durations of, say, two examples of some type. For example, we may if we like suppose that the rate of the earth's rotation is decreasing, so that each day is longer than the preceding by some minute fraction of a second. Such a rule enables us to compare the length of any day with that of any other day. But what is essential is that one series of repetitions, such as successive days, should be taken as the standard series; and, if the various events of that series are not taken as of equal duration, that a rule should be stated which regulates the duration to be assigned to each day in terms of the duration of any other day.
What then are the requisites which such a rule ought to have? In the first place it should lead to the assignment of nearly equal durations to events which common sense judges to possess equal durations. A rule which made days of violently different lengths, and which made the speeds of apparently similar operations vary utterly out of proportion to the apparent minuteness of their differences, would never do. Hence the first requisite is general agreement with common sense. But this is not sufficient absolutely to determine the rule, for common sense is a rough observer and very easily satisfied. The next requisite is that minute adjustments of the rule should be so made as to allow of the simplest possible statements of the laws of nature. For example, astronomers tell us that the earth's rotation is slowing down, so that each day gains in length by some inconceivably minute fraction of a second. Their only reason for their assertion (as stated more fully in the discussion of periodicity) is that without it they would have to abandon the Newtonian laws of motion. In order to keep
the laws of motion simple, they alter the measure of time. This is a perfectly legitimate procedure so long as it is thoroughly understood.
What has been said above about the abstract nature of the mathematical properties
of space applies with appropriate verbal changes to the mathematical properties of time. A sense of the flux of time accompanies all our sensations and perceptions, and practically all that interests us in regard to time can be paralleled by the abstract mathematical properties which we ascribe to it. Conversely what has been said about the two requisites for the rule by which we determine the length of the day, also applies to the rule for determining the length of a yard measure–-namely, the yard measure appears to retain the same length as it moves about. Accordingly, any rule must bring out that, apart from minute changes, it does remain of invariable length; Again, the second requisite is this, a definite rule for minute changes shall be stated which allows of the simplest expression of the laws of nature. For example, in accordance with the second requisite the yard measures are supposed to expand and contract with changes of temperature according to the substances which they are made of.
Apart from the facts that our sensations are accompanied with perceptions of locality and of duration, and that lines, areas, volumes, and durations, are each in their way quantities, the theory of numbers would be of very subordinate use in the exploration of the laws of the Universe, As it is, physical science
reposes on the main ideas of number, quantity, space, and time. The mathematical sciences associated with them do not form the whole of mathematics, but they are the substratum of mathematical physics as at present existing.
Notes
AnoteA (60).–-In reading these equations it must be noted that a bracket is used in mathematical symbolism to mean that the operations within it are to be performed first. Thus ( 1 + 3 ) + 2 directs us first to add 3