between15 and 26 there would die 11,494
25 and 36""" 17,071
35 and 46""" 21,960
45 and 56""" 30,493
55 and 66""" 40,708
65 and upwards""" 44,643

Thus the risk of life from this malady is twice as great at the age of thirty-one as it is at eleven. It is also nearly twice as great at forty-one as it is at twenty-one. It is five times as great at sixty-one as it is at eleven, and nearly four times as great above sixty-five as it is at twenty-one.

From the whole of the foregoing statements, it is manifest that life is a fluctuating quantity. In order to compare this fluctuating quantity under different circumstances, writers on this branch of statistics use several terms, the exact meaning of which it is desirable to explain. It is, for example, very important to have a clear understanding of what is meant by such expressions as the following: the expectation, the probability, the value, the decrement of life, and the law of mortality.

1. The Expectation of Life. It is important to bear in mind that several expressions in common use have a signification perfectly synonymous with this: namely, share of existence; mean duration of life; la vie moyenne.

By these terms is expressed the total number of years, including also the fractional parts of a year, ordinarily attained by human beings from and after any given age. Suppose, for example, that one thousand persons enter on the eighty-sixth year of their age: suppose the number of years and days which each one of them lives afterwards be observed and recorded; suppose the number ultimately attained by each be formed into a sum total; suppose this total be divided equally among the thousand, the quotient of this division is said to be each one's share of existence, or his mean duration of life, or his expectation of life. Thus, of the thousand persons in the present case supposed to commence the age of eighty-five, suppose the number of years they collectively attain amount to 3,500 years: the one-thousandth part of 3,500 is three and a half: three years and a half then is said to be the expectation of life at the age of eighty-five, because, of all the persons originally starting, this is the equal share of existence that falls to the lot of each.

2. Probability of Life; or the probable duration of life, la vie probable. These are synonymous terms, in use chiefly among continental writers as an expression of the comparative duration of life. The tabular methods of setting forth the duration of life consist, for the most part, in assuming that 10,000 infants are born; and that at the age of one, two, three, and each successive year of life, there are so many still remaining in existence. Fix on any age; observe what number remain alive to commence that age; note at what age this number decreases to one-half; the age at which they so come to one-half is called the probable term of life; because, say the continental writers, it is an equal wager whether a person shall or shall not be alive at that period. Thus, suppose one thousand males commence together the age of eighty-four; suppose the table indicate that there will be alive at the age of eighty-five, 817; at the age of eighty-six, 648; at the age of eighty-seven, 493; at the age of eighty-eight, 357, and so on. In the present case, the probable duration of life at eighty-four is said to be very nearly three years, because, at the age of eighty-seven there are left alive 493, very nearly one-half of the thousand that originally started together.

3. Value of Life. This term, when used accurately, expresses the duration of life as measured by one or other of the methods already expounded. But it is sometimes popularly used in a loose and singularly inaccurate sense. Thus it is very commonly said—"Such a man's life is not worth ten years' purchase," which is the same thing as to say, that an annuity, suppose a hundred pounds a year, payable during the life of the person in question, is not worth ten times its magnitude, that is one thousand pounds. If a thousand pounds be put into a bank at some rate of interest to be agreed upon, and if a hundred pounds be drawn every year from the stock, the expression under consideration affirms that the person in question will be dead before the principal and interest are exhausted. For instance, at four per cent., the value of an annuity of one hundred pounds to a man of the age of twenty-five is 1694l., which is 16-9/10 years' purchase; whereas, his expectation of life at that age is 35-9/10 years.

4. Law of Mortality. By this term is expressed the proportion out of any determinate number of human beings who enter on a given year of age, that will die in that year. Every observation on the duration of life presents certain numbers, which, by recorded facts, are found to pass through each year of age, and also shows how many have died or failed to pass through every year of age. Those numbers, by the rule of three, are converted into the proportions who would die at each age out of one million of persons, if such a number had commenced it. Suppose, then, a million of persons to be in existence at the first year of age; suppose a million to be in existence at the second year of age; suppose a million to be in existence at the third year of age; and in this manner suppose an equal number to be in existence at the commencement of each and every year to the extreme term of human life. Now, the proportions that by actual observation are found to die at each and every year out of the million that were alive at the commencement of it, form separately the law of mortality for each year, and collectively for the whole of life.