed calibration of the scale, and does not necessarily involve the comparison of the instrument with any independent standard, but merely the verification of the accuracy of the relative values of its indications. In many cases the process of calibration adopted consists in the comparison of the instrument to be tested with a standard over the whole range of its indications, the relative values of the subdivisions of the standard itself having been previously tested. In this case the distinction of two parts in the process is unnecessary, and the term calibration is for this reason frequently employed to include both. In some cases it is employed to denote the first part only, but for greater clearness and convenience of description we shall restrict the term as far as possible to the second meaning.

The methods of standardization or calibration employed have much in common even in the cases that appear most diverse. They are all founded on the axiom that “things which are equal to the same thing are equal to one another.” Whether it is a question of comparing a scale with a standard, or of testing the equality of two parts of the same scale, the process is essentially one of interchanging or substituting one for the other, the two things to be compared. In addition to the things to be tested there is usually required some form of balance, or comparator, or gauge, by which the equality may be tested. The simplest of such comparators is the instrument known as the callipers, from the same root as calibre, which is in constant use in the workshop for testing equality of linear dimensions, or uniformity of diameter of tubes or rods. The more complicated forms of optical comparators or measuring machines with scales and screw adjustments are essentially similar in principle, being finely adjustable gauges to which the things to be compared can be successively fitted. A still simpler and more accurate comparison is that of volume or capacity, using a given mass of liquid as the gauge or test of equality, which is the basis of many of the most accurate and most important methods of calibration. The common balance for testing equality of mass or weight is so delicate and so easily tested that the process of calibration may frequently with advantage be reduced to a series of weighings, as for instance in the calibration of a burette or measure-glass by weighing the quantities of mercury required to fill it to different marks. The balance may, however, be regarded more broadly as the type of a general method capable of the widest application in accurate testing. It is possible, for instance, to balance two electromotive forces or two electrical resistances against each other, or to measure the refractivity of a gas by balancing it against a column of air adjusted to produce the same retardation in a beam of light. These “equilibrium,” or “null,” or “balance” methods of comparison afford the most accurate measurements, and are generally selected if possible as the basis of any process of calibration. In spite of the great diversity in the nature of things to be compared, the fundamental principles of the methods employed are so essentially similar that it is possible, for instance, to describe the testing of a set of weights, or the calibration of an electrical resistance-box, in almost the same terms, and to represent the calibration correction of a mercury thermometer or of an ammeter by precisely similar curves.

Method of Substitution.—In comparing two units of the same kind and of nearly equal magnitude, some variety of the general method of substitution is invariably adopted. The same method in a more elaborate form is employed in the calibration of a series of multiples or sub multiples of any unit. The details of the method depend on the system of subdivision adopted, which is to some extent a matter of taste. The simplest method of subdivision is that on the binary scale, proceeding by multiples of 2. With a pair of sub multiples of the smallest denomination and one of each of the rest, thus 1, 1, 2, 4, 8, 16, &c., each Weight or multiple is equal to the sum of all the smaller weights, which may be substituted for it, and the small difference, if any, observed. If we call the weights A, B, C, &c., Where each is approximately double the following weight, and if we write a for observed excess of A over the rest of the weights, b for that of B over C+D+&c., and so on, the observations by the method of substitution give the series of equations.

A −rest = a, B−rest = b, C−rest = c, &c. . . (1)

Subtracting the second from the first, the third from the second, and so on, we obtain at once the value of each weight in terms of the preceding, so that all may be expressed in terms of the largest, which is most conveniently taken as the standard

B = A/2+(ba)/2, C = B/2+(cb)/2, &c. . . (2)

The advantages of this method of subdivision and comparison, in addition to its extreme simplicity, are (1) that there is only one possible combination to represent any given weight within the range of the series; (2) that the least possible number of weights is required to cover any given range; (3) that the smallest number of substitutions is required for the complete calibration. These advantages are important in cases where the accuracy of calibration is limited by the constancy of the conditions of observation, as in the case of an electrical resistance-box, but the reverse may be the case when it is a question of accuracy of estimation by an observer.

In the majority of cases the ease of numeration afforded by familiarity with the decimal system is the most important