*p* = the given payment, π = the payment according to T, and S = the sum required; so that π = S.

The problem is, to obtain a value for π which shall be commensurable with W.

In the early treatises on this subject, the mean value assigned to π will be found to be 40.000000. Later writers suspected that the decimal point had been accidentally shifted, and that the proper value was 400.00000: but, as the details of the process for obtaining it had been lost, no further progress was made in the subject till our own time, though several most ingenious methods were tried for solving the problem.

Of these methods we proceed to give some brief account. Those chiefly worthy of note appear to be Rationalisation, the Method of Indifferences, Penrhyn's Method, and the Method of Elimination.

We shall conclude with an account of the great discovery of our own day, the Method of Evaluation under Pressure.