Unit (measurement)

“ 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. ”

“ Although measurement is itself a species of classification, it is distinguished by the adoption of a standard unit of comparison, which, being precise and unvarying, enables us to express the comparisons in precise and unvarying symbols. ”

“ The ratio of two quantities can be determined with unlimited accuracy, if we can multiply both the object of measurement and the standard unit without error, and then observe what multiple of the one coincides or nearly coincides with some multiple of the other. Although perfect coincidence can never be really attained, the error thus arising may be indefinitely reduced. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ It must be remembered that the number expressing any quantity in terms of a unit is inversely proportional to the magnitude of the unit, i.e. the number of new units is to be found by multiplying the number of old units by the ratio of the old unit to the new unit. ”

“ 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.) ”

“ 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. ”

“ 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. ”

“ That these may be capable of enumeration it is needful that they be more or less alike; and before any absolutely true numerical results can be reached, it is requisite that the units be absolutely equal. The only way in which we can establish a numerical relationship between things that do not yield us like impressions, is to divide them into parts that do yield us like impressions. Two unlike magnitudes of extension, force, time, weight, or what not, can have their relative amounts estimated only by means of some small unit that is contained many times in both ”

“ 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. ”

“ 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. ”

“ 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. ”

“ We may even say, with perfect truth, so as to indicate in a word the general range of the science, that if we did not fear to multiply calculations unnecessarily, and if we had not, in consequence, to reserve them for the determination of the quantities which could not be measured directly, the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles. ”

“ 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. ”

“ Number supplies the means of measuring other quantities, by the assumption of a unit of measure of the appropriate kind: but where nature supplies the unit, number is applicable directly and immediately. ”

“ If we derive units of measurement in spelling, manifestly the attempt would be to measure the changes which have been brought about in any group in terms of units which are comparable. If group one shows ability ten, having advanced during the year from ability seven, it will be considered just as satisfactory as the advance which has been made in group two, which has moved from ability eight to ability eleven. In other words, the purpose of measurement is never to attempt to make all alike. ”

“ 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. ”

“ We compare one discrete quantity with another, or one continual quantity with another, and we thus ascertain many important truths: but we select our unit, or our standard of motion and extension, as we please, or according to convenience, subject only to the necessity of adapting our ulterior calculations consistently to this unit, when once selected. ”

“ What is to be the "unit" (i.e. "standard to measure by") in each subject? (2) Are these units to be of equal, or unequal value? The usual "unit" is the knowledge shown by answering the whole paper correctly ”

“ 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. ”

“ It may be asked, as Mill asks, Why spend so much trouble in calculating from imperfect data, when a little trouble would enable us to render a conclusion certain by actual trial? Why calculate the probability of a measurement being correct, when we can try whether it is correct? But I shall fully point out in later parts of this work that in measurement we never can attain perfect coincidence. Two measurements of the same base line in a survey may show a difference of some inches, and there may be no means of knowing which is the better result. ”

“ 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. ”

“ In all scientific calculations there is assumed an unit to start from, though in each branch of science it is a different unit; e.g. in barology, the pound-weight; in harmonics, the quarter-tone; in other branches of science, other units.68 Analytical research teaches us that the corresponding unit in Syllogism is the affirmative or negative proposition which is primary, immediate, indivisible. In Demonstration and Science it is the Noûs or Intellect.69 ”

“ If we are interested, not so much in performing the operations, as in inquiring into what really takes place in a mind when several units are grasped together and made into a new unit,—for example, when twelve units are thought as one dozen,—the mathematician has a right to say: I leave all that to the psychologist or to the metaphysician; every one knows in a general way what is meant by a unit, and knows that units can be added and subtracted, grouped and separated ”

“ 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. ”

“ These inevitably run through the usual series of measures. Even the metric system has to tolerate half-units as a concession to unscientific weakness while refusing quarters otherwise than as 25 hundredths of the unit. But quarters are firmly rooted in the human mind and resist scientific attempt to extirpate them. They are very common in the sexdecimal series, representing a fourth of one unit and four of a lower unit. ”

“ If the unit of the scale is some measure of the variability of a defined group, as it is in the majority of our present educational scales, this median point may well be calculated to the nearest tenth of a unit, or, if there are two hundred or more individual measurements in the distribution, it may be found interesting to calculate the median point to the nearest hundredth of a scale unit. Very seldom will anything be gained by carrying the calculation beyond the second decimal place. ”

“ Having acquired accurate measuring instruments, and decided upon the units in which the results shall be expressed, there remains the question, What use shall be made of our powers of measurement? Our principal object must be to discover general quantitative laws of nature; but a very large amount of preliminary labour is employed in the accurate determination of the dimensions of existing objects, and the numerical relations between diverse forces and phenomena. Step by step every part of the material universe is surveyed and brought into known relations with other parts. ”

“ If we apply the notions of statistical mechanics to such bodies as we usually consider in thermodynamics, for which the kinetic energy is of the same order of magnitude as the unit of energy, but the number of degrees of freedom is enormous, the values of , , and will be of the same order of magnitude as , and the variable part of , , and will be of the same order of magnitude as . ”

“ We may now divide this art of measurement into two parts; placing in the one part all the arts which measure the relative size or number of objects, and in the other all those which depend upon a mean or standard. Many accomplished men say that the art of measurement has to do with all things, but these persons, although in this notion of theirs they may very likely be right, are apt to fail in seeing the differences of classes--they jumble together in one the 'more' and the 'too much,' which are very different things. ”

“ Similarly with the measurements of the ancient buildings. In this case if all our various ‘errors’ were thrown together into one group of statistics we should find that the resultant curve of facility displayed, not two peaks only, but a succession of them; and these of various magnitudes, corresponding to the frequency of occurrence of each particular measurement. We might take an average of the whole, but hardly any rational purpose could be subserved in so doing; whereas each separate point of maximum frequency of occurrence has something significant to teach us. ”