CONCERNING TABLES OF MORTALITY, AND OF MEAN DURATIONS OF LIFE, OF MARRIAGES,

AND OF ASSOCIATIONS.

THE manner

of preparing^{tables of}mortality^isvery simple.

One

takes in the civilregisters a great

num-berof 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 concludedfrom these the

number

ofindividuals living atthe

commencement

of each_year,and this

number

is written inthetable at the side of thatwhich indicatesthe _year.

Thus

one writes atthe side of zero the

number

ofbirths; atthe side of the year ^I the

number

of infants

who

have attainedone_year;atthe sideofthe year²the

number

ofinfants

who

have attainedtwo _years, and so onfor the rest. But since in the first two _years of lifethe mortality^is verygreat, ^{it is} necessary^forthe sakeof greater exactitude to indicate in this first age ^the

number

of survivors atthe endofeach_{half year.}

If

we

divide the

sum

of the years of the ^{life of all} the individuals inscribed inatableof mortality by^the

140

CONCERNING TABLES

OF

MORTALITY,

U*

number

ofthese individuals

we

shall have the

mean

duration of lifewhich corresponds^{to this table.} For

this,

we

will multiply by ^{a half}year the

number

of deaths in the firstyear, a

number

equal ^tothe differ-ence of thenumbersof individuals inscribed atthe side ofthe yearsoand I. Their mortality beingdistributed over the entire year the

mean

durationof theirlifeis

only a half year.

We

will multiplyby a yearand a halfthe

number

ofdeathsin thesecond year;

by

^two yearsandahalfthe

number

ofdeathsinthethirdyear;

andso on.

The sum

ofthese products dividedby^the

number

of births will be the

mean

duration oflife. It iseasy^to conclude from this that

we

will obtain this duration, by

making

^the

sum

ofthe numbersinscribed inthe table atthe sideofeach_year, dividing^itby^the

number

of births and subtracting one halffrom the quotient, the year being taken ^as unity.

The mean

duration of lifethatremains, _startingfrom anyage, ^is determined in the

same

manner, working upon ^the

number

of individuals

who

havearrived atthisage, ^as hasjustbeen donewith the

number

ofbirths. Butit is not atthe

moment

ofbirth thatthe

mean

duration of lifeis the _greatest; itis

when

one has escaped the dangers ^of infancy and it is then about forty-three years.

The

probability of _arriving ata certain age, starting from a given age^isequal^totheratio of the two numbers of individuals indicated in the table at thesetwo_ages.

The

_precision of theseresults

demands

thatforthe formation of tables

we

should employ ^a very great

number

of births. Analysis gives then very simple formulaeforappreciating theprobability thatthe

num-142

A

bers indicated in these tables willvary from ^{the truth} only within narrowlimits.

We

seebythese formulae that the interval ofthe limitsdiminishes andthatthe probability increasesⁱⁿ proportion^as

we

take into con-sideration

more

_births; sothatthetables would repre-sent exactly the^truelaw of mortality^ifthe

number

of birthsemployed were^infinite.

A

^tableofmortality^isthen atableoftheprobability of

human

life.

The

ratioof the individuals inscribed attheside of each year ^tothe

number

ofbirthsis the probability thata

new

birth will attain this year.

As we

estimatethevalueofhope by

making

^a

sum

ofthe products of each ^benefithoped for, by^the probability of obtaining^{it, so}

we

can equally evaluate the

mean

duration of life

by

adding the products ^ofeach year by ^{half the}

sum

of the probabilities of _attaining the

commencement

and the endofit, which leads tothe result found above. But this

manner

ofviewing the

mean

duration of life has the advantage ^of showing thatina_stationary population,^thatisto say, suchthat the

number

ofbirths equals^that of _deaths, the

mean

duration oflife isthe ratioitselfof the population ^to the annual _births; for the populationbeing supposed stationary, the

number

ofindividualsof an age

com-prised 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

ofall these products^will be then the entirepopulation.

Now

itis easy^{tosee thatthis}sum, dividedby^the

number

ofannual births, coincideswith the

mean

durationoflifeas

we

have_just definedit.

It is easyby

means

ofatable of mortality ^to form

MORTALITY, 143 thecorresponding ^{table of the}population supposed ^to bestationary. Forthis

we

take thearithmetical

means

ofthe numbersofthe tableof mortality corresponding tothe ages zero and one_year,one and two_years, two and three _years, etc.

The sum

ofallthese

means

is

the entire population; ^{it iswritten at} the sideof the age ^zero. Thereis subtracted fromthis

sum

thefirst

mean

andtheremainder is the

number

of individuals ofone _yearand upwards; ^{it is written at the side of} the year ^I. There is subtracted from this first re-mainder the second

mean

; this second remainder is the

number

ofindividuals oftwo _years and upwards^;

itiswritten at thesideof the year ^2, andsoon.

So

many

^{variable causesinfluence} mortality^thatthe tableswhich_represent itought^{to be} changed accord-ing ^to place and time.

The

divers states oflifeoffer in this regard appreciable differences relative to the fatigues and the dangers inseparable from each ^state andofwhich itisindispensable ^tokeep ^{account inthe} calculations founded upon ^{the duration of life.} But these differences have not been sufficientlyobserved.

Some

_daytheywill be and thenwill be

known

what

sacrifice of life each _profession

demands

and one will profitby^thisknowledge^todiminish the dangers.

The

_greaterorlesssalubrity ofthe_sun, itselevation, itstemperature, ^thecustoms of theinhabitants,andthe operations ofgovernments havea considerableinfluence uponmortality. Butitis always necessary^to precede theinvestigationof thecause of thedifferencesobserved by^{that ofthe}probabilitywithwhichthiscauseis indi-cated.

Thus

the ratio of the population ^to annual births, which onehas seenraisedin Franceto

twenty-eightand onethird, ^isnot equal^to twenty-five ⁱⁿ the ancientduchy^{of Milan.} Theseratios, bothestablished upona great

number

ofbirths, do_{not permitof calling} into question the existence

among

^the Milanese ofa special cause of_mortality, which itis of

moment

for the government ^{of our} country ^to investigate and remove.

The

ratio of the population ^to the births would increaseagain^if

we

could diminishandremove certain dangerous and _{widely spread} maladies. This has happily been done for the smallpox, ^{at first}

by

^the

inoculation ofthis disease, then in a

manner much more

advantageous, by^the inoculationof_vaccine, the inestimable discovery of Jenner,

who

has thereby

become

oneofthe _greatestbenefactorsof humanity.

The

_smallpox has this in particular, namely, ^that the

same

individual is not twice affected by^{it, or at} leastsuch cases aresorarethatthey

may

^beabstracted

from the calculation. This malady, from which ^few escaped ^before the discovery ofvaccine, ^isoftenfatal

and causes the death ofoneseventh 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

^aproper^diet andina favorableseason.

Then

the ratio of the individuals

who

die to the inoculated ones is not onethree hundredth. This great advan-tageof inoculation, joined^tothose of notaltering the appearance and ^ofpreserving from ^thegrievous conse-quences which the natural smallpox ^often brings, caused ittobe adopted bya great

number

of_persons.

The

practice was _strongly recommended, ^{but it} was

MORTALITY, 145 strongly combated, ^{as is} nearly always the case in things subject to inconvenience. Inthe midstofthis dispute Daniel Bernoulli proposed ^to submit to the calculus of probabilities the influence of inoculation upon ^the

mean

duration oflife. Since _precisedataof the mortalityproducedby^the smallpox ^{at thevarious} ages ^oflifewere _lacking, he _supposedthatthe danger ofhaving ^this malady and^thatof dying^{of it are the} same ateveryage.

By ^means

^{of these}suppositions he succeeded by ^{a delicate} analysis ⁱⁿ converting an ordinary ^table of mortality^{into that} which would be used ifsmallpox ^{did not} exist, or if it caused the death of only a very small

number

ofthoseaffected,and he concludes from it that inoculationwould

augment

by^threeyears^{at least}the

mean

duration oflife, which appeared ^to him beyond doubt ^the advantage ^ofthis operation. D'Alembert attacked the_analysis of Ber-noulli: at first in regard ^to the uncertainty ^{of his} two hypotheses, then ⁱⁿ regard ^{to its} insufficiencyⁱⁿ this,that no comparison was

made

ofthe immediate danger, although verysmall, ofdyingof inoculation, to the very great but veryremote danger ^of succumbing to natural smallpox. This consideration, which dis-appears

when

one considers a great

number

of indi-viduals, ^{is for this} reason immaterial forgovernments andthe advantages^ofinoculationforthemstillremain;

butitisof great weight^for the fatherofa family

who

mustfear, inhaving^hischildren inoculated, toseethat one _perish

whom

he holds most dearand to be the cause ofit.

Many

parentswererestrainedbythis fear,

which thediscovery^of vaccine has happilydissipated.

By

^{one of}those mysterieswhich natureoffersto usso

A ON

frequently, vaccine isa preventive^of smallpoxjust as certain as variolar_virus, and thereis no danger^atall; it does not expose ^to any malady and

demands

_only very^littlecare. Therefore thepractice of^ithas spread quickly; and to renderit universalitremains only^to overcome the natural inertia of the people, against which it is necessary^{to strive} continually, even

when

itis a question of^theirdearest interests.

The

_simplest

means

of calculating the advantage whichthe extinctionofamalady would produce

con-sists indetermining

by

observation the

number

of indi-viduals ofa given age

who

die of it each year and subtracting ^this

number

from the

number

of deathsat the

same

_age.

The

ratioofthe difference tothe total

number

of individuals ofthe givenage would ^{be the} probability of dying ^{in the} year ^{at this} age ^{if the} malady^{did not exist.} Making, ^{then, a}

sum

ofthese probabilities from birthup^to anygiven age, and sub-tracting^this

sum

from_unity, theremainderwill be the probability of living to that age corresponding ^to the extinction ofthe malady.

The

seriesofthese prob-abilitieswill be thetable of mortality^{relative to this} hypothesis, and

we may

^{conclude from it,} by what precedes, the

mean

duration oflife. It is thus that Duvilard hasfound thattheincrease ofthe

mean

dura-tion of _life, due to inoculation with _vaccine, is three years^attheleast.

An

increasesoconsiderablewould produce a very great^increasein the population ^ifthe latter, for other _reasons, were not restrained by ^the relativediminution ofsubsistences.

Itis principallyby^{the lack of}subsistences thatthe progressive march ofthe population ^{is arrested.} In

MORTALITY, 14?

all kinds of animalsandvegetables, ^nature*tends with-out ceasing ^to

augment

^the

number

of individuals until they^are on a levelof the

means

ofsubsistence. In the

human

race moral causes have _{a great} influence upon ^the population. ^Ifeasy clearings ofthe forest can furnish an abundant nourishmentfor

new

genera-tions,thecertainty ofbeing able^tosupport anumerous family encourages marriages and renders

them

more productive.

Upon

^{the same soil the population and}

the births ought^{to increase atthe}

same

time simul-taneouslyⁱⁿ geometricprogression. But

when

clear-ings

become more

difficult and

more

rare then the increase of population diminishes; ^it approaches con-tinually the variable state of subsistences,

making

oscillationsaboutitjust asapendulum whoseperiodicity isretardedby changingthe point of suspension, oscil-latesaboutthis point by^virtueofits

own

_weight. It is difficult to evaluate the

maximum

increase of the population; itappears^after observationsthatin favor-able circumstances the population of the

human

race would be doubled every ^fifteen years.

We

estimate that in North America the period ^ofthisdoubling ^is twenty-twoyears. In thisstate of things, the popula-tion, births, marriages, mortality, ^all increase accord-ing^tothesamegeometric progression ofwhich

we

have the constantratioofconsecutiveterms bythe observa-tion ofannualbirths attwo_epochs.

By

^{means of a table} of mortality representing the probabilities of

human

_life,

we may

^{determine the}

duration ofmarriages. Supposing^inorderto simplify the matter that the mortality^is the sameforthetwo sexes,

we

shallobtainthe probability thatthe marriage

U$ A

PHILOSOPHICAL ESSAY

ON

PROBABILITIES.

will subsistone _year,or two,or three,etc.,byforming aseriesoffractions whose

common

denominatoristhe product of thetwo numbersof thetablecorresponding tothe ages of theconsorts, and whose numeratorsare the successive products of the numbers corresponding to these ages augmented

by

^one, by ^two, by ^three,

etc., years.

The sum

of thesefractions augmented by one half will be the

mean

duration of marriage, the year being taken ^as unity. ^{It is}easy^to extend the

same

ruletothe

mean

duration ofanassociationformed of three orof a greater

number

ofindividuals.

CHAPTER XV.

No documento A PHILOSOPHICAL ESSAY (páginas 152-161)