Maximum

“ Maximum and minimum values of a function. A maximum value of a function is one that is greater than any values immediately preceding or following.
A minimum value of a function is one that is less than any values immediately preceding or following. ”

“ A critical value usually gives a maximum or minimum in the case of a function of one variable, and often in the case of several independent variables, but all maxima and minima, particularly absolutely greatest and least values, are not necessarily critical values. ”

“ In every case we find that the maximum which opens the week is rapidly succeeded by a minimum, which is itself succeeded by a secondary maximum,—usually very secondary, although in 1894 it nearly equals the primary maximum,—followed again by a second minimum—usually nearly identical with the first minimum,—after which there is a rapid rise to the original maximum. ”

“ Now, the quantities being all positive, the product cannot be increased without limit and must somewhere attain a maximum, and no other form of the product than that in which they are all equal can be the maximum, so that the product is a maximum when they are all equal. Its minimum value is obviously zero. If the restriction that all the quantities shall be positive is removed, the product can be made equal to any quantity, positive or negative. ”

“ That is, the sum of the products of the volumes of the masses by their pressures, diminished by the sum of the products of the areas of the surfaces of discontinuity by their tensions, must be a maximum. ”

“ To the resulting function of a single variable apply one of our two rules jor finding maximum and minimum values.
(d) In practical problems it is usually easy to tell which critical value will give a maximum and which a minimum value, so it is not always necessary to apply the fourth step of our rules. ”

“ When the maximum and minimum differ greatly, if no remark is made on the subject, it may be understood that the extremes are so closely connected by intermediate figures that the average is a fair one. ”

“ The function is said to be a maximum at when is greater than for all values of and in the neighborhood of and . Similarly, is said to be a minimum at when is less than for all values of and in the neighborhood of and . ”

“ These values are plotted in Fig. 26, which shows that has apparently a minimum value of , when is made equal to . But are you sure that the minimum occurs at , and not at or at ?
Of course it would be possible with any algebraic expression to work out a lot of values, and in this way arrive gradually at the particular value that may be a maximum or a minimum.
Fig. 27.
Here is another example:
Let
.
Calculate a few values thus:
Plot these values as in Fig. ”

“ As the comparison of numbers with one another, their determination as greater or less, is only possible on the assumption of a unit, a common measure to which each may be referred, so the comparison of bodies with one another, as to quantity and quality alike, demands a corporeal minimum, to which their differences must be reduced. This relation to knowledge carries with it the relativity of the minimum according to the subject-matter with which the knower is for the time being concerned. ”

“ And so the balance sways and the story runs—here a gain, there a loss, and Nature’s great average struck across the two, whereof the sum total forms one of the factors in that mighty equation in which the result will equal the unknown quantity of her purpose. ”

“ It is said in the Bible that God has ordered all things according to weight, number and measure. Omnia in pondere et numero et mensura disposuit Deus. Weight is equilibrium, number is quantity, measure is proportion—these three, and these are the eternal or divine basis of the science of Nature. Here now is the formula of equilibrium: Harmony results from the analogy of contraries. Number is the scale of analogies, the proportion of which is measure. The entire occult philosophy of the Zohar might be termed the science of equilibrium. ”

“ 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. ”

“ Thus, in three diverse ways we have striking indications of the fact that the task of supplanting our present weights and measures with an inferior system is not likely to prove so easy as some people would have us believe. ”

“ But when from the illusion created by this common property we think we shall distinguish something in the minimum, one part on one side and another part on the other side, it must be another minimum equal to the first which catches our eye. ”

“ It is said there can be no uniform standard of weights without weight, or of measure without length or space, and we are asked how anything can be made a uniform standard of value which has itself no value? ”

“ The proportion of unequals is quantity; for the difference, G F, in which it consists, is quantity. But the proportion of equality is not quantity; because, between equals, there is no difference; nor is one equality greater than another, as one inequality is greater than another inequality. ”

“ 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. ”

“ It assumes that the “witnesses” we possess,—meaning thereby every single Codex, Version, Father—, (1) are capable of being weighed: and (2) that every individual Critic is competent to weigh them: neither of which propositions is true.
3. In the very form of the maxim,—“Not to be counted—but to be weighed,”—the undeniable fact is overlooked that “number” is the most ordinary ingredient of weight, and indeed in matters of human testimony, is an element which even cannot be cast away. ”

“ We answer that it is an image in which not one single minimum is in itself reversed, but in which all the minima are transposed merely in relation to one another. The inversion regards merely the relative position of the minima, and not the minima themselves. Thus, the upward part of each minimum in the object must also point upwards in the image on the retina. ”

“ Now when we are referring to averages it must be remembered that the minimum number of different occurrences necessary in order to justify the average is that which enables each of them to present itself once. ”

“ You have no standard to estimate by, no measure to make comparisons with. Given a certain small quantity of higher pleasure in collision with a certain large quantity of lower, how can you decide between them? To work the sum you must reduce the data to the same denomination. You must go to quantity or nothing; you decline to go to quantity, and hence you can not get any result. But if you refuse to work the sum, you abandon the greatest amount of pleasure principle. ”

“ Though proportion be measurable in its own nature, and therefore hath proper quantity, yet as things cannot have proportion which have not quantity of some other kind, it follows that whatever has quantity must have it in one or other of these three kinds, extension, duration, or number. ”

“ WEIGHTS AND MEASURES. Standards of magnitude, of weight, and of value are essential for commercial and scientific purposes, and the facility with which national and international intercourse is carried on depends largely upon the uniformity of these standards. Naturally the units of measure adopted by primitive peoples were varied and imperfect, but as civilization advanced, trade increased, science developed, and more accurate standards became necessary. ”

“ The rise of measurement, the naturalisation of weights and measures brought by commerce, even by conquest, when they are found convenient, the varieties caused by changes of circumstance, the deflections under the constraint of ill-advised rulers, the 309effect of long custom in reconciling to new standards if they can only be arranged conveniently, the shifts by which they can be made endurable, the tendency to resume the old trend along another path—all these traits of human nature are shown in this study. ”

“ That for which it is the way to roughly measure, Yet it is measured in small amounts, It is of secondary importance to the rough amounts, And that for which it is the way to measure small, but was measured roughly, The rough amount is of secondary importance to the small. ”

“ There are few who have any adequate grasp of the momentous issues involved in determining upon what foundation our system of weights and measures should rest. Much has been written upon the subject. To the great bulk of the people it might as well never have been penned—they know nothing, or practically nothing, of it, and the little information which has reached them has often been of a misleading or negative character. ”

“ 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? ”

“ As to equality or any exact proportion, we can only guess at it from a single consideration; except in very short numbers, or very limited portions of extension; which are comprehended in an instant, and where we perceive an impossibility of falling into any considerable error. In all other cases we must settle the proportions with some liberty, or proceed in a more artificial manner. ”

“ By this illustration, then, we obtain a rule to determine what one ought to take from him who has the greater, and what to add to him who has the less. The excess of the mean over the less must be added to the less, and the excess of the greater over the mean be taken from the greater. ”

“ General Remarks In the various names of weights and measures there are many general-utility words which offer no difficulty in the sphere of those who use them habitually, yet which are sometimes puzzling to others, while they are interesting to the student of semantics. They form a chapter in the history of weights and measures, itself a volume in the history of the human mind. ”

“ If any two quantities be made of the mutual multiplication of many quantities, which have proportion to one another, and the efficient quantities on both sides be equal in number, the proportion of the products will be compounded of the several proportions, which the efficient quantities have to one another. ”

“ The process by which an average is obtained, it may be urged, insures this tendency to equalization amongst the magnitudes with which it deals. For instance, let there be a party of ten men, of whom four are tall and four are short, and take the average of any five of them. Since this number cannot be made up of tall men only, or of short men only, it stands to reason that the averages cannot differ so much amongst themselves as the single measures can. ”

“ Now that quantity which is apprehended only as unity, and in which plurality can be represented only by approximation to negation = 0, I call an intensive quantity. Every reality, therefore, in a phenomenon has intensive quantity, that is, a degree. ”

“ If a quantity being given, first one quantity be both added to it and subtracted from it, and then another greater or less, the proportion of the remainder to the aggregate, is greater where the less quantity is added and substracted, than where the greater quantity is added and substracted. ”

“ 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. ”

“ No one will ever be able to determine by an exact enumeration, that an inch has fewer points than a foot, or a foot fewer than an ell, or any greater measure; for which reason, we seldom or never consider this as the standard of equality or inequality. ”

“ Quantities, and the ratio's of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal. ”

“ And we may further notice, how, in this first step of rational mechanics, we see illustrated that truth awhile since referred to, that as magnitudes of linear extension are the only ones of which the equality is exactly ascertainable, the equalities of other magnitudes have at the outset to be determined by means of them. ”

“ And when we deliberately set ourselves to obtain uniformities by taking an average result, derived from comparison of a multitude of subjects, our results are, of course, always mechanical in appearance, because the element of individual purpose and initiative has been excluded by ourselves from our data in the very process of taking the average. ”

“ Just as men talk about inches, and calculate by inches, in thousands of cases in which no actual inch is used as a measure, and in every case of doubt, dispute, or difficulty have recourse to the actual inch, and thus the ideal inch is kept steady in the minds of men by frequent reference to the outward standard; so the mental measure of services, which men insensibly acquire from the use of the objective measure, needs to be kept true by actual and frequent contact with that measure. ”