Prevarication

“ Let us multiply V by a function T of the two variables t and t′ developed like V in ratio of the powers and the products of these variables; the product will be the discriminant function of a derivative of the primitive function; if T, for example, is equal to the variable t plus the variable t′ minus two, this derivative will be the primitive function of which we diminish by unity the index x plus this same primitive function of which we diminish by unity the index x′ less two times the primitive function. ”

“ When two variables are so related that the value of the first variable depends on the value of the second variable, then the first variable is said to be a function of the second variable.
Nearly all scientific problems deal with quantities and relations of this sort, and in the experiences of everyday life we are continually meeting conditions illustrating the dependence of one quantity on another. For instance, the weight a man is able to lift depends on his strength, other things being equal. ”

“ The reason is, that in all these classes of facts, the causes are either not amenable to direct observation at all, or not with the requisite precision, and we have no means of judging of their frequency except from the empirical law afforded by the frequency of the effects. The inference does not the less depend on causation alone. We reason from an effect to a similar effect by passing through the cause. ”

“ Let us suppose that we desire to find the degree of correlation in variation, or deviation from the mean, between an organ A, called subject, and a second organ B, called relative. We first take all the individuals of one (subject) class; that is, individuals whose subject organ deviates from the mean by a constant quantity, p. We next find for those individuals the average deviation-from-the-mean of the organ B, and call it q. We then find the ration q/p; this is the partial index of correlation. ”

“ 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. ”

“ The formation of a final equation by means of the equation of condition amounts to multiplying each one of these by an indeterminate factor and by uniting the products; it is necessary to choose the system of factors which gives the smallest opportunity for error. But it is apparent {75} that if we multiply the possible errors of an element by their respective probabilities, the most advantageous system will be that in which the sum of these products all, taken, positively is a minimum; for a positive or a negative error ought to be considered as a loss. ”

“ The criterion of values which is used in this study is an empirical one. As has already been explained (p. 8) , every word contained in the frequency tables possesses a value of at least 0.1 per cent, and other words have a zero value. With the aid of our method the difficulty of classifying the reactions quoted above is obviated, as it is necessary only to refer to the table to find their proper values: the value of the reaction bath—water is 33.9 per cent, while that of the reaction bath—ink is 0. ”

“ We can measure neither except relatively, because they must be measured one by the other, and no matter how they vary, these variations always compensate one another, leaving us in the same state of ignorance that we were in before. ”

“ When, therefore, such variations in the quantity of the antecedents as we have the means of observing are small in comparison with the total quantities, there is much danger lest we should mistake the numerical law, and be led to miscalculate the variations which would take place beyond the limits; a miscalculation which would vitiate any conclusion respecting the dependence of the effect upon the cause, that could be founded on those variations. Examples are not wanting of such mistakes. ”

“ We can not, therefore, reason respecting causation, without introducing considerations of quantity and extension at every step; and if the nature of the phenomena admits of our obtaining numerical data of sufficient accuracy, the laws of quantity become the grand instrument for calculating forward to an effect, or backward to a cause. That in all other sciences, as well as in geometry, questions of quality are scarcely ever independent of questions of quantity, may be seen from the most familiar phenomena. ”

“ If or is a function of several scalar or vector variables which are themselves functions of the position of a single point, the value of or or will be equal to the sum of the values obtained by making successively all but each one of these variables constant. ”

“ Almost every series of quantitative experiments is directed to obtain the relation between the different values of one quantity which is varied at will, and another quantity which is caused thereby to vary. We may conveniently distinguish these as respectively the variable and the variant. When we are examining the effect of heat in expanding bodies, heat, or one of its dimensions, temperature, is the variable, length the variant. If we compress a body to observe how much it is thereby heated, pressure, or it may be the dimensions of the body, forms the variable, heat the variant. ”

“ So far as regards the direct application of an approximate generalization to an individual instance, this question presents no difficulty. If the proposition, Most A are B, has been established, by a sufficient induction, as an empirical law, we may conclude that any particular A is B with a probability proportioned to the preponderance of the number of affirmative instances over the number of exceptions. If it has been found practicable to attain numerical precision in the data, a corresponding degree of precision may be given to the evaluation of the chances of error in the conclusion. ”

“ Every new natural constant which is recorded brings many fresh inferences within our power. For if n be the number of such constants known, then 1/2 (n2—n) is the number of ratios which are within our powers of calculation, and this increases with the square of n. We thus gradually piece together a map of nature, in which the lines of inference from one phenomenon to another rapidly grow in complexity, and the powers of scientific prediction are correspondingly augmented. ”

“ A variable is said to vary continuously through an interval , when starts with the value and increases until it takes on the value in such a manner as to assume the value of every number between and in the order of their magnitudes. ”

“ In particular we observe that the probability that the energy of an unspecified system of a canonical ensemble lies between the limits and will be represented by the integral [2]
and that the average value in the ensemble of any quantity which only varies with the energy is given by the equation [3] where we may regard the constant as determined by the equation [4] In regard to the lower limit in these integrals, it will be observed that is equivalent to the condition that the value of is the least possible. ”

“ Thus, in so far as physics is an empirical and verifiable science, it must assume or prove that the inference from appearances to matter is, in general, legitimate, and it must be able to tell us, more or less, what appearances to expect. It is through this question of verifiability and empirical applicability to experience that we are led to a theory of matter such as I advocate. ”

“ 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. ”

“ So soon, however, as a result is known to be discoverable, the scientific man ought to proceed to the quantitative inquiry, how great a result follows from a certain amount of the conditions which are supposed to constitute the cause? The possible numbers of experiments are now infinitely great, for every variation in a quantitative condition will usually produce a variation in the amount of the effect. The method of variation which thus arises is no narrow or special method, but it is the general application of experiment to phenomena capable of continuous variation. ”

“ But, as we decrease the number of experiments, the value of the standard deviation found from the sample of experiments becomes itself subject to an increasing error, until judgments reached in this way may become altogether misleading.
In routine work there are two ways of dealing with this difficulty: (1) an experiment may be repeated many times, until such a long series is obtained that the standard deviation is determined once and for all with sufficient accuracy. This value can then be used for subsequent shorter series of similar experiments. ”

“ The number of ratios n are expressible in n equations, which will contain at least n + 1 quantities, so that if we employ them to make known n magnitudes, we must have one magnitude known. Hence, whether we are measuring time, space, density, mass, weight, energy, or any other physical quantity, we must refer to some concrete standard, some actual object, which if once lost and irrecoverable, all our measures lose their absolute meaning. ”

“ In the present case, however, instead of a single instance A B, we have had to study separately the causes A and B, and to infer from the effects which they produce separately, what effect they must produce in the case A B C where they act together. Of the two instances, therefore, which the Method of Difference requires,—the one positive, the other negative,—the negative one, or that in which the given phenomenon is absent, is not the direct result of observation and experiment, but has been arrived at by deduction. ”

“ If one reduces the fraction whose numerator is unity and whose denominator is unity plus a variable, in a series prescribed by the ratio to the powers of this variable, it is easy to see that in supposing the variable equal to unity the fraction becomes 1⁄2, and the series becomes plus one, minus one, plus one, minus one, etc. ”

“ The probable error is the measurement that divides the deviations from the average in either direction into halves, so that there are as many events having a greater deviation as there are events having a less deviation. ”

“ Measure is the qualitative quantum, in the first place as immediate,—a quantum, to which a determinate being or a quality is attached.
Measure, where quality and quantity are in one, is thus the completion of Being. Being, as we first apprehend it, is something utterly abstract and characterless: but it is the very essence of Being to characterise itself, and its complete characterisation is reached in Measure. Measure, like the other stages of Being, may serve as a definition of the Absolute: God, it has been said, is the Measure of all things. ”

“ It ought to be remarked, that these are, of all others, the conjunctures where observations are most available for the determination of data; for, by the same rule that a small change in the data will, in such cases, produce a great one in the thing to be observed; so, vice versâ, any moderate amount of error, committed in an observation undertaken for ascertaining its value, can produce but a very trifling one in the reverse calculation from which the data come to be determined by observation. ”

“ Our knowledge has become scientific to the extent in which it has been able to make use of number and measure. It might be worth while to try and see whether a scientific order of values might not be constructed according to a scale of numbers and measures representing energy.... All other values are matters of prejudice, simplicity, and misunderstanding. They may all be reduced to that scale of numbers and measures representing energy. The ascent in this scale would [Pg 180] represent an increase of value, the descent a diminution. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ In statistics, it is evident that empirical laws may sometimes be traced; and the tracing them forms an important part of that system of indirect observation on which we must often rely for the data of the Deductive Science. ”

“ Analysis, which treats plane curves as equations in two variables, gives the direction of the tangent in terms of the differential coefficient, and declares accordingly that every continuous function has a differential coefficient, that is, may be differentiated, at every point. ”

“ On the other hand when the individuals of the sample lie far apart, the argument of Sheppard’s correction will apply, the real value of the standard deviation being more likely to be smaller than that found owing to the frequency in any group being greater on the side nearer the mode. ”