In treating of the above subject, it cannot be expected that any new principles will be put forward, or that any great amount of novelty will be found in the method of handling the question. All that I have aimed at is, to apply acknowledged principles to the discussion of a question of considerable practical interest at all times, and more peculiarly so at the present time.

On a former occasion I had the honour to lay before the Institute a historical sketch of this subject, from the earliest period of authentic records down to that of the legal establishment of the practice of taking interest, in the reign of Elizabeth.

The facts then adduced showed that very high nominal rates of interest were obtained in the ancient world, and during the middle ages; but such rates are, by no means, to be taken as measures of the profits made upon loans, which must always have been seriously diminished, and often entirely swallowed up, by the exactions and spoliations to which the lender was exposed, and the difficulties he experienced in enforcing his claims, from defective laws, and the prejudices entertained against him by the community.

The calculus of finite differences was created by Taylor, in his celebrated work entitled Methodus Incrementorum, and it consists, essentially, in the consideration of the finite increments which functions receive as a consequence of analogous increments on the part of the corresponding variables. These increments or differences, which take the characteristic Δ to distinguish them from differentials, or infinitely small increments, may be in their turn regarded as new functions, and become the subject of a second similar consideration, and so on; from which results the notion of differences of various successive orders, analogous at least in appearance to the consecutive orders of differentials. Suc h a calculus evidently presents, like the calculus of indirect functions, two general classes of questions:—

1. To determine the successive differences of all the various analytical functions of one or more variables, as the result of a definite manner of increase of the independent variables, which are generally supposed to augment in arithmetical progression.

2. Reciprocally to start from these differences, or, more generally, from any equation established between them, and go back to the primitive functions themselves or to their corresponding relations

In the year 1847, a report by the Special Commission appointed by the Belgian Government to inquire into assurance by the State, led to a project of law, in the first article of which it was proposed that a general system of insurance against fire, hail, and mortality amongst cattle, should be undertaken by the State. The Commission, however, had omitted to state the cost at which the system recommended could be carried out, and it was referred to the Central Commission of Statistics to inquire further into the subject. The report of M. Frére Orban, the Minister of Finance, contains a large amount of very valuable information on each of the subjects referred to; and with regard to fire insurance, he enters upon the general inquiry how far it would be profitable to the State to undertake this branch of business.