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, 52·693, 80·1, is 167·5 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.21 The third part of Sonnenschein and Nesbitt’s excellent book on arithmetic22 describes fully all kinds of approximate calculations, and shows both how to avoid needless labour 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.23
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.24 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.