In the construction of the Tables of Policy Values for the Messenger Prize Essay (1868), the operations were considerably facilitated by the use of different modes of working applicable to the different data at hand; an explanation of which forms the subject of the present paper.

In the formation of Tables the great objects to be attained are accuracy and facility of working. These may be acquired in a high degree if it be possible by means of a simple operation to deduce each value in succession from the one which immediately precedes it, in which case the chances of error are reduced to a minimum, and the construction facilitated by the work not requiring to be checked except at distant intervals.

The theory of Probabilities has been characterized by Laplace, one of those who have contributed most largely to its advance,—as “good sense reduced to a system of calculation;” and such, no doubt, it is. But it must be especially noticed that there is hardly any subject to which thought can be applied, which calls for so continuous an application of that excellent quality, or in which it i s easier to make mistakes from simple want of circumspection. And, moreover, that its reduction to calculation is attended with difficulties of a very peculiar nature, such as occur in no other application of mathematical analysis to practical subjects, arising out of the great magnitudes of the numbers concerned, which defeat the ordinary processes of arithmetical and logarithmic calculation, by exhausting the patience of the computer, and require special methods of approximate evaluation to bring them within the compass of human industry. These methods form a conspicuous feature of the general subject, and have furnished scope for very extraordinary displays of mathematical talent and invention. That very large numbers will inevitably be concerned in questions where numerous and independent contingencies may take place, and in any order or mode of combination, will be apparent to any one who considers the astonishing fecundity of such combinations numerically estimated, when the combining elements are many. For example, the number of possible “hands” at whist (regard being had to the trump) is 1,270,027,119,200.