Publisher's Synopsis
An excerpt from the beginning of CHAPTER I. CHOICE.
We have continually to make our choice among different courses of action open to us, and upon the discretion with which we make it, in little matters and in great, depends our prosperity and our happiness. Of this discretion a higher philosophy treats, and it is not to be supposed that Arithmetic has anything to do with it; but it is the province of Arithmetic, under given circumstances, to measure the choice which we have to exercise, or to determine precisely the number of courses open to us.
Suppose, for instance, that a member is to be returned to parliament for a certain borough, and that four candidates present themselves. Arithmetic has nothing to do with the manner in which we shall exercise our privilege as a voter, which depends on our discretion in judging the qualifications of the different candidates; but it belongs to Arithmetic, as the science of counting and calculation, to tell us that the number of ways in which (if we vote at all) we can exercise our choice, is four.
The operation is, indeed, in this case so simple that we scarcely recognise its arithmetical character at all; but if we pass on to a more complicated case, we shall observe that some thought or calculation is required to determine the number of courses open to us: and thought about numbers is Arithmetic.
Suppose, then, that the borough has to return two members instead of one. And still suppose that we have the same four candidates, whom we will distinguish by names, as A, B, C, D. If we try to note down all the ways in which it is possible for us to vote, we shall find them to be six in number; thus we may vote for any of the following: -
A and B,
A and C, B and C,
A and D, B and D, C and D.
But we can hardly make this experiment without perceiving that the resulting number, six, must in some way depend arithmetically upon the number of candidates and the number of members to be returned, or without suspecting that on some of the principles of arithmetic we ought to be able to arrive at that result without the labour of noting all the possible courses open to us, and then counting them up; a labour which we may observe would be very great if eight or ten candidates offered themselves, instead of four.
In the present chapter we shall establish and explain the principles upon which such calculations are made arithmetically. It will be found that they are very simple in nature as well as few in number. In the next four chapters we shall treat the subject somewhat more largely by algebraical methods; but the reader who is unacquainted with algebra may pass over those chapters, and proceed with the first chapter on Chance (Chap, VI.), in which he will find the principles of Choice applied arithmetically to the solutions of problems in Probability, a subject of great interest and some practical importance.
