Standards of length are all defined on metal bars at present in civilized countries. Various natural standards have been proposed, such as the length of the polar diameter of the earth (inch), the circumference of the earth (metre) in a given longitude, a pendulum vibrating in one second at a fixed distance from the earth, a wave of light emitted by an incandescent gas, &c. But the difficulty of ascertaining the exact value of these lengths prevents any material standard being based upon them with the amount of accuracy that actual measurements, to be taken from the standard, require. A natural standard is therefore only a matter of sentiment.

Standards of length are of two types, the defining points being either at a certain part of two parallel lines engraven in one plane (a line-standard), or else points on two parallel surfaces, which can only be observed by contact (an end-standard). The first type is always used for accurate purposes. Units of surface are always directly related to standards of length, without any separate standards. Volume is either determined by the lineal dimensions of a space or a solid, or, for accurate purposes, by the mass of water contained in a volume at a given temperature, which again is measured either by liquid measure, or, more accurately, by weight. The standard of volume in Britain is a hollow cylinder of bronze, with a plate glass cover, when at 62° (gallon), legally defined as 277⋅274 cubic inches, or containing 10 pounds of water at 62° F.

Comparisons.— Lengths nearly equal are compared accurately by fixing two micrometer microscopes with their axes parallel, and at the required distance apart, on a massive support which will not quickly vary with temperature; then the two lengths to be compared, e.g., the standard yard and another, are alternately placed beneath the microscopes, and their lengths observed several times. The error of a single observation in the Standards Department is stated to be a 100,000th of an inch. For fractional lengths a divided bar is required, the accuracy of which is ascertained by a shorter measure, which can be compared with successive sections of the whole length by micrometers. For ordinary purposes, where not less than⋅001 inch is to be observed, measures may be placed in contact if one is divided on the edge, and the comparison made with a magnifier. In large field-work the ends of a measure are transferred vertically to the ground by a small transit instrument or theodolite. End-measures between surfaces are read by means of a pair of contact pieces bearing line marks, the value of which is ascertained separately, or by a second end-measure; if both measures bear a line for observation, reversals then give the value of each measure.

Volumes are always most accurately defined by their weight of water, as weighing can be more accurately done than measuring. If the volume is hollow, it may be filled with water and closed with a sheet of plate glass, or if solid the body may be weighed in water and out, the difference giving the weight of its volume of water. Unfortunately the relation between water-weight and absolute volume is not yet accurately known. Volumes of liquid are similarly ascertained by their weight. Volumes of gas are measured in a graduated glass vessel inverted over a liquid, or for commercial purposes by some form of registering flow-meter.

Masses are compared by the Balance ( q.v.), which may be made to indicate a 100,000,000th of the mass. They may also be estimated, not by their attractive force being balanced by an equal mass, but by the elasticity of a spring; this, which is the only true weighing-machine showing weight and not mass, is useful for rough purposes, owing to its quick indication; the most accurate form probably is that with angular readings. Temperature and the Atmosphere.— All the serious difficulties of weighing and measuring result from these causes, the effects of which and their corrections we will briefly notice. In measurement, since all bodies expand by heat, the temperature at which any measure or standard bar represents the abstract unit requires to be accurately stated and observed, the accuracy of optical observation being about equal to 1 / 100 of a degree F. of expansion in a standard. Great accuracy is therefore needed in the manufacture and reading of thermometers, and care that the standard and the thermometer shall be at the same temperature. Another method is to attach a parallel bar of very expansible metal to one end of the standard, and read its length on the standard at the other end; this ensures a more thorough uniformity of mean temperature between the standard and the heat-measurer. The most accurate method is by immersing the measures in a liquid, of which the temperature is read by several thermometers; but this is scarcely needful unless high or low temperatures are required to ascertain the rate of expansion. A room with thick walls, double windows, and the temperature regulated by a gas stove is practically sufficiently equable for comparisons.

The temperature adopted for the standards is not the same in different countries. In Britain 62° F. has been adopted since the revision of the standards in 1822, as being a convenient average temperature for work; but, as it is purely a temperature of convenience, the rather higher point of 68° F. would be better. In any case an aliquot part of the thermal unit from freezing to boiling of water should be adopted; 62° is ⅙ and 68° is ⅕ of this interval. Whether a much higher temperature would not be more conducive to accuracy is a question; 92°, or ⅓ of the thermal unit, would be so near the temperature of the observer’s skin and breath that measures and balances could be approached with less production of error; and such a heat does not at all hinder accurate observing. The French temperature of 32° F. for standards has abandoned all other considerations in favour of readily fixing the temperature in practice by melting ice. This is a ready means of regulation, but a point so far from ordinary working temperatures has two great disadvantages: the observer’s warmth produces more error, and the corrections for all observations not iced are so large that the rates of expansion require to be known very accurately for every substance employed. For water their standard temperature is 39°⋅2 F., when it is at its maximum density; this has the advantage that the density varies less with temperature than at any other point, but it is very doubtful if this is much used for actual work.

No substance expands uniformly with temperature, most materials expanding more rapidly at higher temperatures. The expansion of rods of the following metals, of 100 inches long, is given in decimals of an inch for the 90° from 32° to 122° F. (0 to 50° C.), and from 122° to 212° F. (50° to 100° C.): — [table]

But variations of 3 or 4 per cent. may easily be found in the rates of different specimens apparently alike; hence the individual expansion of every important measure needs to be ascertained.

Weighing is complicated by being done in a dense and variable atmosphere, unless—as in the most refined work —the whole balance is placed in a vacuum. When in the air all bodies placed in the balance must, for accurate purposes, have their volume known; and the weight of an equal volume of such air as they are weighed in must be added to their apparent weight to get their true weight. The weight of air displaced by a pound of the following materials is given in grains, at temperature 62° F., barometer 30 inches,—also with barometer 29 inches (temperature 62°), and with temperature 32° (barometer 30 inches), to illustrate the variation (allowing for contraction of the material as well):— [table]

The above is for London at sea-level; but where the force of gravity is less 30 inches height of mercury will weigh less, and will therefore balance a less weight of air; the air allowance must therefore be less for 30 inches of mercury barometer in lower latitudes and greater heights over sea-level. The change, for instance, in the allowance of air equal to the brass pound will make it, instead of

1Ό47 grains, become 1⋅046 when 15,000 feet above the sea, or 10° S. of London. Hence this reduction need rarely be noticed. The composition of the air also varies, and most seriously in the amount of aqueous vapour; the above is ordinary air, but if quite dry the 1⋅047 grains would become 1⋅052 grains; the change in carbonic acid is quite immaterial, unless in very close rooms, so that it may be concluded that the moisture of the air is the main point to be noted, after its temperature and pressure,— small errors in any of these three data making far more difference than any other compensation that can be made in the weight of air.

The more complex allowances for the expansion of water in glass, brass, or other vessels we need not enter on here; the principles are simple, but the data require to be accurately determined for the material in question. The expansion of water is, however, so often in question, especially for taking specific gravities, that it is here given. A constant volume which contains or displaces 10,000 grains of water at 62° will contain — [table]

Hence if a specific gravity is observed at any of these temperatures it must be × the corresponding weight ÷ 10,000 to reduce it to a comparison with water at 62°; the expansion of the body observed is another question altogether, and must be compensated also.

The weight of a cubic inch, or other linearly measured volume, of water is not yet very accurately known. The observations have been made by weighing closed hollow metal cases in and out of water (thus obtaining the weight of an equal volume of water), and then gauging the size of the case with exactitude. Cubes, cylinders, and spheres have been employed. The results are: — [table]

National Standards and Copies.— Having now noticed the principles and constants involved, we will consider the British and metric standards, the only ones now used in scientific work.

The imperial standard yard is a bronze bar 38 inches long, 1 inch square; the defining lines, 36 inches apart, are cut on gold studs, sunk in holes, so that their surface passes through the axis of the bar. Thus flexure does not tend to tip the engraved surfaces nearer or farther apart. This bar when in use rests on a lever frame, which supports it at 8 points, 4⋅78 inches apart, on rollers which divide the pressure exactly equally. This standard is in actual use for all important comparisons at the Standards Office. Four copies, which are all equal to it, within ⅙° of temperature, are deposited in other places in case of injury or loss of the standard. The standard pound is a thick disk of platinum about 1⅙ inches across, and 1 inch high, with a shallow groove around it near the top. Four copies are deposited with the above copies of the yard. For public use there are a series of end-standards exposed on the outer wall of Greenwich observatory; and a length of 100 feet, and another of 66 feet (1 chain), marked on brass plates let into the granite step along the back of Trafalgar Square. As this is a practically invariable earth-fast standard, and most convenient for reference, it is important to know the minute errors of it, as determined by the Standards Department. Starting from 0 the errors are in inches—

The mean uncertainty in these values is Ό03, and the greatest uncertainty⋅01. The total length of the chain standard is - ⋅019 inch from the truth. There is also a public balance provided at Greenwich observatory, which shows the accuracy of any pound weight placed upon it. For important scientific standards comparisons are made gratuitously, as a matter of courtesy, by the officials of the Standards Office, 7 Old Palace Yard, Westminster. The most delicate weighings are all performed in a vacuum case with glass sides, which is so constructed that the weights can be exchanged from one arm to the other without opening the case, so as to obtain double weighings.

The toleration of error in copies for scientific purposes, by the Standards Department, is ⋅0005 inch on the yard or lesser lengths, about equal to 15 divisions of the micrometer; on the pound ⋅0025 grain, about ½ a division of the official balances; on the ounce ⋅001 grain; on the gallon 1 grain; and on the cubic foot 4 grains. The toleration for commercial copies is ⋅005 on the yard, ⋅001 on the foot and under, and Ί grain on weights of 1 ounce to 1 pound. The Standards Commission of 1851 recommended a limit of 1 in 20,000.

For practical work of moderate accuracy the most convenient forms of measures are—for lengths under a foot, feather-edge metal scales divided to 1 / 50 inch (finer divisions are only confusing, and 1 / 1000 of an inch can be safely read by estimation); for lengths of 1 to 10 feet, metal tubes or deep bars bearing line-divisions, and with permanent feet attached at 21 per cent. from either end, so that the deformation by flexure is always the same; for long distances a steel tape with fine divisions scratched across it and numbered by etching. The most accurate way of using such a tape is not to prepare a flat bed for it, but to support it at points not more than 50 feet apart; then by observing the distances and levels of these points, and knowing the weight of the tape, the correction for the sloping distance between the points and the difference of the catenary length of the tape from the straight distance can be precisely calculated; the corrections for stretching of the tape (best done by a lever arm with fixed weight), and for temperature, are all that are needful besides.

The first French standard metre (of 1799) is a platinum bar end-standard of about 1 inch wide and 1 / 7 inch thick; the new standard of the International Metric Commission is a line-standard of platino-iridium, 40 inches long and ⋅8 inches square, grooved out on all four sides so that its section is between × and H form; this provides the greatest rigidity, and also a surface in the axis of the bar to bear the lines of the standard. The new standard kilogramme, like the old one, is a cylinder of platinum of equal diameter and height. These new standards are preserved in the International Metric Bureau at Paris, to which seventeen nations contribute in support and direction, and in which the most refined methods of comparison are adopted. For lineal comparisons the alternate substitution of the measures on a sliding bed beneath fixed micrometer microscopes is provided as in the British office, and a bath for the heating of one measure in a liquid to ascertain its expansion. For weighing four balances are provided, each with mechanism for the transposition of the weights, and the lowering of the balance into play on its bearings, so that weighings can be performed at 13 feet distance from the balance, thus avoiding the disturbance caused by the warmth of the observer. The readings of the balance scale are made by a fixed telescope, the motion being observed by the reflexion of a fixed scale from a mirror attached to the beam of the balance. In this bureau are also an equally fine hydrostatic balance for taking specific gravities by water weighing, a standard barometer, and an air thermometer, with all subsidiary apparatus. The special work of the bureau is the construction and comparison of metric standards for all the countries supporting it, and for scientific work of all kinds.

The legal theory of the British system of weights and measures is—(A) the standard yard, with all lineal measures and their squares and cubes based upon that; (B) the standard pound of 7000 grains, with all weights based upon that, with the troy pound of 5760 grains for trade purposes; (C) the standard gallon (and multiples and fractions of it), declared to contain 10 lb of water at 62° F., being in volume 277⋅274 cubic inches, which contain each 252⋅724 grains of water in a vacuum at 62°, or 252⋅458 grains of water weighed with brass weights in air of 62° with the barometer at 30 inches.

The legal theory of the metric system of weights and measures is—(A) the standard metre, with decimal fractions and multiples thereof; (B) the standard kilogramme, with decimal fractions and multiples thereof; (C) the litre (with decimal fractions and multiples), declared to be a cube of 1 / 10 metre, and to contain a kilogramme of water at 4° C. in a vacuum. No standard litre exists, all liquid measures being legally fixed by weight. The metre was supposed, when established in 1799, to be a ten-millionth of the quadrant of the earth through Paris; it differs from this theoretical amount by about 1 in 4000.

The legal equivalents between the British and French systems are—metre = 39⋅37079 inches; kilogramme = 15432⋅34874 grains. By the more exact comparisons of Captain Clarke (1866) the metre (at 0° C., 32° F.) is equal to 39⋅37043196 inches of the yard at 62° F.; but Rogers in 1882 compared the metre as 39⋅37027. It must always be remembered that a French metre of perfect legal exactitude will, by expanding from 32° to 62° F., become equal to a greater number of inches when the two measures are placed together; thus a brass metre is equal to 39⋅382 inches when compared with British measures at the same temperature, and this is its true commercial equivalent. The kilogramme determination above is that of Professor Miller (1844), against the kilogramme des Archives, but in 1884 the international kilogramme yielded 15432⋅35639.

For further details, see H. W. Chisholm, Weighing and Measuring (Nature series), 1877, and Reports of the Warden of the Standards, subsequently of the Standards Department (all in British Museum Newspaper Room), for all practical details,—especially reports on metre (1868-9), errors of grain weights (1872), principles of measuring (long paper of German Standards Commission, translated 1872), Trafalgar Square standards (1876), density of water (1883), toleration of error, British and international (1883), standard wire and plate gauges in inches (1883), besides numerous practical tables, mainly in the earlier numbers before the wardenship was merged in the Board of Trade.

II. Historical.

Though no line can be drawn between ancient and modern metrology, yet, owing to neglect, and partly to the scarcity of materials, there is a gap of more than a thousand years over which the connexion of units ] of measure is mostly guess-work. Hence, except in a few cases, we shall not here consider any units of the Middle Ages. A constant...