When our mutual life companies began their business in 1843, they had noAmerican table of mortality to guide them in determining the premiums ofinsurance that ought to be charged at the different periods of life. Therewere no American statistics public or private, good, bad, or indifferent, towhich they could refer, except the mortuary reports of cities, and thesewere so imperfect and unreliable as to be utterly useless, except toencourage the opinion that the chances of long life were about the same hereas in the countries from which our people had emigrated. It was known thatthe numbers of the dying, as reported by our city registers, were below thereal deaths; that the ages were full of errors; that the boundaries of themortuary limits were constantly changing; that residents of the city wereoften buried in the country, and sometimes country people were interred intown; that the population was fluctuating; that the immigration from therural districts and from foreign countries was large and irregular; that thecensus of the population, whether taken by the United States, or by thestates, or by the cities themselves, was full of errors; that the ages ofthe living, both among males and females, were wrongly reported, sometimesintentionally, but always carelessly and thoughtlessly; and that theseerrors in the numbers and ages of the people and of the deaths were sonumerous that no confidence could be placed in the ratio of the living andthe dying at any particular age, while this ratio at all ages is anindispensable element in determining the proper premiums to be charged inany of the contracts made by our life companies.

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

Mr. James Smith will, I have no doubt, be the most uneclipsed circle-squarerof our day. He will not owe this distinction to his being an influential andrespected member of the commercial world of Liverpool, even though the powerof publishing which his means give him should induce him to issue a wholelibrary upon one paradox. Neither will he owe it to the pains taken with himby a mathematician, who corresponded with him until the joint letters filledan octavo volume. Neither will he owe it to the notice taken of him by SirWilliam Hamilton, of Dublin, who refuted him in a manner intelligible to anordinary student of Euclid, which refutation he calls a remarkable paradoxeasily explainable, but without explaining it. What he will owe it to Iproceed to show.