By absolute equality we signify that which is complete and perfect to the last degree; but it is obvious that we can only know such equality in a theoretical or hypothetical manner. The areas of two triangles standing upon the same base and between the same parallels are absolutely equal. Hippocrates beautifully proved that the area of a lunula or figure contained between two segments of circles was absolutely equal to that of a certain right-angled triangle. As a general rule all geometrical and other elementary mathematical theorems involve absolute equality.

De Morgan proposed to describe as sub-equal those quantities which are equal within an infinitely small. quantity, so that x is sub-equal to x + dx. The differential calculus may be said to arise out of the neglect of infinitely small quantities, and in mathematical science other subtle distinctions may have to be drawn between kinds of equality, as De Morgan has shown in a remarkable memoir "On Infinity; and on the sign of Equality."1

Apparent equality is that with which physical science deals. Those magnitudes are apparently equal which differ only by an imperceptible quantity. To the carpenter anything less than the hundredth part of an inch is nonexistent; there are few arts or artists to which the hundredthousandth of an inch is of any account. Since all coincidence between physical magnitudes is judged by one or other sense, we must be restricted to a knowledge of apparent equality.

In reality even apparent equality is rarely to be expected. More commonly experiments will give only probable equality, that is results will come so near to each other that the difference may be ascribed to unimportant disturbing causes. Physicists often assume quantities to be equal provided that they fall within the limits of probable error of the processes employed. We cannot expect observations to agree with theory more closely than they agree with each other, as Newton remarked of his investigations concerning Halley's Comet.

1 Cambridge Philosophical Transactions (1865), vol. xi. Part L.

Arithmetic of Approximate Quantities.

Considering that almost all the quantities which we treat in physical and social science are approximate only, it seems desirable that attention should be paid in the teaching of arithmetic to the correct interpretation and treatment of approximate numerical statements. We seem to need notation for expressing the approximateness or exactness of decimal numbers. The fraction 025 may mean either precisely one 40th part, or it may mean anything between 0245 and 0255. I propose that when a decimal fraction is completely and exactly given, a small cipher or circle should be added to indicate that there is nothing more to come, as in 0250. When the first figure of the decimals rejected is 5 or more, the first figure retained should be raised by a unit, according to a rule approved by De Morgan, and now generally recognised. To indicate that the fraction thus retained is more than the truth, a point has been placed over the last figure in some tables of logarithms; but a similar point is used to denote the period of a repeating decimal, and I should therefore propose to employ a colon after the figure; thus 025: would mean that the true quantity lies between 0245° and 025° inclusive of the lower but not the higher limit. When the fraction is less than the truth, two dots might be placed horizontally as in '025.. which would mean anything between 025° and '0255° not inclusive.

When approximate numbers are added, subtracted, multiplied, or divided, it becomes a matter of some complexity to determine the degree of accuracy of the result. There are few persons who could assert off-hand that the sum of the approximate numbers 34'70, 52693, 801, is 1675 within less than 07. Mr. Sandeman has traced out the rules of approximate arithmetic in a very thorough manner, and his directions are worthy of careful attention.1 The third part of Sonnenschein and Nesbitt's excellent book on arithmetic 2 describes fully all kinds of approximate calculations, and shows both how to avoid needless labour

1 Sandeman, Pelicotetics, p. 214.

2 The Science and Art of Arithmetic for the Use of Schools. (Whitaker and Co.)

and how to take proper account of inaccuracy in operating with approximate decimal fractions. A simple investigation of the subject is to be found in Sonnet's Algèbre Elémentaire (Paris, 1848) chap. xiv., "Des Approximations Absolues et Relatives." There is also an American work on the subject.1

Although the accuracy of measurement has so much advanced since the time of Leslie, it is not superfluous to repeat his protest against the unfairness of affecting by a display of decimal fractions a greater degree of accuracy than the nature of the case requires and admits. I have known a scientific man to register the barometer to a second of time when the nearest quarter of an hour would have been amply sufficient. Chemists often publish results of analysis to the ten-thousandth or even the millionth part of the whole, when in all probability the processes employed cannot be depended on beyond the hundredth part. It is seldom desirable to give more than one place of figures of uncertain amount; but it must be allowed that a nice perception of the degree of accuracy possible and desirable is requisite to save misapprehension and needless computation on the one hand, and to secure all attainable exactness on the other hand.

1 Principles of Approximate Calculations, by J. J. Skinner, C.E (New York, Henry Holt), 1876.

2 Leslie, Inquiry into the Nature of Heat, p. 505.

CHAPTER XXII

QUANTITATIVE INDUCTION.

We have not yet formally considered any processes of reasoning which have for their object to disclose laws of nature expressed in quantitative equations: We have been inquiring into the modes by which a phenomenon may be measured, and, if it be a composite phenomenon, may be resolved, by the aid of several measurements, into its component parts. We have also considered the precautions to be taken in the performance of observations and experiments in order that we may know what phenomena we really do measure, but we must remember that no number of facts and observations can by themselves constitute science. Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the principles of nature. It is by an inverse process of reasoning that we can alone discover the mathematical laws to which varying quantities conform. By wellconducted experiments we gain a series of values of a variable, and a corresponding series of values of a variant, and we now want to know what mathematical function the variant is as regards the variable. In the usual progress of a science three questions will have to be answered as regards every important quantitative phenomenon :(1) Is there any constant relation between a variable and a variant?

(2) What is the empirical formula expressing this relation? (3) What is the rational formula expressing the law of nature involved?

Probable Connection of Varying Quantities.

We find it stated by Mill, that "Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation." This assertion may be considered true when it is interpreted with sufficient caution; but it might otherwise lead us into error. There is nothing whatever in the nature of things to prevent the existence of two variations which should apparently follow the same law, and yet have no connection with each other. One binary star might be going through a revolution which, so far as we could tell, was of equal period with that of another binary star, and according to the above rule the motion of one would be the cause of the motion of the other, which would not be really the case. Two astronomical clocks might conceivably be made so nearly perfect that, for several years, no difference could be detected, and we might then infer that the motion of one clock was the cause or effect of the motion of the other. This matter requires careful discrimination. We must bear in mind that the continuous quantities of space, time, force, &c., which we measure, are made up of an infinite number of infinitely small units. We may then meet with two variable phenomena which follow laws so nearly the same, that in no part of the variations open to our observation can any discrepancy be discovered I grant that if two clocks could be shown to have kept exactly the same time during any finite interval, the probability would become infinitely high that there was a connection between their motions. But we can never absolutely prove such coincidences to exist. Allow that we may observe a difference of one-tenth of a second in their time, yet it is possible that they were independently regulated so as to go together within less than that quantity of time. In short, it would require either an infinitely long time of observation, or infinitely acute powers of measuring discrepancy, to decide positively whether two clocks were or were not in relation with each other.

1 System of Logic, bk. iii. chap. viii. § 6.