between 15 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.