five years. Each of the first five rates in col. 1 was therefore multiplied by 93014, reducing each of them by 6986 per cent., and thus constituting the 'corrected rate of mortality' for each of the first five years of age appearing in col. 2 of Table XI. The yearly rates in each group of ages were dealt with in a similar manner, the rates for the five years 5-10 being reduced 29.955 per cent.; those for the five years 10-15, 32155 per cent.; and so on through each of the twelve groups of ages. These corrected rates of mortality at each year of age constitute the ma of the new life table. There is an obvious defect in this series of corrected annual rates of mortality, viz. the irregularities due to the assumption that the changes in the proportions of decrease in the mean rates of succeeding age-periods, took place suddenly at the commencement of each period, instead of coming gradually into operation, as was certainly the case. The series would undoubtedly look better, and would probably be more technically correct, if these irregularities had been graduated and smoothed away, but having regard to the intended purpose of the table, this process has been omitted. "From this mortality (m) column the probability of living at each year of age (p) has been obtained by the formula p Thus 2-17046 = = 2 2+mx 84293, which represents the corrected proba bility of a male living one year from birth. By this process, the probability of living one year is obtained for each age in the series. "The next column required is one showing the numbers born and living at each age (). We start for reasons that have already been described, with 509,208 males at birth, and having ascertained the probability at birth of living one year to be 84293, the survivors attaining one year of age or (1) may be obtained by multiplying 509,208 by 84293. Carrying on this process of multiplying the numbers surviving to each year of age by the probability at that age of living one year, the survivors to the next age are successively obtained, until the generation becomes extinct. "Thus the formula used for obtaining the l, column is :-( x p1 = × P x l+1). This process, laborious by common arithmetic, is shortened by the use of logarithms. By adding the logarithm of Po to the logarithm of 1, the logarithm of 1, or the number calculated to complete the first year of life, is obtained. The addition of the logarithm of p1 to the logarithm of, in like manner gives the logarithm of 12, and so on to the end of the series. TABLE XI. A PORTION OF MR. NOEL HUMPHREY'S ENGLISH LIFE TABLE, BASED ON MORTALITY IN FIVE YEARS 1876-80: MALES. 1 This number shows how many years the 79 males who attain the age of 100 years will live, calculated by the English Life Table No. 3. D "The next necessary column for our purpose is one showing the mean numbers living in each year of age, which is obviously less than the number surviving to, or living at the commencement of each year. This column is described in a life table as P1, meaning the mean population living during the year following each age in the series. It is assumed that the number living in each year of age is the arithmetical mean of the numbers living at the beginning 1⁄2 + 1⁄2 + 1 and at the end of the year. P, is therefore equal to This also indicates the number of years of life lived in the from age x to age x + 1. 2 year, "The last column given in the table is what is technically called the column. The number against any age in this column is the sum of all the numbers in the P, from that age to the end of the table. It therefore shows the aggregate number of years which the males at each age in the table will live, until their extinction by death. Thus 2, is equal to 21,347,889, showing that according to this life table the generation of 509,208 male infants at birth, which was assumed as the radix of the male table, would live this aggregate number of years before final extinction by death; and gives the mean future lifetime of all 1. the persons living at age x in the table, and as the mean duration of life of a generation of males subject to the mean rates of mortality that prevailed at twelve groups of ages during the five years 1876-80 would be 41-92 years." CHAPTER III. AIDS TO CALCULATION. (19) Tables. FIRST in order are suitable tables, e.g., logarithms to four or five figures; of these Mr. Hy. Law's and Dr. C. J. Woodward's will be found most convenient. Hoüel's reprint of Lalande is also to be commended. Crelles' multiplication tables, by the aid of which three figures may be dealt with at once, and reciprocals such as Lieut.-Col. Oakes's tables (footnote, p. 15) which reduce division to the short multiplication of decimals and render easy the addition of fractions are all useful. The greatest saving in labour and avoidance of monotony is however obtained from the use of mechanical aids such as Professor Fuller's Spiral Rule and the Arithmometer (20) Professor Fuller's Spiral Rule1 The rule (Figs. 1 and 2) consists of a cylinder jacket (d) that can be moved up and down upon, and turned round, an axis (ƒ), which is held by a handle (e). Upon this cylinder is wound in a spiral a single logarithmic scale. Fixed to the handle is an index (1) which for distinction may be called "the indicator," for the answer to a sum is always read from it. Two other indices (c) and (a), whose distance apart is the axial length of the complete spiral, are fixed to the cylinder (g). This cylinder slides in (ƒ) like a telescope tube, and thus enables the operator to place these indices on any part of the scale. Two stops (o) and (p) 1 Messrs. Stanley, Great Turnstile, are the agents. The price is 31. |