Measure for Measure

“ 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.) ”

“ You cannot avail yourself of the argument that a foot may measure the variable height of a man, although the variable height of a man cannot truly measure the foot, because you have agreed that under certain circumstances the man's height is not variable, and it is to those circumstances that I always refer. You say of my measure, and say truly, that if all commodities were produced under the same circumstances of time, etc., as itself, it would be a perfect measure, and you say further that it is now a perfect measure for all commodities produced under such circumstances. ”

“ In quantity (a translation of the Greek ‘posotes’—‘how-muchness’—and seemingly formed by Aristotle on the analogy of Plato’s ‘poiotēs’, from which we have quality) we can perhaps see the beginning of that interest in the calculable aspect of the objects of the visible world from which the exact sciences have arisen. The human mind had now begun to weigh and measure, to examine and compare; and that weighing and measuring has gone on—with intervals—for twenty-three centuries. ”

“ 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. ”

“ What is their easiest way of settling the point? It is not necessary to use stick, string, or implement of any kind for measuring. I Pg 110give this merely as one of the simplest possible examples of the value of ordinary sagacity in the solving of puzzles. What are apparently very difficult problems may frequently be solved in a similarly easy manner if we only use a little common sense.
365.—NEW MEASURING PUZZLE.
Here is a new poser in measuring liquids that will be found interesting. A man has two ten-quart vessels full of wine, and a five-quart and a four-quart measure. ”

“ For some bodies we count, some we measure, and some we weigh; [8] those which are by nature continuous we apprehend by measure; to those which are divided we apply number (with the exception of those which on account of their fineness are measured) ; while heavy objects are distinguished by the inclination of the balance. ”

“ If we differ in opinion about two quantities, we can have recourse to a common measure, which may decide the question with the utmost exactness; and this, I take it, is what gives mathematical knowledge a greater certainty than any other. But in things whose excess is not judged by greater or smaller, as smoothness and roughness, hardness and softness, darkness and light, the shades of colors, all these are very easily distinguished when the difference is any way considerable, but not when it is minute, for want of some common measures, which perhaps may never come to be discovered. ”

“ But it helps little, in getting at the aggregate weight of a collection of objects, to know that A among them is heavier than B, while D is lighter than F. To get a knowledge of the situation adequate for quantitative manipulation, it is best to compare all of the objects with some one object, chosen as the standard of weight, or common measure of weights. ”

“ A measure, then, is a divisor which leaves no remainder.
92. When one number is a measure of two others, it is called a common measure of the two. Thus, 15 is a common measure of 180 and 75. Two numbers may have several common measures. For example, 360 and 168 have the common measures 2, 3, 4, 6, 24, and several others. Now, this question maybe asked: Of all the common measures of 360 and 168, which is the greatest? The answer to this question is derived from a rule of arithmetic, called the rule for finding the greatest common measure, which we proceed to consider. ”

“ We must also admit, in taking for our guide, the reasoning which discourses to us things which are invisible to the senses, that the most minute magnitudes, those which are not compound magnitudes, and which from the limit of sensible extent, are the first measure of the other magnitudes which are only called greater or less in their relation to the others. ”

“ It is not easy to say how many kinds of improper quantity may in time be introduced into mathematics, or to what new subjects measures may be applied; but this, I think, we may conclude, that there is no foundation in nature for, nor can any valuable end be served by, applying measure to anything but what has these two properties. ”

“ For one must aim at attaining a certain measure, and yet this measure admits neither weight nor calculation of any kind, by which it may be accurately determined, unless it be the sensation of the body ”

“ 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. ”

“ If we first examine, for example, the process of measuring a length, it consists in our applying to the distance to be measured a length conceived as unchangeable, the unit of measure, until we have passed over the distance. ”

“ 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. ”

“ 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. ”

“ Meas′uring, that measures, or fitted for measuring.—Measure one's length, to fall or be thrown down at full length; Measure strength, to engage in a contest; Measure swords, to fight with swords: to try one's skill against.—Above, or Beyond, measure, to an exceedingly great degree ”

“ The use of units of measure in drills which will never be used in bona fide measuring is like the use of fractions like sevenths, elevenths, and thirteenths. A very little of it is perhaps desirable to test the appreciation of certain general principles, but for regular training it should give place to the use of units of practical significance. ”

“ As a measure of length, it is unknown to the mass of our citizens; and if retained for the purpose of superficial measure, the foot will supply its place, and fix the acre as in the fourth resolution. ”

“ All these relations are numerically expressed and are determinations of number, and so in another way are the equal and the like and the same, for all refer to unity. Those things are the same whose substance is one; those are like whose quality is one; those are equal whose quantity is one; and 1 is the beginning and measure of number, so that all these relations imply number, though not in the same way.
(2) The active and the passive imply an active and a passive potency and the actualization of the potencies, e.g. ”

“ 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. ”

“ 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! ”

“ 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. ”

“ And since we are assured that the all-wise Creator has observed the most exact proportions of number, weight, and measure in the make of all things, the most likely way, therefore, to get any insight into the nature of those parts of the creation which come within our [Pg 30] observation must in all reason be to number, weigh, and measure. ”

“ What language was for the evolution of primitive man, measurement is for the advance of modern science. As a word selects similarities and ignores differences, so a measurement selects certain similarities from the concrete manifoldness of things. That such a great part of the world can be described in terms of a few units of measurement, and that this description should lead to such useful applications, is truly marvelous and admirable. ”

“ When an object, in which magnitude is a desirable attribute, leaves an impression, on a practised eye, of less magnitude than it really possesses, we should place objects beside it, of whose magnitude we can satisfy ourselves, of larger size than that which we are accustomed to; for, by finding these large objects in precisely the proportion to the grand object, to which we are accustomed, while we know their actual size to be one to which we are not accustomed, we become aware of the true magnitude of the principal feature. ”

“ No! We are mere prosaic mortals: it is beyond us!
In p. 49 I see some strange remarks about ratios. First look at Def. 44. 'When a magnitude is contained an exact number of times in two magnitudes of its kind, it is said to be their common measure.' (The wording is awkward, and suggests the idea of their having only one 'common measure'; but let that pass.) 'The ratio of two magnitudes of the same kind is the number which would express the measure of the first, if the second were taken as unity.'
'The measure of the first'! Do you understand that? ”

“ But how are measures of light to give us fractions of a magnitude? How can the vague, qualitative relation, the brighter the light the higher the magnitude, become an exact and quantitative one? The discussion of this question may be of use by showing that, even with matters of so uncertain a nature, science does not proceed by guesswork. It is a general law, that the human senses measure ratios and not differences. ”

“ 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. ”

“ Through the science of mathematics, the relation between magnitudes only conceivable to the mathematician by his employment of calculative symbols, can be correctly ascertained, and a working hypothesis for practical purposes thereby obtained. ”

“ That, however, is apart from the question I have been debating in this work, and my purpose will have been amply served if I have been successful in throwing some light upon a little known subject, and, above all, in leading my readers to the conclusion I have long since arrived at—that our system is, fundamentally, the best in existence, the most convenient for the needs of man, and that it would be an act of stupendous and costly folly, to change to the metric system. ”

“ 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 ”

“ Mathematics contain properly the doctrine of measure; and the object of this science is commonly said to be quantity; therefore quantity ought to be defined, what may be measured. Those who have defined quantity to be whatever is capable of more or less, have given too wide a notion of it, which, I apprehend, has led some persons to apply mathematical reasoning to subjects that do not admit of it. ”