Coefficient

“ We may, however, often do a little better than this without using a more complicated equation. For may be estimated very roughly as equal to Whenever, therefore, and are about as large as as is often the case, it may be a little better to use the coefficient instead of in the last term. ”

“ In all cases which we are now considering, the energy, or any function of the energy, is such a function. will therefore satisfy the equation, as indeed appears identically if we write it in the form There are, however, other conditions to which is subject, which are not so much conditions of statistical equilibrium, as conditions implicitly involved in the definition of the coefficient of probability, whether the case is one of equilibrium or not. ”

“ It will be observed that since the 's are linear functions of the 's, and the 's linear functions of the 's, the 's will be linear functions of the 's, so that a differential coefficient of the form will be independent of the 's, and function of the 's alone. ”

“ Multiplying conjugate pairs together, attributing proper signs to each product, and adding, we find from the properties of the minors of a determinant that the coefficient of a term such as , is zero, unless the quantities s,t,p,q,r,... are all different. When these quantities are different the coefficient is simply the determinant with a proper sign. ”

“ The new ideal entities are called Derived Functions, or Differential Coefficients, and for a long time it was assumed as almost an axiom that every function continuous within given intervals must have, within those intervals, a derived function, or differential coefficient. ”

“ The great reason why is regarded of importance is that possesses a property, not possessed by any other function of , that when you differentiate it its value remains unchanged; or, in other words, its differential coefficient is the same as itself. This can be instantly seen by differentiating it with respect to , thus ”

“ Doubtless different parameters are suited to different purposes and contexts; c when we consult the common tables, and in connexion with the operator, as below, par. 160; k ( = 1⁄2c2) when we investigate the formation of the probability-curve out of independent elements (below, par. 104) ; h ( = 1/c2) when we are concerned with weights or precisions (below, par. 134) . If one form of the coefficient must be uniformly adhered to, probably, σ ( = c/√2) , for which Professor Pearson expresses a preference, appears the best. It is called by him the “standard deviation.” ”

“ Generally, since every analytical expression is susceptible of being expressed in a series ordered according to certain functions of the variable, we perceive that the machine will include all analytical calculations which can be definitively reduced to the formation of coefficients according to certain laws, and to the distribution of these with respect to the variables. ”

“ The other extreme would be the state of affairs in which men of a certain type (that is to say men differing from the general average by a definite amount) always chose wives of a definite type; the resemblance would then be perfect and the correlation, as we call it, would be expressed by a coefficient of 1. ”

“ To get each equation we select a library that we are willing to accept as being conservatively and properly operated, and substitute for x, y, etc., its reported circulation, number of books, and so on, putting in place of R its total cost of administration. Solution of this system of equations gives the coefficients, A, B, C, etc., and furnishes the working formula required. ”

“ In order to determine the best values of the coefficients Determination of Constants by the Inverse Method. involved in the law of error, and to test the worth of the results obtained by using any values, recourse must be had to inverse probability.
130. The simplest problem under this head is where the quaesitum is a single real object and the data consist of a large number of observations, x1, x2, . . . xn, such that if the number were indefinitely increased, the completed series would form a normal probability-curve with the true point as its centre, and having a given modulus c. ”

“ Suppose for a moment that t1, t2, t3 are all known; then the equations being linear in y1, y2, y3 these can be expressed rationally in terms of the coefficients and of t1, t2, t3; that is, y1, y2, y3 will be known. ”

“ The Ultimate Values of the Coefficients in Fourier’s Series.—If ƒ (x) is everywhere finite within the given interval −π to +π, it can be shown that an, bn, the coefficients of cos nx, sin nx in the series which represent the function, are such that nan, nbn, however great n is, are each less than a fixed finite quantity. ”

“ A very much closer approximation may be obtained, if we regard at any given point of the medium and for light of a given period, as linear functions of and the nine differential coefficients We shall write and diff. ”

“ Those who are unacquainted with the details of scientific investigation have no idea of the amount of labor expended in the determination of those numbers on which important calculations or inferences depend. They have no idea of the patience shown by a Berzelius in determining atomic weights; by a Regnault in determining coefficients of expansion; or by a Joule in determining the mechanical equivalent of heat. ”

“ Whatever has quantity, or is measurable, must be made up of parts, which bear proportion to one another and to the whole; so that it may be increased by addition of like parts, and diminished by subtraction, may be multiplied and divided, and, in a word, may bear any proportion to another quantity of the same kind, that one line or number can bear to another. That this is essential to all mathematical quantity, is evident from the first elements of algebra, which treats of quantity in general, or of those relations and properties which are common to all kinds of quantity. ”

“ 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. ”

“ 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. ”

“ Of the proportion, which the product of any region bears to the people, an estimate is commonly made according to the pecuniary price of the necessaries of life; a principle of judgment which is never certain, because it supposes what is far from truth, that the value of money is always the same, and so measures an unknown quantity by an uncertain standard. ”

“ Consequently, the value of the variable quantity of produce which, under different circumstances, forms the wages of a given number of men, being composed of the values of the two elements, labour and profits, varying as above described, must be constant, and may therefore, with propriety, be proposed as a standard measure. ”

“ When in an imperfect balance, we reverse the weights to eliminate error, the true weight will be the geometric mean of the two apparent weights. In almost all the calculations of statistics and commerce the geometric mean ought, strictly speaking, to be used. If a commodity rises in price 100 per cent. and another remains unaltered, the mean rise of a price is not 50 per cent. because the ratio 150 : 200 is not the same as 100 : 150. The mean ratio is as unity to √1·00 × 2·00 or 1 to 1·41. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ Bearer of options.
The common measure of value function rests in the intellectual needs of man. It grows out of the necessity for calculation, for bookkeeping, for understanding what is going on. Any object of value may be used to measure [Pg 419] the value of anything else, just as any object of weight—say an irregular mass of iron—may be put in the balance against some other object, and the relation between the absolute weights of the two objects thus more or less definitely ascertained. ”

“ Neither shall we make it a part of our present task to study the equally interesting and more important problem of measurement, which is concerned with quantitative determinations of the variation of the ordinal characteristic; it will here suffice if we say that the possibility of measurement of a given continuous aggregate is dependent upon the existence of a certain relation between every two elements of the continuum, which relation it is possible to express numerically, and to compare as to equality or inequality between various pairs of elements ”

“ 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. ”

“ An important remark which pertains to great generality of analysis, and which permits us to extend this method to formulae and to equations of difference which the theory of probability presents most frequently, is that the series to which one comes by supposing the limits of the definite integrals to be real and positive take place equally in the case where the equation which determines these limits has only negative or imaginary roots. ”

“ 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? ”

“ Among the subjects which should be curtailed or omitted are compound proportion, cube root, abstract mensuration and the greater part of commercial arithmetic. Percentage should be reduced, and the needs of practical life—profit and loss, bank discount, compound interest, with such complications as result from fractional periods of time—are useless and undesirable. The metric system should be taught in application to actual measurements, and the weights and measures handled. ”

“ When we consider farther, that value itself is relative, and that two things are necessary to constitute it, independently of the third thing which is to measure it; we may define a Measure of Value to be something, by comparing with which any two other things, we may infer their value in relation to one another.
In this sense, any commodity will serve as a measure of value at a given time and place; since we can always infer the proportion in which things exchange for one another, when we know the proportion in which each exchanges for any third thing. ”