Physical Measurements

The greater number of the physical experiments of the present day and the whole of those described in this book consist in, or involve, measurement in some form or other. Now a physical measurement - a measurement, that is to say, of a physical quantity - consists essentially in the comparison of the quantity to be measured with a unit quantity of the same kind. By comparison we mean here the determination of the number of times that the unit is contained in the quantity measured, and the number in question may be an integer or a fraction, or be composed of an integral part and a fractional part. In one sense the unit quantity must remain from the nature of the case perfectly arbitrary, although by general agreement of scientific men the choice of the unit quantities may be determined in accordance with certain general principles which, once accepted for a series of units, establish certain relations between the units thus chosen, so that they form members of a system known as an absolute system of units. For example, to measure energy we must take as our unit the energy of some body under certain conditions, but when we agree that it shall always be the energy of a body on which a unit force has acted through unit space, our choice has been exercised, and the unit of energy is no longer arbitrary, but defined, as soon as the units of force and space are agreed upon; we have thus substituted the right of selection of the general principle for the right of selection of the particular unit.

We see, then, that the number of physical units is at least as great as the number of physical quantities to be measured, and indeed under different circumstances several different units may be used for the measurement of the same quantity. The physical quantities may be suggested by or related to phenomena grouped under the different headings of Mechanics, Hydro-mechanics, Heat, Acoustics, Light, Electricity or Magnetism, some being related to phenomena on the common ground of two or more such subjects. We must expect, therefore, to have to deal with a very large number of physical quantities and a correspondingly large number of units.

The process of comparing a quantity with its unit - the measurement of the quantity - may be either direct or indirect, although the direct method is available perhaps in one class of measurements only, namely, in that of length measurements. This, however, occurs so frequently in the different physical experiments, as scale readings for lengths and heights, circle readings for angles, scale readings for galvanometer deflections, and so on, that it will be well to consider it carefully.

The process consists in laying off standards against the length to be measured. The unit, or standard length, in this case is the distance under certain conditions of temperature between two marks on a bar kept in the Standards Office of the Board of Trade. This, of course, cannot be moved from place to place, but a portable bar may be obtained and compared with the standard, the difference between the two being expressed as a fraction of the standard. Then we may apply the portable bar to the length to be measured, determining the number of times the length of the bar is contained in the given length, with due allowance for temperature, and thus express the given length in terms of the standard by means of successive direct applications of the fundamental method of measurement. Such a bar is known as a scale . or rule. In case the given length does not contain the length of the bar an exact number of times, we must be able to determine the excess as a fraction of the length of the bar; for this purpose the length of the bar is divided by transverse marks into a number of equal parts - say 10 - each of these again into 10 equal parts, and perhaps each of these still further into 10 equal parts. Each of these smallest parts will then be 1/1000 of the bar and we can thus determine the number of tenths, hundredths, and thousandths of the bar contained in the excess. But the end of the length to be measured may still lie between two consecutive thousandths, and we may wish to carry the comparison to a still greater accuracy, although the divisions may be now so small that we cannot further subdivide by marks. We must adopt some different plan of estimating the fraction of the thousandth. The one most usually employed is that of the 'vernier.' An account of this method of increasing the accuracy of length measurements is given in §1.

This is, as already stated, the only instance usually occurring in practice of a direct comparison of a quantity with its unit. The method of determining the mass of a body by double weighing (see §13), in which we determine the number of units and fractions of a unit of mass, which together produce the same effect as was previously produced by the mass to be measured, approaches very nearly to a direct comparison. And the strictly analogous method of substitution of units and fractions of a unit of electrical resistance, until their effect is equal to that previously produced by the resistance to be measured, may also be mentioned, as well as the measurement of time by the method of coincidences.

But in the great majority of cases the comparison is far from direct. The usual method of proceeding is as follows: An experiment is made the result of which depends upon the relative magnitude of the quantity and its unit, and the numerical relation is then deduced by a train of reasoning which may, indeed, be strictly or only approximately accurate. In the measurement, for instance, of a resistance by Wheatstone's Bridge, the method consists in arranging the unknown resistance with three standard resistances so chosen that under certain conditions no disturbance of a galvanometer is produced. We can then determine the resistance by reasoning based on Ohm's law and certain properties of electric currents. These indirect methods of comparison do not always afford perfectly satisfactory methods of measurement, though they are sometimes the only ones available. It is with these indirect methods of comparing quantities with their units that we shall be mostly concerned in the experiments detailed in the present work.

We may mention in passing that the consideration of the experimental basis of the reasoning on which the various methods depend forms a very valuable exercise for the student As an example, let us consider the determination of a quantity of heat by the method of mixture (§39). It is usual in the rougher experiments to assume (1) that the heat absorbed by water is proportional to the rise of temperature; (2) that no heat is lost from the vessel or calorimeter; (3) that in case two thermometers are used, their indications are identical for the same temperature. All these three points may be considered with advantage by those who wish to get clear ideas about the measurement of heat.

Let us now turn our attention to the actual process in which the measurement of the various physical quantities consists. A little consideration will show that, whether the quantity be mechanical, optical, acoustical, magnetic or electric, the process really and truly resolves itself into measuring certain lengths, or masses.⁽¹⁾ Some examples will make this sufficiently clear. Angles are measured by readings of length along certain arcs; the ordinary measurement of time is the reading of an angle on a clock face or the space described by a revolving drum; force is measured by longitudinal extension of an elastic body or by weighing; pressure by reading the height of a column of fluid supported by it; differences of temperature by the lengths of a thermometer scale passed over by a mercury thread; heat by measuring a mass and a difference of temperature; luminous intensity by the distances of certain screens and sources of light; electric currents by the angular deflection of a galvanometer needle; coefficients of electro-magnetic induction also by the angular throw of a galvanometer needle.

Again, a consideration of the definitions of the various physical quantities leads in the same direction. Each physical quantity has been defined in some way for the purpose of its measurement, and the definition is insufficient and practically useless unless it indicates the basis upon which the measurement of the quantity depends. A definition of force, for instance, is for the physicist a mere arrangement of words unless it states that a force is measured by the quantity of momentum it generates in the unit of time; and in the same way, while it may be interesting to know that' electrical resistance of a body is the opposition it offers to the passage of an electric current,' yet we have not made much progress towards understanding the precise meaning intended to be conveyed by the words ' a resistance of 10 ohms,' until we have acknowledged that the ratio of the electromotive force between two points of a conductor to the current passing between those points is a quantity which is constant for the same conductor in the same physical state, and is called and is the ' resistance' of the conductor; and, further, this only conveys a definite meaning to our minds when we understand the bases of measurement suggested by the definitions of electromotive force and electric current.

When the quantity is once defined, we may possibly be able to choose a unit and make a direct comparison; but such a method is very seldom, if ever, adopted, and the measurements really made in any experiment are often suggested by the definitions of the quantities measured.

The following table gives some instances of indirect methods of measurement suggested by the definitions of the quantities to be measured. The student may consult the descriptions of the actual processes of measurement detailed in subsequent chapters:

The quantities given in the second column of the table are often such as are not measured directly, but the basis of measurement has, in each case, already been given higher up in the table. If the measurement of any quantity be reduced to its ultimate form it will be found to consist always in measurements of length or mass.⁽²⁾ The measurement of time by counting ' ticks' may seem at first sight an exception to this statement, but further consideration will show that it, also, depends ultimately upon length measurement.

As far as the apparatus for making the actual observations is concerned, many experiments, belonging to different subjects, often bear a striking similarity. The observing apparatus used in a determination of a coefficient of torsion, the earth's horizontal magnetic intensity, and a coefficient of electro-magnetic induction, are practically identical in each case, namely, a heavy swinging needle and a telescope and scale; the difference between the experiments consists in the difference in the origin of the forces which set the moving needle in motion. Many similar instances might be quoted. Maxwell, in the work already referred to ('Scientific Apparatus,' p. 15), has laid down the grounds on which this analogy between the experiments in different branches of the subject is based. 'All the physical sciences relate to the passage of energy under its various forms from one body to another,' and, accordingly, all instruments, or arrangements of apparatus, possess the following functions:

1. The Source of energy. The energy involved in the phenomenon we are studying is not, of course, produced from nothing, but enters the apparatus at a particular place which we may call the Source.
2. The channels or distributors of energy, which carry it to the places where it is required to do work.
3. The restraints which prevent it from doing work when it is not required.
4. The reservoirs in which energy is stored up when it is not required
5. Apparatus for allowing superfluous energy to escape.
6. Regulators for equalising the rate at which work is done.
7. Indicators or movable pieces which are acted upon by the forces under investigation.
8. Fixed scales on which the position of the indicator is read off.

The various experiments differ in respect of the functions included under the first six headings, while those under the headings numbered 7 and 8 will be much the same for all instruments, and these are the parts with which the actual observations for measurement are made. In some experiments, as in optical measurements, the observations are simply those of length and angles, and we do not compare forces at all, the whole of the measurements being ultimately, length measurements. In others we are concerned with forces either mechanical, hydrostatic, electric or magnetic, and an experiment consists in observations of the magnitude of these forces under certain conditions; while, again, the ultimate measurements will be measurements of length and of mass. In all these experiments, then, we find a foundation in the fundamental principles of the measurement of length and of the measurements of force and mass. The knowledge of the first involves an acquaintance with some of the elementary properties of space, and to understand the latter we must have some acquaintance with the properties of matter, the medium by which we are able to realise the existence of force and energy, and with the properties of motion, since all energy is more or less connected with the motion of matter. We cannot, then, do better than urge those who intend making physical experiments to begin by obtaining a sound knowledge of those principles of dynamics, which are included in an elementary account of the science of matter and motion. The opportunity has been laid before them by one - to whom, indeed, many other debts of gratitude are owed by the authors of this work - who was well known as being foremost in scientific book-writing, as well as a great master of the subject For us it will be sufficient to refer to Maxwell's work on 'Matter and Motion' as the model of what an introduction to the study of physics should be.

(1)	See articles by Clifford and Maxwell: Scientific Apparatus. Handbook to the Special Loan Collection, 1876, p. 55.
(2)	The measurement of mass may frequently be resolved into that of length. The method of double weighing, however, is a fundamental measurement sui generis.