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On Rates of Mortality and their Causes. (Continued from page 61.)
Published online by Cambridge University Press: 18 August 2016
Extract
We are so accustomed, even when we do not aim at scientific exactness, to regard every phenomenon as produced by causes,—as the resultant of forces,—that we cannot help looking on what we call “life” as the effect of certain definite forces, and on death as the consequence of the extinction of those forces. But when we thus contrast life and death, we must always bear in mind not only that what we call “life” is a condition which presents itself under circumstances of unexhaustable variety, but that in all probability quite as many differences in its manifestation entirely elude our observation. Given the effect, we reason as to the cause; when the effects are different, we infer that either the causes are different, or the conditions on which the effect depends. But if the differences are so numerous that we can scarcely find two cases exactly alike, we can draw no conclusion either as to the causes or as to the conditions. Since in no single case can we recognize the various conditions under which the cause of life works, we cannot hope to decide how the effect is connected with the conditions and how with the causes. Death too is a phenomenon as to which we have not the least idea why it happens; whether because the vital force in the individual has sunk below a certain degree which is necessary for the maintenance of life, or whether as a result of wholly different forces. On this point we scarcely know more than that the possibility of organic life depends upon certain conditions, that it can exist so long as these conditions remain within limits more or less wide, and that a transgression of the limits causes death. We do not, however, know exactly what these conditions are, and still less do we know the limits within which human life can exist; and if we reflect how complicated are the circumstances to be taken into account, we can scarcely hope to see these relations settled for some time to come. It is further to be observed that life, being regarded as the effect of a force, and death as its extinction, there must be a gradual transition from the one to the other.
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- Copyright © Institute and Faculty of Actuaries 1875
References
page 215 note * We remind the reader of a very closely allied method of investigation applied in the calculus of probabilities.
page 218 note * Since the above was written, I have become aware that my deduction is not quite correct. Different causes of death, although they may be quite independent of each other, constantly disturb one another; and their joint effect is not equal to the sum of effects of every cause taken separately. On the other hand, the analytical expression of a cause of death acting concurrently with other causes, differs from the expression of the cause acting alone. This ought to have been taken into consideration before I tried to explain the constants of the formula; v and w being the causes, each acting concurrently with the other.
Nov. 1873.
W.L.
With reference to this, a correspondent writes.
Does not the modification of his hypothesis, which Mr. L. here finds himself constrained to admit, strike at the root of his whole argument ? In Vital Statistics, as in other branches of Social Science, in Physiology, and even in Chemistry, it is not true, but very far from being true, that the effects of causes acting in combination are equal to the sum of the effects of the same causes acting separately. But if this is so, must we not regard Vital Statistics as a strictly experimental or empirical science, the complex and often fugitive phenomena of which cannot be legitimately brought within the boundaries of formulæ which are applicable to dynamics or the strictly mathematical branches of physics?
Mr. L. takes as an illustration of the value of his hypothesis, drawn from another branch of science, the Epicycles of Ptolemy. But were the epicycles really a help or a hindrance to the progress of astronomical discovery ? Did they not rather stand in the way of true investigation, unti l their increasing number and complexity showed clearly that they did not answer to anything in nature; but were mere arbitrary hypotheses invented to cover facts which they did not truly explain, and they gave way before the simple and natural theory of Copernicus? Without absolutely denying the value of a temporary hypothesis—clearly recognized as such—as a mode of arranging and classifying facts, it is scarcely necessary to say that any practical use of it in such a field as Vital Statistics might prove misleading and hazardous.
W. S.
page 219 note *
Let r denote the period, then the probability of living thro' it is 
. The logarithm of this, lx+1
−loglx
, is easily seen to be equal to c + qx+1
logk+(x+1)log h-c-qx
log k-xlog h =qx
(qr
-1)log k=rlog h; and if we add to this the constant quantity r log 
=-rlog h, we get qr(qr-1)log k, the values of which for equidistant values of x, form a geometrical series.
page 220 note * Disregarding the mortality of the first period, the constants in the formula may be approximately determined from the mortality table as follows:—
In the same manner we find
Forming now the second differences, we get
and dividing the latter of these equations by the former,
whence q can be immediately determined, and with the greater accuracy the greater n is taken. We require, for this purpose, to take four equidistant terms of the mortality table, lx, lx+n, lx+2n, lx+3n If we put
also 
,
then 
.
From the same four values of the mortality table h is most simply determined as follows:
Consequently 
,
and 
.
The determination of k and c presents no difficulty.
page 223 note * The constants in this case are, if we start with l20, l40, l60, l80 , and use the uncorrected observed numbers, so that
To complete the table at the younger ages, I put
