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.
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 ·025◦. 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.