CONCERNING THE VARIOUS MEANS OF APPROACHING CERTAINTY

INDUCTION,analogy,hypotheses foundedupon ^facts and rectifiedcontinuallyby

new

observations, a happy tact given

by

^{nature and} strengthened by numerous comparisons ^{of its} indications with experience, such arethe _principal

means

forarriving at truth.

If one considers a series of _objects of the

same

natureone _perceives

among

^{them andintheir} changes ratios which manifest themselves more and more in proportion ^as the series is prolonged, and _which, extendingand generalizing continually, leadfinallyto the _principlefromwhich_theywerederived. Butthese ratios areenveloped by^so

many

strange circumstances thatitrequiresgreat sagacity ^todisentangle them and to recur to this principle: ^{it is in this} that the true genius of sciences consists. Analysis and natural philosophy

owe

theirmost_importantdiscoveries tothis fruitfulmeans, which ^iscalledinditction.

Newton

was indebted to itforhistheorem ofthe binomialandthe principle of universal gravity. ^{Itisdifficult}to appre-ciatetheprobabilityof theresultsof_induction,whichis

176

CERTAINTY. i?7 based upon ^{this thatthe} simplest ^{ratios are}the most

common

;thisisverifiedin the formulaeof_analysisand

isfound again^innaturalphenomena,ⁱⁿ crystallization, and in chemical combinations. This _simplicity of ratios will not appear astonishing ^if

we

consider that allthe effects ofnature areonly mathematical ^results of a small

number

ofimmutable laws.

Yet _induction, in leading ^{to the} discovery of the general principles ofthe _sciences, does not sufficeto establish them absolutely. ^{It is} always necessary^to confirm

them

_by demonstrationsorby^decisive experi-ences; ^for the _history ofthe sciences shows us that inductionhas sometimesled toinexactresults. I shall cite, for example, ^a theorem ofFermat in regard ^to primary ^numbers. This great geometrician,

who

had meditated, profoundly upon ^this theorem, sought a formula which, containing only primarynumbers, gave directly a primary

number

_greater than any ^other

number

assignable. Induction led him tothink that two, raisedtoapower which was^itselfa powerof two, formed with unity a primary number. _Thus, two raised to the square plus one, forms theprimary

num-ber _five; two raised to the second power^{of two, or} sixteen,forms withonethe primary

number

seventeen.

He

foundthatthiswas stilltrueforthe eighthandthe sixteenth power^of two augmented

by

unity; andthis induction,based upon ^several arithmetical considera-tions, caused him to regard ^{this result as} general.

However, he avowedthathehadnot demonstratedit.

Indeed, Euler recognized ^{that this} does not hold for the thirty-second power^oftwo, which, augmentedby unity, gives4,294,967.297, a

number

divisibleby641.

I? 8

A

PHILOSOPHICAL ESSAY

ON

We

_judgebyinduction thatifvarious_events, move-ments,^for example, appear constantlyand have been long connected bya simple ratio, they ^will continue to be _subjected to it; and

we

conclude from_this, by the theory ofprobabilities, that^{this ratiois}due, not^to hazard, but to a regular ^cause. Thus the equality of the

movements

of the rotation and the revolution of the

moon

_; thatof the

movements

of the nodes ofthe orbitand of the lunar_equator, andthe coincidence of these nodes; the singular ratio ofthe

movements

of the firstthree satellites of Jupiter, according ^towhich the

mean

_longitudeofthefirstsatellite, lessthreetimes thatof the second, plustwo timesthat of the third, is

equal^totwo _rightangles; the _equalityof the interval ofthetides to thatofthe passage ^of the

moon

tothe meridian; the return of the _greatest tides_.with the syzygies, andofthe smallest with the quadratures; all these _things, which have been maintained sincethey were first observed, indicate with an extreme

prob-ability,the existenceofconstant causes which

geome-tricianshavehappily succeeded ⁱⁿattaching^tothe law of universal gravity, and the knowledge ^of which renderscertain the perpetuity oftheseratios.

The

chancellorBacon, the eloquentpromoter^ofthe true philosophical method, has

made

a very strange misuse of induction inordertoprove the immobility ^of theearth.

He

reasons thus inthe

Novum _Organum,

his finest work^{: ''}

The movement

ofthe stars from the orient to the Occident increases in swiftness, ⁱⁿ proportion ^{to their distance} from the earth. This

movement

isswiftestwith the stars

;itslackens alittle

with Saturn, a ^little

more

with _Jupiter, and so on to

'i?9 the

moon

andthe highest comets. ^{Itis still} percepti-ble inthe atmosphere, especiallybetweenthe _tropics, on accountofthe great^circleswhichthe molecules of theairdescribe there_;finally,itisalmost inappreciable with the ocean; ^itisthen nilforthe earth." Butthis induction proves only^thatSaturn, andthe starswhich areinferiortoit, have their

own

movements, contrary to the real or apparent

movement

which sweeps ^the wholecelestialsphere from the ^{orient to}the _Occident, and thatthese

movements

appear slower with the

more

remote _stars, which is conformable to the laws of optics. Bacon ought ^to have been struck

by

^the inconceivableswiftnesswhich thestarsrequireⁱⁿ order to accomplish ^{their diurnal} revolution, ^if the earthis

immovable, and by^the extreme _simplicitywithwhich

its rotation explains

how

bodies so distant, the ones from the _{others, as}the stars, the _sun, the_planets, and the moon, ^all seem _subjected to this revolution.

As

tothe ocean andto the atmosphere, he ought^{not to} compare ^their

movement

with thatofthe starswhich aredetached from the earth_; butsincethe airandthe sea

make

_{part of} the terrestrial globe, theyought ^to participate ^{in its}

movement

or in its repose. ^{It is} singular that Bacon, ^{carried to}great prospectsby^his genius, was not

won

over by^the majestic ideawhich the Copernican system^ofthe universe offers.

He

was able, however, ^{to find in favor ofthat}system, strong analogies ⁱⁿ the discoveries of _Galileo, which were continued by^him.

He

_{has given}forthe searchafter truth the _precept, but not the example. But by

insisting, with allthe force ofreasonandof eloquence, upon ^the necessity of abandoning ^the insignificant

A

subtilities of the _school, in order to apply ^{oneself to} observationsand to experiences, and byindicating the true method of ascending ^to the general causes of phenomena, ^this great philosopher contributed ^tothe

immense

strideswhich the

human mind made

in the grandcenturyⁱⁿwhich he terminated his career.

Analogy

^{is based} upon ^the probability, that similar things have causesof the

same

kind and produce the same effects. This probability increase as the simili-tude becomes

more

perfect.

Thus we

_judge without doubt that beings provided with the same _organs, doing thesame_things, experience the

same

sensations, andare

moved

_bythe

same

^desires.

The

probability that the animals which resemble us have sensations analogous ^to ours, although a little inferior to that which is relative to individuals of our _species, is still

exceedinglygreat; andithas required^all theinfluence of _{religious prejudices} to

make

us think with

some

philosophers ^thatanimalsaremereautomatons.

The

probability ofthe existence of_feeling decreases in the

same

_proportion as the similitude of the organs with ours diminishes, but ^itisalways verygreat, evenwith insects. In seeing those of the same _speciesexecute very complicated things exactly ⁱⁿ the

same manner

from _generation to generation, and without having learnedthem, one ^{isledto believe that} they^act

by

^a kind of_affinityanalogous^tothatwhichbrings together the molecules ofcrystals, but which, together with the sensation attachedtoallanimalorganization,produces, with the_regularityofchemical combinations, combina-tions that are

much more

_singular; one might, perhaps,

name

this minglingof electiveaffinitiesandsensations

animal_affinity. Althoughthere existsa greatanalogy betweenthe organization ofplantsandthatof animals,

itdoes notseemto

me

sufficienttoextendtovegetables the sense of_feeling;but nothingauthorizes usin deny-ing^ittothem.

Since the sun brings forth, bythe beneficent ^action ofits light and of its heat, the animals and _plants Avhich cover the earth,

we

_{judge by analogy} that it

produces ^{similar effects} upon ^theother planets^{; forit} isnotnatural tothinkthatthe causewhoseactivity

we

seedeveloped^{in so}

many

ways ^{shouldbe sterile} upon sogreat aplanetas Jupiter, which, ^like the terrestrial globe, has itsdays, ^itsnights, anditsyears, and upon which observations indicate changes which _suppose veryactive forces. Yetthiswould begiving too great an extension toanalogy^to conclude fromitthe simili-tude of theinhabitantsof the planetsandof the earth.

Man,

made

for the temperaturewhich he _enjoys, and

for theelementwhich he _breathes, wouldnotbeable, according ^{to all} appearance, ^{to live} upon ^{the other} planets. But ought ^{there not to be an} infinity of organizationrelative to thevarious constitutions ofthe globes of^thisuniverse? Ifthesingle difference ofthe elements andof the climates

make

so

much

_varietyin terrestrialproductions,

how much

greater the^difference ought^tobe

among

^thoseofthe various planetsandof their satellites!

The

mostactiveimagination can form noidea ofit; buttheirexistence isvery probable.

We

are ledby^a strong analogy^to regard the^stars as so

many

^{suns endowed, likeours, withanattractive} powerproportional ^to the mass and _reciprocal tothe square of the distances; for this power being

demon-PHILOSOPHICAL ESSAY

ON

PROBABILITIES.

stratedforall the bodies ofthe solar system, and for theirsmallest molecules, ^itappears ^to appertain^toall matter. Already ^the

movements

ofthe small _stars, which have been called double, on account of their conjunction,appear^{to indicate}it; a century^atmostof precise observations,

by

verifying their

movements

of revolution,theonesabout the _others, willplacebeyond doubttheir reciprocal attractions.

The

analogy which^{leads us to}

make

each starthe centre of a planetary system ^{is far less} strong than the preceding one; but it acquires probability

by

^the hypothesis which has been proposed ⁱⁿ regard to the formation of thestars andof the sun; ^{for in this} hypothesis each star, having been ^likethesun, primi-tively environed by^avastatmosphere, ^{it isnatural to} attribute to this atmosphere^the

same

effectsas tothe solar atmosphere,andto suppose^thatithas produced, in condensing, planets andsatellites.

A

_great

number

ofdiscoveriesinthe sciencesisdue toanalogy. ^Ishallciteasoneofthemostremarkable, the discovery of atmospheric electricity, towhich one has been led

by

^the analogy ^{of electric}

phenomena

with the effectsofthunder.

The

surest

method

which_{can guide} usin thesearch fortruth, consistsinrisingby^{inductionfrom}

phenomena

to lawsand fromlawsto forces.

Laws

arethe ratios

which connect _particular

phenomena

together:

when

they have

shown

the general principle of the forces from which _theyare derived, one verifies it either by direct experiences,

when

this is possible, orby exami-nation ifit agrees with

known

phenomena; and ^if by a rigorous analysis

we

see them _proceed from this

183

principle, evenintheirsmall _details, andif, moreover, they^are quite variedand_very numerous, ^{then science} acquiresthe highest degree ofcertaintyand_of perfec-tion that it is able to attain. Such, astronomy ^has

become

_by the discovery of ^universal gravity.

The

historyofthesciencesshowsthattheslowandlaborious path^ofinduction has not always been^that ofinventors.

The

imagination, impatient ^{to arrive at} the _causes, takes pleasure ⁱⁿ creating hypotheses, and often it

changes ^{thefacts in orderto} adapt

them

toits

work

;

then the hypotheses ^are dangerous. But

when

one regards them _only as the

means

of connecting the

phenomena

^{in order to} discover the laws; when, by refusing to attribute

them

to a _reality, one rectifies

them

continually by

new

observations, they ^are able toleadtothe veritable causes, orat least put ^{us in} a position to conclude from the

phenomena

^observed thosewhichgiven circumstances ought^toproduce.

If

we

should _try all the hypotheses which can be formedinregard^tothecause of

phenomena we

should arrive, by ^a process of exclusion, at the true one.

This

means

has been employed with success; some-times

we

have arrived at several hypotheses which explain equally well ^all the facts known, and

among

whichscholars are divided, ^until decisiveobservations have

made known

thetrueone.

Then

itisinteresting, forthe _historyof the

human

_mind, to return tothese hypotheses, ^{to see}

how

they succeed ⁱⁿ explaining a great

number

of facts, andto investigatethe changes which_theyought^toundergo^{in orderto}agree with the history ^of nature. It is thus that the system ⁰¹ Ptolemy, which is only ^the realization of celestial

84 PHILOSOPHICAL ESSAY PROBABILITIES.

appearances, ^is transformed into the hypothesis^ofthe

movement

of the planets about the sun, byrendering equalandparallel tothesolar orbitthe circles andthe epicycles which he causes to be described annually, and the magnitude ^ofwhich heleaves undetermined.

It suffices,then,ⁱⁿ ordertochange^thishypothesis^into the truesystemof the world,^totransport theapparent

movement

ofthe sunin a sense contrary^tothe earth.

Itisalmost alwaysimpossible^to submittocalculus the probabilityof theresultsobtainedbythese various

means

; this is true likewise for historical facts. But the _totality of the

phenomena

explained, or of the testimonies, ^issometimes such thatwithout being able to appreciate the probability

we

cannot reasonably permit ourselvesany doubt ^{inregard to}them. In the other cases itisprudent^toadmit

them

only with great reserve.

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