A

p

from B be such that in one second it would move A to p, and similarly let the second force, acting alone, move A to r. The question arises, then, whether their joint action will urge A to q along the diagonal of the parallelogram. May we say that A will move the distance Ap in the direction AB, and Ar in the direction

AC, or, what is the same thing, along the parallel line pq? In all strictness we cannot say so; for when A has moved towards P, the force from C will no longer act along the line AC, and similarly the motion of A towards r will modify the action of the force from B. This interference of one force with the line of action of the other will evidently be greater the larger is the extent of motion considered; on the other hand, as we reduce the parallelogram Apqr, compared with the distances AB and AC, the less will be the interference of the forces. Accordingly mathematicians avoid all error by considering the motions as infinitely small, so that the interference becomes of a still higher order of infinite smallness, and may be entirely neglected. By the resources of the Differential Calculus it is possible to calculate the motion of the particle A, as if it went through an infinite number of infinitely small diagonals of parallelograms. The great discoveries of Newton really arose from applying this method of calculation to the movements of the moon round the earth, which, while constantly tending to move onward in a straight line, is also deflected towards the earth by gravity, and moves through an elliptic curve, composed as it were of the infinitely small diagonals of infinitely small parallelograms. The mathematician, in his investigation of a curve, always in fact treats it as made up of a great number of short straight lines, and it

may even be doubtful whether he could treat it in any other manner. Nevertheless there is no error in the final results, because having obtained the formula flowing from this supposition, each straight line is then regarded as becoming infinitely small, and the polygonal line becomes undistinguishable from a perfect curve z.

In

In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible. Nevertheless, while carefully distinguishing between these two different cases, we may fearlessly apply to both the principle of the superposition of small motions or effects. physical science we have only to take care that the effects really are so small that any joint effect will be unquestionably imperceptible. Suppose, for instance, that there is some cause which alters the dimensions of a body in the ratio of 1 to 1 + a, and another cause which produces an alteration in the ratio of 1 to 1 + B. If they both act at once the change will be in the ratio of 1 to (1+a) (1 + ß), or as 1 to 1+a+B+aß. But if a and ẞ be both very small fractions of the total dimensions, aß will be yet far smaller and may be disregarded; the ratio of change is then approximately that of 1 to 1+a+B, or the joint effect is the sum of the separate effects. Thus if a body were subjected to three strains at right angles to each her, the total change in the volume of the body would be approximately equal to the sum of the changes pro

by the separate strains, provided that these are of Tal amount. In like manner not only is the exevery solid and liquid substance by heat ey proportional to the change of temperature, age is very small in amount, but the cubic -Yos on the Principles of Pure and Applied Calculation,'

expansion may also be considered as being three times as great as the linear expansion. For if the increase of temperature expands a bar of metal in the ratio of 1 to 1 +α, and the expansion be equal in all directions, then a cube of the same metal would expand as I to (1+a), or as I to 1 + 3a + 3a2 + a3. When a is a very small quantity the third term 3a2 will be imperceptible, and still more so the fourth term a3. The coefficients of expansion of solids are in fact so small, and so imperfectly determined, that physicists seldom take into account their second and higher powers.

It is an universal and important result of these principles that all very small errors may be assumed to vary in simple proportion to their causes; a new reason why, in eliminating errors, we should first of all make them as small as possible. Let us suppose, with De Morgan, that there is a right-angled triangle of which the two sides containing the right angle are really of the lengths 3 and 4, so that the hypothenuse is / 32 +42 or 5. Now if in two measurements of the first side we commit slight errors, making it successively 4'001 and 4'002, then calculation will give the lengths of the hypothenuse as almost exactly 50008 and 5'00016, so that the error in the hypothenuse will seem to vary in simple proportion to that of the side, although it does not really do so with perfect exactness a The logarithm of a number does not vary in proportion to that number-nevertheless we should find the difference between the logarithms of the numbers 100000 and 100001 to be almost exactly equal to that between the numbers 100001 and 100002. It is thus a general rule that very small differences between successive values of a function are approximately proportional to the small differences of the variable quantity.

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186

Four Meanings of Equality.

Although it might seem that there are few terms more free from ambiguity than the term equal, yet scientific men do as a matter of fact employ it with four meanings, which it is very desirable to distinguish carefully. These meanings I may briefly describe as

(1) Absolute Equality.

(2) Sub-equality.
(3) Apparent Equality.

(4) Probable Equality.

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 whole of e differential calculus may, as I apprehend it, be said arse out of the neglect of infinitely small quantities;

his subject however we are not in this place much ened In mathematical science many other subtle s may have to be drawn between kinds of De Morgan has shown in a remarkable memoir : and on the Sign of Equality' b.

wality is that with which physical science The magnitudes are practically equal which

Pal Transactions,' [1865] vol. xi, Part I.

differ only by an imperceptible quantity. To the carpenter anything less than the hundredth part of an inch is non-existent; there are few arts or artists to which the hundred-thousandth 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. Thus 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.

Arithmetic of Approximate Quantities.

We

Considering that almost all the quantities which we treat in physical and social science are approximate only, it seems desirable that some attention should be paid in the teaching of arithmetic to the correct interpretation and treatment of approximate numerical statements. ought carefully to distinguish between 25 when it means exactly two and a half, and when it means, as it usually does, anything between 2:45 and 2'55 It would be better in the latter case to write the number as 2'5.... and we might then distinguish 2'50.... as meaning anything between 2'495 .... and 2'505. 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, for instance, who could assert straightway that