A Formula for Expressing the Decrement of Human Life

XX. A formula for expressing the decrement of human life. In a letter addressed to Sir Edward Hyde East, Bart. M. P. F. R. S. By Thomas Young, M. D. For. Sec. R. S. Communicated February 2, 1826. Read April 19, 1826. My Dear Sir, The investigation of the laws, by which the general mortality of the human species appears to be governed, is of equal importance to the statesman, the physician, the natural philosopher, and the mathematician; and as you have had occasion to pay particular attention to the subject, I trust that it will not be disagreeable to you to receive the results of an inquiry, into which I have entered, for the purpose of appreciating, if not of reconciling, the many discordant opinions that have been advanced, respecting the comparative mortality of mankind, at different times, and under different circumstances. Of late years, there is little doubt, that, whether from the protective effects of vaccination in infancy, or from the increase of the comforts of the poorer, and of the temperance of the more affluent classes of society, or in some measure also from the simplification of the practice of physic and surgery, there is a decided increase in the mean duration of life in many parts of Europe: but it is also extremely probable that this improvement has been greatly exaggerated; partly on account of the limited description of the persons on whom Dr. Young's formula for expressing the observations have been made, and partly from an erroneous opinion respecting the profits of certain establishments, which have been attributed to the employment of too low an estimate of mortality, while they have, in fact, been principally derived from the high rate of interest which the state of public credit has afforded. A very laborious and well informed actuary has lately asserted, before a Committee of the House of Commons, that "the duration of existence now, compared with what it was a hundred years ago, is as four to three, in round numbers." (Parl. Rep. N. 522, p. 44.) It does indeed happen, that this particular result may in one sense be very correctly deduced from the immediate comparison of the annual mortality of a certain number of persons of the same description, that is, annuitants, at the periods in question; nor is it possible to deny that some importance must be attached to the remark: but the mortality of the same class of persons in France, at the earlier period, was no greater, according to Mr. Deparcieux's estimate of their longevity, than in England at the later, while the general mortality in France has never been materially less than in England, and appears at present to be even somewhat greater: and it can only be conjectured, that the annuitants of the tontine of King William were in general most injudiciously selected, while those who were the subjects of Mr. Deparcieux's observations, like the annuitants of the modern tontines, were chosen with more care, or with greater success. Mr. Finlaison's tables, therefore, though they may be extremely just and valuable for the purpose of setting a price upon annuities to be granted on the lives of the proposers, cannot, with any prudence, be adopted where the parties concerned have an interest in offering the worst lives that they can find; notwithstanding any partial security that might be afforded by the exercise of medical skill in their rejection; and if it is true, that some of the tontines were principally filled by lot (Rep. p. 16), with the children of country clergymen and magistrates, it must still be supposed that the families of such persons may have been more healthy than the average of the population of London and the country taken together. For the comparison of the general characters of different tables of mortality, the simplest and most obvious criterion is perhaps the number of individuals out of which one dies annually, which is also the number of years expressing the expectation of life at the time of birth. But this test is liable to material objections with regard to the most usual application of the table, which depends more on the comparative expectations at later periods than in early infancy. For example; the Northampton table affords results, throughout the whole of middle and advanced life, agreeing almost exactly with Demoivre's hypothesis of equal decrements, although the annual mortality is supposed to be nearly 1 in 25 at Northampton, instead of 1 in 43, as assumed by Demoivre. It would therefore be very unjust for a person allowing the truth of Demoivre's hypothesis, to condemn the practical employment of Dr. Price's tables in common cases, on account of this variation only. A less exceptionable test will be, to find the mean of the numbers expressing, for different ages, the full term of life, or the sum of the age and twice the expectation, taking the decades from 10 to 80 as the most important. Another standard of comparison may be the age Dr. Young's formula for expressing which is equal to the expectation of life, and which, in Demoivre's arithmetical hypothesis, is the mean age of all the population, and probably very near it in all tables formed from actual observation. In this manner a general comparison of the most remarkable tables may be instituted. Characteristics of Mortality. | Roman estimate of Ulpian, probably with some deduction for present value | 74 (+ disc) | 26 (+ disc) | |-----------------------------|-------------|-------------| | Deparcieux's Tontines, beginning 1689 | 47.67 | 94.17 | 32.5 | | Halley's table for Breslau, 1690 | 33.50 | 87.15 | 28.1 | | Tontine of 1695, Finlaison, males | 37.61 | 83.42 | 27.25 | | females | (43.0) | 87.50 | 29.32 | | Simpson's table for London, about 1730 | 19.2 | 82.30 | 25.7 | | Dupré, in Buffon, about 1750 | 33.0 | 85.30 | 28.67 | | Northampton tables, about 1760 | 25.18 | 87.39 | 28.86 | | Swedish tables, about 1785 | 36.12 | 91.86 | 31.3 | | France, before the Revolution, Duvillard | 28.76 | 86.96 | 29.0 | | Finlaison's tontine and annuitants, about 1800, males | 50.16 | 93.25 | 32.0 | | females | 55.51 | 100.7 | 34.6 | | Finlaison's Chelsea Pensioners | 90.0 | 29.65 | | Carlisle tables, about 1810, Milne | 37.14 | 95.47 | 32.6 | | Returns for all England, 1811 | 49 | The order of the mortalities expressed by the first column of this table, is, Simpson, Northampton, France, Dupre', Halley, Sweden, Carlisle, tontine 1695 males, females, Deparcieux, returns of 1811, tontines of 1800, males, and females; the order of the second column is Simpson, tontine of 1695, males, Dupre', France, Halley, Northampton, females 1695, pensioners, Sweden, tontines of 1800, males, Carlisle, females of 1800: but besides this difference in the order, the disproportion exhibited in this column is less enormous than in the former; the numbers of the Carlisle tables, for example, exceeding those of the Northampton by one half in the former, and by one tenth only in the latter. The proportion of Mr. Finlaison's tontines also stands as 3 to 4 in the first, and as 7 to 8, or 8 to 9 only in the second: the latter comparison giving a much fairer practical estimate of the comparative longevity, indicated by the tables, than the former. Another mode of easily appreciating the regularity and the analogies of different tables is, to construct a diagram, in the form of a curve, of which the absciss represents the age, and the ordinates the corresponding decrements of life. (Plate XI.) The inspection of such a diagram is sufficient to convince us of the great irregularity of the Carlisle tables of mortality, which must obviously have been formed, as they confessedly were, from observations on a very limited number of individuals, so that they exhibit a succession of different climacterics, after which the mortality is diminished, while about the age of 74 the curve that represents them towers to an incredible height, affording an expectation of longevity which some of the strongest advocates of those tables have abandoned in their practical applications, since they take their estimate of life, in advanced age, even lower than it is represented in the Northampton tables. It appears therefore to be highly probable, that the fairest basis for general computations, to be applied throughout Great Britain, may be obtained by a proper combination of the tables of Northampton, which have been long known and very generally approved, with the Carlisle tables, corrected however in their extravagant values of old lives, by some other documents; and with the mortality of London as MDCCCXXVI. Dr. Young's formula for expressing derived from the parish registers, which, when thus incorporated with tables formed in the country, will be freed from the objections that have been made to the observations of burials in great cities only. The Carlisle table agrees in the earlier parts pretty nearly with the observations of Mr. Morgan on the experience of the Equitable Office from 1768 to 1810, as it appears from Mr. Milne's comparison, as well as from the reduction and interpolation of those observations published by Mr. Gompertz in the Philosophical Transactions for 1825: but for correcting the later portions of the Carlisle table, it may be allowable to employ a subsequent register of the experience of the Equitable Office, so far as it is possible to make any inferences from it with safety. The numbers of deaths occurring in 20 years, as recorded by Mr. Morgan, might have been made the foundation of a very valuable determination of the mortality occurring in a certain class of persons, if the number of the Equitable Society had become stationary before the commencement of the record: but in order to deduce from it a just estimate of the value of life, it would then be necessary to alter the numbers of deaths at each age, in the inverse proportion of the numbers of the living compared: that is to say, not simply of the sums of the persons admitted under that age, but of the numbers of persons born whom they represent: since, in comparing the joint mortalities of any two lists of persons, we must obviously add together the deaths belonging, not to a given number of persons of various ages, but of a number proportionate to the survivors at the respective ages out of a given number of births, so that in this manner the apparent mortality in the earlier portions of the register would require to be augmented, not only on account of the smaller number of persons who have actually contributed to furnish it, but also on account of the greater proportion that these persons bear to the corresponding number at birth, when compared with the survivors at more advanced ages, who represent a population still more exceeding their own numbers. On the other hand, since the register in question relates only to a limited number of years, immediately following a very rapid increase of the Society, it is evident that the deaths must have occurred at earlier ages than if it had been continued until all the lives had dropped. Of these three modifications, it may be sufficiently accurate for the present purpose to omit the two latter as nearly counterbalancing each other, and to augment the earlier numbers in the proportion only of the members of the Society to whom they must necessarily have belonged, supposing that the admissions had taken place about the same ages at all periods; assuming also the number of survivors at 45 to be in the same proportion to the births as in the Carlisle table. We may then proceed to take a mean between the mortality thus obtained, with proper interpolations, and the observations at Carlisle, as the second of the three principal bases to be afterwards incorporated with the mortality of Northampton and of London. Further than this, it is impossible to place any great reliance on Mr. Morgan's document, which makes the annual deaths, in "a population exceeding 150000," not quite 1 in 1500. Of the mortality of London, taken for the ten years from 1811 to 1820, it may be observed, that its results bear the Dr. Young's formula for expressing internal evidence of greater apparent correctness than either of the other bases, exhibiting a curve less irregular in its flexures, and generally intermediate between the others: it has also the advantage of exhibiting the duration of life as prolonged by the general introduction of vaccination: and when thus incorporated with the registers of two places in the country, each reduced to an equal supposed population, it must probably be sufficiently corrected for the errors that may be attributed to the effect of an afflux of settlers at an early age. The mean obtained in this manner might be employed at once as a standard table without much inconvenience, but it still exhibits some minute but obvious irregularities, as an inspection of the line of stars in the diagram will show, principally perhaps from the want of skill or care with which the interpolations have been made by Dr. Price and others. The most effectual of all interpolations for harmonizing the various orders of differences, is to obtain a formula which shall extend with sufficient accuracy throughout the whole curve. It may be easily believed that it must be extremely difficult to find such an expression; and that its form must be too complicated to be applied to any practical purpose throughout its extent. I have however drawn a curve which comes extremely near to the line of stars, and crosses it in 10 or 12 different points, by means of the equation \[ y = 368 + 10x - 11 \left( 156 + 20x - xx \right)^{\frac{3}{2}} + \frac{1}{285 + 2.05xx + 2\left(\frac{x}{10}\right)^6} - 5.5 \left(\frac{x}{50}\right)^{10} + \frac{5.5^2}{4000} \left(\frac{x}{50}\right)^{20} - 5500 \left(\frac{x}{100}\right)^{40} : y \text{ being the number of deaths among } 100000 \text{ persons, in the year that completes the age } x. \] The terms of this formula have some remarkable relations to the different periods of life. Halley's first approximation was $y = 1000$, throughout life. Demoivre's arithmetical hypothesis was $y = \frac{100000}{86} = 1163$: but of the present formula the principal foundation, as extending to the whole of life, is, $y = 368 + 10x$. In infancy the term containing the reciprocal of the powers of $x$ has a preponderating value: in youth, the term $-(156 + 20x - xx)^{\frac{3}{2}}$, which diminishes the mortality, ends somewhat abruptly at 25, and would be incapable of being employed with safety in algebraical calculations, from its having a negative as well as a positive value. Old age is expressed almost exclusively by the high powers at the end of the formula, which terminate the series with great and increasing rapidity. It is obvious that for many purposes of calculation, the terms belonging to youth and to old age might be neglected without inconvenience, and that, for the middle portion of life, the terms $368 + 10x$ alone, with some little modification, might be employed as sufficiently correct; or certainly as much nearer to the truth than either the arithmetical or geometrical hypothesis of Demoivre. The relations of the different parts of the formula will be best appreciated from their development in the following tables. Dr. Young's formula for expressing **DOCUMENTS.** Deaths among persons assured by the Equitable Society, for 20 years, from 1800 to 1821.—Morgan on Assurances, 1821, p. 325. | Age | Deaths | Assured | Deaths cor. | Interpolations | |---------|--------|---------|-------------|----------------| | 10 to 20| 7 | 1494 | 251 | 514 | | 20 to 30| 37 | 8996 | 275 | 566 | | 30 to 40| 166 | 33850 | 299 | 606 | | 40 to 50| 299 | 45429 | 323 | 616 | | 50 to 60| 458 | 36489 | 347 | 618 | | 60 to 70| 536 | 19042 | 371 | 620 | | 70 to 80| 345 | 6454 | 395 | 622 | | above 80| 82 | | | | Total: 151754 Mortality of London from 1811 to 1820.—Gent. Mag. | Between 0 & 2 | 2 & 5 | 5 & 10 | 10 & 20 | 20 & 30 | 30 & 40 | 40 & 50 | 50 & 60 | 60 & 70 | 70 & 80 | 80 & 90 | 90 & 100 | 100 | 101 | 102 | 103 | 104 | 105 | 108 | 109 | 111 | 113 | |--------------|-------|--------|---------|---------|---------|---------|---------|---------|---------|---------|---------|------|-----|-----|-----|-----|-----|-----|-----|-----|-----| | 1811 | 5106 | 1638 | 654 | 509 | 1231 | 1641 | 1741 | 1591 | 1385 | 1038 | 449 | 56 | 1 | 1 | 1 | | | | | | | 1812 | 5636 | 1907 | 655 | 620 | 1226 | 1685 | 1841 | 1543 | 1425 | 1193 | 492 | 71 | 0 | 0 | 1 | | | | | | | 1813 | 5167 | 1733 | 604 | 526 | 1108 | 1501 | 1751 | 1606 | 1559 | 1211 | 489 | 61 | 1 | 1 | 0 | | | | | | | 1814 | 5845 | 2038 | 770 | 649 | 1268 | 1678 | 1950 | 1810 | 1747 | 1343 | 592 | 88 | 1 | 1 | 0 | | | | | | | 1815 | 5200 | 1916 | 870 | 677 | 1425 | 1824 | 2075 | 1886 | 1021 | 1221 | 674 | 167 | 2 | 1 | 0 | | | | | | | 1816 | 5400 | 1960 | 845 | 675 | 1464 | 1912 | 2123 | 1955 | 1720 | 1308 | 781 | 168 | 3 | 0 | 1 | | | | | | | 1817 | 5698 | 2019 | 929 | 706 | 1364 | 1795 | 1983 | 1788 | 1614 | 1224 | 683 | 156 | 7 | 0 | 0 | 2 | | | | | | | 1818 | 5381 | 1815 | 808 | 703 | 1453 | 1884 | 2040 | 1864 | 1585 | 1271 | 722 | 175 | 1 | 1 | 0 | | | | | | | 1819 | 4779 | 1771 | 826 | 631 | 1577 | 1990 | 2095 | 1918 | 1600 | 1230 | 666 | 144 | 0 | 0 | 0 | 1 | | | | | | | 1820 | 4758 | 1975 | 887 | 667 | 1484 | 2006 | 2069 | 1878 | 1632 | 1208 | 662 | 119 | | | | | | | | | | | Total | 52970 | 18772 | 7848 | 6363 | 13600 | 17916 | 19668 | 17839 | 15888 | 12247 | 6210 | 1205 | 16 | 5 | 4 | 1 | 2 | 2 | 2 | 1 | 1 | Total: 190565 ### Interpolations | Age | Deaths | |-----|--------| | 0 | 32970 | | 1 | 20000 | | 2 | 8500 | | 3 | 6000 | | 4 | 4272 | | 5 | 2800 | | 6 | 1800 | | 7 | 1400 | | 8 | 1008 | | 9 | 840 | | 10 | 740 | | 11 | 660 | | 12 | 615 | | 13 | 605 | | 14 | 600 | | 15 | 603 | | 16 | 610 | | 17 | 620 | | 18 | 640 | | 19 | 670 | | Age | Deaths | |-----|--------| | 20 | 710 | | 21 | 770 | | 22 | 960 | | 23 | 1160 | | 24 | 1460 | | 25 | 1680 | | 26 | 1700 | | 27 | 1710 | | 28 | 1720 | | 29 | 1730 | | 30 | 1741 | | 31 | 1752 | | 32 | 1763 | | 33 | 1774 | | 34 | 1786 | | 35 | 1798 | | 36 | 1809 | | 37 | 1820 | | 38 | 1831 | | 39 | 1842 | | 40 | 1855 | | 41 | 1885 | | 42 | 1925 | | 43 | 1963 | | 44 | 1990 | | 45 | 2010 | | 46 | 2020 | | 47 | 2020 | | 48 | 2010 | | 49 | 1990 | | 50 | 1959 | | 51 | 1920 | | 52 | 1880 | | 53 | 1840 | | 54 | 1800 | | 55 | 1760 | | 56 | 1720 | | 57 | 1680 | | 58 | 1650 | | 59 | 1630 | | 60 | 1622 | | 61 | 1615 | | 62 | 1607 | | 63 | 1600 | | 64 | 1592 | | 65 | 1585 | | 66 | 1577 | | 67 | 1570 | | 68 | 1563 | | 69 | 1557 | | 70 | 1550 | | 71 | 1478 | | 72 | 1405 | | 73 | 1333 | | 74 | 1261 | | 75 | 1188 | | 76 | 1116 | | 77 | 1044 | | 78 | 972 | | 79 | 900 | | 80 | 529 | | 81 | 768 | | 82 | 718 | | 83 | 678 | | 84 | 638 | | 85 | 597 | | 86 | 556 | | 87 | 516 | | 88 | 475 | | 89 | 435 | | 90 | 380 | | 91 | 290 | | 92 | 180 | | 93 | 90 | | 94 | 70 | | 95 | 57 | | 96 | 48 | | 97 | 40 | | 98 | 30 | | 99 | 20 | | 100 | 16 | ### Comparative Decrements from various Tables | Age | Northampton | Carlisle | Equitable Office Red | Mean of Carlisle and Eq. Office | London Bills | General Mean | Living | |-----|-------------|----------|----------------------|---------------------------------|--------------|--------------|--------| | 0 | 25751 | 15390 | | | | | | | 1 | 11734 | 6820 | | | | | | | 2 | 4309 | 5050 | | | | | | | 3 | 2876 | 2760 | | | | | | | 4 | 1691 | 2010 | | | | | | | 5 | 1579 | 1210 | | | | | | | 6 | 1202 | 820 | | | | | | | 7 | 944 | 580 | | | | | | | 8 | 687 | 430 | | | | | | | 9 | 515 | 330 | | | | | | | 10 | 446 | 290 | | | | | | | 11 | 429 | 310 | | | | | | | 12 | 429 | 320 | | | | | | | 13 | 429 | 330 | | | | | | | 14 | 429 | 350 | | | | | | ### Dr. Young's formula for expressing **Comparative Decrements from various Tables.** | Age | Northampton | Carlisle | Equitable Office Red. | Mean of Carlisle and Equi. Office. | London Bills | General Mean | Living | |-----|-------------|----------|-----------------------|-----------------------------------|--------------|--------------|--------| | 15 | 429 | 390 | | | 317 | 379 | 54302 | | 16 | 455 | 420 | | | 320 | 398 | 53923 | | 17 | 497 | 430 | | | 325 | 417 | 53525 | | 18 | 541 | 430 | | | 335 | 435 | 53108 | | 19 | 575 | 430 | | | 352 | 452 | 52673 | | 20 | 618 | 430 | | | 372 | 473 | 52221 | | 21 | 644 | 420 | | | 404 | 489 | 51748 | | 22 | 644 | 420 | | | 503 | 522 | 51259 | | 23 | 644 | 420 | | | 608 | 557 | 50737 | | 24 | 644 | 420 | | | 766 | 610 | 50180 | | 25 | 644 | 430 | | | 882 | 652 | 49570 | | 26 | 644 | 430 | | | 892 | 655 | 48918 | | 27 | 644 | 450 | | | 897 | 664 | 48263 | | 28 | 644 | 500 | | | 902 | 682 | 47599 | | 29 | 644 | 500 | | | 907 | 704 | 46917 | | 30 | 644 | 570 | | | 913 | 709 | 46213 | | 31 | 644 | 570 | | | 919 | 711 | 45594 | | 32 | 644 | 560 | | | 925 | 710 | 44793 | | 33 | 644 | 550 | | | 931 | 708 | 44083 | | 34 | 644 | 550 | | | 937 | 710 | 43375 | | 35 | 644 | 550 | | | 943 | 712 | 42665 | | 36 | 644 | 560 | | | 950 | 718 | 41953 | | 37 | 644 | 570 | | | 955 | 723 | 41235 | | 38 | 644 | 580 | | | 961 | 728 | 40512 | | 39 | 644 | 610 | | | 967 | 740 | 39784 | | 40 | 652 | 660 | | | 974 | 762 | 39044 | | 41 | 661 | 690 | | | 990 | 780 | 38282 | | 42 | 669 | 710 | | | 1010 | 796 | 37502 | | 43 | 669 | 710 | | | 1030 | 803 | 36706 | | 44 | 669 | 710 | | | 1044 | 808 | 35903 | | 45 | 669 | 700 | 1346 | (765) | 1055 | 830 | 35095 | | 46 | 669 | 690 | 1346 | (821) | 1059 | 850 | 34205 | | 47 | 669 | 670 | 1346 | (873) | 1059 | 867 | 33415 | | 48 | 669 | 630 | 1346 | (916) | 1055 | 880 | 32548 | | 49 | 678 | 610 | 1346 | 978 | 1044 | 900 | 31668 | | 50 | 695 | 590 | 1346 | 968 | 1028 | 897 | 30768 | | 51 | 704 | 620 | 1375 | 997 | 1007 | 903 | 29871 | | 52 | 704 | 650 | 1404 | 1027 | 987 | 906 | 28968 | | 53 | 704 | 680 | 1416 | 1048 | 966 | 906 | 28062 | | 54 | 704 | 700 | 1421 | 1060 | 945 | 903 | 27156 | | 55 | 704 | 730 | 1416 | 1073 | 924 | 900 | 26253 | | 56 | 704 | 760 | 1399 | | 902 | 894 | 25353 | | 57 | 704 | 820 | 1381 | | 881 | 895 | 24459 | | 58 | 704 | 930 | 1359 | | 866 | 904 | 23564 | | 59 | 704 | 1060 | 1348 | | 856 | 921 | 22660 | ### Comparative Decrements from various Tables. | Age | Northampton | Carlisle | Equitable Office Red. | Mean of Carlisle and Equit.Office | London Bills | General Mean | Living | |-----|-------------|----------|-----------------------|----------------------------------|--------------|--------------|--------| | 60 | 704 | 1220 | 1342 | | 851 | 945 | 21739 | | 61 | 704 | 1260 | 1338 | | 848 | 950 | 20794 | | 62 | 695 | 1270 | 1333 | | 844 | 948 | 19844 | | 63 | 695 | 1250 | 1329 | | 840 | 941 | 18996 | | 64 | 687 | 1250 | 1325 | | 836 | 936 | 18055 | | 65 | 687 | 1240 | 1321 | | 832 | 936 | 17119 | | 66 | 687 | 1230 | 1317 | | 828 | 929 | 16183 | | 67 | 687 | 1230 | 1295 | | 824 | 925 | 15254 | | 68 | 687 | 1230 | 1210 | | 819 | 909 | 14329 | | 69 | 687 | 1240 | 1098 | | 817 | 889 | 13420 | | 70 | 687 | 1240 | 1005 | | 813 | 874 | 12531 | | 71 | 687 | 1340 | 940 | | 776 | 867 | 11657 | | 72 | 687 | 1460 | 895 | | 738 | 868 | 10790 | | 73 | 687 | 1560 | 844 | | 700 | 863 | 9922 | | 74 | 687 | 1660 | 792 | | 661 | 858 | 9059 | | 75 | 687 | 1600 | 742 | | 623 | 839 | 8201 | | 76 | 661 | 1560 | 690 | | 585 | 790 | 7362 | | 77 | 627 | 1460 | 639 | | 548 | 741 | 6572 | | 78 | 584 | 1320 | 587 | | 510 | 683 | 5831 | | 79 | 548 | 1280 | 536 | | 472 | 643 | 5148 | | 80 | 541 | 1160 | 480 | | 435 | 599 | 4505 | | 81 | 515 | 1120 | 440 | | 402 | 549 | 3906 | | 82 | 489 | 1020 | 300 | | 376 | 508 | 3357 | | 83 | 472 | 940 | 200 | | 356 | 466 | 2849 | | 84 | 412 | 840 | 100 | | 335 | 406 | 2383 | | 85 | 351 | 780 | 60 | | 313 | 361 | 1977 | | 86 | 291 | 710 | 40 | | 292 | 319 | 1616 | | 87 | 241 | 640 | 30 | | 271 | 284 | 1297 | | 88 | 180 | 510 | 20 | | 251 | 234 | 1013 | | 89 | 137 | 390 | 10 | | 230 | 190 | 779 | | 90 | 103 | 370 | 8 | | 199 | 164 | 589 | | 91 | 86 | 300 | 7 | | 152 | 130 | 425 | | 92 | 69 | 210 | 7 | | 94 | 87 | 295 | | 93 | 60 | 140 | 6 | | 47 | 60 | 208 | | 94 | 43 | 100 | 6 | | 37 | 44 | 148 | | 95 | 26 | 70 | 5 | | 30 | 31 | 104 | | 96 | 9 | 50 | 5 | | 25 | 19 | 73 | | 97 | 0 | 40 | 5 | | 21 | 14 | 54 | | 98 | | 30 | 4 | | 16 | 9 | 40 | | 99 | | 20 | 4 | | 10 | 6 | 31 | MDCCCXXVI. ### Comparative Decrements from various Tables. | Age | Northampton | Carlisle | Equitable Office Red | Mean of Carlisle and Eq. Office | London Bills | General Mean | Living | |-----|-------------|----------|----------------------|---------------------------------|--------------|--------------|--------| | 100 | ... | 20 | 4 | ... | 8 | 6 | 25 | | 101 | ... | 20 | 3 | ... | 3 | 5 | 19 | | 102 | ... | 20 | 3 | ... | 3 | 5 | 14 | | 103 | ... | 20 | 2 | ... | 2 | 4 | 9 | | 104 | ... | 10 | 2 | [1] | 2 | | 5 | | 105 | ... | 0 | 1 | ... | 1 | 1 | 3 | | 106 | ... | 0 | | | .5 | (.25) | (2.00) | | 107 | ... | | | | .5 | .25 | 1.75 | | 108 | ... | | | | .5 | .25 | 1.50 | | 109 | ... | | | | .5 | .25 | 1.25 | | 110 | ... | | | | .5 | .25 | 1.00 | | 111 | ... | | | | .25 | | .75 | | 112 | ... | | | | .25 | | .50 | | 113 | ... | | | | .25 | | .25 | | 114 | ... | | | | .0 | | .00 | ### Decrements of Mortality computed from the Formula. | Age \((x-1)\) | \(+ 368 + 10 x\) | \(-.11 (156 + 20 x - xx)\) | \(+\frac{1}{2.85 + 2.05 xx + 2 (\frac{x}{10})^6}\) | Decrement | |---------------|------------------|----------------------------|-------------------------------------------------|-----------| | 0 | 378 | -255 | + 20408 | 20531 | | 1 | 388 | 241 | 9009 | 9106 | | 2 | 398 | 313 | 4695 | 4780 | | 3 | 408 | 359 | 2805 | 2854 | | 4 | 418 | 386 | 1848 | 1880 | | 5 | 428 | 409 | 1322 | 1341 | | 6 | 438 | 427 | 968 | 979 | | 7 | 448 | 440 | 746 | 752 | | 8 | 458 | 447 | 592 | 603 | | 9 | 468 | 451 | 477 | 494 | | 10 | 478 | 447 | 392 | 423 | | 11 | 488 | 440 | 329 | 377 | | 12 | 498 | 427 | 278 | 349 | | 13 | 508 | 409 | 238 | 337 | | 14 | 518 | 386 | 205 | 337 | Decrement of Mortality computed from the Formula. | Age \((x-1)\) | \(368 + 10x\) | \(-.11(156+20x - xx)^3\) | \(+\frac{1}{2.85+2.05xx+2(x/10)^6}\) | \(-5.5(x/50)^{10}\) | Decrement | |---------------|---------------|-----------------------------|----------------------------------|-------------------|-----------| | 15 | 528 | 359 | 178 | | 347 | | 16 | 538 | 313 | 156 | | 381 | | 17 | 548 | 291 | 136 | | 393 | | 18 | 558 | 255 | 119 | | 422 | | 19 | 568 | 214 | 104 | | 458 | | 20 | 578 | 174 | 93 | | 497 | | 21 | 588 | 130 | 82 | | 540 | | 22 | 598 | 89 | 72 | | 581 | | 23 | 608 | 51 | 64 | | 621 | | 24 | 618 | 19 | 57 | | 656 | | 25 | 628 | 0 | 50 | | 678 | | 26 | 638 | | 44 | | 682 | | 27 | 648 | | 39 | | 687 | | 28 | 658 | | 34 | | 692 | | 29 | 668 | | 30 | | 698 | | 30 | 678 | | 27 | | 705 | | 31 | 688 | | 24 | | 712 | | 32 | 698 | | 21 | | 719 | | 33 | 708 | | 18 | | 726 | | 34 | 718 | | 16 | | 734 | | 35 | 728 | | 14 | | 742 | | 36 | 738 | | 13 | | 751 | | 37 | 748 | | 11 | | 759 | | 38 | 758 | | 10 | | 768 | | 39 | 768 | | 9 | | 776 | | 40 | 778 | | 8 | | 785 | | 41 | 788 | | 8 | | 795 | | 42 | 798 | | 7 | | 804 | | 43 | 808 | | 6 | | 813 | | 44 | 818 | | 5 | | 821 | | 45 | 828 | | 5 | | 831 | | 46 | 838 | | 4 | | 839 | | 47 | 848 | | 4 | | 848 | | 48 | 858 | | 3 | | 857 | | 49 | 868 | | 3 | | 866 | | 50 | 878 | | 3 | | 874 | | 51 | 888 | | 2 | | 882 | | 52 | 898 | | 2 | | 890 | | 53 | 908 | | 2 | | 898 | | 54 | 918 | | 2 | | 906 | Dr. Young's formula for expressing Decrements of Mortality computed from the Formula. | Age (x-1) | 368 + 10 x | \( \frac{1}{2.85 + 2.05 x} \times \left( \frac{x}{10} \right)^6 - 5.5 \left( \frac{x}{50} \right)^{10} + .001 \left( \frac{5.5 \left( \frac{x}{50} \right)^9}{2} \right) - 5500 \left( \frac{x}{100} \right)^4 = \text{Decrement} \) | |-----------|------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | 55 | 928 | ... | | 56 | 938 | ... | | 57 | 948 | ... | | 58 | 958 | ... | | 59 | 968 | ... | | 60 | 978 | ... | | 61 | 988 | ... | | 62 | 998 | ... | | 63 | 1008 | ... | | 64 | 1018 | ... | | 65 | 1028 | ... | | 66 | 1038 | ... | | 67 | 1048 | ... | | 68 | 1058 | ... | | 69 | 1068 | ... | | 70 | 1078 | ... | | 71 | 1088 | ... | | 72 | 1098 | ... | | 73 | 1108 | ... | | 74 | 1118 | ... | | 75 | 1128 | ... | | 76 | 1138 | ... | | 77 | 1148 | ... | | 78 | 1158 | ... | | 79 | 1168 | ... | | 80 | 1178 | ... | | 81 | 1188 | ... | | 82 | 1198 | ... | | 83 | 1208 | ... | | 84 | 1218 | ... | | 85 | 1228 | ... | | 86 | 1238 | ... | | 87 | 1248 | ... | | 88 | 1258 | ... | | 89 | 1268 | ... | | 90 | 1278 | ... | | 91 | 1288 | ... | | 92 | 1298 | ... | | 93 | 1308 | ... | | 94 | 1318 | ... | | 95 | 1328 | ... | | 96 | 1338 | ... | | 97 | 1348 | ... | | 98 | 1358 | ... | | 99 | 1368 | ... | | 100 | 1378 | ... | MEAN STANDARD TABLE OF THE DECREMENTS OF LIFE IN GREAT BRITAIN, 1824. | Age | Decrement | Living | Age | Decrement | Living | Age | Decrement | Living | Age | Decrement | Living | |-----|-----------|--------|-----|-----------|--------|-----|-----------|--------|-----|-----------|--------| | 0 | 20531 | 100003 | 30 | 795 | 46527 | 60 | 938 | 21810 | 90 | 164 | 589 | | 1 | 9106 | 79472 | 31 | 712 | 45822 | 61 | 942 | 20872 | 91 | 130 | 425 | | 2 | 4780 | 70366 | 32 | 719 | 45110 | 62 | 943 | 19930 | 92 | 87 | 295 | | 3 | 2854 | 65586 | 33 | 726 | 44391 | 63 | 944 | 18987 | 93 | 60 | 208 | | 4 | 1880 | 62732 | 34 | 734 | 43665 | 64 | 943 | 18043 | 94 | 44 | 148 | | 5 | 1341 | 60852 | 35 | 742 | 42931 | 65 | 942 | 17100 | 95 | 31 | 104 | | 6 | 979 | 59511 | 36 | 751 | 42189 | 66 | 939 | 16158 | 96 | 19 | 73 | | 7 | 752 | 58532 | 37 | 759 | 41438 | 67 | 933 | 15219 | 97 | 14 | 54 | | 8 | 603 | 57780 | 38 | 768 | 40679 | 68 | 926 | 14286 | 98 | 9 | 40 | | 9 | 494 | 57177 | 39 | 776 | 39911 | 69 | 915 | 13360 | 99 | 6 | 31 | | 10 | 423 | 56683 | 40 | 785 | 39135 | 70 | 903 | 12445 | 100 | 6 | 25 | | 11 | 377 | 56260 | 41 | 795 | 38350 | 71 | 888 | 11542 | 101 | 5 | 19 | | 12 | 349 | 55883 | 42 | 804 | 37555 | 72 | 871 | 10654 | 102 | 5 | 14 | | 13 | 337 | 55334 | 43 | 813 | 36751 | 73 | 850 | 9783 | 103 | 4 | 9 | | 14 | 337 | 55197 | 44 | 821 | 35938 | 74 | 826 | 8933 | 104 | 2 | 5 | | 15 | 347 | 54860 | 45 | 831 | 35117 | 75 | 801 | 8107 | 105 | 1 | 3 | | 16 | 381 | 54513 | 46 | 839 | 34286 | 76 | 768 | 7306 | 106 | .25 | 2 | | 17 | 393 | 54132 | 47 | 848 | 33447 | 77 | 733 | 6538 | 107 | .25 | 1.75 | | 18 | 422 | 53739 | 48 | 857 | 32599 | 78 | 697 | 5805 | 108 | .25 | 1.50 | | 19 | 458 | 53317 | 49 | 866 | 31742 | 79 | 654 | 5108 | 109 | .25 | 1.25 | | 20 | 497 | 52859 | 50 | 874 | 30876 | 80 | 610 | 4454 | 110 | .25 | 1.0 | | 21 | 540 | 52362 | 51 | 882 | 30002 | 81 | 559 | 3844 | 111 | .25 | .75 | | 22 | 581 | 51822 | 52 | 890 | 29120 | 82 | 513 | 3285 | 112 | .25 | .50 | | 23 | 621 | 51241 | 53 | 898 | 28230 | 83 | 460 | 2772 | 113 | .25 | .25 | | 24 | 656 | 50620 | 54 | 906 | 27332 | 84 | 408 | 2312 | 114 | 0 | 0 | | 25 | 678 | 49964 | 55 | 913 | 26426 | 85 | 357 | 1904 | | | | | 26 | 682 | 49286 | 56 | 917 | 25513 | 86 | 307 | 1547 | | | | | 27 | 687 | 48604 | 57 | 923 | 24596 | 87 | 258 | 1240 | | | | | 28 | 692 | 47917 | 58 | 929 | 23673 | 88 | 215 | 982 | | | | | 29 | 698 | 47225 | 59 | 934 | 22744 | 89 | 178 | 767 | | | | I shall take this opportunity of endeavouring to demonstrate, in a simple and undeniable manner, the error into which Dr. Price and his followers have fallen, in consequence, as it appears, of their adopting the legal restraints on usury as essential steps in the mathematical calculation of the amount of compound interest. The error has indeed of late Dr. Young's formula for expressing years been very commonly admitted; but its effects have not been so satisfactorily rectified as could be desired. In the 66th volume of the Philosophical Transactions, for the year 1776, we find a Paper of Dr. Price, in which he lays down these theorems, \( r \) denoting the interest of £1. for a year, and \( n \) the term or number of years during which any annuity will be paid, \( p \) the perpetuity, or \( \frac{1}{r} \), \( y \) the value of an annuity paid yearly, and \( h \) half yearly: then, I, \( y = p - \frac{1}{r(1+r)^n} \); and, II, \( h = p - \frac{1}{r\left(\frac{1}{2}\right)^n} \): and as examples, taking \( r = .04 \), and \( n = 5 \), we have \( y = 4.4518 \), and \( p = 4.4913 \). Now, if we analyse the results thus obtained, by dividing them into the present values of the separate payments, they will stand thus: I. Present value of £1. payable at the end of | Year | Value | |------|-------| | 1 | £.961538 | | 2 | .924556 | | 3 | .888996 | | 4 | .854804 | | 5 | .821927 | Total: 4.451821 II. Present value of 10 shillings, payable at the end of half a year | Year | Value | |------|-------| | 1 | £.49020 | | 1\(\frac{1}{2}\) | .48058 | | 2 | .47127 | | 2\(\frac{1}{2}\) | .46192 | | 3 | .45286 | | 3\(\frac{1}{2}\) | .44398 | | 4 | .43528 | | 4\(\frac{1}{2}\) | .42674 | | 5 | .41837 | | 5\(\frac{1}{2}\) | .41018 | Total: 4.49138 The present values of 10 shillings are therefore assumed; I, at 1 year .48077; II, .48058 2 years .46228 .46192 3 years .44450 .44398 4 years .42740 .42674 5 years .41096 .41018 The latter column exhibiting obviously a larger deduction for discount than the former; so that the rate of interest in the two calculations is by no means the same: although in the case of $r = .05$, they would respectively represent the highest rate of interest allowed by our laws to be received without a new investment or engagement: but this arbitrary restraint ought certainly not to affect the mathematical consideration of the question. The difficulty, if any person thinks it such, may be avoided by a mode of investigation which I have lately had occasion to point out. "An annuity, of which a payment is due on a given day, is more valuable than an annuity purchased on that day, and to commence a year after, by the amount of a year's payment: and the value of a life annuity, becoming payable at any intermediate time between the day of purchase and its first anniversary, will be greater than the simple tabular value of the annuity by a sum proportional to the anticipation of the payment;" the increase of the value being very nearly uniform, when we suppose the anticipation to be gradually increased: this increase of the value comprehending obviously the greater probability as well as the greater proximity of each payment, and proceeding from day to day by very nearly equal increments. Thus, if we wished to purchase an annuity of £100. a year, and its value were £1000., upon the ordinary supposition of the payments commencing after the end of a year; supposing that we desired to have the first payment made at the end of nine months, and the subsequent payments at annual intervals as usual, we should have to add £25. to the purchase money, making it £1025. at whatever rate of interest the value might have been computed. If we began at six months, £50., and if at three months, £75. must be added to the purchase: it being obvious that an additional £100. would be equivalent to an anticipation of twelve months, or to an immediate payment of a year's annuity. From this simple and incontestable principle, in which the second differences only are neglected, it is very easy to deduce the values of annuities, payable at intervals shorter than a year. An annuity of 1, payable half yearly, is equal to two annuities of $\frac{1}{2}$, the one beginning as usual at the end of the year, the other anticipated by half a year; and the value of this portion is greater than the other by half of one of the payments, that is, by $\frac{1}{4}$: so that "We may always find the value of a life annuity payable half yearly, by adding a quarter of a year to the tabular value of the same annuity." In a similar manner it is very easily shown, that "for quarterly payments, we must add $\frac{3}{8}$ of a year's value to the computation made on the supposition of annual payments;" and "the continual bisection of the interval would at last afford us the addition of half a yearly payment for the value of a daily or hourly payment of a proportional part of the given annuity." "It may also be observed, that when we reckon at 3 per cent. interest, an annuity payable half yearly is the same, throughout the middle of life, that would be granted on the life of a person a year older, if payable annually." If it is required to ascertain the value of a reversionary annuity payable half yearly or quarterly, the calculation becomes in appearance a little paradoxical; for since the true value of a reversionary annuity for the life of one person, for example, after the death of another, is the difference between the values of two annuities on the single life and the joint lives, and since an equal addition must be made to these values in consideration of the period of payment being shortened, it follows that the reversionary annuity must be of equal value in either form. This conclusion would indeed be strictly true if the periodical times of payment remained unaltered, according to the supposition from which the value of the annuities is deduced; while in fact it is usual to grant such an annuity to commence at the first quarterly, half yearly, or annual period after the contingent event: a variation which would have no sensible effect in the case of daily payments, but which lessens the value of reversionary annuities at other periods by that of half a payment for the given period, reduced to the present time in the same manner as any other sum assured as payable upon the same contingency of survivorship. The simplicity observable in the progression of the values of annuities, calculated according to the values of lives here supposed, and at 3 per cent. interest, leads us to inquire what would be the exact law of mortality required to make that progression strictly uniform throughout life; and it will appear on investigation, that in order to have the value $24.45 - \frac{1}{4}x$, Dr. Young's formula for expressing $x$ being the age of the person, which is nearly true between 20 and 70, the annual mortality must be expressed by $$\frac{.03x + .066}{100.8 - x}$$ a fraction which at 20 becomes $\frac{1}{121}$, at 40, $\frac{1}{48}$, at 60, $\frac{1}{22}$, and at 80, $\frac{1}{8.4}$. Our table gives respectively $\frac{1}{103}$, $\frac{1}{50}$; $\frac{1}{23}$, and $\frac{1}{7.3}$: the Northampton $\frac{1}{71}$, $\frac{1}{48}$, $\frac{1}{25}$, and $\frac{1}{7.4}$. Mr. Finlaison's male annuitants $\frac{1}{87}$, $\frac{1}{73}$, $\frac{1}{32}$, and $\frac{1}{8.3}$. The healthiness of Mr. Finlaison's annuitants about 40 and 50 is one of the most remarkable features of his table: he observes (p. 58), that out of 10,000 persons at 23, 141 will die in a year, and 141 will die out of the same number at the age of 48; but at the age of 34 there will only die 124. The curve marked by obelisks, $+$, in the diagram, will show the comparative progress of mortality in this system; which, however valuable the data may be, appears to exhibit too many novelties, if not anomalies, to be generally adopted with confidence: while the line of crosses, $x$, representing the tontine of Deparcieux, will serve to show how little difference the lapse of a century has made in the results of these two similar cases. I shall conclude, my dear Sir, with a comparison of the climacteric years, as they may be called without impropriety, in which the greatest numbers of adults die, as taken from different tables. I sincerely hope that these considerations may help to undeceive the too credulous public, who have of late not only received some hints that tend to insinuate the probability of an occasional recurrence of a patriarchal longevity, but who have been required to believe, upon the authority of a most respectable mathematician, that the true and unerring value of life is not to be obtained by taking an average of various decrements, but by adopting the extreme of all conceivable estimates, founded only on a hasty assertion of Mr. Morgan, and unsupported by any detailed report; an estimate which makes the grand climacteric of mankind in this country, not a paltry fifty four, or the too much dreaded sixty three; but no less than eighty two! An age to which nearly one sixth of the survivors at ten are supposed to attain! **Climacterics, or greatest Decrements.** | Berlin, formerly | 38 | |-----------------|----| | London, about 1733 | 40 | | Paris, formerly | 40 | | Stockholm, 1762 | 42 | | London, 1764 | 43 | | London, 1815 | 46 | | Northampton, 1757 | 56 | | Breslau, 1695 | 61 | | Formula | 63 | | Brandenburg | 65 | | Warrington, 1777 | 65 | | Norwich, 1765 | 66 | | Montpellier, 1782 | 67 | | Duvillard, France | 67 | | Sweden, 1762 | 68 | | Chester, 1776 | 68 | | Sweden, 1785 | 69 | | Holycross, 1760 | 70 | | Deparcieux | 73 | | Carlisle | 74 | | Ackworth, 1752 | 75 | | Kersseboom | 77 | | Finlaison | 78 | | E. O. Mr. B | 82 | Believe me, dear Sir, Your faithful and obedient Servant, THOMAS YOUNG. Park Square, 28 Feb. 1826.