Measurement is the process of obtaining quantitative information about the physical world. Methods for the collection of numerical data and for the inaccuracy of measurements are intimately associated with the growth of technology. This article discusses the methods that technology has developed for measuring particular fundamental quantities. UNITS AND STANDARDS Any measurement must involve the comparison of the measured quantity with a known standard unit. In absolute measurement, the unit may be the official unit for the quantity considered, such as the meter or the ampere. In relative measurement, a special reference unit is chosen for a given measurement; for example, the brightness of a star is expressed in terms of the brightness of another star. A length of 3.6 meters means that the measured length is 3.6 times as large as a standard length, in this case the meter. Until 1960, the standard meter was equal to the length of a prototype meter bar kept in Paris. It was then redefined as 1,650,763.73 times the wavelength of the radiation emitted at a specified energy level by krypton-86. In 1983 it was redefined as the length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second. This definition has the great advantage of being reproducible in any well-equipped laboratory, rather than depending on an actual object. The wide variety of units and standards employed worldwide are similarly based on physical quantities. The presently agreed-on system of units used for scientific work in many countries, known as the International System of Units, or simply the SI system, is based on the mks (meter-kilogram-second) system and contains seven base units of length, mass, time, temperature (kelvin), luminous intensity (candela), amount of substance (mole), and electric current (ampere). The mole is a dimensionless chemical unit that cannot be measured directly; the others are directly measurable. MEASUREMENT OF BASIC SI NONELECTRICAL QUANTITIES In addition to the basic dimensionally independent SI quantities, there are many other measurements closely related to these basic quantities. Thus, measurement of thermal conductivity is related to measurement of temperature, even though the two have differing dimensions. In order to measure the heat conduction in a copper rod, for example, one must determine both the rod length and the temperature at each end of the rod. The quantity of heat transmitted from one end to the other within a certain time is then determined. This quantity may be determined by measuring the initial and final temperatures of a known amount of water into which one end of the rod is immersed. The survey below deals with the principles of measuring the standard quantities as well as certain derived quantities. Length Measurements. Length measurements play a special role in measurement technology, because nearly all other analog measurements (those involving continuous--rather than stepwise, or digital--monitoring) may be reduced to measurements of length. The simplest measurement of this kind is carried out with a ruler, in which case an accuracy of approximately 1 mm can be achieved. A VERNIER caliper will correctly measure to 0.05 mm, while an accuracy of 0.01 mm is possible with a screw \Tmicrometer\t. Accurate length comparisons to 0.001 mm are possible with \Lcaliper\ls or end gauges. These are then used as length substandards, with which it is only possible to determine whether an object has the same dimensions as the substandard. Measurements of even smaller lengths are performed by the optical enlargement of an image with a microscope; the enlargement may then be measured against an ordinary scale. In this case, absolute measurement requires a knowledge of the magnification of the microscope, and since the precision with which this is known is never very great, the measurement itself cannot be considered very accurate. The most accurate length measurements require the use of an \Tinterferometer\t. With this instrument, a measured distance may be compared to a given wavelength of light, which is accurately known. The standard meter is now based on such measurements. The determination of such quantities as angles, areas, deformation, and velocity depends on accurate measurement of length. Angular measurements can be derived from length measurements if a circular ruler is used. If a straight ruler is used, the values found must be converted by using trigonometric functions. The most sensitive systems for measuring angles use mirrors in which light rays are reflected on scales. It is remarkable that there are no direct methods of measuring the areas of arbitrary surfaces, although the determination of areas for regular surfaces through mathematical relationships (such as length times width for the rectangle) again depends on a knowledge of length. Similarly, the deformation of an object by a force can be measured as a displacement, or a change in length. A well-known example is the stretched-spring principle used in many pointer-type instruments. The pointer rests at a position of equilibrium between an acting force and the restoring force exerted by the spring. Deformations are also used in various pressure gauges and \Lmanometer\ls. Finally, velocity measurements often involve measurements of a path traversed by an object during a known time interval. This is not only true for the classical methods for determining the velocity of light, but also for many more commonplace velocity measurements. Mass Measurements. The mass m of an object is measured by means of the force W, or weight, exerted upon it by the Earth. This force is related to the mass through the expression W = mg, where g is the known acceleration of gravity. This acceleration may in turn be measured as the distance a falling object covers within a given time frame. The simplest means for measuring mass is to use a spring balance; however, such a device is inherently inaccurate. The measurement of mass is therefore usually carried out as a null measurement, in which one mass is compared to another, known mass. A BALANCE is used for this purpose. Comparison is made through the use of several standard weights, which generally cannot be lighter than several milligrams. Smaller differences in weight are compared to a force exerted upon the balance in one of three ways. In a beam balance, the counterforce is related to the inclination of the balance beam. In a torsion balance, the balance beam is returned to its equilibrium position by the torque exerted by a wire. This torsion wire can also be used as the suspension for the beam in a very sensitive system. The principle of a third type, the electrical balance, is similar to that of the torsion balance, but the force (known here as the Lorentz force) is exerted by a specific current that flows through a coil placed in a magnetic field. A wide variety of measurements are performed as weight determinations. Forces can be measured by letting them act upon one arm of a balance. Accelerations can be measured by determining the forces acting on a known mass in a given time frame. This method also permits the measurement of rotational speeds, since the number of revolutions per second can be determined from a known centrifugal acceleration. Measurements of volume and density (mass per unit volume) are also closely related to determinations of mass. Knowledge of the density of an object permits direct computation of its volume by weighing. By the same token, density can be determined by weighing a known volume of the substance considered. It is always possible to measure the volume of gases and liquids; if the volume of a solid cannot be determined directly, one can use a pycnometer, in which the solid is immersed in a liquid and the displaced volume of liquid is measured. Finally, the pressure (force per unit area) that causes a column of liquid to rise in a manometer can be used to compute the weight of the raised liquid, if the surface area over which the pressure acts is known and the level difference between the two legs of the manometer is measured. Time Measurements. Time measurements are always based on counting periodic phenomena, such as oscillations of atoms and molecules, electromagnetic oscillations in oscillators, and sound or mechanical vibrations. The use of these time standards results in a variety of clocks (see \Tclocks and watches\t), including the \Tatomic clock\t, the quartz crystal clock, and the common watch and pendulum clock. Temperature Measurements. There are two fundamental laws on which temperature measurements may be based. The best-known law is that of Robert Boyle and J. L. Gay-Lussac (see \Tgas laws\t), according to which gas pressure depends on temperature. Temperature measurement thus becomes a pressure measurement that can be performed as a length measurement with the aid of a manometer. The radiation law, according to which the quantity of radiation emitted by a substance depends on temperature, is less well known. The measurement in this case is that of the radiation intensity as determined by a \Tbolometer\t. Several other methods of temperature measurement are also known. These make use of \Lthermocouple\ls, \Lthermometer\ls, bimetals, and vapor-pressure thermometers. The field of calorimetry is closely related to temperature measurement. It involves determination of the initial and final temperatures of a given, weighed amount of a liquid whose specific heat is known. Calorimetry forms the basis for a determination of specific heat, thermal conductivity, and energy, through the conversion of energy, or work, into heat in a \Tcalorimeter\t. Measurement of Luminous Intensity. Luminous intensity may be determined by either absolute or relative measurements. Relative measurements involve comparison of the strength of an unknown light source to that of a known (and variable) light source. This can be done very accurately through a visual null measurement by attenuating the light from the known source until both sources appear to be equally bright. Attenuation is possible through the use of a diaphragm or a light-absorbing prism, or by varying the distance to one of the light sources. Determination of the null point is then followed by the actual measurement. Absolute light measurements are carried out by means of visual null comparison to a radiation standard, such as a piece of tungsten wire heated electrically to a specified temperature; the amount of radiation thus emitted is known from the radiation laws. Another type of absolute measurement makes use of calibrated radiation-absorption meters. \Tpolarized light\t is used with many measuring methods. The measurement of wavelengths combined with a determination of the corresponding intensities form the basis of \Tspectroscopy\t, which is of great importance in atomic physics, chemistry, and astronomy. MEASUREMENT OF ELECTRICAL QUANTITIES The measurement of electric current is chiefly based on the 19th-century discovery by Hans Christian \TOersted\t and Michael \TFaraday\t of the relationship between electricity and magnetism. The fundamental association is the phenomenon, first reported by Oersted in 1820, that an electric current passing through a conductor produces a magnetic field, which in turn exerts a force on other currents near it. Two currents thus always exert a force on each other (the law of Biot and Savart), and the measurement of the force exerted by one on the other gives a measure of the current. This fact is also used in the definition of the unit of current: one ampere is the current that gives rise to a force of 2 X (10 to the power of -7) newtons between two perfect conductors of infinite length, in vacuo, carrying the current at a distance of one meter from each other. In practical measuring instruments that make use of this phenomenon, the conductors consist of two coils with a large number of turns. By connecting these coils in different ways, such electrodynamometers as \Lammeter\ls, \Lvoltmeter\ls, and wattmeters may be constructed, as well as a class of instruments known as Ferraris meters. Measurement with an electrodynamometer is based on determining the force acting between the two coils through which current flows. This is effected by measuring the deviation from equilibrium of a free-turning coil opposed by a small spring. In Ferraris meters the two coils are fixed with respect to a piece of metal that is free to turn, and the measurement is based on the eddy currents induced in the metal. In all of these two-coil instruments, the magnetic field is weak when the current intensity is low, so that these meters are not very sensitive. However, they can measure both direct and alternating currents. Another widely used electromagnetic current-measuring instrument is the moving-coil meter, or \Tgalvanometer\t, in which the current to be measured flows through a coil suspended in a strong magnetic field induced by a permanent magnet rather than by the current itself. Such instruments measure only direct current. The operation of a hot-wire meter is not based on the action of electromagnetic forces. The current in such an instrument flows through a resistance wire, causing it to heat and expand. The actual measurement is thus not an electrical but a thermal one, related to the electrical quantity. An oscilloscope, based on the electrostatic deflection of a beam of electrons sent between a pair of oppositely charged plates, can also be adapted for current and voltage measurements, but it is primarily of importance in investigations of periodic phenomena. While certain electrical phenomena such as the photoelectric effect allow for direct measurement of an emitted electric current, other electrical measurements are performed with the aid of an external current or voltage source, so that the resistance, self-inductance, or capacitance can be determined. These measurements are in the end also based on determining the current intensity or voltage. They are performed chiefly with a \TWheatstone bridge\t, a circuit that requires a current source, a number of comparison resistances, and a calibrated \Tpotentiometer\t. The measurement is based on a null measurement of the voltage between two tapping points. TRANSDUCER MEASUREMENTS When direct measurements of a particular quantity are impossible, some other measurable quantity can often be found that is linked to the former by some law. Many measuring instruments convert one form of energy into another. This is called transduction; the converter itself is called a \Ttransducer\t. Although transducers as a group involve many forms of energy, in practice it is often convenient to convert a quantity into an electrically measurable one. The measurement signal is then available in the form of a current-to-voltage signal that can be processed in a number of different ways that are either impossible or very difficult to do with mechanical signals. Common electrical transducers include the photoelectric cell, the Geiger counter, the thermocouple, and the piezoelectric crystal. MEASUREMENT ERRORS The result of a measurement and the actual value of the quantity to be measured are often not precisely equal. The difference between these two values may be due either to random errors or to systematic errors. Random errors are those that occur in the act of measurement itself; systematic errors occur as a result of instrument faults and calibration mistakes. Random Errors. In order to obtain a meaningful measurement, one must always specify the precision with which it is made--that is, the limits between which the measured quantity lies. The interval within which the real measurement value lies determines the absolute error of the measurement. For instance, the absolute error is approximately 1 mm when a length is measured with a ruler 1 m long. The relative error is equal to the absolute error divided by the value measured, and is usually expressed as a percentage. Two values, the arithmetic mean and the dispersion (the width, or spread, of the distribution curve), characterize the distribution of the possible measurement results. The arithmetic mean of the probability distribution coincides with the real value of the quantity measured when there are no systematic errors. Two guidelines can be established for random errors: (1) repeating a measurement yields information on the magnitude of the random errors, and (2) repeating a measurement reduces the error in the final result in proportion to the square root of n where n is the number of measurements taken. The random error decreases rapidly at the beginning, but then more slowly. Once systematic errors begin to predominate, random error cannot be further reduced. Systematic Errors. It is more difficult to estimate and reduce the magnitude of systematic errors. The measuring process must be analyzed carefully in each case. Every type of measurement has its own characteristic systematic errors, but some of the most prominent may be enumerated here: 1. Null-point errors, caused by a measuring error in the null condition or by a faulty null setting of the instrument, will often result in a constant shift of all measured values. 2. Calibration errors result when the conditions under which the reference measurement (calibration) is taken do not approximate the conditions of the actual measurement as nearly as possible. For instance, a ruler calibrated at a certain temperature will give a constant relative deviation if it is used at a different temperature. 3. The measuring instrument itself nearly always influences the magnitude of the signal to be measured. Measuring an electric potential difference lowers the actual voltage reading, because the measuring instrument places a load on the voltage source. 4. \Thysteresis\t and lost motion are phenomena in which the indication of a measuring instrument depends on its previous reading. 5. Parallax errors result from the fact that the pointer in most dial instruments is located at a slight distance from the scale, and the reading thus depends on the angle from which it is taken. The methods for dealing with random errors are easier to use than those for dealing with systematic ones. The best way to correct for systematic errors is therefore to convert them to random ones. This can be done by introducing as many variations as possible into the measuring method and instrument. For instance, it is easy to overlook a systematic error that is introduced when the same slow stopwatch is used for several time measurements (an example of calibration error). However, repeating the measurements with a number of independently calibrated stopwatches will cause the calibration errors to assume a random distribution. Application of a completely different measuring method is a time-consuming but thorough way of detecting systematic errors. REVIEWED BY \TSteven\t J. \TDick\t Bibliography: Anthony, D. M., Engineering Metrology (1987); Bailey, Harold J., Measurements and the Metric System (1976); Bottaccini, M. R., Instruments and Measurement (1975); Dilke, O. A., Mathematics and Measurement (1987); Doeblin, E. O., Measurement Systems, 3d ed. (1982); Drazil, J. V., Quantities and Units of Measurement (1983); Geczy, Steven, Basic Electrical Measurements (1984); Hewitt, P. L., Modern Techniques in Metrology (1984); Johnstone, William, For Good Measure (1976); Klein, A. Arthur, The World of Measurements (1974); Liebman, J. F., and Greenberg, A., eds., Physical Measurements, vol. 2 (1986); Sirohi, R. S., and Krishna, R., Mechanical Measurements (1983). See also: LABORATORY \Ttechnology\t; \Tprocess control\t; \Tweights and measures\t.