_CONCERNING TABLES OF MORTALITY, AND OF MEAN DURATIONS OF LIFE, OF
MARRIAGES, AND OF ASSOCIATIONS._
The manner of preparing tables of mortality is very simple. One takes in
the civil registers a great number of individuals whose birth and death
are indicated. One determines how many of these individuals have died in
the first year of their age, how many in the second year, and so on. It
is concluded from these the number of individuals living at the
commencement of each year, and this number is written in the table at
the side of that which indicates the year. Thus one writes at the side
of zero the number of births; at the side of the year 1 the number of
infants who have attained one year; at the side of the year 2 the number
of infants who have attained two years, and so on for the rest. But
since in the first two years of life the mortality is very great, it is
necessary for the sake of greater exactitude to indicate in this first
age the number of survivors at the end of each half year.
If we divide the sum of the years of the life of all the individuals
inscribed in a table of mortality by the number of these individuals we
shall have the mean duration of life which corresponds to this table.
For this, we will multiply by a half year the number of deaths in the
first year, a number equal to the difference of the numbers of
individuals inscribed at the side of the years 0 and 1. Their mortality
being distributed over the entire year the mean duration of their life
is only a half year. We will multiply by a year and a half the number of
deaths in the second year; by two years and a half the number of deaths
in the third year; and so on. The sum of these products divided by the
number of births will be the mean duration of life. It is easy to
conclude from this that we will obtain this duration, by making the sum
of the numbers inscribed in the table at the side of each year, dividing
it by the number of births and subtracting one half from the quotient,
the year being taken as unity. The mean duration of life that remains,
starting from any age, is determined in the same manner, working upon
the number of individuals who have arrived at this age, as has just been
done with the number of births. But it is not at the moment of birth
that the mean duration of life is the greatest; it is when one has
escaped the dangers of infancy and it is then about forty-three years.
The probability of arriving at a certain age, starting from a given age
is equal to the ratio of the two numbers of individuals indicated in the
table at these two ages.
The precision of these results demands that for the formation of tables
we should employ a very great number of births. Analysis gives then very
simple formulæ for appreciating the probability that the numbers
indicated in these tables will vary from the truth only within narrow
limits. We see by these formulæ that the interval of the limits
diminishes and that the probability increases in proportion as we take
into consideration more births; so that the tables would represent
exactly the true law of mortality if the number of births employed were
infinite.
A table of mortality is then a table of the probability of human life.
The ratio of the individuals inscribed at the side of each year to the
number of births is the probability that a new birth will attain this
year. As we estimate the value of hope by making a sum of the products
of each benefit hoped for, by the probability of obtaining it, so we can
equally evaluate the mean duration of life by adding the products of
each year by half the sum of the probabilities of attaining the
commencement and the end of it, which leads to the result found above.
But this manner of viewing the mean duration of life has the advantage
of showing that in a stationary population, that is to say, such that
the number of births equals that of deaths, the mean duration of life is
the ratio itself of the population to the annual births; for the
population being supposed stationary, the number of individuals of an
age comprised between two consecutive years of the table is equal to the
number of annual births, multiplied by half the sum of the probabilities
of attaining these years; the sum of all these products will be then the
entire population. Now it is easy to see that this sum, divided by the
number of annual births, coincides with the mean duration of life as we
have just defined it.
It is easy by means of a table of mortality to form the corresponding
table of the population supposed to be stationary. For this we take the
arithmetical means of the numbers of the table of mortality
corresponding to the ages zero and one year, one and two years, two and
three years, etc. The sum of all these means is the entire population;
it is written at the side of the age zero. There is subtracted from this
sum the first mean and the remainder is the number of individuals of one
year and upwards; it is written at the side of the year 1. There is
subtracted from this first remainder the second mean; this second
remainder is the number of individuals of two years and upwards; it is
written at the side of the year 2, and so on.
So many variable causes influence mortality that the tables which
represent it ought to be changed according to place and time. The divers
states of life offer in this regard appreciable differences relative to
the fatigues and the dangers inseparable from each state and of which it
is indispensable to keep account in the calculations founded upon the
duration of life. But these differences have not been sufficiently
observed. Some day they will be and then will be known what sacrifice of
life each profession demands and one will profit by this knowledge to
diminish the dangers.
The greater or less salubrity of the soil, its elevation, its
temperature, the customs of the inhabitants, and the operations of
governments have a considerable influence upon mortality. But it is
always necessary to precede the investigation of the cause of the
differences observed by that of the probability with which this cause is
indicated. Thus the ratio of the population to annual births, which one
has seen raised in France to twenty-eight and one third, is not equal to
twenty-five in the ancient duchy of Milan. These ratios, both
established upon a great number of births, do not permit of calling into
question the existence among the Milanese of a special cause of
mortality, which it is of moment for the government of our country to
investigate and remove.
The ratio of the population to the births would increase again if we
could diminish and remove certain dangerous and widely spread maladies.
This has happily been done for the smallpox, at first by the inoculation
of this disease, then in a manner much more advantageous, by the
inoculation of vaccine, the inestimable discovery of Jenner, who has
thereby become one of the greatest benefactors of humanity.
The smallpox has this in particular, namely, that the same individual is
not twice affected by it, or at least such cases are so rare that they
may be abstracted from the calculation. This malady, from which few
escaped before the discovery of vaccine, is often fatal and causes the
death of one seventh of those whom it attacks. Sometimes it is mild, and
experience has taught that it can be given this latter character by
inoculating it upon healthy persons, prepared for it by a proper diet
and in a favorable season. Then the ratio of the individuals who die to
the inoculated ones is not one three hundredth. This great advantage of
inoculation, joined to those of not altering the appearance and of
preserving from the grievous consequences which the natural smallpox
often brings, caused it to be adopted by a great number of persons. The
practice was strongly recommended, but it was strongly combated, as is
nearly always the case in things subject to inconvenience. In the midst
of this dispute Daniel Bernoulli proposed to submit to the calculus of
probabilities the influence of inoculation upon the mean duration of
life. Since precise data of the mortality produced by the smallpox at
the various ages of life were lacking, he supposed that the danger of
having this malady and that of dying of it are the same at every age. By
means of these suppositions he succeeded by a delicate analysis in
converting an ordinary table of mortality into that which would be used
if smallpox did not exist, or if it caused the death of only a very
small number of those affected, and he concludes from it that
inoculation would augment by three years at least the mean duration of
life, which appeared to him beyond doubt the advantage of this
operation. D'Alembert attacked the analysis of Bernoulli: at first in
regard to the uncertainty of his two hypotheses, then in regard to its
insufficiency in this, that no comparison was made of the immediate
danger, although very small, of dying of inoculation, to the very great
but very remote danger of succumbing to natural smallpox. This
consideration, which disappears when one considers a great number of
individuals, is for this reason immaterial for governments and the
advantages of inoculation for them still remain; but it is of great
weight for the father of a family who must fear, in having his children
inoculated, to see that one perish whom he holds most dear and to be the
cause of it. Many parents were restrained by this fear, which the
discovery of vaccine has happily dissipated. By one of those mysteries
which nature offers to us so frequently, vaccine is a preventive of
smallpox just as certain as variolar virus, and there is no danger at
all; it does not expose to any malady and demands only very little care.
Therefore the practice of it has spread quickly; and to render it
universal it remains only to overcome the natural inertia of the people,
against which it is necessary to strive continually, even when it is a
question of their dearest interests.
The simplest means of calculating the advantage which the extinction of
a malady would produce consists in determining by observation the number
of individuals of a given age who die of it each year and subtracting
this number from the number of deaths at the same age. The ratio of the
difference to the total number of individuals of the given age would be
the probability of dying in the year at this age if the malady did not
exist. Making, then, a sum of these probabilities from birth up to any
given age, and subtracting this sum from unity, the remainder will be
the probability of living to that age corresponding to the extinction of
the malady. The series of these probabilities will be the table of
mortality relative to this hypothesis, and we may conclude from it, by
what precedes, the mean duration of life. It is thus that Duvilard has
found that the increase of the mean duration of life, due to inoculation
with vaccine, is three years at the least. An increase so considerable
would produce a very great increase in the population if the latter, for
other reasons, were not restrained by the relative diminution of
subsistences.
It is principally by the lack of subsistences that the progressive march
of the population is arrested. In all kinds of animals and vegetables,
nature tends without ceasing to augment the number of individuals until
they are on a level of the means of subsistence. In the human race moral
causes have a great influence upon the population. If easy clearings of
the forest can furnish an abundant nourishment for new generations, the
certainty of being able to support a numerous family encourages
marriages and renders them more productive. Upon the same soil the
population and the births ought to increase at the same time
simultaneously in geometric progression. But when clearings become more
difficult and more rare then the increase of population diminishes; it
approaches continually the variable state of subsistences, making
oscillations about it just as a pendulum whose periodicity is retarded
by changing the point of suspension, oscillates about this point by
virtue of its own weight. It is difficult to evaluate the _maximum_
increase of the population; it appears after observations that in
favorable circumstances the population of the human race would be
doubled every fifteen years. We estimate that in North America the
period of this doubling is twenty-two years. In this state of things,
the population, births, marriages, mortality, all increase according to
the same geometric progression of which we have the constant ratio of
consecutive terms by the observation of annual births at two epochs.
By means of a table of mortality representing the probabilities of human
life, we may determine the duration of marriages. Supposing in order to
simplify the matter that the mortality is the same for the two sexes, we
shall obtain the probability that the marriage will subsist one year, or
two, or three, etc., by forming a series of fractions whose common
denominator is the product of the two numbers of the table corresponding
to the ages of the consorts, and whose numerators are the successive
products of the numbers corresponding to these ages augmented by one, by
two, by three, etc., years. The sum of these fractions augmented by one
half will be the mean duration of marriage, the year being taken as
unity. It is easy to extend the same rule to the mean duration of an
association formed of three or of a greater number of individuals.