Unit of measurement

“ A measurement is a comparison. We measure the length of a lot by comparing it with the standard of length, the yard or foot. We measure a farm by comparing its area with the standard unit of surface measure, the acre, square rod, or square yard.
In every measurement we must first have an accepted standard unit. The history of units of measurement is a very interesting one, and its difficulty arises from the fact that no two things in nature are the same. ”

“ All numbers, for instance, which represent physical measurements, are limited in their accuracy not only by our powers of measurement but also by the accuracy of the measure we use as our unit. Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations. ”

“ Accordingly the unit of energy will be in the English system, that required to lift one pound through about the thirty-second part of a foot; in terms of metric units, it will be that required to lift a kilogram through about one tenth part of a metre.
Every person is at liberty to measure and record quantities in terms of any unit which he likes. He may use the yard for linear measurement and the litre for cubic measurement, only there will then be a complicated relation between his different results. ”

“ Solar time.—To measure any quantity we need a unit in terms of which it must be expressed. Angles are measured in degrees, and the degree is the unit for angular measurement. For most scientific purposes the centimeter is adopted as the unit with which to measure distances, and similarly a day is the fundamental unit for the measurement of time. ”

“ We are almost constantly obliged, in physical science, to measure the quantities with which we deal. We measure a quantity when we compare it with some standard of the same kind. A simple number expresses the result of the comparison.
If we adopt arbitrary units of length, time, and mass (or quantity of matter) , we can express the measure of all other quantities in terms of these so-called fundamental units. A unit of any other quantity, thus expressed, is called a derived unit. ”

“ The metre is a length of 40 inches nearly, and by applying to it exclusively the principle of decimal division, no measure corresponding to the ancient foot was provided. A unit of that denomination, though with slightly varied differences of length, was in universal use among all civilised nations, and the type of it is found in the dimensions of the human body. Perhaps for half of the occasions which arise in the life of every individual for a linear measure, the instrument to suit his purpose must be portable, and fit to be carried in his pocket. ”

“ Standards of length are preserved by a Bureau of Weights and Measures in most countries. Delambre, a French authority on the decimal system of measures, taught at the beginning of the nineteenth century that magnitudes as small as the hundredth of a millimeter are incapable of observation. The International Bureau of Weights and Measures now guarantees to determine two or three ten-thousandths of a millimeter. So much has the science of measurement progressed in a century. ”

“ In almost every case a measuring instrument serves, and should serve only as a means of comparison between two or more magnitudes. As a general rule, we should not attempt to make the divisions of the measuring scale exact multiples or submultiples of the unit, but, regarding them as arbitrary marks, should determine their values by comparison with the standard itself. ”

“ From the unit of linear measure of metre is derived the unit of the measure of capacity or Litre. This represents the cube of one-tenth part of a metre, or a cubic decimetre, and its multiples and sub-divisions with their corresponding equivalents in Imperial fluid measure are as follows:—
The unit of weight in the metric system is the Gramme. ”

“ UNITS AND STANDARDS OF MEASUREMENT. As we have seen, instruments of measurement are only means of comparison between one magnitude and another, and as a general rule we must assume some one arbitrary magnitude, in terms of which all results of measurement are to be expressed. Mere ratios between any series of objects will never tell us their absolute magnitudes; we must have at least one ratio for each, and we must have one absolute magnitude. ”

“ So in like manner we must choose our unit of work, but in order to do so we must first of all choose our units of weight and of length, and for these we will take the kilogramme and the metre, these being the units of the metrical system. ”

“ Analysing the question, and his words have not since been improved upon, he said—"The French metrology, in the ardent and exclusive search for an universal standard from nature, seems to have viewed the subject too much with reference to the nature of things, and not enough to the nature of man. Its authors do not appear to have considered the proportions dictated by nature between the physical organisation of man and the unit of his weights and measures. The standard taken from the admeasurement of the earth, has no reference to the admeasurement and power of the human body. ”

“ He broadly outlined the committee's ideas as follows:
In all physical magnitudes that are made use of, we have: (1) the physical magnitude itself, aside from the units that serve to measure it; (2) the C.G.S. unit that serves to measure such grandeur (granted the adoption of the C.G.S. system) ; (3) practical units, which, in general, have a special name for each kind of magnitude, and are a decimal multiple or sub-multiple of the C.G.S. unit, except for time and angles; (4) finally, decimal multiples and sub-multiples of these practical units, that are in current use. ”

“ Hence, as there is but one kind of quantity in geometry, lines, surfaces, and solids must have identically the same unit of comparison, and must be always perfect measures of each other. The unit may be infinitely varied in size—it being the name or representative of any assumed magnitude to which it is applied—but it always represents a magnitude of a definite form, and hence a magnitude which has an extension in every direction from its centre, and consequently represents not only one in length, but also one in breadth, and one in thickness. ”

“ The metric system is the only system of weights and measures that can be called international. Moreover, only the very best available system should be good enough for Americans.
It is sometimes complained that the meter as a unit should be set aside because it is inaccurate. ”

“ Of course I do not call in question the great advantages to be derived from the ability to carry the method of decimal calculation into quantities and values, and of course I do not call in question the desirableness of having some rationally originated unit from which all measures of lengths, weights, forces, etc., shall be derived. That as promising to end the present chaos the metric system has merits goes without saying. ”

“ A standard measure of length at first sight appears to be very simple—merely a bar of metal of any length, according to the unit of any country; and comparisons of different standards do not seem to present any difficulty. But on looking further into the thing, we find that standards are referred to some natural invariable length, and we are at once confronted with a mass of scientific reductions giving different values to the same thing, according to successively improved means of observation. ”

“ When the standard quantity is greater than that to be measured, we often adopt the first mode, and subdivide the unit until we get a magnitude equal to that measured. The angles observed in surveying, in astronomy, or in goniometry are usually smaller than a whole revolution, and the measuring circle is divided by the use of the screw and microscope, until we obtain an angle undistinguishable from that observed. The dimensions of minute objects are determined by subdividing the inch or centimetre, the screw micrometer being the most accurate means of subdivision. ”

“ A unit is an abstract quantity, represented by a certain standard, and more or less perfectly by copies of the standard.
A unit of mass is the matter of a standard of mass, or a multiple of that. For instance, in Britain the unit of the pound is defined by a piece of platinum preserved in the Standards Office. ”

“ The relationships between physical magnitudes are established by measurements. When these are accurately ascertained, questions regarding their variable functions can be solved by mathematical principles. Physics is thus linked with mathematics through measurements. The more science advances, the greater is the accuracy needed in physical measurements. The strictness and clearness of experimentation which has been attained in physics has given birth to a science of measurement, which has its own instruments, rules, methods, and formulæ. ”

“ The measure is always homogeneous with the thing measured; the measure of spatial magnitudes is a spatial magnitude, and in particular that of length is a length, that of breadth a breadth, that of articulate sound an articulate sound, that of weight a weight, that of units a unit. (For we must state the matter so, and not say that the measure of numbers is a number; we ought indeed to say this if we were to use the corresponding form of words, but the claim does not really correspond-it is as if one claimed that the measure of units is units and not a unit; number is a plurality of units.) ”

“ For machine and shop work the English unit is the inch and fractions, and countries having the metrical standard have universally adopted the millimètre.
The inch is well fixed in the minds of all mechanics by constant use, and the ease with which the fractions are had by halving only renders the system very convenient. ”

“ The standards of measurement which every nation possesses are part of its history, and their long survival is at least prima facie evidence of their utility and convenience. If we take the particular instance which the word "metric" itself suggests, it seems to us we are better off with such familiar and convenient measures as the yard, the foot, the inch, and the subdivisions of the latter into halves, quarters, eighths, sixteenths, etc., than we should be with the metre divided into centimetres and millimetres—that is, into hundredths and thousandths. ”

“ For instance, the scientific unit of volume is always the cube whose side is the unit of length. If the unit of length varies, the unit of volume will vary as its third power, and the unit of volume is said to be of three dimensions with respect to the unit of length. ”

“ To establish a system of measurement on a line we must assign a distinct number to each point of it, and for this purpose we shall plainly have to suppose the numbers carried out into an infinite number of places of decimals. ”

“ Again, within the aspects selected for attention, all that experiment can establish is that the deviation from uniform law, if there is any deviation, is not sufficiently great to affect our measurements and calculations. But how far our standards of measurement are from rigid precision may be readily learned from the chapter on physical standards in any good work on the logic of the inductive sciences. ”

“ Besides which, the cubit being the unit, the half cubit might serve the purpose of the foot; but the metre, divisible only by two and ten, gave no measure practically corresponding to the foot whatever. It appears to have been considered that decimal arithmetic, although affording great facilities for the computation of numbers, is not equally well suited for the division of material substances. A glance of the eye is sufficient to divide material substances into successive halves, fourths, eighths, and sixteenths. A slight attention will give thirds, sixths, and twelfths. ”

“ The question of correct measurement is in immediate connexion with another, which will repay all the attention that can be given to it, and I think there is no subject that can be more profitably discussed amongst us;—I mean that of proper gradations of size in all the various branches of the mechanical arts. I think no estimate can be formed of our national loss from the over-multiplication of sizes. ”

“ The nature of measurement is well understood except in so far as it relates to psychological phenomena; in this realm metaphysicians seem wholly to misconceive its nature. I cannot measure the number of the ultimate particles of a body by counting them, but I may measure the relative number of its atoms by weighing it. I do not determine its force, but its mass only, when I weigh the body, for the total force in the body is the sum of its motion in all its incorporations. A pound of powder has much more force than that which is measured as a pound. ”

“ Weights and Measures.—The French metric system is the official standard of weights and measures and is in use in the custom-houses of the republic and in foreign trade, but the old units are still commonly used among the people. ”

“ 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 1/16 metre, and to contain a kilogramme of water at 4° C. in a vacuum. ”

“ Every mathematician knows that scores of methods, differing altogether from each other in process, all end in this mysterious 3.14159..., which insists on calling itself the circumference to a unit of diameter. A reader who is competent to follow processes of arithmetic may be easily satisfied that such methods do actually exist. I will give a sketch, carried out to a few figures, of three: the first two I never met with in my reading; the third is the old method of Vieta. ”