The paper which I have the honour to submit is so elementary in its details that I have to claim the indulgence of the meeting; but, as the practical application of the simple formulæ to which I shall have occasion to refer may be interesting, I trust I may not unworthily occupy a short portion of your time.

A few months ago it occurred to me that tables to facilitate the process of multiplication, based upon the identity

would be of extensive use in actuarial and other calculations, and would effect a considerable saving of labour—to an extent even greater than that acquired by the use of logarithms. This led me to give attention to the subject; and after a trial on a small scale, I deemed it of sufficient importance to construct for my own use a series of tables of the fourth parts of the squares of all integer numbers up to 100,000.

The progressive rates of mortality, from birth to extreme old age, in a given number of any population, are known from the proportionate numbers who die at the different ages. This is the only mode of estimating correctly the expectation or conceived “security” of life at any given age—viz., by showing the chance which an individual has of living a certain number of years or shorter periods.

The application of the doctrine of chances to questions connected with the money values of contingencies depending on human life is now become so common, in this and many other countries, that we may he apt to forget its origin and the process by which it arrived at its present degree of improvement. Although the history of this calculus may be well known to some members of this Institute, there may be others whose time is, perhaps, too occupied to allow of more than the mere study of the calculations and formulæ necessary for the daily routine of a Life Assurance Office, and who may not be aware of the number of other subjects to which, at an early period, the calculus of probabilities was suggested to be applied. I hope, therefore, that in taking a rapid glance at the changes and improvements effected since the first crude ideas on the subject, I may not only be able to excite an increased interest in the science, but induce a further extension of its usefulness.

As the reasonings of the actuary, when extended beyond the mere official routine of a Life Assurance Office, depend upon the proper collection of data relating to population, and the other important elements of the condition and progress of a country, we need make no apology to our readers for presenting to them the following very interesting supplement to the valuable Report of Dr. Farr on the Statistical Congress at Paris in 1855:—

“There are seven great powers in the world.

“England, France, Turkey, and Austria, have existed as great powers for several centuries. Prussia, Russia, and the United States of America, have entered this class within the last hundred years.