a great number of straight lines, and it may be doubted whether he could treat it in any other manner. There is no error in the final results, because having obtained the formulæ flowing from this supposition, each straight line is then regarded as becoming infinitely small, and the polygonal line becomes undistinguishable from a perfect curve.18
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 effects. In 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 + α, and another cause which produces an alteration in the ratio of 1 to 1 + β. If they both act at once the change will be in the ratio of 1 to (1 + α)(1 + β), or as 1 to 1 + α + β + αβ. But if α and β be both very small fractions of the total dimensions, αβ will be yet far smaller and may be disregarded; the ratio of change is then approximately that of 1 to 1 + α + β, 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 other, the total change in the volume of the body would be approximately equal to the sum of the changes produced by the separate strains, provided that these are very small. In like manner not only is the expansion of every solid and liquid substance by heat approximately proportional to the change of temperature, when this change is very small in amount, but the cubic 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 1 to (1 + α)3, or as 1 to 1 + 3α + 3α2 + α3. When α is a very small quantity the third term 3α2 will be imperceptible, and still more so the fourth term α3. 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 a result of these principles that all 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 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 5·0008 and 5·0016, 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. The logarithm of a number does not vary in proportion to that number—nevertheless we 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.
On these principles it is easy to draw up a series of rules such as those given by Kohlrausch19 for performing calculations in an abbreviated form when the variable quantity is very small compared with unity. Thus for 1 ÷ (1 + α) we may substitute 1 – α; for 1 ÷ (1 – α) we may put 1 + α; 1 ÷ √1 + α becomes 1 – 1/2α, and so forth.