When our mutual life companies began their business in 1843, they had no American table of mortality to guide them in determining the premiums of insurance that ought to be charged at the different periods of life. There were no American statistics public or private, good, bad, or indifferent, to which they could refer, except the mortuary reports of cities, and these were so imperfect and unreliable as to be utterly useless, except to encourage the opinion that the chances of long life were about the same here as in the countries from which our people had emigrated. It was known that the numbers of the dying, as reported by our city registers, were below the real deaths; that the ages were full of errors; that the boundaries of the mortuary limits were constantly changing; that residents of the city were often buried in the country, and sometimes country people were interred in town; that the population was fluctuating; that the immigration from the rural districts and from foreign countries was large and irregular; that the census of the population, whether taken by the United States, or by the states, or by the cities themselves, was full of errors; that the ages of the living, both among males and females, were wrongly reported, sometimes intentionally, but always carelessly and thoughtlessly; and that these errors in the numbers and ages of the people and of the deaths were so numerous that no confidence could be placed in the ratio of the living and the dying at any particular age, while this ratio at all ages is an indispensable element in determining the proper premiums to be charged in any of the contracts made by our life companies.

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.

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.