Measurement

“ Measurement owes its existence to Earth; Estimation of quantity to Measurement; Calculation to Estimation of quantity; Balancing of chances to Calculation; and Victory to Balancing of chances. ”

“ You know how, when you contemplate the purchase of a horse, you lead him up to the measuring-bar, and there ascertain the precise number of hands and inches which he stands. How have I longed for the means of subjecting the mental stature of human beings to an analogous process of measurement! ”

“ I would, however, have his first estimates tested by measurement, so that he may correct his errors, and if there is a false impression left upon the senses he may correct it by a better judgment. The same natural standards of measurement are in use almost everywhere, the man's foot, the extent of his outstretched arms, his height. When the child wants to measure the height of a room, his tutor may serve as a measuring rod; if he is estimating the height of a steeple let him measure it by the house ”

“ 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. ”

“ 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. ”

“ A FEW WORDS ABOUT THE MÈTRE. The idea of an invariable and constant system of measurement, of which nature herself should furnish the exact value, may be said to have existed in the mind of man from the earliest ages. It was of the highest importance, however, that this measurement should be accurately determined, whatever had been the cataclysms of which our earth had been the scene, and it is certain that the ancients felt the same, though they failed in methods and appliances for carrying out the work with sufficient accuracy. ”

“ Modern views of measurement have a philosophical aspect. There is an indefinite number of systems of measuring along a line; thus, a perspective representation of a scale on one line may be taken to measure another, although of course such measurements will not agree with what we call the distances of points on the latter line. 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. ”

“ 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. ”

“ If we take the word measurement in its direct and general mathematical acceptation, which signifies simply the determination of the value of the ratios between any homogeneous magnitudes, we must consider, in geometry, that the measurement of surfaces and of volumes, unlike that of lines, is never conceived, even in the most simple and the most favourable cases, as being effected directly. ”

“ 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 result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude, or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. ”

“ But having determined just what qualities and abilities it is desired to measure, we must set up a standard of measurement by which to compare the indicated ability of each individual examined, or we shall have nothing as a result of our test but a mass of information, of the significance of which we cannot possibly be aware. ”

“ 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. ”

“ 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. ”

“ 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 real purpose of a measurement is to tell facts about a situation more exactly and with greater objectiveness than they could be told in a description. A child may seem, on first appearance, to be under weight, but in order to know definitely whether or not that is true it is necessary to measure his age in terms of years, months, and days, to measure his weight in terms of pounds and ounces, and to measure his height in terms of feet and inches. ”

“ Next in importance to a true plane is the power of measurement. I have brought with me, for your inspection at the close of the meeting, a small machine, by which a difference in length of the one-millionth part of an inch is at once detected. ”

“ As science has been developed, the domain of quantity has everywhere encroached on that of quality, till the process of scientific inquiry seems to have become simply the measurement and registration of quantities, combined with a mathematical discussion of the numbers thus obtained. It is this scientific method of directing our attention to those features of phenomena which may be regarded as quantities which brings physical research under the influence of mathematical reasoning. ”

“ 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. ”

“ Ordinary measurement by a foot rule, a surveyor’s chain, or the excessively careful measurements of the base line of a trigonometrical survey by standard bars, are sufficient instances of this procedure.
In the second case, where p y/x = q, we multiply or divide a magnitude until we get what is equal to the unit, or to some magnitude easily comparable with it. As a general rule the quantities which we desire to measure in physical science are too small rather than too great for easy determination, and the problem consists in multiplying them without introducing error. ”

“ Accurate measurement.—Accurate measurement is the foundation of exact science, and at the very beginning of his study in astronomy the student should learn something of the astronomer's kind of measurement. He should practice measuring the stars with all possible care, and should seek to attain the most accurate results of which his instruments and apparatus are capable. ”

“ In the case of the physicist, observation always takes the strict form which is called measurement. The relationships between what is actually observed or measured and the quantities which finally appear in the equations expressing the laws of nature are extremely complicated, but we do not have to concern ourselves with them. ”

“ Three such generalizations as the foregoing afford such facilities for the indirect measurement of magnitudes, (by supplying us with known lines or angles which are marks of the magnitude of unknown ones, and thereby of the spaces which they inclose,) that it is easily intelligible how from a few data we can go on to ascertain the magnitude of an indefinite multitude of lines, angles, and spaces, which we could not easily, or could not at all, measure by any more direct process. ”

“ Measurements must be selected in accordance with the problem that we are trying to investigate. The proportion of length and breadth of head may be a very desirable measurement in one case, while in another case it may be of no value whatever. Measurements should always have a biological significance. ”

“ The course may begin with the careful measurement of lengths, employing different methods, such as the direct application of the rule to the object, the transference of distances by means of compasses, and obtaining the lengths of curves by means of a string laid along them and afterwards measured; and the children should be taught to make measurements on the metrical system as well as in feet and inches, especially if they already possess any knowledge of decimals. ”

“ 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. ”

“ But let us reduce the unit of measure from the mile to the inch, and then let us take the reciprocals of these enormous values obtained by the astronomer in his study of the planetary composition of the universe, and we shall have before us the order of measurements which engage the attention of the physicist who studies the molecular composition of matter. If we are not dismayed by the one, let us not be by the other. ”

“ 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. ”

“ That the members and dimensions of our own body should have been the original standards of measurement is most natural, and the terms in which they are conveyed afford a sufficient illustration of the fact. ”

“ What an important influence may be exercised over the progress of a single branch of science by the invention of a ready and convenient mode of executing a definite measurement, and the construction and common introduction of an instrument adapted for it cannot be better exemplified than by the instance of the reflecting goniometer. ”

“ If we do, we would want to keep an iron bedstead to measure every person according to our own measurement and dimensions; and if persons were too long we would cut them off, and if too short draw them out. ”

“ This characteristic of modern experiments—that they consist principally of measurements,—is so prominent, that the opinion seems to have got abroad, that in a few years all the great physical constants will have been approximately estimated, and that the only occupation which will then be left to men of science will be to carry on these measurements to another place of decimals. ”

“ Balances have been constructed which have rendered sensible the millionth part of the whole quantity weighed; and to turn with the thousandth part of a grain is the performance of balances pretending to no very extraordinary degree of merit. The elegant invention of the sphærometer, by substituting the sense of touch for that of sight in the measurement of minute objects, permits the determination of their dimensions with a degree of precision which is fully adequate to the nicest purposes of scientific enquiry. ”