The Insurance Times of New York having reprinted the greater part of Mr. Sprague's paper On the proper method of estimating the liability of a Life Insurance Company under its policies, and Mr. Makeham's letter which appeared in the same number of this Journal, the Hon. Elizur Wright has thought it desirable to explain the aim of the American Insurance Law in a letter to that periodical, from which we make the following extract. We do this not only on account of the general interest with which Mr. Wright's views will be read, but still more because he corrects a misconception as to the object and purpose of the American Life Insurance legislation.—Ed. J. I. A.

The theory of the construction of Tables of Mortality offers no difficulty when applied either to the members of a Society, or to a collection of individuals, who can be traced so that all the necessary particulars as to them are known. But in the population of a country or a town, several incidental causes, such as epidemics and migrations, exercise an influence; some of them on the total numbers of the population, and others on the members living at certain ages. The mobility of a population or the constant change taking place, from one residence to another, or from one country to another, prevents us from following each individual or each group of individuals from the moment of birth to that of death.

We have received through the courtesy of Mr. William Barnes, late Superintendent of the Insurance Department of New York, and now “Consulting Counsel and Actuary,” a pamphlet of 63 pages, purporting to be a report on the condition and prospects of the above Insurance Company for the year 1870, but which may with perhaps greater propriety be called a review of the general position of Life Insurance in America, England, Germany, and France.

If any excuse be wanting for once more bringing this question forward for your consideration, it is not because the question itself is unimportant. The frequency of the cases involving it upon which the opinion of actuaries is sought, and the magnitude of the interests concerned, will be allowed by all to furnish ample grounds for the utmost possible care and pains in determining the principles by which the advice given should be regulated. Upwards of 15 years have elapsed since Mr. Jellicoe's well-known paper on this subject was read before the Institute. How far the practice of actuaries has been influenced by the conclusions laid down in that paper I do not know; but from the fact that (so far as I am aware, at least,) the subject has not been reopened since, it would seem that these conclusions have been accepted by the profession, and that in the opinion of those most competent to form a judgment there is nothing more to be said about them. If this be so, I am afraid that I shall incur the suspicion of rashness when I say that the excuse I have to urge for reopening the question is, that the conclusions laid down by Mr. Jellicoe do not commend themselves to my mind as perfectly satisfactory. I propose very briefly to state my reasons for differing from him; and should I fail, as I well may, in convincing others of the correctness of my views, I shall be consoled by the reflection, that however firmly established, and apparently beyond discussion, any truth may be, it is a profitable thing for most, perhaps for all of us, if a rash sceptic now and then compels us to review the grounds of our confidence, and to restate to ourselves, if not to others, the reasons for the faith that is in us.

Mr. James Smith will, I have no doubt, be the most uneclipsed circle-squarer of our day. He will not owe this distinction to his being an influential and respected member of the commercial world of Liverpool, even though the power of publishing which his means give him should induce him to issue a whole library upon one paradox. Neither will he owe it to the pains taken with him by a mathematician, who corresponded with him until the joint letters filled an octavo volume. Neither will he owe it to the notice taken of him by Sir William Hamilton, of Dublin, who refuted him in a manner intelligible to an ordinary student of Euclid, which refutation he calls a remarkable paradox easily explainable, but without explaining it. What he will owe it to I proceed to show.

Having thus, as I believe, demonstrated that life insurance calculations have nothing to do with probabilities, I come back to the idea of risk. This, as I pointed out at starting, must be taken from the theory of probabilities, or more precisely, from that part of it which has been cultivated since the beginning of this century, by Lagrange, Gauss, Laplace, and others, viz., the method of least squares. In that method is defined the idea of the “mean error,” which is considered as the measure of the danger to which we are exposed in a single case. This “mean error” is the square root of the sum of all the squares of the errors divided by their number; and the squares of the errors themselves are formed from the deviations of all the single cases from the average or most probable value. In insurances depending upon life and death, the value is also calculated according to the average, so that when all the assured are dead, if the mortality has followed the mean numbers given by the table of mortality, and the additions to the premiums for the expenses of management are disregarded, there will be neither surplus nor deficiency. This average value is the so-called net premium, which may be either a single premium or may be payable for a term of years agreed on beforehand. But we can calculate beforehand from the mortality table all the deviations, or the gains and losses, which can arise from the earlier or later death of the lives assured. Squaring all these deviations, and dividing the sum of the squares by their number, and taking the square root of this sum, we get the value of the mean danger or the risk attaching to a single insurance. For further elucidation some applications of this process will now be given.