The first question that must be considered in connection with this subject is,—When is a Life Insurance Company insolvent? This question has recently acquired greater practical importance in consequence of the passing of the “Life Assurance Companies Act, 1870,” by which it is for the first time in effect enacted that an insolvent Life Office may be wound up, altho it has not committed any act of bankruptcy. Under the old law, even if such a Company were notoriously insolvent, it may be said that practically there was no means of putting a stop to its operations until it failed to pay an accrued claim. This has now been altered, and a Company that is proved to be insolvent can be wound up. The 21st section of the above Act provides that the Court of Chancery “may order “the winding up of any Company in accordance with the Companies “Act, 1862, on the application of one or more policyholders or “shareholders, upon its being proved to the satisfaction of the “Court that the Company is insolvent, and in determining whether “or not the Company is insolvent the Court shall take into account “its contingent or prospective liability under policies and annuity “and other existing contracts.”

The accompanying illustration gives a top view of the instrument. The following description is taken from the Engineer, 20th May, 1870.

It is constructed chiefly of a brass plate A A furnished with eight slots as shown; directly under these slots are mounted eight drums, each having nine elongated cog-teeth of successively decreasing length; over each drum, and between it and the slot, is mounted a square shaft on which slides a pinion wheel, so as to catch any number of teeth on the drum. Each of these pinion wheels is moved by a button a, of which there is one in each slot, the figures at the sides of the slots showing the proper position of each button a for any work to be performed by the instrument, so that not the least trouble is encountered in arriving at the result. The cogged drums gear by bevel wheels with a long horizontal shaft which is also in gear with the vertical shaft moved by the handle b, by which the instrument is worked. B B is a moveable brass plate, which can turn and slide on a round bar hinge at the back; in this plate there are sixteen holes, c, under each of which is a moveable disk numbered from 0 to 9, and arranged so that any one figure of each disk may be brought under its corresponding hole c.

In every branch of business, even in the very rudest stage of barter, men have to count. A skin is exchanged for so many cowries, a rifle for so many skins. Do what we will, go where we will, the necessity for counting meets us. Yet, though numbers be essential to all our operations, we do not easily form an idea of large numbers. Thus in no language do the separate names for numbers reach to twenty. In one, I believe, the numeration goes as far as fifteen; in the northern languages of Europe it reaches to twelve, but in most languages it stops short at ten.

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.

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.