On Rates of Mortality and their Causes. By W. LAZARUS, of Hamburg, Actuary of the General Insurance Company of Trieste. Translated by T. B. SPRAGUE, M.A., Manager of the Scottish Equitable Life Assurance Society.

A CORRECT knowledge of the rate of mortality among mankind

is of such importance for answering many interesting questions, that we might hav expected it would hav been diligently studied from the earliest times. This, however, appears not to hav been the case, the reason probably being that questions as to the expectation of life arise in the first instance with reference to particular persons, and not to men in general. The proverbial uncertainty of human life-so often seen in the occurrence of death when least expected —is unfavourable to the notion of a general law of mortality, but naturally suggests the search after a means of predicting the duration of individual life. As thus understood, the question of the duration of life has at all times attracted great attention. Soothsayers and magicians claimd to hav the power of discerning from signs and omens of many kinds every event of life, its termination not excepted, and astrologers pretended to read in the stars the precise hour of each man's death.

It was reservd for a comparativly recent period to perceiv that the prediction of future events cannot be the result of a special prophetic gift; but that the only means we hav of lifting the veil of the future consists in a knowledge of the laws of nature that determin that future. If we wish to predict the length of human life, we must investigate the natural law which regulates life and death.

But so long as enquirers attempted to construct a law of mortality from observations on individuals, they must necessarily be unsuccessful, because the human organism thro' its complexity modifies a thousand-fold every law of nature that influences it. Whenever we hav recognized in nature the regular sequence of cause and effect, so as to refer the effect to a cause that has calld it into existence, we hav always found this effect to depend upon a definit set of conditions, which accompany the cause in such a way that a change in the conditions is followd by a change in the effect, the conditions themselves thus really constituting a part of the cause. If, however, we mentally distinguish between the conditions and the cause, this is solely for the sake of simplicity of conception. The various states and relations of the human organism, deviating on both sides from a certain medium (or normal) state, form such conditions; and when we have observd the effect that a given cause has

produced on the normal organism, we hav no right to expect that it will produce the same phenomenons on an organism that deviates from the medium; for the conditions in the two cases are different. Now in making observations as to mortality, we find it utterly impossible to select for the purpose individuals who can be considerd as completely normal and agreeing in all respects with the medium; for not only is that medium itself imperfectly known to us, but even if it were known, agreement with it would be extremely difficult to define and to test, because the living organism is in a constant state of development and change. In order, therefore, to determin the typical man to which our observations are to refer, we must adopt the expedient of classing together a very large number of individuals who are as nearly as possible in the same circumstances, and observing this aggregat. The larger the number of the persons combined in such a group, the greater becomes the probability that deviations in one direction among the individuals composing it will be compensated by equal deviations in the contrary direction; and altho' the individual men observd will not coincide with the typical or normal man, yet, if the group is sufficiently large, the average represents an ideal type, so that by observing the aggregat of the phenomenons of this group, we may trace the law they follow under normal conditions. Various circumstances concur to make the collection of a sufficiently extensiv number of exact observations on the rate of mortality, one of the most difficult of statistical undertakings; and thus it happens that the knowledge of human mortality is even at the present time far from complete. The results hitherto obtaind are, as regards their details, far from certain and well establisht; much remains to be done in this direction, and the work is still being carried on with the greatest zeal.

Altho' the determination of the numerical death-rate is certainly the most important part of the enquiry, the following pages are devoted to a totally different branch of it, their aim being to consider the manner in which the observd facts of mortality may be referd back to the causes that produce them. No apology, however, will be required for the investigations, if they either lead to a satisfactory theoretical conclusion, covering the whole observd phenomenons, or are used for the discovery of new truths. Either of those results, tho' of no present practical utility, would be sufficient justification of the course of reasoning that led to it. Moreover, experience teaches us that, sooner or later, such labours are rewarded by useful practical results.

The author of these remarks, conscious that he has gained no such end, and that his work therefore requires some apology, hopes that a sufficient one may be found in his wish to direct the attention of more able enquirers to some noteworthy points well deserving their consideration; and that, if his remarks should giv an impulse to more successful investigations in the same direction, and to the attainment of more valuable results, they may lay claim to indulgent consideration.

The very slightest consideration of the rate of mortality shows that it exhibits a certain regularity and conformity to law; and even in early times it appears to have been seen that the death rate among mankind must stand in some relation to the age. The fact that the death rate, which is very considerable at birth, diminishes pretty rapidly until it reaches a minimum; and then increases, at first slowly, but afterwards with constantly increasing rapidity; is so obvious that it could not long remain undiscoverd. It required, however, closer observation to investigate the numerical changes in the rate of mortality in passing from one age to another, and the actual rate at each age, and at the same time to determin whether the above is a complete description of the general course of the curve of mortality, or whether there are, besides the minimum mortality just mentiond, other less conspicuous minimums and maximums. We hav already seen that this problem can be solvd only by means of observing a large body of persons. If we wish to determin the rate of mortality at the various ages, we must collect into a group, as large a number as possible of persons of the same age, and we must observ how many of these persons die in a given time, making this observation for each age separatly.

But when observation has furnisht the answer to this enquiry, the further question arises, whether it is possible to survey from a common point of view, and to combine into a whole, the series of separat results so obtaind, in such a way that the observd ratios shall appear as the consequences of a law which governs the mortality. This question is quite independent of the former, and, to answer it, we must attempt an explanation of the observd facts and argue back from them to their causes. The following remarks hav to do with this enquiry, which, it will be seen, belongs to the recognized domain of speculativ investigation. Taking for granted the facts, as determind by observation, we shall seek to elucidate them, to combine them into a whole, and explain them as a consequence of a law of nature, which is the real object of our search.

We hav only to bear in mind how difficult it is to survey and take in at a glance the results of observations on the rate of mortality, and we at once see that the solution of the above problem would not only be of the greatest importance to the more thoro' understanding of the law of mortality, but still more as a stepping-stone to further conclusions. The rate of mortality being different for each year of age, so that if we wish to represent the mortality of a particular country we must state the rate at every age, and consequently employ a numerical series of more than 90 terms; it is obvious that such a series is neither a good representation of the actual rate of mortality there prevailing, nor a convenient standard of comparison with the mortality prevailing elsewhere. That, for such purposes, we shall not make much progress with the results of observation, is shown very clearly by the numerous attempts that have been made to condense these serieses of figures into some simple quantity, such as the "average lifetime", the "years of life", the "mean age", and other auxiliary quantities. These, however, are unsatisfactory substitutes for the serieses of numbers from which they are deduced, inasmuch as they are averages, in which the properties of the original numbers are not reproduced, since identical average values may result from elements of the most different kind. If, on the contrary, the problem should ever be solvd of representing the whole series by a simple formula in which its peculiarities are preservd, this would present a clear picture of the law of mortality, and could be easily compared with the formula similarly got from any other series of a similar nature.

If, however, we are still very far from possessing an exact and complete knowledge of the rate of mortality; if we may rather say it is known to us only in its general outlines and more prominent features should not this be a reason for abstaining for the present from more general investigations? We think not; but certainly it greatly increases the difficulties to be encounterd. It should also lead to double caution in our remarks; for while we must never allow our theory to contradict facts establisht by observation, we cannot consider its complete agreement with them, as a proof of its truth and accuracy. For the results of observation can never be considerd absolutely correct, but must rather be regarded as approximations to the truth; and we must constantly bear in mind the possibility of their deviating from it, sometimes on one side and sometimes on the other.

Another branch of science may be referd to in support of this

opinion, that the fact of our observations not being absolutely trustworthy is no reason for postponing our investigations. It is very well known that if the orbits of the heavenly bodies are calculated from the law of gravitation, they differ greatly from the actual orbits, because the mutual attractions have been left out of account; indeed, the disturbances which the orbits of comets and asteroids suffer thro' these attractions, may be so great as completely to change their form. Nevertheless, it is of inestimable value to us to know the law of gravitation and to be able to deduce from it the elliptical form of the orbit. We can during a long series of investigations wholly disregard the disturbances; indeed, we are often compeld to do so, because the present resources of analysis do not permit of our completely taking account of their effect. The law of gravitation, however, was discoverd when the observations in existence, tho' good, were far from absolutely

correct.

Going back, however, to an earlier time, when the law of gravitation and the elliptical form of the planetary orbits were not yet known; and when the observations on the latter were represented by epicycles; it must be admitted that this representation materially contributed to simplify men's ideas, and was an important step towards the discovery of the truth. But at that time observations were faulty; and the hypothesis in question

was erroneous.

The difficulties of collecting large numbers of observations as to mortality are greater than they appear at first sight. In these observations we require to know, not merely the number of persons of a given age who hav died in a certain period, say a year, but the ratio of that number to the number of persons of the same age who were alive at the beginning of the period, because the mortality is measured by the ratio of these two numbers. It is but seldom that we can succeed in ascertaining both terms of this ratio directly and wholly by means of observation; and we are mostly compeld to calculate one or other of them by the help of some hypothesis, which detracts from the exactness of the results.

The results of observations on mortality are embodied in mortality tables. But these tables do not give the values of the ratio we hav just explaind; they usually give in one column the number of survivers at each age out of a certain number of persons born; and in a second, the differences, that is to say, the number of deaths at each age. Thus the numbers given by observation are reduced