,transcrib]'s ,note3 ,app>5t typographical errors h be5 correct$4 ,italics are indicat$ by und]scores4 ,small capitals h be5 replac$ by full capitals4 ,! corrections not$ 9 ! ,errata section h be5 incorporat$ 9 ! text4 ,two fur!r corrections h also be5 made3 #i_/aj 9 place ( #a_/aj1 & #i_/aj 9 place ( #f_/aj1 on page #aaj2 & ex voto 9 place ( ex veto on page #agc4 ,a ,,philosophical ,,essay ,,on ,,probabilities4 ,,by ,,pi]re ,,simon1 ,,m>quis ,,de ,,laplace4 ,,translat$ ,,f ,,! ,,six? ,,fr5* ,,edition ,,by ,,fr$]ick ,,wilson ,,truscott1 ,,ph4,d4 7,,h>v471 ,pr(essor ( ,g]manic ,languages 9 ! ,we/ ,virg9ia ,univ]sity1 ,,& ,,fr$]ick ,,l9coln ,,emory1 ,m4,e4 7,,wor4 ,,poly4 ,,in/471 ,pr(essor ( ,me*anics & ,appli$ ,ma!matics 9 ! ,we/ ,virg9ia ,univ]sity2 ,mem4 ,am]4 ,soc4 ,me*4 ,eng4 ,,fir/ ,,edition4 ,,fir/ ,,?\s&4 ,,new ,,york3 ,,john ,,wiley ,,sons4 ,,london3 ,,*apman ,,hall1 ,,limit$4 #aijb4 ,copyright1 #aijb1 ,,by ,f4 ,w4 ,,truscott ,,& ,f4 ,l4 ,,emory4 ,,rob]t ,,drummond ,,pr9t]1 ,,new ,,york ,,table ,,( ,,cont5ts4 ,,page ,,"p ,i4 ,a ,,philosophical ,,essay ,,on ,,probabilities4 ,,*apt] ,i4 ,introduction #a ,,*apt] ,,ii4 ,conc]n+ ,probability #c ,,*apt] ,,iii4 ,g5]al ,pr9ciples ( ! ,calculus ( ,probabilities #aa ,,*apt] ,,iv4 ,conc]n+ ,hope #bj ,,*apt] ,v4 ,analytical ,me?ods ( ! ,calculus ( ,probabilities #bf ,,"p ,,ii4 ,,application ,,( ,,! ,,calculus ,,( ,,probabilities4 ,,*apt] ,,vi4 ,games ( ,*ance #ec ,,*apt] ,,vii4 ,conc]n+ ! ,unkn[n ,inequalities :i* may ,exi/ among ,*ances ,suppos$ to be ,equal #ef ,,*apt] ,,viii4 ,conc]n+ ! ,laws ( ,probability :i* result f ! ,indef9ite ,multiplication ( ,ev5ts #fj ,,*apt] ,,ix4 ,application ( ! ,calculus ( ,probabilities to ,natural ,philosophy #gc ,,*apt] ,x4 ,application ( ! ,calculus ( ,probabilities to ! ,moral ,sci5ces #ajg ,,*apt] ,,xi4 ,conc]n+ ! ,probability ( ,te/imonies #aji ,,*apt] ,,xii4 ,conc]n+ ! ,selections & ,decisions ( ,assemblies #abf ,,*apt] ,,xiii4 ,conc]n+ ! ,probability ( ! ,judgm5ts ( ,tribunals #acb ,,*apt] ,,xiv4 ,conc]n+ ,tables ( ,mortality1 & ! ,mean ,durations ( ,life1 ,m>riage1 & ,"s ,associations #adj ,,*apt] ,,xv4 ,conc]n+ ! ,b5efits ( ,in/itutions :i* ,dep5d ^u ! ,probability ( ,ev5ts #adi ,,*apt] ,,xvi4 ,conc]n+ ,illusions 9 ! ,e/imation ( ,probabilities #afj ,,*apt] ,,xvii4 ,conc]n+ ! ,v>i\s ,means ( ,approa*+ ,c]ta9ty #agf ,,*apt] ,,xviii4 ,hi/orical ,notice ( ! ,calculus ( ,probabilities to #ahaf #ahe ,,errata4 ,page #hi1 l9e #bb1 = ,pl9e read ,pl9y #ajb1 l9es #ad1 #af1 m9utes days #adc1 l9e #be1 sun soil #agg1 l9es #ae1 #ag1 #ah1 #ba1 #bb1 #bd1 = prim>y read prime #ahb1 l9e #e1 = conjunctions read be+ b9>y ,a ,,philosophical ,,essay ,,on ,,probabilities4 ,,*apt] ,i4 ,,introduction4 ,?is philosophical essay is ! developm5t ( a lecture on probabilities :i* ,i deliv]$ 9 #agie to ! normal s*ools :i!r ,i _h be5 call$1 by a decree ( ! national conv5tion1 z pr(essor ( ma!matics ) ,lagrange4 ,i h rec5tly publi%$ ^u ! same subject a "w entitl$ ,! ,analytical ,!ory ( ,probabilities4 ,i pres5t "h )\t ! aid ( analysis ! pr9ciples & g5]al results ( ?is !ory1 apply+ !m to ! mo/ important que/ions ( life1 :i* are inde$ = ! mo/ "p only problems ( probability4 ,/rictly speak+ x may ev5 be said t ne>ly all our k is problematical2 & 9 ! small numb] ( ?+s :i* we are able to "k ) c]ta9ty1 ev5 9 ! ma!matical sci5ces !mselves1 ! pr9cipal means = asc]ta9+ tru?--induction & analogy--are bas$ on probabilities2 s t ! entire sy/em ( human k is connect$ ) ! !ory set =? 9 ?is essay4 ,d\btless x w be se5 "h ) int]e/ t 9 consid]+1 ev5 9 ! et]nal pr9ciples ( reason1 ju/ice1 & humanity1 only ! favorable *ances :i* are con/antly atta*$ to !m1 !re is a great advantage 9 foll[+ ^! pr9ciples & s]i\s inconv5i5ce 9 dep>t+ f !m3 _! *ances1 l ^? favorable to lott]ies1 always end by prevail+ 9 ! mid/ ( ! vacillations ( haz>d4 ,i hope t ! reflections giv5 9 ?is essay may m]it ! att5tion ( philosoph]s & direct x to a subject s wor?y ( engag+ _! m9ds4 ,,*apt] ,,ii4 ,,conc]n+ ,,probability4 ,all ev5ts1 ev5 ^? :i* on acc\nt ( _! insignificance d n seem to foll[ ! great laws ( nature1 are a result ( x j z necess>ily z ! revolutions ( ! sun4 ,9 ignorance ( ! ties :i* unite su* ev5ts to ! entire sy/em ( ! univ]se1 !y h be5 made to dep5d ^u f9al causes or ^u haz>d1 accord+ z !y occur & are repeat$ ) regul>ity1 or appe> )\t reg>d to ord]2 b ^! imag9>y causes h gradually rec$$ ) ! wid5+ b\nds ( k & disappe> entirely be=e s\nd philosophy1 :i* sees 9 !m only ! expression ( our ignorance ( ! true causes4 ,pres5t ev5ts are connect$ ) prec$+ ones by a tie bas$ ^u ! evid5t pr9ciple t a ?+ _c occur )\t a cause :i* produces x4 ,?is axiom1 kn[n by ! "n ( ! pr9ciple ( suffici5t reason1 ext5ds ev5 to actions :i* are consid]$ indiff]5t2 ! free/ w is unable )\t a det]m9ative motive to give !m bir?2 if we assume two positions ) exactly simil> circum/ances & f9d t ! w is active 9 ! "o & inactive 9 ! o!r1 we say t its *oice is an effect )\t a cause4 ,x is !n1 says ,leibnitz1 ! bl9d *ance ( ! ,epicureans4 ,! contr>y op9ion is an illusion ( ! m9d1 :i*1 los+ sight ( ! evasive reasons ( ! *oice ( ! w 9 indiff]5t ?+s1 believes t *oice is det]m9$ ( itself & )\t motives4 ,we ought !n to reg>d ! pres5t /ate ( ! univ]se z ! effect ( its ant]ior /ate & z ! cause ( ! "o :i* is to foll[4 ,giv5 = "o in/ant an intellig5ce :i* c\ld compreh5d all ! =ces by :i* nature is animat$ & ! respective situation ( ! be+s :o compose x--an intellig5ce suffici5tly va/ to submit ^! data to analysis--x w\ld embrace 9 ! same =mula ! movem5ts ( ! greate/ bodies ( ! univ]se & ^? ( ! lighte/ atom2 = x1 no?+ w\ld be unc]ta9 & ! future1 z ! pa/1 w\ld be pres5t to its eyes4 ,! human m9d (f]s1 9 ! p]fection :i* x has be5 able to give to a/ronomy1 a feeble idea ( ?is intellig5ce4 ,its discov]ies 9 me*anics & geometry1 add$ to t ( univ]sal gravity1 h enabl$ x to compreh5d 9 ! same analytical expressions ! pa/ & future /ates ( ! sy/em ( ! _w4 ,apply+ ! same me?od to "s o!r objects ( its k1 x has succe$$ 9 ref]r+ to g5]al laws obs]v$ ph5om5a & 9 =esee+ ^? :i* giv5 circum/ances ought to produce4 ,all ^! ef=ts 9 ! se>* = tru? t5d to lead x back cont9ually to ! va/ intellig5ce :i* we h j m5tion$1 b f :i* x w always rema9 inf9itely remov$4 ,?is t5d5cy1 peculi> to ! human race1 is t :i* r5d]s x sup]ior to animals2 & _! progress 9 ?is respect di/+ui%es nations & ages & con/itutes _! true glory4 ,let u recall t =m]ly1 & at no remote epo*1 an unusual ra9 or an extreme dr\ght1 a comet hav+ 9 tra9 a v long tail1 ! eclipses1 ! aurora borealis1 & 9 g5]al all ! unusual ph5om5a 7 reg>d$ z s _m signs ( cele/ial wra?4 ,heav5 0 invok$ 9 ord] to av]t _! baneful influ5ce4 ,no "o pray$ to h ! planets & ! sun arre/$ 9 _! c\rses3 obs]vation _h soon made app>5t ! futility ( su* pray]s4 ,b z ^! ph5om5a1 occurr+ & disappe>+ at long int]vals1 seem$ to oppose ! ord] ( nature1 x 0 suppos$ t ,heav51 irritat$ by ! crimes ( ! e>?1 _h creat$ !m to ann\nce its v5geance4 ,?us ! long tail ( ! comet ( #adef spread t]ror ?r\gh ,europe1 already ?r[n into con/]nation by ! rapid successes ( ! ,turks1 :o _h j ov]?r[n ! ,l[] ,empire4 ,?is /> aft] f\r revolutions has excit$ among u a v diff]5t int]e/4 ,! k ( ! laws ( ! sy/em ( ! _w acquir$ 9 ! int]val _h dissipat$ ! fe>s begott5 by ! ignorance ( ! true relation%ip ( man to ! univ]se2 & ,halley1 hav+ recogniz$ ! id5tity ( ?is comet ) ^? ( ! ye>s #aeca1 #afjg1 & #afhb1 ann\nc$ its next return = ! end ( ! ye> #ageh or ! beg9n+ ( ! ye> #agei4 ,! le>n$ _w await$ ) impati5ce ?is return :i* 0 to confirm "o ( ! greate/ discov]ies t h be5 made 9 ! sci5ces1 & fulfil ! pr$iction ( ,s5eca :5 he said1 9 speak+ ( ! revolutions ( ^? />s :i* fall f an enorm\s height3 ,! day w come :51 by /udy pursu$ ?r\gh sev]al ages1 ! ?+s n[ conceal$ w appe> ) evid5ce2 & po/]ity w be a/oni%$ t tru?s s cle> _h escap$ u4 ,clairaut !n und]took to submit to analysis ! p]turbations :i* ! comet _h exp]i5c$ by ! action ( ! two great planets1 ,jupit] & ,saturn2 aft] imm5se calculations he fix$ its next passage at ! p]ihelion t[>d ! beg9n+ ( ,april1 #agei1 :i* 0 actually v]ifi$ by obs]vation4 ,! regul>ity :i* a/ronomy %[s u 9 ! movem5ts ( ! comets d\btless exi/s also 9 all ph5om5a4 ,! curve describ$ by a simple molecule ( air or vapor is regulat$ 9 a mann] j z c]ta9 z ! planet>y orbits2 ! only diff]5ce betwe5 !m is t :i* comes f our ignorance4 ,probability is relative1 9 "p to ?is ignorance1 9 "p to our k4 ,we "k t ( ?ree or a great] numb] ( ev5ts a s+le "o ought to occur2 b no?+ induces u to believe t "o ( !m w occur r ?an ! o!rs4 ,9 ?is /ate ( indecision x is impossible = u to ann\nce _! occurr5ce ) c]ta9ty4 ,x is1 h[ev]1 probable t "o ( ^! ev5ts1 *os5 at w1 w n occur because we see sev]al cases equally possible :i* exclude its occurr5ce1 :ile only a s+le "o favors x4 ,! !ory ( *ance consi/s 9 r$uc+ all ! ev5ts ( ! same k9d to a c]ta9 numb] ( cases equally possible1 t is to say1 to su* z we may be equally undecid$ ab\t 9 reg>d to _! exi/5ce1 & 9 det]m9+ ! numb] ( cases favorable to ! ev5t ^: probability is s\ght4 ,! ratio ( ?is numb] to t ( all ! cases possible is ! measure ( ?is probability1 :i* is ?us simply a fraction ^: num]ator is ! numb] ( favorable cases & ^: d5om9ator is ! numb] ( all ! cases possible4 ,! prec$+ notion ( probability supposes t1 9 increas+ 9 ! same ratio ! numb] ( favorable cases & t ( all ! cases possible1 ! probability rema9s ! same4 ,9 ord] to conv9ce ourselves let u take two urns1 ,a & ,b1 ! fir/ conta9+ f\r :ite & two black balls1 & ! second conta9+ only two :ite balls & "o black "o4 ,we may imag9e ! two black balls ( ! fir/ urn atta*$ by a ?read :i* breaks at ! mom5t :5 "o ( !m is seiz$ 9 ord] to be drawn out1 & ! f\r :ite balls ?us =m+ two simil> sy/ems4 ,all ! *ances :i* w favor ! seizure ( "o ( ! balls ( ! black sy/em w lead to a black ball4 ,if we conceive n[ t ! ?reads :i* unite ! balls d n break at all1 x is cle> t ! numb] ( possible *ances w n *ange any m ?an t ( ! *ances favorable to ! extraction ( ! black balls2 b two balls w be drawn f ! urn at ! same "t2 ! probability ( draw+ a black ball f ! urn ,a w !n be ! same z at fir/4 ,b !n we h obvi\sly ! case ( urn ,b ) ! s+le diff]5ce t ! ?ree balls ( ?is la/ urn w\ld be replac$ by ?ree sy/ems ( two balls inv>iably connect$4 ,:5 all ! cases are favorable to an ev5t ! probability *anges to c]ta9ty & its expression becomes equal to unity4 ,^u ?is condition1 c]ta9ty & probability are comp>able1 al?\gh !re may be an ess5tial diff]5ce betwe5 ! two /ates ( ! m9d :5 a tru? is rigor\sly demon/rat$ to x1 or :5 x /ill p]ceives a small s\rce ( error4 ,9 ?+s :i* are only probable ! diff]5ce ( ! data1 :i* ea* man has 9 reg>d to !m1 is "o ( ! pr9cipal causes ( ! div]sity ( op9ions :i* prevail 9 reg>d to ! same objects4 ,let u suppose1 = example1 t we h ?ree urns1 ,a1 ,b1 ,c1 "o ( :i* conta9s only black balls :ile ! two o!rs conta9 only :ite balls2 a ball is to be drawn f ! urn ,c & ! probability is dem&$ t ?is ball w be black4 ,if we d n "k :i* ( ! ?ree urns conta9s black balls only1 s t !re is no reason to believe t x is ,c r ?an ,b or ,a1 ^! ?ree hypo!ses w appe> equally possible1 & s9ce a black ball c be drawn only 9 ! fir/ hypo!sis1 ! probability ( draw+ x is equal to "o ?ird4 ,if x is kn[n t ! urn ,a conta9s :ite balls only1 ! indecision !n ext5ds only to ! urns ,b & ,c1 & ! probability t ! ball drawn f ! urn ,c w be black is "o half4 ,f9ally ?is probability *anges to c]ta9ty if we are assur$ t ! urns ,a & ,b conta9 :ite balls only4 ,x is ?us t an incid5t relat$ to a num]\s assembly f9ds v>i\s degrees ( cr$5ce1 accord+ to ! ext5t ( k ( ! auditors4 ,if ! man :o reports x is fully conv9c$ ( x & if1 by 8 position & "*1 he inspires great confid5ce1 8 /atem5t1 h[ev] extraord9>y x may be1 w h = ! auditors :o lack in=mation ! same degree ( probability z an ord9>y /atem5t made by ! same man1 & !y w h entire fai? 9 x4 ,b if "s "o ( !m kn[s t ! same incid5t is reject$ by o!r equally tru/wor?y m51 he w be 9 d\bt & ! incid5t w be discr$it$ by ! enlight5$ auditors1 :o w reject x :e!r x be 9 reg>d to facts well av]r$ or ! immutable laws ( nature4 ,x is to ! influ5ce ( ! op9ion ( ^? :om ! multitude judges be/ in=m$ & to :om x has be5 accu/om$ to give its confid5ce 9 reg>d to ! mo/ important matt]s ( life t ! propagation ( ^? errors is due :i* 9 times ( ignorance h cov]$ ! face ( ! e>?4 ,magic & a/rology (f] u two great examples4 ,^! errors inculcat$ 9 infancy1 adopt$ )\t exam9ation1 & hav+ = a basis only univ]sal cr$5ce1 h ma9ta9$ !mselves dur+ a v long "t2 b at la/ ! progress ( sci5ce has de/roy$ !m 9 ! m9ds ( enlight5$ m51 ^: op9ion consequ5tly has caus$ !m to disappe> ev5 among ! common p1 ?r\gh ! p[] ( imitation & habit :i* _h s g5]ally spread !m abroad4 ,?is p[]1 ! ri*e/ res\rce ( ! moral _w1 e/abli%es & cons]ves 9 a :ole nation ideas entirely contr>y to ^? :i* x upholds else:]e ) ! same au?ority4 ,:at indulg5ce ought we n !n to h = op9ions diff]5t f ours1 :5 ?is diff]5ce (t5 dep5ds only ^u ! v>i\s po9ts ( view :]e circum/ances h plac$ u6 ,let u enlight5 ^? :om we judge insuffici5tly in/ruct$2 b fir/ let u exam9e critically our own op9ions & weigh ) imp>tiality _! respective probabilities4 ,! diff]5ce ( op9ions dep5ds1 h[ev]1 ^u ! mann] 9 :i* ! influ5ce ( kn[n data is det]m9$4 ,! !ory ( probabilities holds to consid]ations s delicate t x is n surpris+ t ) ! same data two p]sons arrive at diff]5t results1 especially 9 v complicat$ que/ions4 ,let u exam9e n[ ! g5]al pr9ciples ( ?is !ory4 ,,*apt] ,,iii4 ,,! ,,g5]al ,,pr9ciples ,,( ,,! ,,calculus ,,( ,,probabilities4 ,fir/ ,pr9ciple4--,! fir/ ( ^! pr9ciples is ! def9ition itself ( probability1 :i*1 z has be5 se51 is ! ratio ( ! numb] ( favorable cases to t ( all ! cases possible4 ,second ,pr9ciple4--,b t supposes ! v>i\s cases equally possible4 ,if !y are n s1 we w det]m9e fir/ _! respective possibilities1 ^: exact appreciation is "o ( ! mo/ delicate po9ts ( ! !ory ( *ance4 ,!n ! probability w be ! sum ( ! possibilities ( ea* favorable case4 ,let u illu/rate ?is pr9ciple by an example4 ,let u suppose t we ?r[ into ! air a l>ge & v ?9 co9 ^: two l>ge opposite faces1 :i* we w call heads & tails1 are p]fectly simil>4 ,let u f9d ! probability ( ?r[+ heads at lea/ "o "t 9 two ?r[s4 ,x is cle> t f\r equally possible cases may arise1 namely1 heads at ! fir/ & at ! second ?r[2 heads at ! fir/ ?r[ & tails at ! second2 tails at ! fir/ ?r[ & heads at ! second2 f9ally1 tails at bo? ?r[s4 ,! fir/ ?ree cases are favorable to ! ev5t ^: probability is s\ght2 consequ5tly ?is probability is equal to 2 s t x is a bet ( ?ree to "o t heads w be ?r[n at lea/ once 9 two ?r[s4 ,we c c\nt at ?is game only ?ree diff]5t cases1 namely1 heads at ! fir/ ?r[1 :i* disp5ses ) ?r[+ a second "t2 tails at ! fir/ ?r[ & heads at ! second2 f9ally1 tails at ! fir/ & at ! second ?r[4 ,?is w\ld r$uce ! probability to if we %\ld consid] ) d',alemb]t ^! ?ree cases z equally possible4 ,b x is app>5t t ! probability ( ?r[+ heads at ! fir/ ?r[ is 1 :ile t ( ! two o!r cases is 1 ! fir/ case be+ a simple ev5t :i* corresponds to two ev5ts comb9$3 heads at ! fir/ & at ! second ?r[1 & heads at ! fir/ ?r[1 tails at ! second4 ,if we !n1 con=m+ to ! second pr9ciple1 add ! possibility ( heads at ! fir/ ?r[ to ! possibility ( tails at ! fir/ ?r[ & heads at ! second1 we %all h = ! probability s\ght1 :i* agrees ) :at is f\nd 9 ! supposition :5 we play ! two ?r[s4 ,?is supposition does n *ange at all ! *ance ( t "o :o bets on ?is ev5t2 x simply s]ves to r$uce ! v>i\s cases to ! cases equally possible4 ,?ird ,pr9ciple4--,"o ( ! mo/ important po9ts ( ! !ory ( probabilities & t :i* l5ds ! mo/ to illusions is ! mann] 9 :i* ^! probabilities increase or dim9i% by _! mutual comb9ation4 ,if ! ev5ts are indep5d5t ( "o ano!r1 ! probability ( _! comb9$ exi/5ce is ! product ( _! respective probabilities4 ,?us ! probability ( ?r[+ "o ace ) a s+le die is 2 t ( ?r[+ two aces 9 ?r[+ two dice at ! same "t is #a_/cf4 ,ea* face ( ! "o be+ able to comb9e ) ! six faces ( ! o!r1 !re are 9 fact ?irty-six equally possible cases1 among :i* "o s+le case gives two aces4 ,g5]ally ! probability t a simple ev5t 9 ! same circum/ances w occur consecutively a giv5 numb] ( times is equal to ! probability ( ?is simple ev5t rais$ to ! p[] indicat$ by ?is numb]4 ,hav+ ?us ! successive p[]s ( a fraction less ?an unity dim9i%+ )\t ceas+1 an ev5t :i* dep5ds ^u a s]ies ( v great probabilities may become extremely improbable4 ,suppose !n an incid5t be transmitt$ to u by tw5ty witnesses 9 su* mann] t ! fir/ has transmitt$ x to ! second1 ! second to ! ?ird1 & s on4 ,suppose aga9 t ! probability ( ea* te/imony be equal to ! fraction #i_/aj2 t ( ! incid5t result+ f ! te/imonies w be less ?an 4 ,we _c bett] comp>e ?is dim9ution ( ! probability ?an ) ! ext9ction ( ! light ( objects by ! int]position ( sev]al pieces ( glass4 ,a relatively small numb] ( pieces suffices to take away ! view ( an object t a s+le piece all[s u to p]ceive 9 a di/9ct mann]4 ,! hi/orians d n appe> to h paid suffici5t att5tion to ?is degradation ( ! probability ( ev5ts :5 se5 across a great numb] ( successive g5]ations2 _m hi/orical ev5ts reput$ z c]ta9 w\ld be at lea/ d\btful if !y 7 submitt$ to ?is te/4 ,9 ! purely ma!matical sci5ces ! mo/ di/ant consequ5ces p>ticipate 9 ! c]ta9ty ( ! pr9ciple f :i* !y are d]iv$4 ,9 ! applications ( analysis to physics ! results h all ! c]ta9ty ( facts or exp]i5ces4 ,b 9 ! moral sci5ces1 :]e ea* inf]5ce is d$uc$ f t :i* prec$es x only 9 a probable mann]1 h[ev] probable ^! d$uctions may be1 ! *ance ( error increases ) _! numb] & ultimately surpasses ! *ance ( tru? 9 ! consequ5ces v remote f ! pr9ciple4 ,f\r? ,pr9ciple4--,:5 two ev5ts dep5d ^u ea* o!r1 ! probability ( ! comp\nd ev5t is ! product ( ! probability ( ! fir/ ev5t & ! probability t1 ?is ev5t hav+ occurr$1 ! second w occur4 ,?us 9 ! prec$+ case ( ! ?ree urns ,a1 ,b1 ,c1 ( :i* two conta9 only :ite balls & "o conta9s only black balls1 ! probability ( draw+ a :ite ball f ! urn ,c is 1 s9ce ( ! ?ree urns only two conta9 balls ( t color4 ,b :5 a :ite ball has be5 drawn f ! urn ,c1 ! indecision relative to t "o ( ! urns :i* conta9 only black balls ext5ds only to ! urns ,a & ,b2 ! probability ( draw+ a :ite ball f ! urn ,b is 2 ! product ( by 1 or 1 is !n ! probability ( draw+ two :ite balls at "o "t f ! urns ,b & ,c4 ,we see by ?is example ! influ5ce ( pa/ ev5ts ^u ! probability ( future ev5ts4 ,= ! probability ( draw+ a :ite ball f ! urn ,b1 :i* prim>ily is 1 becomes :5 a :ite ball has be5 drawn f ! urn ,c2 x w\ld *ange to c]ta9ty if a black ball _h be5 drawn f ! same urn4 ,we w det]m9e ?is influ5ce by means ( ! foll[+ pr9ciple1 :i* is a coroll>y ( ! prec$+ "o4 ,fif? ,pr9ciple4--,if we calculate priori ! probability ( ! occurr$ ev5t & ! probability ( an ev5t compos$ ( t "o & a second "o :i* is expect$1 ! second probability divid$ by ! fir/ w be ! probability ( ! ev5t expect$1 drawn f ! obs]v$ ev5t4 ,"h is pres5t$ ! "q rais$ by "s philosoph]s t\*+ ! influ5ce ( ! pa/ ^u ! probability ( ! future4 ,let u suppose at ! play ( heads & tails t heads has occurr$ (t5] ?an tails4 ,by ?is alone we %all be l$ to believe t 9 ! con/itution ( ! co9 !re is a secret cause :i* favors x4 ,?us 9 ! conduct ( life con/ant happ9ess is a pro( ( compet5cy :i* %\ld induce u to employ pref]ably happy p]sons4 ,b if by ! unreliability ( circum/ances we are con/antly br\ght back to a /ate ( absolute indecision1 if1 = example1 we *ange ! co9 at ea* ?r[ at ! play ( heads & tails1 ! pa/ c %$ no light ^u ! future & x w\ld be absurd to take acc\nt ( x4 ,six? ,pr9ciple4--,ea* ( ! causes to :i* an obs]v$ ev5t may be attribut$ is indicat$ ) j z mu* likelihood z !re is probability t ! ev5t w take place1 suppos+ ! ev5t to be con/ant4 ,! probability ( ! exi/5ce ( any "o ( ^! causes is !n a fraction ^: num]ator is ! probability ( ! ev5t result+ f ?is cause & ^: d5om9ator is ! sum ( ! simil> probabilities relative to all ! causes2 if ^! v>i\s causes1 consid]$ priori1 are unequally probable1 x is necess>y1 9 place ( ! probability ( ! ev5t result+ f ea* cause1 to employ ! product ( ?is probability by ! possibility ( ! cause itself4 ,?is is ! fundam5tal pr9ciple ( ?is bran* ( ! analysis ( *ances :i* consi/s 9 pass+ f ev5ts to causes4 ,?is pr9ciple gives ! reason :y we attribute regul> ev5ts to a p>ticul> cause4 ,"s philosoph]s h ?\ght t ^! ev5ts are less possible ?an o!rs & t at ! play ( heads & tails1 = example1 ! comb9ation 9 :i* heads occurs tw5ty successive times is less easy 9 its nature ?an ^? :]e heads & tails are mix$ 9 an irregul> mann]4 ,b ?is op9ion supposes t pa/ ev5ts h an influ5ce on ! possibility ( future ev5ts1 :i* is n at all admissible4 ,! regul> comb9ations occur m r>ely only because !y are less num]\s4 ,if we seek a cause :]ev] we p]ceive symmetry1 x is n t we reg>d a symmetrical ev5t z less possible ?an ! o!rs1 b1 s9ce ?is ev5t ought to be ! effect ( a regul> cause or t ( *ance1 ! fir/ ( ^! suppositions is m probable ?an ! second4 ,on a table we see lett]s arrang$ 9 ?is ord]1 ,c o n s t a n t i n o p l e1 & we judge t ?is arrangem5t is n ! result ( *ance1 n because x is less possible ?an ! o!rs1 = if ?is ^w 7 n employ$ 9 any language we %\ld n suspect x came f any p>ticul> cause1 b ?is ^w be+ 9 use among u1 x is incomp>ably m probable t "s p]son has ?us arrang$ ! a=esaid lett]s ?an t ?is arrangem5t is due to *ance4 ,?is is ! place to def9e ! ^w extraord9>y4 ,we arrange 9 our ?\ght all possible ev5ts 9 v>i\s classes2 & we reg>d z extraord9>y ^? classes :i* include a v small numb]4 ,?us at ! play ( heads & tails ! occurr5ce ( heads a hundr$ successive times appe>s to u extraord9>y because ( ! almo/ inf9ite numb] ( comb9ations :i* may occur 9 a hundr$ ?r[s2 & if we divide ! comb9ations into regul> s]ies conta9+ an ord] easy to compreh5d1 & into irregul> s]ies1 ! latt] are incomp>ably m num]\s4 ,! draw+ ( a :ite ball f an urn :i* among a million balls conta9s only "o ( ?is color1 ! o!rs be+ black1 w\ld appe> to u likewise extraord9>y1 because we =m only two classes ( ev5ts relative to ! two colors4 ,b ! draw+ ( ! numb] #dgehac1 = example1 f an urn t conta9s a million numb]s seems to u an ord9>y ev5t2 because1 comp>+ individually ! numb]s ) "o ano!r )\t divid+ !m into classes1 we h no reason to believe t "o ( !m w appe> soon] ?an ! o!rs4 ,f :at prec$es1 we ought g5]ally to conclude t ! m extraord9>y ! ev5t1 ! great] ! ne$ ( its be+ support$ by /rong pro(s4 ,= ^? :o atte/ x1 be+ able to deceive or to h be5 deceiv$1 ^! two causes are z mu* m probable z ! reality ( ! ev5t is less4 ,we %all see ?is p>ticul>ly :5 we come to speak ( ! probability ( te/imony4 ,sev5? ,pr9ciple4--,! probability ( a future ev5t is ! sum ( ! products ( ! probability ( ea* cause1 drawn f ! ev5t obs]v$1 by ! probability t1 ?is cause exi/+1 ! future ev5t w occur4 ,! foll[+ example w illu/rate ?is pr9ciple4 ,let u imag9e an urn :i* conta9s only two balls1 ea* ( :i* may be ei!r :ite or black4 ,"o ( ^! balls is drawn & is put back into ! urn be=e proce$+ to a new draw4 ,suppose t 9 ! fir/ two draws :ite balls h be5 drawn2 ! probability ( aga9 draw+ a :ite ball at ! ?ird draw is requir$4 ,only two hypo!ses c be made "h3 ei!r "o ( ! balls is :ite & ! o!r black1 or bo? are :ite4 ,9 ! fir/ hypo!sis ! probability ( ! ev5t obs]v$ is 2 x is unity or c]ta9ty 9 ! second4 ,?us 9 reg>d+ ^! hypo!ses z s _m causes1 we %all h = ! six? pr9ciple & = _! respective probabilities4 ,b if ! fir/ hypo!sis occurs1 ! probability ( draw+ a :ite ball at ! ?ird draw is 2 x is equal to c]ta9ty 9 ! second hypo!sis2 multiply+ !n ! la/ probabilities by ^? ( ! correspond+ hypo!ses1 ! sum ( ! products1 or #i_/aj1 w be ! probability ( draw+ a :ite ball at ! ?ird draw4 ,:5 ! probability ( a s+le ev5t is unkn[n we may suppose x equal to any value f z]o to unity4 ,! probability ( ea* ( ^! hypo!ses1 drawn f ! ev5t obs]v$1 is1 by ! six? pr9ciple1 a fraction ^: num]ator is ! probability ( ! ev5t 9 ?is hypo!sis & ^: d5om9ator is ! sum ( ! simil> probabilities relative to all ! hypo!ses4 ,?us ! probability t ! possibility ( ! ev5t is compris$ )9 giv5 limits is ! sum ( ! fractions compris$ )9 ^! limits4 ,n[ if we multiply ea* fraction by ! probability ( ! future ev5t1 det]m9$ 9 ! correspond+ hypo!sis1 ! sum ( ! products relative to all ! hypo!ses w be1 by ! sev5? pr9ciple1 ! probability ( ! future ev5t drawn f ! ev5t obs]v$4 ,?us we f9d t an ev5t hav+ occurr$ successively any numb] ( times1 ! probability t x w happ5 aga9 ! next "t is equal to ?is numb] increas$ by unity divid$ by ! same numb]1 increas$ by two units4 ,plac+ ! mo/ anci5t epo* ( hi/ory at five ?\s& ye>s ago1 or at #ahbfbac days1 & ! sun hav+ ris5 con/antly 9 ! int]val at ea* revolution ( tw5ty-f\r h\rs1 x is a bet ( #ahbfbad to "o t x w rise aga9 to-morr[4 ,b ?is numb] is incomp>ably great] = him :o1 recogniz+ 9 ! totality ( ph5om5a ! pr9cipal regulator ( days & seasons1 sees t no?+ at ! pres5t mom5t c arre/ ! c\rse ( x4 ,buffon 9 8 ,political ,ari?metic calculates diff]5tly ! prec$+ probability4 ,he supposes t x diff]s f unity only by a fraction ^: num]ator is unity & ^: d5om9ator is ! numb] #b rais$ to a p[] equal to ! numb] ( days :i* h elaps$ s9ce ! epo*4 ,b ! true mann] ( relat+ pa/ ev5ts ) ! probability ( causes & ( future ev5ts 0 unkn[n to ?is illu/ri\s writ]4 ,,*apt] ,,iv4 ,,conc]n+ ,,hope4 ,! probability ( ev5ts s]ves to det]m9e ! hope or ! fe> ( p]sons int]e/$ 9 _! exi/5ce4 ,! ^w hope has v>i\s acceptations2 x expresses g5]ally ! advantage ( t "o :o expects a c]ta9 b5efit 9 suppositions :i* are only probable4 ,?is advantage 9 ! !ory ( *ance is a product ( ! sum hop$ = by ! probability ( obta9+ x2 x is ! p>tial sum :i* ought to result :5 we d n wi% to run ! risks ( ! ev5t 9 suppos+ t ! division is made proportional to ! probabilities4 ,?is division is ! only equitable "o :5 all /range circum/ances are elim9at$2 because an equal degree ( probability gives an equal "r to ! sum hop$ =4 ,we w call ?is advantage ma!matical hope4 ,eigh? ,pr9ciple4--,:5 ! advantage dep5ds on sev]al ev5ts x is obta9$ by tak+ ! sum ( ! products ( ! probability ( ea* ev5t by ! b5efit atta*$ to its occurr5ce4 ,let u apply ?is pr9ciple to "s examples4 ,let u suppose t at ! play ( heads & tails ,paul receives two francs if he ?r[s heads at ! fir/ ?r[ & five francs if he ?r[s x only at ! second4 ,multiply+ two francs by ! probability ( ! fir/ case1 & five francs by ! probability ( ! second case1 ! sum ( ! products1 or two & a qu>t] francs1 w be ,paul's advantage4 ,x is ! sum :i* he ought to give 9 advance to t "o :o has giv5 him ?is advantage2 =1 9 ord] to ma9ta9 ! equality ( ! play1 ! ?r[ ought to be equal to ! advantage :i* x procures4 ,if ,paul receives two francs by ?r[+ heads at ! fir/ & five francs by ?r[+ x at ! second ?r[1 :e!r he has ?r[n x or n at ! fir/1 ! probability ( ?r[+ heads at ! second ?r[ be+ 1 multiply+ two francs & five francs by ! sum ( ^! products w give ?ree & "o half francs = ,paul's advantage & consequ5tly = 8 /ake at ! game4 ,n9? ,pr9ciple4--,9 a s]ies ( probable ev5ts ( :i* ! ones produce a b5efit & ! o!rs a loss1 we %all h ! advantage :i* results f x by mak+ a sum ( ! products ( ! probability ( ea* favorable ev5t by ! b5efit :i* x procures1 & subtract+ f ?is sum t ( ! products ( ! probability ( ea* unfavorable ev5t by ! loss :i* is atta*$ to x4 ,if ! second sum is great] ?an ! fir/1 ! b5efit becomes a loss & hope is *ang$ to fe>4 ,consequ5tly we ought always 9 ! conduct ( life to make ! product ( ! b5efit hop$ =1 by its probability1 at lea/ equal to ! simil> product relative to ! loss4 ,b x is necess>y1 9 ord] to atta9 ?is1 to appreciate exactly ! advantages1 ! losses1 & _! respective probabilities4 ,= ?is a great accuracy ( m9d1 a delicate judgm5t1 & a great exp]i5ce 9 affairs is necess>y2 x is necess>y to "k h[ to gu>d "o's self aga9/ prejudices1 illusions ( fe> or hope1 & errone\s ideas1 ideas ( =tune & happ9ess1 ) :i* ! majority ( p fe$ _! self-love4 ,! application ( ! prec$+ pr9ciples to ! foll[+ "q has greatly ex]cis$ ! geometricians4 ,paul plays at heads & tails ) ! condition ( receiv+ two francs if he ?r[s heads at ! fir/ ?r[1 f\r francs if he ?r[s x only at ! second ?r[1 eight francs if he ?r[s x only at ! ?ird1 & s on4 ,8 /ake at ! play ought to be1 accord+ to ! eigh? pr9ciple1 equal to ! numb] ( ?r[s1 s t if ! game cont9ues to inf9ity ! /ake ought to be inf9ite4 ,h[ev]1 no reasonable man w\ld wi% to risk at ?is game ev5 a small sum1 = example five francs4 ,:5ce comes ?is diff]5ce betwe5 ! result ( calculation & ! indication ( common s5se8 ,we soon recognize t x am\nts to ?is3 t ! moral advantage :i* a b5efit procures = u is n proportional to ?is b5efit & t x dep5ds ^u a ?\s& circum/ances1 (t5 v difficult to def9e1 b ( :i* ! mo/ g5]al & mo/ important is t ( =tune4 ,inde$ x is app>5t t "o franc has mu* great] value = him :o possesses only a hundr$ ?an = a millionaire4 ,we ought !n to di/+ui% 9 ! hop$-= b5efit its absolute f its relative value4 ,b ! latt] is regulat$ by ! motives :i* make x desirable1 :]eas ! fir/ is indep5d5t ( !m4 ,! g5]al pr9ciple = appreciat+ ?is relative value _c be giv51 b "h is "o propos$ by ,daniel ,b]n\lli :i* w s]ve 9 _m cases4 ,t5? ,pr9ciple4--,! relative value ( an inf9itely small sum is equal to its absolute value divid$ by ! total b5efit ( ! p]son int]e/$4 ,?is supposes t e "o has a c]ta9 b5efit ^: value c nev] be e/imat$ z z]o4 ,inde$ ev5 t "o :o possesses no?+ always gives to ! product ( 8 labor & to 8 hopes a value at lea/ equal to t :i* is absolutely necess>y to su/a9 him4 ,if we apply analysis to ! pr9ciple j prop\nd$1 we obta9 ! foll[+ rule3 ,let u designate by unity ! "p ( ! =tune ( an individual1 indep5d5t ( 8 expectations4 ,if we det]m9e ! diff]5t values t ?is =tune may h by virtue ( ^! expectations & _! probabilities1 ! product ( ^! values rais$ respectively to ! p[]s indicat$ by _! probabilities w be ! physical =tune :i* w\ld procure = ! individual ! same moral advantage :i* he receives f ! "p ( 8 =tune tak5 z unity & f 8 expectations2 by subtract+ unity f ! product1 ! diff]5ce w be ! increase ( ! physical =tune due to expectations3 we w call ?is increase moral hope4 ,x is easy to see t x co9cides ) ma!matical hope :5 ! =tune tak5 z unity becomes inf9ite 9 ref]5ce to ! v>iations :i* x receives f ! expectations4 ,b :5 ^! v>iations are an appreciable "p ( ?is unity ! two hopes may diff] v mat]ially among !mselves4 ,?is rule conduces to results con=mable to ! indications ( common s5se :i* c by ?is means be appreciat$ ) "s exactitude4 ,?us 9 ! prec$+ "q x is f\nd t if ! =tune ( ,paul is two hundr$ francs1 he ought n reasonably to /ake m ?an n9e francs4 ,! same rule leads u aga9 to di/ribute ! dang] ov] sev]al p>ts ( a b5efit expect$ r ?an to expose ! entire b5efit to ?is dang]4 ,x results simil>ly t at ! faire/ game ! loss is always great] ?an ! ga94 ,let u suppose1 = example1 t a play] hav+ a =tune ( "o hundr$ francs risks fifty at ! play ( heads & tails2 8 =tune aft] 8 /ake at ! play w be r$uc$ to eighty-sev5 francs1 t is to say1 ?is la/ sum w\ld procure = ! play] ! same moral advantage z ! /ate ( 8 =tune aft] ! /ake4 ,! play is !n disadvantage\s ev5 9 ! case :]e ! /ake is equal to ! product ( ! sum hop$ =1 by its probability4 ,we c judge by ?is ( ! immorality ( games 9 :i* ! sum hop$ = is bel[ ?is product4 ,!y subsi/ only by false reason+s & by ! cupidity :i* !y excite & :i*1 lead+ ! p to sacrifice _! necess>ies to *im]ical hopes ^: improbability !y are n 9 condition to appreciate1 are ! s\rce ( an inf9ity ( evils4 ,! disadvantage ( games ( *ance1 ! advantage ( n expos+ to ! same dang] ! :ole b5efit t is expect$1 & all ! simil> results indicat$ by common s5se1 subsi/1 :atev] may be ! function ( ! physical =tune :i* = ea* individual expresses 8 moral =tune4 ,x is en\gh t ! proportion ( ! increase ( ?is function to ! increase ( ! physical =tune dim9i%es 9 ! measure t ! latt] increases4 ,,*apt] ,v4 ,,conc]n+ ,,! ,,analytical ,,me?ods ,,( ,,! ,,calculus ,,( ,,probabilities4 ,! application ( ! pr9ciple :i* we h j exp\nd$ to ! v>i\s que/ions ( probability requires me?ods ^: inve/igation has giv5 bir? to sev]al me?ods ( analysis & especially to ! !ory ( comb9ations & to ! calculus ( f9ite diff]5ces4 ,if we =m ! product ( ! b9omials1 unity plus ! fir/ lett]1 unity plus ! second lett]1 unity plus ! ?ird lett]1 & s on up to n lett]s1 & subtract unity f ?is develop$ product1 ! result w be ! sum ( ! comb9ation ( all ^! lett]s tak5 "o by "o1 two by two1 ?ree by ?ree1 etc41 ea* comb9ation hav+ unity = a coeffici5t4 ,9 ord] to h ! numb] ( comb9ations ( ^! n lett]s tak5 s by s times1 we %all obs]ve t if we suppose ^! lett]s equal among !mselves1 ! prec$+ product w become ! n? p[] ( ! b9omial "o plus ! fir/ lett]2 ?us ! numb] ( comb9ations ( n lett]s tak5 s by s times w be ! coeffici5t ( ! s? p[] ( ! fir/ lett] 9 ! developm5t 9 ?is b9omial2 & ?is numb] is obta9$ by means ( ! kn[n b9omial =mula4 ,att5tion mu/ be paid to ! respective situations ( ! lett]s 9 ea* comb9ation1 obs]v+ t if a second lett] is jo9$ to ! fir/ x may be plac$ 9 ! fir/ or second position :i* gives two comb9ations4 ,if we jo9 to ^! comb9ations a ?ird lett]1 we c give x 9 ea* comb9ation ! fir/1 ! second1 & ! ?ird rank :i* =ms ?ree comb9ations relative to ea* ( ! two o!rs1 9 all six comb9ations4 ,f ?is x is easy to conclude t ! numb] ( arrangem5ts ( :i* s lett]s are susceptible is ! product ( ! numb]s f unity to s4 ,9 ord] to pay reg>d to ! respective positions ( ! lett]s x is necess>y !n to multiply by ?is product ! numb] ( comb9ations ( n lett]s s by s times1 :i* is tantam\nt to tak+ away ! d5om9ator ( ! coeffici5t ( ! b9omial :i* expresses ?is numb]4 ,let u imag9e a lott]y compos$ ( n numb]s1 ( :i* r are drawn at ea* draw4 ,! probability is dem&$ ( ! draw+ ( s giv5 numb]s 9 "o draw4 ,to arrive at ?is let u =m a fraction ^: d5om9ator w be ! numb] ( all ! cases possible or ( ! comb9ations ( n lett]s tak5 r by r times1 & ^: num]ator w be ! numb] ( all ! comb9ations :i* conta9 ! giv5 s numb]s4 ,?is la/ numb] is evid5tly t ( ! comb9ations ( ! o!r numb]s tak5 n less s by n less s times4 ,?is fraction w be ! requir$ probability1 & we %all easily f9d t x c be r$uc$ to a fraction ^: num]ator is ! numb] ( comb9ations ( r numb]s tak5 s by s times1 & ^: d5om9ator is ! numb] ( comb9ations ( n numb]s tak5 simil>ly s by s times4 ,?us 9 ! lott]y ( ,france1 =m$ z is kn[n ( #ij numb]s ( :i* five are drawn at ea* draw1 ! probability ( draw+ a giv5 comb9ation is #e_/ij1 or #a_/ah2 ! lott]y ought !n = ! equality ( ! play to give eighte5 times ! /ake4 ,! total numb] ( comb9ations two by two ( ! #ij numb]s is #djje1 & t ( ! comb9ations two by two ( #e numb]s is #aj4 ,! probability ( ! draw+ ( a giv5 pair is !n #a_/djje1 & ! lott]y ought to give f\r hundr$ & a half times ! /ake2 x ought to give #aagdh times = a giv5 tray1 #eaajch times = a quat]n>y1 & #dcidibfh times = a qu9t4 ,! lott]y is f> f giv+ ! play] ^! advantages4 ,suppose 9 an urn a :ite balls1 b black balls1 & aft] hav+ drawn a ball x is put back into ! urn2 ! probability is ask$ t 9 n numb] ( draws m :ite balls & n - m black balls w be drawn4 ,x is cle> t ! numb] ( cases t may occur at ea* draw+ is a b4 ,ea* case ( ! second draw+ be+ able to comb9e ) all ! cases ( ! fir/1 ! numb] ( possible cases 9 two draw+s is ! squ>e ( ! b9omial a b4 ,9 ! developm5t ( ?is squ>e1 ! squ>e ( a expresses ! numb] ( cases 9 :i* a :ite ball is twice drawn1 ! d\ble product ( a by b expresses ! numb] ( cases 9 :i* a :ite ball & a black ball are drawn4 ,f9ally1 ! squ>e ( b expresses ! numb] ( cases 9 :i* two black balls are drawn4 ,cont9u+ ?us1 we see g5]ally t ! n? p[] ( ! b9omial a b expresses ! numb] ( all ! cases possible 9 n draws2 & t 9 ! developm5t ( ?is p[] ! t]m multipli$ by ! m? p[] ( a expresses ! numb] ( cases 9 :i* m :ite balls & n - m black balls may be drawn4 ,divid+ !n ?is t]m by ! entire p[] ( ! b9omial1 we %all h ! probability ( draw+ m :ite balls & n - m black balls4 ,! ratio ( ! numb]s a & a b be+ ! probability ( draw+ "o :ite ball at "o draw2 & ! ratio ( ! numb]s b & a b be+ ! probability ( draw+ "o black ball2 if we call ^! probabilities p & q1 ! probability ( draw+ m :ite balls 9 n draws w be ! t]m multipli$ by ! m? p[] ( p 9 ! developm5t ( ! n? p[] ( ! b9omial p q2 we may see t ! sum p q is unity4 ,?is rem>kable prop]ty ( ! b9omial is v useful 9 ! !ory ( probabilities4 ,b ! mo/ g5]al & direct me?od ( resolv+ que/ions ( probability consi/s 9 mak+ !m dep5d ^u equations ( diff]5ces4 ,comp>+ ! successive conditions ( ! function :i* expresses ! probability :5 we increase ! v>iables by _! respective diff]5ces1 ! propos$ "q (t5 furni%es a v simple proportion betwe5 ! conditions4 ,?is proportion is :at is call$ equation ( ord9>y or p>tial diff]5tials2 ord9>y :5 !re is only "o v>iable1 p>tial :5 !re are sev]al4 ,let u consid] "s examples ( ?is4 ,?ree play]s ( suppos$ equal ability play toge!r on ! foll[+ conditions3 t "o ( ! fir/ two play]s :o beats 8 adv]s>y plays ! ?ird1 & if he beats him ! game is f9i%$4 ,if he is beat51 ! victor plays aga9/ ! second until "o ( ! play]s has defeat$ consecutively ! two o!rs1 :i* ends ! game4 ,! probability is dem&$ t ! game w be f9i%$ 9 a c]ta9 numb] n ( plays4 ,let u f9d ! probability t x w end precisely at ! n? play4 ,= t ! play] :o w9s ought to ent] ! game at ! play n - #a & w9 x ?us at ! foll[+ play4 ,b if 9 place ( w9n+ ! play n - #a he %\ld be beat5 by 8 adv]s>y :o _h j beat5 ! o!r play]1 ! game w\ld end at ?is play4 ,?us ! probability t "o ( ! play]s w ent] ! game at ! play n - #a & w w9 x is equal to ! probability t ! game w end precisely ) ?is play2 & z ?is play] ought to w9 ! foll[+ play 9 ord] t ! game may be f9i%$ at ! n? play1 ! probability ( ?is la/ case w be only "o half ( ! prec$+ "o4 ,?is probability is evid5tly a function ( ! numb] n2 ?is function is !n equal to ! half ( ! same function :5 n is dim9i%$ by unity4 ,?is equality =ms "o ( ^? equations call$ ord9>y f9ite diff]5tial equations4 ,we may easily det]m9e by its use ! probability t ! game w end precisely at a c]ta9 play4 ,x is evid5t t ! play _c end soon] ?an at ! second play2 & = ?is x is necess>y t t "o ( ! fir/ two play]s :o has beat5 8 adv]s>y %\ld beat at ! second play ! ?ird play]2 ! probability t ! game w end at ?is play is 4 ,h5ce by virtue ( ! prec$+ equation we conclude t ! successive probabilities ( ! end ( ! game are = ! ?ird play1 = ! f\r? play1 & s on2 & 9 g5]al rais$ to ! p[] n - #a = ! n? play4 ,! sum ( all ^! p[]s ( is unity less ! la/ ( ^! p[]s2 x is ! probability t ! game w end at ! late/ 9 n plays4 ,let u consid] aga9 ! fir/ problem m difficult :i* may be solv$ by probabilities & :i* ,pascal propos$ to ,f]mat to solve4 ,two play]s1 ,a & ,b1 ( equal skill play toge!r on ! conditions t ! "o :o fir/ %all beat ! o!r a giv5 numb] ( times %all w9 ! game & %all take ! sum ( ! /akes at ! game2 aft] "s ?r[s ! play]s agree to quit )\t hav+ f9i%$ ! game3 we ask 9 :at mann] ! sum ought to be divid$ betwe5 !m4 ,x is evid5t t ! p>ts ought to be proportional to ! respective probabilities ( w9n+ ! game4 ,! "q is r$uc$ !n to ! det]m9ation ( ^! probabilities4 ,!y dep5d evid5tly ^u ! numb] ( po9ts :i* ea* play] lacks ( hav+ atta9$ ! giv5 numb]4 ,h5ce ! probability ( ,a is a function ( ! two numb]s :i* we w call indices4 ,if ! two play]s %\ld agree to play "o ?r[ m 7an agreem5t :i* does n *ange _! condition1 provid$ t aft] ?is new ?r[ ! division is always made proportionally to ! new probabilities ( w9n+ ! game71 !n ei!r ,a w\ld w9 ?is ?r[ & 9 t case ! numb] ( po9ts :i* he lacks w\ld be dim9i%$ by unity1 or ! play] ,b w\ld w9 x & 9 t case ! numb] ( po9ts lack+ to ?is la/ play] w\ld be less by unity4 ,b ! probability ( ea* ( ^! cases is 2 ! function s\ght is !n equal to "o half ( ?is function 9 :i* we dim9i% by unity ! fir/ index plus ! half ( ! same function 9 :i* ! second v>iable is dim9i%$ by unity4 ,?is equality is "o ( ^? equations call$ equations ( p>tial diff]5tials4 ,we are able to det]m9e by its use ! probabilities ( ,a by divid+ ! smalle/ numb]s1 & by obs]v+ t ! probability or ! function :i* expresses x is equal to unity :5 ! play] ,a does n lack a s+le po9t1 or :5 ! fir/ index is z]o1 & t ?is function becomes z]o ) ! second index4 ,suppos+ ?us t ! play] ,a lacks only "o po9t1 we f9d t 8 probability is 1 1 1 etc41 accord+ z ,b lacks "o po9t1 two1 ?ree1 etc4 ,g5]ally x is !n unity less ! p[] ( 1 equal to ! numb] ( po9ts :i* ,b lacks4 ,we w suppose !n t ! play] ,a lacks two po9ts & 8 probability w be f\nd equal to 1 1 #aa_/af1 etc41 accord+ z ,b lacks "o po9t1 two po9ts1 ?ree po9ts1 etc4 ,we w suppose aga9 t ! play] ,a lacks ?ree po9ts1 & s on4 ,?is mann] ( obta9+ ! successive values ( a quantity by means ( its equation ( diff]5ces is long & labori\s4 ,! geometricians h s\ght me?ods to obta9 ! g5]al function ( indices t satisfies ?is equation1 s t = any p>ticul> case we ne$ only to sub/itute 9 ?is function ! correspond+ values ( ! indices4 ,let u consid] ?is subject 9 a g5]al way4 ,= ?is purpose let u conceive a s]ies ( t]ms arrang$ along a horizontal l9e s t ea* ( !m is d]iv$ f ! prec$+ "o accord+ to a giv5 law4 ,let u suppose ?is law express$ by an equation among sev]al consecutive t]ms & _! index1 or ! numb] :i* indicates ! rank t !y occupy 9 ! s]ies4 ,?is equation ,i call ! equation ( f9ite diff]5ces by a s+le index4 ,! ord] or ! degree ( ?is equation is ! diff]5ce ( rank ( its two extreme t]ms4 ,we are able by its use to det]m9e successively ! t]ms ( ! s]ies & to cont9ue x indef9itely2 b = t x is necess>y to "k a numb] ( t]ms ( ! s]ies equal to ! degree ( ! equation4 ,^! t]ms are ! arbitr>y con/ants ( ! expression ( ! g5]al t]m ( ! s]ies or ( ! integral ( ! equation ( diff]5ces4 ,let u imag9e n[ bel[ ! t]ms ( ! prec$+ s]ies a second s]ies ( t]ms arrang$ horizontally2 let u imag9e aga9 bel[ ! t]ms ( ! second s]ies a ?ird horizontal s]ies1 & s on to inf9ity2 & let u suppose ! t]ms ( all ^! s]ies connect$ by a g5]al equation among sev]al consecutive t]ms1 tak5 z mu* 9 ! horizontal z 9 ! v]tical s5se1 & ! numb]s :i* indicate _! rank 9 ! two s5ses4 ,?is equation is call$ ! equation ( p>tial f9ite diff]5ces by two indices4 ,let u imag9e 9 ! same way bel[ ! plan ( ! prec$+ s]ies a second plan ( simil> s]ies1 ^: t]ms %\ld be plac$ respectively bel[ ^? ( ! fir/ plan2 let u imag9e aga9 bel[ ?is second plan a ?ird plan ( simil> s]ies1 & s on to inf9ity2 let u suppose all ! t]ms ( ^! s]ies connect$ by an equation among sev]al consecutive t]ms tak5 9 ! s5se ( l5g?1 wid?1 & dep?1 & ! ?ree numb]s :i* indicate _! rank 9 ^! ?ree s5ses4 ,?is equation ,i call ! equation ( p>tial f9ite diff]5ces by ?ree indices4 ,f9ally1 consid]+ ! matt] 9 an ab/ract way & indep5d5tly ( ! dim5sions ( space1 let u imag9e g5]ally a sy/em ( magnitudes1 :i* %\ld be functions ( a c]ta9 numb] ( indices1 & let u suppose among ^! magnitudes1 _! relative diff]5ces to ^! indices & ! indices !mselves1 z _m equations z !re are magnitudes2 ^! equations w be p>tial f9ite diff]5ces by a c]ta9 numb] ( indices4 ,we are able by _! use to det]m9e successively ^! magnitudes4 ,b 9 ! same mann] z ! equation by a s+le index requires = x t we "k a c]ta9 numb] ( t]ms ( ! s]ies1 s ! equation by two indices requires t we "k "o or sev]al l9es ( s]ies ^: g5]al t]ms %\ld be express$ ea* by an arbitr>y function ( "o ( ! indices4 ,simil>ly ! equation by ?ree indices requires t we "k "o or sev]al plans ( s]ies1 ! g5]al t]ms ( :i* %\ld be express$ ea* by an arbitr>y function ( two indices1 & s on4 ,9 all ^! cases we %all be able by successive elim9ations to det]m9e a c]ta9 t]m ( ! s]ies4 ,b all ! equations among :i* we elim9ate be+ compris$ 9 ! same sy/em ( equations1 all ! expressions ( ! successive t]ms :i* we obta9 by ^! elim9ations ought to be compris$ 9 "o g5]al expression1 a function ( ! indices :i* det]m9e ! rank ( ! t]m4 ,?is expression is ! integral ( ! propos$ equation ( diff]5ces1 & ! se>* = x is ! object ( integral calculus4 ,taylor is ! fir/ :o 9 8 "w entitl$ ,metodus increm5torum has consid]$ l9e> equations ( f9ite diff]5ces4 ,he gives ! mann] ( integrat+ ^? ( ! fir/ ord] ) a coeffici5t & a la/ t]m1 functions ( ! index4 ,9 tru? ! relations ( ! t]ms ( ! ari?metical & geometrical progressions :i* h always be5 tak5 into consid]ation are ! simple/ cases ( l9e> equations ( diff]5ces2 b !y _h n be5 consid]$ f ?is po9t ( view4 ,x 0 "o ( ^? :i*1 atta*+ !mselves to g5]al !ories1 lead to ^! !ories & are consequ5tly v]itable discov]ies4 ,ab\t ! same "t ,moivre 0 consid]+ "u ! "n ( recurr+ s]ies ! equations ( f9ite diff]5ces ( a c]ta9 ord] hav+ a con/ant coeffici5t4 ,he succe$$ 9 integrat+ !m 9 a v +5i\s mann]4 ,z x is always int]e/+ to foll[ ! progress ( inv5tors1 ,i %all exp\nd ! me?od ( ,moivre by apply+ x to a recurr+ s]ies ^: relation among ?ree consecutive t]ms is giv54 ,fir/ he consid]s ! relation among ! consecutive t]ms ( a geometrical progression or ! equation ( two t]ms :i* expresses x4 ,ref]r+ x to t]ms less ?an unity1 he multiplies x 9 ?is /ate by a con/ant factor & subtracts ! product f ! fir/ equation4 ,?us he obta9s an equation among ?ree consecutive t]ms ( ! geometrical progression4 ,moivre consid]s next a second progression ^: ratio ( t]ms is ! same factor :i* he has j us$4 ,he dim9i%es simil>ly by unity ! index ( ! t]ms ( ! equation ( ?is new progression4 ,9 ?is condition he multiplies x by ! ratio ( ! t]ms ( ! fir/ progression1 & he subtracts ! product f ! equation ( ! second progression1 :i* gives him among ?ree consecutive t]ms ( ?is progression a relation entirely simil> to t :i* he has f\nd = ! fir/ progression4 ,!n he obs]ves t if "o adds t]m by t]m ! two progressions1 ! same ratio exi/s among any ?ree ( ^! consecutive t]ms4 ,he comp>es ! coeffici5ts ( ?is ratio to ^? ( ! relation ( ! t]ms ( ! propos$ recurr5t s]ies1 & he f9ds = det]m9+ ! ratios ( ! two geometrical progressions an equation ( ! second degree1 ^: roots are ^! ratios4 ,?us ,moivre decomposes ! recurr5t s]ies into two geometrical progressions1 ea* multipli$ by an arbitr>y con/ant :i* he det]m9es by means ( ! fir/ two t]ms ( ! recurr5t s]ies4 ,?is +5i\s process is 9 fact ! "o t d',alemb]t has s9ce employ$ = ! integration ( l9e> equations ( inf9itely small diff]5ces ) con/ant coeffici5ts1 & ,lagrange has trans=m$ into simil> equations ( f9ite diff]5ces4 ,f9ally1 ,i h consid]$ ! l9e> equations ( p>tial f9ite diff]5ces1 fir/ "u ! "n ( recurro-recurr5t s]ies & aft]w>ds "u _! own "n4 ,! mo/ g5]al & simple/ mann] ( integrat+ all ^! equations appe>s to me t :i* ,i h bas$ ^u ! consid]ation ( discrim9ant functions1 ! idea ( :i* is "h giv54 ,if we conceive a function ,v ( a v>iable t develop$ accord+ to ! p[]s ( ?is v>iable1 ! coeffici5t ( any "o ( ^! p[]s w be a function ( ! expon5t or index ( ?is p[]1 :i* index ,i %all call x4 ,v is :at ,i call ! discrim9ant function ( ?is coeffici5t or ( ! function ( ! index4 ,n[ if we multiply ! s]ies ( ! developm5t ( ,v by a function ( ! same v>iable1 su*1 = example1 z unity plus two times ?is v>iable1 ! product w be a new discrim9ant function 9 :i* ! coeffici5t ( ! p[] x ( ! v>iable t w be equal to ! coeffici5t ( ! same p[] 9 ,v plus twice ! coeffici5t ( ! p[] less unity4 ,?us ! function ( ! index x 9 ! product w be equal to ! function ( ! index x 9 ,v plus twice ! same function 9 :i* ! index is dim9i%$ by unity4 ,?is function ( ! index x is ?us a d]ivative ( ! function ( ! same index 9 ! developm5t ( ,v1 a function :i* ,i %all call ! primitive function ( ! index4 ,let u designate ! d]ivative function by ! lett] ,alemb]t plac$ be=e ! primitive function4 ,! d]ivation indicat$ by ?is lett] w dep5d ^u ! multipli] ( ,v1 :i* we w call ,t & :i* we w suppose develop$ l ,v by ! ratio to ! p[]s ( ! v>iable t4 ,if we multiply anew by ,t ! product ( ,v by ,t1 :i* is equival5t to multiply+ ,v by ,t1 we %all =m a ?ird discrim9ant function1 9 :i* ! coeffici5t ( ! x? p[] ( t w be a d]ivative simil> to ! correspond+ coeffici5t ( ! prec$+ product2 x may be express$ by ! same "* plac$ be=e ! prec$+ d]ivative1 & !n ?is "* w be writt5 twice be=e ! primitive function ( x4 ,b 9 place ( writ+ x ?us twice we give x #b = an expon5t4 ,cont9u+ ?us1 we see g5]ally t if we multiply ,v by ! n? p[] ( ,t1 we %all h ! coeffici5t ( ! x? p[] ( t 9 ! product ( ,v by ! n? p[] ( ,t by plac+ be=e ! primitive function ! "* ) n = an expon5t4 ,let u suppose1 = example1 t ,t be unity divid$ by t2 !n 9 ! product ( ,v by ,t ! coeffici5t ( ! x? p[] ( t w be ! coeffici5t ( ! p[] great] by unity 9 ,v2 ?is coeffici5t 9 ! product ( ,v by ! n? p[] ( ,t w !n be ! primitive function 9 :i* x is augm5t$ by n units4 ,let u consid] n[ a new function ,z ( t1 develop$ l ,v & ,t accord+ to ! p[]s ( t2 let u designate by ! "* plac$ be=e ! primitive function ! coeffici5t ( ! x? p[] ( t 9 ! product ( ,v by ,z2 ?is coeffici5t 9 ! product ( ,v by ! n? p[] ( ,z w be express$ by ! "* affect$ by ! expon5t n & plac$ be=e ! primitive function ( x4 ,if1 = example1 ,z is equal to unity divid$ by t less "o1 ! coeffici5t ( ! x? p[] ( t 9 ! product ( ,v by ,z w be ! coeffici5t ( ! x #a p[] ( t 9 ,v less ! coeffici5t ( ! x? p[]4 ,x w be !n ! f9ite diff]5ce ( ! primitive function ( ! index x4 ,!n ! "* indicates a f9ite diff]5ce ( ! primitive function 9 ! case :]e ! index v>ies by unity2 & ! n? p[] ( ?is "* plac$ be=e ! primitive function w indicate ! f9ite n? diff]5ce ( ?is function4 ,if we suppose t ,t be unity divid$ by t1 we %all h ,t equal to ! b9omial ,z #a4 ,! product ( ,v by ! n? p[] ( ,t w !n be equal to ! product ( ,v by ! n? p[] ( ! b9omial ,z #a4 ,develop+ ?is p[] 9 ! ratio ( ! p[]s ( ,z1 ! product ( ,v by ! v>i\s t]ms ( ?is developm5t w be ! discrim9ant functions ( ^! same t]ms 9 :i* we sub/itute 9 place ( ! p[]s ( ,z ! correspond+ f9ite diff]5ces ( ! primitive function ( ! index4 ,n[ ! product ( ,v by ! n? p[] ( ,t is ! primitive function 9 :i* ! index x is augm5t$ by n units2 repass+ f ! discrim9ant functions to _! coeffici5ts1 we %all h ?is primitive function ?us augm5t$ equal to ! developm5t ( ! n? p[] ( ! b9omial ,z #a1 provid$ t 9 ?is developm5t we sub/itute 9 place ( ! p[]s ( ,z ! correspond+ diff]5ces ( ! primitive function & t we multiply ! indep5d5t t]m ( ^! p[]s by ! primitive function4 ,we %all ?us obta9 ! primitive function ^: index is augm5t$ by any numb] n by means ( its diff]5ces4 ,suppos+ t ,t & ,z always h ! prec$+ values1 we %all h ,z equal to ! b9omial ,t - #a2 ! product ( ,v by ! n? p[] ( ,z w !n be equal to ! product ( ,v by ! developm5t ( ! n? p[] ( ! b9omial ,t - #a4 ,repass+ f ! discrim9ant functions to _! coeffici5ts z has j be5 done1 we %all h ! n? diff]5ce ( ! primitive function express$ by ! developm5t ( ! n? p[] ( ! b9omial ,t - #a1 9 :i* we sub/itute = ! p[]s ( ,t ?is same function ^: index is augm5t$ by ! expon5t ( ! p[]1 & = ! indep5d5t t]m ( t1 :i* is unity1 ! primitive function1 :i* gives ?is diff]5ce by means ( ! consecutive t]ms ( ?is function4 ,plac+ be=e ! primitive function express+ ! d]ivative ( ?is function1 :i* multiplies ! x p[] ( t 9 ! product ( ,v by ,t1 & express+ ! same d]ivative 9 ! product ( ,v by ,z1 we are l$ by t :i* prec$es to ?is g5]al result3 :atev] may be ! function ( ! v>iable t repres5t$ by ,t & ,z1 we may1 9 ! developm5t ( all ! id5tical equations susceptible ( be+ =m$ among ^! functions1 sub/itute ! *>act]s & 9 place ( ,t & ,z1 provid$ t we write ! primitive function ( ! index 9 s]ies ) ! p[]s & ) ! products ( ! p[]s ( ! *>act]s1 & t we multiply by ?is function ! indep5d5t t]ms ( ^! *>act]s4 ,we are able by means ( ?is g5]al result to trans=m any c]ta9 p[] ( a diff]5ce ( ! primitive function ( ! index x1 9 :i* x v>ies by unity1 into a s]ies ( diff]5ces ( ! same function 9 :i* x v>ies by a c]ta9 numb] ( units & reciprocally4 ,let u suppose t ,t be ! i p[] ( unity divid$ by t - #a1 & t ,z be always unity divid$ by t - #a2 !n ! coeffici5t ( ! x p[] ( t 9 ! product ( ,v by ,t w be ! coeffici5t ( ! x i p[] ( t 9 ,v less ! coeffici5t ( ! x p[] ( t2 x w !n be ! f9ite diff]5ce ( ! primitive function ( ! index x 9 :i* we v>y ?is index by ! numb] i4 ,x is easy to see t ,t is equal to ! diff]5ce betwe5 ! i p[] ( ! b9omial ,z #a & unity4 ,! n? p[] ( ,t is equal to ! n? p[] ( ?is diff]5ce4 ,if 9 ?is equality we sub/itute 9 place ( ,t & ,z ! *>act]s & 1 & aft] ! developm5t we place at ! end ( ea* t]m ! primitive function ( ! index x1 we %all h ! n? diff]5ce ( ?is function 9 :i* x v>ies by i units express$ by a s]ies ( diff]5ces ( ! same function 9 :i* x v>ies by unity4 ,?is s]ies is only a trans=mation ( ! diff]5ce :i* x expresses & :i* is id5tical ) x2 b x is 9 simil> trans=mations t ! p[] ( analysis resides4 ,! g5]ality ( analysis p]mits u to suppose 9 ?is expression t n is negative4 ,!n ! negative p[]s ( & indicate ! integrals4 ,inde$ ! n? diff]5ce ( ! primitive function hav+ = a discrim9ant function ! product ( ,v by ! n? p[] ( ! b9omial "o divid$ by t less unity1 ! primitive function :i* is ! n? integral ( ?is diff]5ce has = a discrim9ant function t ( ! same diff]5ce multipli$ by ! n? p[] tak5 less ?an ! b9omial "o divid$ by t m9us "o1 a p[] to :i* ! same p[] ( ! "* corresponds2 ?is p[] indicates !n an integral ( ! same ord]1 ! index x v>y+ by unity2 & ! negative p[]s ( indicate equally ! integrals x v>y+ by i units4 ,we see1 ?us1 9 ! cle>e/ & simple/ mann] ! rationality ( ! analysis obs]v$ among ! positive p[]s & diff]5ces1 & among ! negative p[]s & ! integrals4 ,if ! function indicat$ by plac$ be=e ! primitive function is z]o1 we %all h an equation ( f9ite diff]5ces1 & ,v w be ! discrim9ant function ( its integral4 ,9 ord] to obta9 ?is discrim9ant function we %all obs]ve t 9 ! product ( ,v by ,t all ! p[]s ( t ought to disappe> except ! p[]s inf]ior to ! ord] ( ! equation ( diff]5ces2 ,v is !n equal to a fraction ^: d5om9ator is ,t & ^: num]ator is a polynomial 9 :i* ! highe/ p[] ( t is less by unity ?an ! ord] ( ! equation ( diff]5ces4 ,! arbitr>y coeffici5ts ( ! v>i\s p[]s ( t 9 ?is polynomial1 includ+ ! p[] z]o1 w be det]m9$ by z _m values ( ! primitive function ( ! index :5 we make successively x equal to z]o1 to "o1 to two1 etc4 ,:5 ! equation ( diff]5ces is giv5 we det]m9e ,t by putt+ all its t]ms 9 ! fir/ memb] & z]o 9 ! second2 by sub/itut+ 9 ! fir/ memb] unity 9 place ( ! function :i* has ! l>ge/ index2 ! fir/ p[] ( t 9 place ( ! primitive function 9 :i* ?is index is dim9i%$ by unity2 ! second p[] ( t = ! primitive function :]e ?is index is dim9i%$ by two units1 & s on4 ,! coeffici5t ( ! x? p[] ( t 9 ! developm5t ( ! prec$+ expression ( ,v w be ! primitive function ( x or ! integral ( ! equation ( f9ite diff]5ces4 ,analysis furni%es = ?is developm5t v>i\s means1 among :i* we may *oose t "o :i* is mo/ suitable = ! "q propos$2 ?is is an advantage ( ?is me?od ( integration4 ,let u conceive n[ t ,v be a function ( ! two v>iables t & t develop$ accord+ to ! p[]s & products ( ^! v>iables2 ! coeffici5t ( any product ( ! p[]s x & x ( t & t w be a function ( ! expon5ts or indices x & x ( ^! p[]s2 ?is function ,i %all call ! primitive function ( :i* ,v is ! discrim9ant function4 ,let u multiply ,v by a function ,t ( ! two v>iables t & t develop$ l ,v 9 ratio ( ! p[]s & ! products ( ^! v>iables2 ! product w be ! discrim9ant function ( a d]ivative ( ! primitive function2 if ,t1 = example1 is equal to ! v>iable t plus ! v>iable t m9us two1 ?is d]ivative w be ! primitive function ( :i* we dim9i% by unity ! index x plus ?is same primitive function ( :i* we dim9i% by unity ! index x less two times ! primitive function4 ,designat+ :atev] ,t may be by ! "* plac$ be=e ! primitive function1 ?is d]ivative1 ! product ( ,v by ! n? p[] ( ,t1 w be ! discrim9ant function ( ! d]ivative ( ! primitive function be=e :i* "o places ! n? p[] ( ! "* 4 ,h5ce result ! !orems analog\s to ^? :i* are relative to functions ( a s+le v>iable4 ,suppose ! function indicat$ by ! "* be z]o2 "o w h an equation ( p>tial diff]5ces4 ,if1 = example1 we make z be=e ,t equal to ! v>iable t plus ! v>iable t - #b1 we h z]o equal to ! primitive function ( :i* we dim9i% by unity ! index x plus ! same function ( :i* we dim9i% by unity ! index x m9us two times ! primitive function4 ,! discrim9ant function ,v ( ! primitive function or ( ! integral ( ?is equation ought !n to be su* t its product by ,t does n include at all ! products ( t by t2 b ,v may include sep>ately ! p[]s ( t & ^? ( t1 t is to say1 an arbitr>y function ( t & an arbitr>y function ( t2 ,v is !n a fraction ^: num]ator is ! sum ( ^! two arbitr>y functions & ^: d5om9ator is ,t4 ,! coeffici5t ( ! product ( ! x? p[] ( t by ! x p[] ( t 9 ! developm5t ( ?is fraction w !n be ! integral ( ! prec$+ equation ( p>tial diff]5ces4 ,?is me?od ( integrat+ ?is k9d ( equations seems to me ! simple/ & ! easie/ by ! employm5t ( ! v>i\s analytical processes = ! developm5t ( rational fractions4 ,m ample details 9 ?is matt] w\ld be sc>cely und]/ood )\t ! aid ( calculus4 ,consid]+ equations ( inf9itely small p>tial diff]5ces z equations ( f9ite p>tial diff]5ces 9 :i* no?+ is neglect$1 we are able to ?r[ light ^u ! obscure po9ts ( _! calculus1 :i* h be5 ! subject ( great discussions among geometricians4 ,x is ?us t ,i h demon/rat$ ! possibility ( introduc+ discont9u$ functions 9 _! integrals1 provid$ t ! discont9uity takes place only = ! diff]5tials ( ! ord] ( ^! equations or ( a sup]ior ord]4 ,! transc5d5t results ( calculus are1 l all ! ab/ractions ( ! und]/&+1 g5]al signs ^: true mean+ may be asc]ta9$ only by repass+ by metaphysical analysis to ! elem5t>y ideas :i* h l$ to !m2 ?is (t5 pres5ts great difficulties1 = ! human m9d tries /ill less to transport itself into ! future ?an to retire )9 itself4 ,! comp>ison ( inf9itely small diff]5ces ) f9ite diff]5ces is able simil>ly to %$ great light ^u ! metaphysics ( inf9itesimal calculus4 ,x is easily prov5 t ! f9ite n? diff]5ce ( a function 9 :i* ! increase ( ! v>iable is ,e be+ divid$ by ! n? p[] ( ,e1 ! quoti5t r$uc$ 9 s]ies by ratio to ! p[]s ( ! increase ,e is =m$ by a fir/ t]m indep5d5t ( ,e4 ,9 ! measure t ,e dim9i%es1 ! s]ies approa*es m & m ?is fir/ t]m f :i* x c diff] only by quantities less ?an any assignable magnitude4 ,?is t]m is !n ! limit ( ! s]ies & expresses 9 diff]5tial calculus ! inf9itely small n? diff]5ce ( ! function divid$ by ! n? p[] ( ! inf9itely small increase4 ,consid]+ f ?is po9t ( view ! inf9itely small diff]5ces1 we see t ! v>i\s op]ations ( diff]5tial calculus am\nt to comp>+ sep>ately 9 ! developm5t ( id5tical expressions ! f9ite t]ms or ^? indep5d5t ( ! increm5ts ( ! v>iables :i* are reg>d$ z inf9itely small2 ?is is rigor\sly exact1 ^! increm5ts be+ indet]m9ant4 ,?us diff]5tial calculus has all ! exactitude ( o!r algebraic op]ations4 ,! same exactitude is f\nd 9 ! applications ( diff]5tial calculus to geometry & me*anics4 ,if we imag9e a curve cut by a secant at two adjac5t po9ts1 nam+ ,e ! int]val ( ! ord9ates ( ^! two po9ts1 ,e w be ! increm5t ( ! abscissa f ! fir/ to ! second ord9ate4 ,x is easy to see t ! correspond+ increm5t ( ! ord9ate w be ! product ( ,e by ! fir/ ord9ate divid$ by its subsecant2 augm5t+ !n 9 ?is equation ( ! curve ! fir/ ord9ate by ?is increm5t1 we %all h ! equation relative to ! second ord9ate4 ,! diff]5ce ( ^! two equations w be a ?ird equation :i*1 develop$ by ! ratio ( ! p[]s ( ,e & divid$ by ,e1 w h its fir/ t]m indep5d5t ( ,e1 :i* w be ! limit ( ?is developm5t4 ,?is t]m1 equal to z]o1 w give !n ! limit ( ! subsecants1 a limit :i* is evid5tly ! subtang5t4 ,?is s+ul>ly happy me?od ( obta9+ ! subtang5t is due to ,f]mat1 :o has ext5d$ x to transc5d5t curves4 ,?is great geometrician expresses by ! "* ,e ! increm5t ( ! abscissa2 & consid]+ only ! fir/ p[] ( ?is increm5t1 he det]m9es exactly z we d by diff]5tial calculus ! subtang5ts ( ! curves1 _! po9ts ( inflection1 ! maxima & m9ima ( _! ord9ates1 & 9 g5]al ^? ( rational functions4 ,we see likewise by 8 beautiful solution ( ! problem ( ! refraction ( light ins]t$ 9 ! ,collection ( ! ,lett]s ( ,desc>tes t he kn[s h[ to ext5d 8 me?ods to irrational functions 9 free+ !m f irrationalities by ! elevation ( ! roots to p[]s4 ,f]mat %\ld be reg>d$1 !n1 z ! true discov]] ( ,diff]5tial ,calculus4 ,newton has s9ce r5d]$ ?is calculus m analytical 9 8 ,me?od ( ,fluxions1 & simplifi$ & g5]aliz$ ! processes by 8 beautiful !orem ( ! b9omial4 ,f9ally1 ab\t ! same "t ,leibnitz has enri*$ diff]5tial calculus by a notation :i*1 by indicat+ ! passage f ! f9ite to ! inf9itely small1 adds to ! advantage ( express+ ! g5]al results ( calculus t ( giv+ ! fir/ approximate values ( ! diff]5ces & ( ! sums ( ! quantities2 ?is notation is adapt$ ( itself to ! calculus ( p>tial diff]5tials4 ,we are (t5 l$ to expressions :i* conta9 s _m t]ms & factors t ! num]ical sub/itutions are impracticable4 ,?is takes place 9 que/ions ( probability :5 we consid] a great numb] ( ev5ts4 ,mean:ile x is necess>y to h ! num]ical value ( ! =mul 9 ord] to "k ) :at probability ! results are indicat$1 :i* ! ev5ts develop by multiplication4 ,x is necess>y especially to h ! law accord+ to :i* ?is probability cont9ually approa*es c]ta9ty1 :i* x w f9ally atta9 if ! numb] ( ev5ts 7 inf9ite4 ,9 ord] to obta9 ?is law ,i consid]$ t ! def9ite integrals ( diff]5tials multipli$ by ! factors rais$ to great p[]s w\ld give by integration ! =mul compos$ ( a great numb] ( t]ms & factors4 ,?is rem>k br\ght me to ! idea ( trans=m+ into simil> integrals ! complicat$ expressions ( analysis & ! integrals ( ! equation ( diff]5ces4 ,i fulfill$ ?is condition by a me?od :i* gives at ! same "t ! function compris$ "u ! integral sign & ! limits ( ! integration4 ,x (f]s ?is rem>kable ?+1 t ! function is ! same discrim9ant function ( ! expressions & ! propos$ equations2 ?is atta*es ?is me?od to ! !ory ( discrim9ant functions ( :i* x is ?us ! complem5t4 ,fur!r1 x w\ld only be a "q ( r$uc+ ! def9ite integral to a conv]g+ s]ies4 ,?is ,i h obta9$ by a process :i* makes ! s]ies conv]ge ) z mu* m rapidity z ! =mula :i* x repres5ts is m complicat$1 s t x is m exact z x becomes m necess>y4 ,frequ5tly ! s]ies has = a factor ! squ>e root ( ! ratio ( ! circumf]5ce to ! diamet]2 sometimes x dep5ds ^u o!r transc5d5ts ^: numb] is inf9ite4 ,an important rem>k :i* p]ta9s to great g5]ality ( analysis1 & :i* p]mits u to ext5d ?is me?od to =mul & to equations ( diff]5ce :i* ! !ory ( probability pres5ts mo/ frequ5tly1 is t ! s]ies to :i* "o comes by suppos+ ! limits ( ! def9ite integrals to be real & positive take place equally 9 ! case :]e ! equation :i* det]m9es ^! limits has only negative or imag9>y roots4 ,^! passages f ! positive to ! negative & f ! real to ! imag9>y1 ( :i* ,i fir/ h made use1 h l$ me fur!r to ! values ( _m s+ul> def9ite integrals1 :i* ,i h accord+ly demon/rat$ directly4 ,we may !n consid] ^! passages z a means ( discov]y p>allel to induction & analogy long employ$ by geometricians1 at fir/ ) an extreme res]ve1 aft]w>ds ) entire confid5ce1 s9ce a great numb] ( examples has ju/ifi$ its use4 ,9 ! mean "t x is always necess>y to confirm by direct demon/rations ! results obta9$ by ^! div]s means4 ,i h nam$ ! ensemble ( ! prec$+ me?ods ! ,calculus ( ,discrim9ant ,functions2 ?is calculus s]ves z a basis = ! "w :i* ,i h publi%$ "u ! title ( ! ,analytical ,!ory ( ,probabilities4 ,x is connect$ ) ! simple idea ( indicat+ ! repeat$ multiplications ( a quantity by itself or its entire & positive p[]s by writ+ t[>d ! top ( ! lett] :i* expresses x ! numb]s :i* m>k ! degrees ( ^! p[]s4 ,?is notation1 employ$ by ,desc>tes 9 8 ,geometry & g5]ally adopt$ s9ce ! publication ( ?is important "w1 is a little ?+1 especially :5 comp>$ ) ! !ory ( curves & v>iable functions by :i* ?is great geometrician has e/abli%$ ! f\ndations ( mod]n calculus4 ,b ! language ( analysis1 mo/ p]fect ( all1 be+ 9 itself a p[]ful in/rum5t ( discov]ies1 its notations1 especially :5 !y are necess>y & happily conceiv$1 are s _m g]ms ( new calculi4 ,?is is r5d]$ appreciable by ?is example4 ,wallis1 :o 9 8 "w entitl$ ,ari?metica ,inf9itorum1 "o ( ^? :i* h mo/ contribut$ to ! progress ( analysis1 has int]e/$ himself especially 9 foll[+ ! ?read ( induction & analogy1 consid]$ t if "o divides ! expon5t ( a lett] by two1 ?ree1 etc41 ! quoti5t w be accord+ly ! ,c>tesian notation1 & :5 division is possible ! expon5t ( ! squ>e1 cube1 etc41 root ( ! quantity :i* repres5ts ! lett] rais$ to ! divid5d expon5t4 ,ext5d+ by analogy ?is result to ! case :]e division is impossible1 he consid]$ a quantity rais$ to a fractional expon5t z ! root ( ! degree indicat$ by ! d5om9ator ( ?is fraction--namely1 ( ! quantity rais$ to a p[] indicat$ by ! num]ator4 ,he obs]v$ !n t1 accord+ to ! ,c>tesian notation1 ! multiplication ( two p[]s ( ! same lett] am\nts to add+ _! expon5ts1 & t _! division am\nts to subtract+ ! expon5ts ( ! p[] ( ! divisor f t ( ! p[] ( ! divid5d1 :5 ! second ( ^! expon5ts is great] ?an ! fir/4 ,wallis ext5d$ ?is result to ! case :]e ! fir/ expon5t is equal to or great] ?an ! second1 :i* makes ! diff]5ce z]o or negative4 ,he suppos$ !n t a negative expon5t indicates unity divid$ by ! quantity rais$ to ! same expon5t tak5 positively4 ,^! rem>ks l$ him to integrate g5]ally ! monomial diff]5tials1 :5ce he inf]r$ ! def9ite integrals ( a p>ticul> k9d ( b9omial diff]5tials ^: expon5t is a positive integral numb]4 ,! obs]vation !n ( ! law ( ! numb]s :i* express ^! integrals1 a s]ies ( int]polations & happy inductions :]e "o p]ceives ! g]m ( ! calculus ( def9ite integrals :i* has s mu* ex]cis$ geometricians & :i* is "o ( ! fundam5ts ( my new ,!ory ( ,probabilities1 gave him ! ratio ( ! area ( ! circle to ! squ>e ( its diamet] express$ by an inf9ite product1 :i*1 :5 "o /ops x1 conf9es ?is ratio to limits m & m conv]g+2 ?is is "o ( ! mo/ s+ul> results 9 analysis4 ,b x is rem>kable t ,wallis1 :o _h s well consid]$ ! fractional expon5ts ( radical p[]s1 %\ld h cont9u$ to note ^! p[]s z _h be5 done be=e him4 ,newton 9 8 ,lett]s to ,oldemb\rg1 if ,i am n mi/ak51 0 ! fir/ to employ ! notation ( ^! p[]s by fractional expon5ts4 ,comp>+ by ! way ( induction1 ( :i* ,wallis _h made su* a beautiful use1 ! expon5ts ( ! p[]s ( ! b9omial ) ! coeffici5ts ( ! t]ms ( its developm5t 9 ! case :]e ?is expon5t is integral & positive1 he det]m9$ ! law ( ^! coeffici5ts & ext5d$ x by analogy to fractional & negative p[]s4 ,^! v>i\s results1 bas$ ^u ! notation ( ,desc>tes1 %[ 8 influ5ce on ! progress ( analysis4 ,x has /ill ! advantage ( giv+ ! simple/ & faire/ idea ( log>i?ms1 :i* are inde$ only ! expon5ts ( a magnitude ^: successive p[]s1 increas+ by inf9itely small degrees1 c repres5t all numb]s4 ,b ! mo/ important ext5sion t ?is notation has receiv$ is t ( v>iable expon5ts1 :i* con/itutes expon5tial calculus1 "o ( ! mo/ fruitful bran*es ( mod]n analysis4 ,leibnitz 0 ! fir/ to indicate ! transc5d5ts by v>iable expon5ts1 & !reby he has complet$ ! sy/em ( elem5ts ( :i* a f9ite function c be compos$2 = e f9ite explicit function ( a v>iable may be r$uc$ 9 ! la/ analysis to simple magnitudes1 comb9$ by ! me?od ( addition1 subtraction1 multiplication1 & division & rais$ to con/ant or v>iable p[]s4 ,! roots ( ! equations =m$ f ^! elem5ts are ! implicit functions ( ! v>iable4 ,x is ?us t a v>iable has = a log>i?m ! expon5t ( ! p[] :i* is equal to x 9 ! s]ies ( ! p[]s ( ! numb] ^: hyp]bolic log>i?m is unity1 & ! log>i?m ( a v>iable ( x is an implicit function4 ,leibnitz ?\ght to give to 8 diff]5tial "* ! same expon5ts z to magnitudes2 b !n 9 place ( indicat+ ! repeat$ multiplications ( ! same magnitude ^! expon5ts indicate ! repeat$ diff]5tiations ( ! same function4 ,?is new ext5sion ( ! ,c>tesian notation l$ ,leibnitz to ! analogy ( positive p[]s ) ! diff]5tials1 & ! negative p[]s ) ! integrals4 ,lagrange has foll[$ ?is s+ul> analogy 9 all its developm5ts2 & by s]ies ( inductions :i* may be reg>d$ z "o ( ! mo/ beautiful applications :i* h ev] be5 made ( ! me?od ( induction he has arriv$ at g5]al =mul :i* are z curi\s z useful on ! trans=mations ( diff]5ces & ( integrals ! ones into ! o!rs :5 ! v>iables h div]s f9ite increm5ts & :5 ^! increm5ts are inf9itely small4 ,b he has n giv5 ! demon/rations ( x :i* appe> to him difficult4 ,! !ory ( discrim9ant functions ext5ds ! ,c>tesian notations to "s ( its *>act]s2 x %[s ) pro( ! analogy ( ! p[]s & op]ations indicat$ by ^! *>act]s2 s t x may /ill be reg>d$ z ! expon5tial calculus ( *>act]s4 ,all t conc]ns ! s]ies & ! integration ( equations ( diff]5ces spr+s f x ) an extreme facility4 ,,"p ,,ii4 ,,applications ,,( ,,! ,,calculus ,,( ,,probabilities4 ,,*apt] ,,vi4 ,,games ,,( ,,*ance4 ,! comb9ations :i* games pres5t 7 ! object ( ! fir/ inve/igations ( probabilities4 ,9 an inf9ite v>iety ( ^! comb9ations _m ( !m l5d !mselves readily to calculus2 o!rs require m difficult calculi2 & ! difficulties increas+ 9 ! measure t ! comb9ations become m complicat$1 ! desire to surm\nt !m & curiosity h excit$ geometricians to p]fect m & m ?is k9d ( analysis4 ,x has be5 se5 already t ! b5efits ( a lott]y are easily det]m9$ by ! !ory ( comb9ations4 ,b x is m difficult to "k 9 h[ _m draws "o c bet "o aga9/ "o1 = example t all ! numb]s w be drawn1 n be+ ! numb] ( numb]s1 r t ( ! numb]s drawn at ea* draw1 & i ! unkn[n numb] ( draws4 ,! expression ( ! probability ( draw+ all ! numb]s dep5ds ^u ! n? f9ite diff]5ce ( ! i p[] ( a product ( r consecutive numb]s4 ,:5 ! numb] n is consid]able ! se>* = ! value ( i :i* r5d]s ?is probability equal to becomes impossible at lea/ unless ?is diff]5ce is conv]t$ into a v conv]g+ s]ies4 ,?is is easily done by ! me?od "h bel[ indicat$ by ! approximations ( functions ( v l>ge numb]s4 ,x is f\nd ?us s9ce ! lott]y is compos$ ( t5 ?\s& numb]s1 "o ( :i* is drawn at ea* draw1 t !re is a disadvantage 9 bett+ "o aga9/ "o t all ! numb]s w be drawn 9 #iegfg draws & an advantage 9 mak+ ! same bet = #iegfh draws4 ,9 ! lott]y ( ,france ?is bet is disadvantage\s = #he draws & advantage\s = #hf draws4 ,let u consid] aga9 two play]s1 ,a & ,b1 play+ toge!r at heads & tails 9 su* a mann] t at ea* ?r[ if heads turns up ,a gives "o c\nt] to ,b1 :o gives him "o if tails turns up2 ! numb] ( c\nt]s ( ,b is limit$1 :ile t ( ,a is unlimit$1 & ! game is to end only :5 ,b %all h no m c\nt]s4 ,we ask 9 h[ _m ?r[s "o %\ld bet "o to "o t ! game w end4 ,! expression ( ! probability t ! game w end 9 an i numb] ( ?r[s is giv5 by a s]ies :i* comprises a great numb] ( t]ms & factors if ! numb] ( c\nt]s ( ,b is consid]able2 ! se>* = ! value ( ! unkn[n i :i* r5d]s ?is s]ies w\ld !n be impossible if we did n r$uce ! same to a v conv]g5t s]ies4 ,9 apply+ to x ! me?od ( :i* we h j spok51 we f9d a v simple expression = ! unkn[n f :i* x results t if1 = example1 ,b has a hundr$ c\nt]s1 x is a bet ( a little less ?an "o aga9/ "o t ! game w end 9 #bcghj ?r[s1 & a bet ( a little m ?an "o aga9/ "o t x w end 9 #bcgha ?r[s4 ,^! two examples add$ to ^? we h already giv5 are suffici5t to %[s h[ ! problems ( games h contribut$ to ! p]fection ( analysis4 ,,*apt] ,,vii4 ,,conc]n+ ,,! ,,unkn[n ,,inequalities ,,:i* ,,may ,,exi/ ,,among ,,*ances ,,:i* ,,are ,,suppos$ ,,equal4 ,inequalities ( ?is k9d h ^u ! results ( ! calculation ( probabilities a s5sible influ5ce :i* des]ves p>ticul> att5tion4 ,let u take ! game ( heads & tails1 & let u suppose t x is equally easy to ?r[ ! "o or ! o!r side ( ! co94 ,!n ! probability ( ?r[+ heads at ! fir/ ?r[ is & t ( ?r[+ x twice 9 succession is 4 ,b if !re exi/ 9 ! co9 an inequality :i* causes "o ( ! faces to appe> r ?an ! o!r )\t kn[+ :i* side is favor$ by ?is inequality1 ! probability ( ?r[+ heads at ! fir/ ?r[ w always 2 because ( our ignorance ( :i* face is favor$ by ! inequality ! probability ( ! simple ev5t is increas$ if ?is inequality is favorable to x1 j s mu* is x dim9i%$ if ! inequality is contr>y to x4 ,b 9 ?is same ignorance ! probability ( ?r[+ heads twice 9 succession is increas$4 ,inde$ ?is probability is t ( ?r[+ heads at ! fir/ ?r[ multipli$ by ! probability t hav+ ?r[n x at ! fir/ ?r[ x w be ?r[n at ! second2 b its happ5+ at ! fir/ ?r[ is a reason = belief t ! inequality ( ! co9 favors x2 ! unkn[n inequality increases1 !n1 ! probability ( ?r[+ heads at ! second ?r[2 x consequ5tly increases ! product ( ^! two probabilities4 ,9 ord] to submit ?is matt] to calculus let u suppose t ?is inequality increases by a tw5tie? ! probability ( ! simple ev5t :i* x favors4 ,if ?is ev5t is heads1 its probability w be plus #a_/bj1 or #aa_/bj1 & ! probability ( ?r[+ x twice 9 succession w be ! squ>e ( #aa_/bj1 or #aba_/djj4 ,if ! favor$ ev5t is tails1 ! probability ( heads1 w be m9us #a_/bj1 or #i_/bj1 & ! probability ( ?r[+ x twice 9 succession w be #ha_/djj4 ,s9ce we h at fir/ no reason = believ+ t ! inequality favors "o ( ^! ev5ts r ?an ! o!r1 x is cle> t 9 ord] to h ! probability ( ! comp\nd ev5t heads heads x is necess>y to add ! two prec$+ probabilities & take ! half ( _! sum1 :i* gives #aja_/djj = ?is probability1 :i* exce$s by #a_/djj or by ! squ>e ( ! favor #a_/bj t ! inequality adds to ! possibilities ( ! ev5t :i* x favors4 ,! probability ( ?r[+ tails tails is simil>ly #aja_/djj1 b ! probability ( ?r[+ heads tails or tails heads is ea* #ii_/djj2 = ! sum ( ^! f\r probabilities ought to equal c]ta9ty or unity4 ,we f9d ?us g5]ally t ! con/ant & unkn[n causes :i* favor simple ev5ts :i* are judg$ equally possible always increase ! probability ( ! repetition ( ! same simple ev5t4 ,9 an ev5 numb] ( ?r[s heads & tails ought bo? to happ5 ei!r an ev5 numb] ( times or odd numb] ( times4 ,! probability ( ea* ( ^! cases is if ! possibilities ( ! two faces are equal2 b if !re is betwe5 !m an unkn[n inequality1 ?is inequality is always favorable to ! fir/ case4 ,two play]s ^: skill is suppos$ to be equal play on ! conditions t at ea* ?r[ t "o :o loses gives a c\nt] to 8 adv]s>y1 & t ! game cont9ues until "o ( ! play]s has no m c\nt]s4 ,! calculation ( ! probabilities %[s u t = ! equality ( ! play ! ?r[s ( ! play]s ought to be an inv]se ratio to _! c\nt]s4 ,b if !re is betwe5 ! play]s a small unkn[n inequality1 x favors t "o ( ! play]s :o has ! smalle/ numb] ( c\nt]s4 ,8 probability ( w9n+ ! game increases if ! play]s agree to d\ble or triple _! c\nt]s2 & x w be or ! same z ! probability ( ! o!r play] 9 ! case :]e ! numb] ( _! c\nt]s %\ld become inf9ite1 pres]v+ always ! same ratio4 ,"o may correct ! influ5ce ( ^! unkn[n inequalities by submitt+ !m !mselves to ! *ances ( haz>d4 ,?us at ! play ( heads & tails1 if "o has a second co9 :i* is ?r[n ea* "t ) ! fir/ & "o agrees to "n con/antly heads ! face turn$ up by ! second co91 ! probability ( ?r[+ heads twice 9 succession ) ! fir/ co9 w approa* mu* ne>] ?an 9 ! case ( a s+le co94 ,9 ?is la/ case ! diff]5ce is ! squ>e ( ! small increm5t ( possibility t ! unkn[n inequality gives to ! face ( ! fir/ co9 :i* x favors2 9 ! o!r case ?is diff]5ce is ! quadruple product ( ?is squ>e by ! correspond+ squ>e relative to ! second co94 ,let !re be ?r[n into an urn a hundr$ numb]s f #a to #ajj 9 ! ord] ( num]ation1 & aft] hav+ %ak5 ! urn 9 ord] to mix ! numb]s "o is drawn2 x is cle> t if ! mix+ has be5 well done ! probabilities ( ! draw+ ( ! numb]s w be ! same4 ,b if we fe> t !re is among !m small diff]5ces dep5d5t ^u ! ord] accord+ to :i* ! numb]s h be5 ?r[n into ! urn1 we %all dim9i% consid]ably ^! diff]5ces by ?r[+ into a second urn ! numb]s accord+ to ! ord] ( _! draw+ f ! fir/ urn1 & by %ak+ !n ?is second urn 9 ord] to mix ! numb]s4 ,a ?ird urn1 a f\r? urn1 etc41 w\ld dim9i% m & m ^! diff]5ces already inappreciable 9 ! second urn4 ,,*apt] ,,viii4 ,,conc]n+ ,,! ,,laws ,,( ,,probability ,,:i* ,,result ,,f ,,! ,,indef9ite ,,multiplication ,,( ,,ev5ts4 ,amid ! v>iable & unkn[n causes :i* we compreh5d "u ! "n ( *ance1 & :i* r5d] unc]ta9 & irregul> ! m>* ( ev5ts1 we see appe>+1 9 ! measure t !y multiply1 a /rik+ regul>ity :i* seems to hold to a design & :i* has be5 consid]$ z a pro( ( ,provid5ce4 ,b 9 reflect+ ^u ?is we soon recognize t ?is regul>ity is only ! developm5t ( ! respective possibilities ( simple ev5ts :i* ought to pres5t !mselves m (t5 :5 !y are m probable4 ,let u imag9e1 = example1 an urn :i* conta9s :ite balls & black balls2 & let u suppose t ea* "t a ball is drawn x is put back into ! urn be=e proce$+ to a new draw4 ,! ratio ( ! numb] ( ! :ite balls drawn to ! numb] ( black balls drawn w be mo/ (t5 v irregul> 9 ! fir/ draw+s2 b ! v>iable causes ( ?is irregul>ity produce effects alt]nately favorable & unfavorable to ! regul> m>* ( ev5ts :i* de/roy ea* o!r mutually 9 ! totality ( a great numb] ( draws1 all[+ u to p]ceive m & m ! ratio ( :ite balls to ! black balls conta9$ 9 ! urn1 or ! respective possibilities ( draw+ a :ite ball or black ball at ea* draw4 ,f ?is results ! foll[+ !orem4 ,! probability t ! ratio ( ! numb] ( :ite balls drawn to ! total numb] ( balls drawn does n deviate beyond a giv5 int]val f ! ratio ( ! numb] ( :ite balls to ! total numb] ( balls conta9$ 9 ! urn1 approa*es indef9itely to c]ta9ty by ! indef9ite multiplication ( ev5ts1 h[ev] small ?is int]val4 ,?is !orem indicat$ by common s5se 0 difficult to demon/rate by analysis4 ,accord+ly ! illu/ri\s geometrician ,jacques ,b]n\lli1 :o fir/ has occupi$ himself ) x1 atta*es great importance to ! demon/rations he has giv54 ,! calculus ( discrim9ant functions appli$ to ?is matt] n only demon/rates ) facility ?is !orem1 b /ill m x gives ! probability t ! ratio ( ! ev5ts obs]v$ deviates only 9 c]ta9 limits f ! true ratio ( _! respective possibilities4 ,"o may draw f ! prec$+ !orem ?is consequ5ce :i* ought to be reg>d$ z a g5]al law1 namely1 t ! ratios ( ! acts ( nature are v ne>ly con/ant :5 ^! acts are consid]$ 9 great numb]4 ,?us 9 spite ( ! v>iety ( ye>s ! sum ( ! productions dur+ a consid]able numb] ( ye>s is s5sibly ! same2 s t man by useful =esight is able to provide aga9/ ! irregul>ity ( ! seasons by spread+ out equally ov] all ! seasons ! goods :i* nature di/ributes 9 an unequal mann]4 ,i d n except f ! above law results due to moral causes4 ,! ratio ( annual bir?s to ! population1 & t ( m>riages to bir?s1 %[ only small v>iations2 at ,p>is ! numb] ( annual bir?s is almo/ ! same1 & ,i h he>d x said at ! po/-(fice 9 ord9>y seasons ! numb] ( lett]s ?r[n aside on acc\nt ( defective addresses *anges little ea* ye>2 ?is has likewise be5 obs]v$ at ,london4 ,x foll[s aga9 f ?is !orem t 9 a s]ies ( ev5ts indef9itely prolong$ ! action ( regul> & con/ant causes ought to prevail 9 ! long run ov] t ( irregul> causes4 ,x is ?is :i* r5d]s ! ga9s ( ! lott]ies j z c]ta9 z ! products ( agriculture2 ! *ances :i* !y res]ve assure !m a b5efit 9 ! totality ( a great numb] ( ?r[s4 ,?us favorable & num]\s *ances be+ con/antly atta*$ to ! obs]vation ( ! et]nal pr9ciples ( reason1 ( ju/ice1 & ( humanity :i* e/abli% & ma9ta9 societies1 !re is a great advantage 9 con=m+ to ^! pr9ciples & ( grave inconv5i5ce 9 dep>t+ f !m4 ,if "o consult hi/ories & 8 own exp]i5ce1 "o w see all ! facts come to ! aid ( ?is result ( calculus4 ,consid] ! happy effects ( in/itutions f\nd$ ^u reason & ! natural rights ( man among ! peoples :o h kn[n h[ to e/abli% & pres]ve !m4 ,consid] aga9 ! advantages :i* good fai? has procur$ = ! gov]nm5ts :o h made x ! basis ( _! conduct & h[ !y h be5 indemnifi$ = ! sacrifices :i* a scrupul\s exactitude 9 keep+ _! engagem5ts has co/ !m4 ,:at imm5se cr$it at home6 ,:at prepond]ance abroad6 ,on ! contr>y1 look into :at an abyss ( mis=tunes nations h (t5 be5 precipitat$ by ! ambition & ! p]fidy ( _! *iefs4 ,e "t t a great p[] intoxicat$ by ! love ( conque/ aspires to univ]sal dom9ation ! s5tim5t ( indep5d5ce produces among ! m5ac$ nations a coalition ( :i* x becomes almo/ always ! victim4 ,simil>ly 9 ! mid/ ( ! v>iable causes :i* ext5d or re/ra9 ! div]s /ates1 ! natural limits act+ z con/ant causes ought to end by prevail+4 ,x is important !n to ! /ability z well z to ! happ9ess ( empires n to ext5d !m beyond ^? limits into :i* !y are l$ aga9 )\t cessation by ! action ( ! causes2 j z ! wat]s ( ! seas rais$ by viol5t tempe/s fall aga9 into _! bas9s by ! =ce ( gravity4 ,x is aga9 a result ( ! calculus ( probabilities confirm$ by num]\s & melan*oly exp]i5ces4 ,hi/ory treat$ f ! po9t ( view ( ! influ5ce ( con/ant causes w\ld unite to ! int]e/ ( curiosity t ( (f]+ to man mo/ useful lessons4 ,sometimes we attribute ! inevitable results ( ^! causes to ! accid5tal circum/ances :i* h produc$ _! action4 ,x is1 = example1 aga9/ ! nature ( ?+s t "o p %\ld ev] be gov]n$ by ano!r :5 a va/ sea or a great di/ance sep>ates !m4 ,x may be affirm$ t 9 ! long run ?is con/ant cause1 jo9+ itself )\t ceas+ to ! v>iable causes :i* act 9 ! same way & :i* ! c\rse ( "t develops1 w end by f9d+ !m suffici5tly /rong to give to a subjugat$ p its natural indep5d5ce or to unite x to a p[]ful /ate :i* may be contigu\s4 ,9 a great numb] ( cases1 & ^! are ! mo/ important ( ! analysis ( haz>ds1 ! possibilities ( simple ev5ts are unkn[n & we are =c$ to se>* 9 pa/ ev5ts = ! indices :i* c guide u 9 our conjectures ab\t ! causes ^u :i* !y dep5d4 ,9 apply+ ! analysis ( discrim9ant functions to ! pr9ciple elucidat$ above on ! probability ( ! causes drawn f ! ev5ts obs]v$1 we are l$ to ! foll[+ !orem4 ,:5 a simple ev5t or "o compos$ ( sev]al simple ev5ts1 z1 = in/ance1 9 a game1 has be5 repeat$ a great numb] ( times ! possibilities ( ! simple ev5ts :i* r5d] mo/ probable t :i* has be5 obs]v$ are ^? t obs]vation indicates ) ! greate/ probability2 9 ! measure t ! obs]v$ ev5t is repeat$ ?is probability increases & w\ld end by am\nt+ to c]ta9ty if ! numb]s ( repetitions %\ld become inf9ite4 ,!re are two k9ds ( approximations3 ! "o is relative to ! limits tak5 on all sides ( ! possibilities :i* give to ! pa/ ! greate/ probability2 ! o!r approximation is relat$ to ! probability t ^! possibilities fall )9 ^! limits4 ,! repetition ( ! comp\nd ev5t increases m & m ?is probability1 ! limits rema9+ ! same2 x r$uces m & m ! int]val ( ^! limits1 ! probability rema9+ ! same2 9 inf9ity ?is int]val becomes z]o & ! probability *anges to c]ta9ty4 ,if we apply ?is !orem to ! ratio ( ! bir?s ( boys to t ( girls obs]v$ 9 ! diff]5t c\ntries ( ,europe1 we f9d t ?is ratio1 :i* is ev]y:]e ab\t equal to t ( #bb to #ba1 indicates ) an extreme probability a great] facility 9 ! bir? ( boys4 ,consid]+ fur!r t x is ! same at ,naples & at ,/4 ,pet]sburg1 we %all see t 9 ?is reg>d ! influ5ce ( climate is )\t effect4 ,we might !n suspect1 contr>y to ! common belief1 t ?is pr$om9ance ( mascul9e bir?s exi/s ev5 9 ! ,ori5t4 ,i h consequ5tly invit$ ! ,fr5* s*ol>s s5t to ,egypt to occupy !mselves ) ?is int]e/+ "q2 b ! difficulty 9 obta9+ exact in=mation ab\t ! bir?s has n p]mitt$ !m to solve x4 ,happily1 ,m4 de ,humboldt has n neglect$ ?is matt] among ! innum]able new ?+s :i* he has obs]v$ & collect$ 9 ,am]ica ) s mu* sagacity1 con/ancy1 & c\rage4 ,he has f\nd 9 ! tropics ! same ratio ( ! bir?s z we obs]ve 9 ,p>is2 ?is ought to make u reg>d ! great] numb] ( mascul9e bir?s z a g5]al law ( ! human race4 ,! laws :i* ! diff]5t k9ds ( animals foll[ 9 ?is reg>d seem to me wor?y ( ! att5tion ( naturali/s4 ,! fact t ! ratio ( bir?s ( boys to t ( girls diff]s v little f unity ev5 9 ! great numb] ( ! bir?s obs]v$ 9 a place w\ld (f] 9 ?is reg>d a result contr>y to ! g5]al law1 )\t :i* we %\ld be "r 9 conclud+ t ?is law did n exi/4 ,9 ord] to arrive at ?is result x is necess>y to employ great numb]s & to be sure t x is indicat$ by great probability4 ,buffon cites1 = example1 9 8 ,political ,ari?metic sev]al communities ( ,b\rgogne :]e ! bir?s ( girls h surpass$ ^? ( boys4 ,among ^! communities t ( ,c>celle-le-,grignon pres5ts 9 #bjji bir?s dur+ five ye>s #ajbf girls & #ihc boys4 ,al?\gh ^! numb]s are consid]able1 !y indicate1 h[ev]1 only a great] possibility 9 ! bir?s ( girls ) a probability ( #i_/aj1 & ?is probability1 small] ?an t ( n ?r[+ heads f\r times 9 succession 9 ! game ( heads & tails1 is n suffici5t to inve/igate ! cause = ?is anomaly1 :i*1 accord+ to all probability1 w\ld disappe> if "o %\ld foll[ dur+ a c5tury ! bir?s 9 ?is community4 ,! regi/]s ( bir?s1 :i* are kept ) c>e 9 ord] to assure ! condition ( ! citiz5s1 may s]ve 9 det]m9+ ! population ( a great empire )\t recurr+ to ! enum]ation ( its inhabitants--a labori\s op]ation & "o difficult to make ) exactitude4 ,b = ?is x is necess>y to "k ! ratio ( ! population to ! annual bir?s4 ,! mo/ precise means ( obta9+ x consi/s1 fir/1 9 *oos+ 9 ! empire di/ricts di/ribut$ 9 an almo/ equal mann] ov] its :ole surface1 s z to r5d] ! g5]al result indep5d5t ( local circum/ances2 second1 9 enum]at+ ) c>e = a giv5 epo* ! inhabitants ( sev]al communities 9 ea* ( ^! di/ricts2 ?ird1 by det]m9+ f ! /atem5t ( ! bir?s dur+ sev]al ye>s :i* prec$e & foll[ ?is epo* ! mean numb] correspond+ to ! annual bir?s4 ,?is numb]1 divid$ by t ( ! inhabitants1 w give ! ratio ( ! annual bir?s to ! population 9 a mann] m & m accurate z ! enum]ation becomes m consid]able4 ,! gov]nm5t1 conv9c$ ( ! utility ( a simil> enum]ation1 has decid$ at my reque/ to ord] its execution4 ,9 ?irty di/ricts spread out equally ov] ! :ole ( ,france1 communities h be5 *os5 :i* w\ld be able to furni% ! mo/ exact in=mation4 ,_! enum]ations h giv5 #bjcgfae individuals z ! total numb] ( _! inhabitants on ! #bc;d ( ,septemb]1 #ahjb4 ,! /atem5t ( ! bir?s 9 ^! communities dur+ ! ye>s #ahjj1 #ahja1 & #ahjb h giv53 ,bir?s4 ,m>riages4 ,dea?s4 #aajcab boys #dfjcg #ajcfei m5 #ajebhg girls #iiddc wom5 ,! ratio ( ! population to annual bir?s is !n #bh 7#cebhde_/ajjjjjj72 x is great] ?an _h be5 e/imat$ up to ?is "t4 ,multiply+ ! numb] ( annual bir?s 9 ,france by ?is ratio1 we %all h ! population ( ?is k+dom4 ,b :at is ! probability t ! population ?us det]m9$ w n deviate f ! true population beyond a giv5 limit8 ,resolv+ ?is problem & apply+ to its solution ! prec$+ data1 ,i h f\nd t1 ! numb] ( annual bir?s 9 ,france be+ suppos$ to be #ajjjjjj1 :i* br+s ! population to #bhcebhde inhabitants1 x is a bet ( almo/ #cjjjjj aga9/ #a t ! error ( ?is result is n half a million4 ,! ratio ( ! bir?s ( boys to t ( girls :i* ! prec$+ /atem5t (f]s is t ( #bb to #ba2 & ! m>riages are to ! bir?s z #c is to #d4 ,at ,p>is ! baptisms ( *ildr5 ( bo? sexes v>y a little f ! ratio ( #bb to #ba4 ,s9ce #agde1 ! epo* 9 :i* "o has comm5c$ to di/+ui% ! sexes ^u ! bir?-regi/]s1 up to ! end ( #aghd1 !re h be5 baptiz$ 9 ?is capital #cicchf boys & #cggeee girls4 ,! ratio ( ! two numb]s is almo/ t ( #be to #bd2 x appe>s !n at ,p>is t a p>ticul> cause approximates an equality ( baptisms ( ! two sexes4 ,if we apply to ?is matt] ! calculus ( probabilities1 we f9d t x is a bet ( #bch to #a 9 favor ( ! exi/5ce ( ?is cause1 :i* is suffici5t to au?orize ! inve/igation4 ,^u reflection x has appe>$ to me t ! diff]5ce obs]v$ holds to ?is1 t ! p>5ts 9 ! c\ntry & ! prov9ces1 f9d+ "s advantage 9 keep+ ! boys at home1 h s5t to ! ,hospital = ,f\ndl+s 9 ,p>is few] ( !m relative to ! numb] ( girls accord+ to ! ratio ( bir?s ( ! two sexes4 ,?is is prov$ by ! /atem5t ( ! regi/]s ( ?is hospital4 ,f ! beg9n+ ( #agde to ! end ( #ahji !re 7 ent]$ #afcdii boys & #aeidje girls4 ,! fir/ ( ^! numb]s exce$s only by #a_/ch ! second1 :i* x ought to h surpass$ at lea/ by #a_/bd4 ,?is confirms ! exi/5ce ( ! assign$ cause1 namely1 t ! ratio ( bir?s ( boys to ^? ( girls is at ,p>is t ( #bb to #ba1 no att5tion hav+ be5 paid to f\ndl+s4 ,! prec$+ results suppose t we may comp>e ! bir?s to ! draw+s ( balls f an urn :i* conta9s an inf9ite numb] ( :ite balls & black balls s mix$ t at ea* draw ! *ances ( draw+ ought to be ! same = ea* ball2 b x is possible t ! v>iations ( ! same seasons 9 diff]5t ye>s may h "s influ5ce ^u ! annual ratio ( ! bir?s ( boys to ^? ( girls4 ,! ,bureau ( ,longitudes ( ,france publi%es ea* ye> 9 its annual ! tables ( ! annual movem5t ( ! population ( ! k+dom4 ,! tables already publi%$ comm5ce 9 #ahag2 9 t ye> & 9 ! five foll[+ ye>s !re 7 born #bifbcfa boys & #bghaiig girls1 :i* gives ab\t #af_/ae = ! ratio ( ! bir?s ( boys to t ( girls4 ,! ratios ( ea* ye> v>y little f ?is mean result2 ! smalle/ ratio is t ( #ahbb1 :]e x 0 only #ag_/af2 ! greate/ is ( ! ye> #ahag1 :5 x 0 #ae_/ad4 ,^! ratios v>y appreciably f ! ratio ( #bb_/ba f\nd above4 ,apply+ to ?is deviation ! analysis ( probabilities 9 ! hypo!sis ( ! comp>ison ( bir?s to ! draw+s ( balls f an urn1 we f9d t x w\ld be sc>cely probable4 ,x appe>s1 !n1 to indicate t ?is hypo!sis1 al?\gh closely approximat$1 is n rigor\sly exact4 ,9 ! numb] ( bir?s :i* we h j /at$ !re are ( natural *ildr5 #bjjdid boys & #aijfih girls4 ,! ratio ( mascul9e & fem99e bir?s 0 !n 9 ?is reg>d #bj_/ai1 small] ?an ! mean ratio ( #af_/ae4 ,?is result is 9 ! same s5se z t ( ! bir?s ( f\ndl+s2 & x seems to prove t 9 ! class ( natural *ildr5 ! bir?s ( ! two sexes approa* m ne>ly equality ?an 9 ! class ( legitimate *ildr54 ,! diff]5ce ( ! climates f ! nor? to ! s\? ( ,france does n appe> to influ5ce appreciably ! ratio ( ! bir?s ( boys & girls4 ,! ?irty mo/ s\!rn di/ricts h giv5 #af_/ae = ?is ratio1 ! same z t ( entire ,france4 ,! con/ancy ( ! sup]iority ( ! bir?s ( boys ov] girls at ,p>is & at ,london s9ce !y h be5 obs]v$ has appe>$ to "s s*ol>s to be a pro( ( ,provid5ce1 )\t :i* !y h ?\ght t ! irregul> causes :i* di/urb )\t ceas+ ! c\rse ( ev5ts ought sev]al times to h r5d]$ ! annual bir?s ( girls sup]ior to ^? ( boys4 ,b ?is pro( is a new example ( ! abuse :i* has be5 s (t5 made ( f9al causes :i* always disappe> on a se>*+ exam9ation ( ! que/ions :5 we h ! necess>y data to solve !m4 ,! con/ancy 9 "q is a result ( regul> causes :i* give ! sup]iority to ! bir?s ( boys & :i* ext5d x to ! anomalies due to haz>d :5 ! numb] ( annual bir?s is consid]able4 ,! inve/igation ( ! probability t ?is con/ancy w ma9ta9 itself = a long "t belongs to t bran* ( ! analysis ( haz>ds :i* passes f pa/ ev5ts to ! probability ( future ev5ts2 & tak+ z a basis ! bir?s obs]v$ f #agde to #aghd1 x is a bet ( almo/ #d aga9/ #a t at ,p>is ! annual bir?s ( boys w con/antly surpass = a c5tury ! bir?s ( girls2 !re is !n no reason to be a/oni%$ t ?is has tak5 place = a half-c5tury4 ,let u take ano!r example ( ! developm5t ( con/ant ratios :i* ev5ts pres5t 9 ! measure t !y are multipli$4 ,let u imag9e a s]ies ( urns arrang$ circul>ly1 & ea* conta9+ a v great numb] ( :ite balls & black balls2 ! ratio ( :ite balls to ! black 9 ! urns be+ orig9ally v diff]5t & su*1 = example1 t "o ( ^! urns conta9s only :ite balls1 :ile ano!r conta9s only black balls4 ,if "o draws a ball f ! fir/ urn 9 ord] to put x into ! second1 &1 aft] hav+ %ak5 ! second urn 9 ord] to mix well ! new ball ) ! o!rs1 "o draws a ball to put x into ! ?ird urn1 & s on to ! la/ urn1 f :i* is drawn a ball to put into ! fir/1 & if ?is s]ies is recomm5c$ cont9ually1 ! analysis ( probability %[s u t ! ratios ( ! :ite balls to ! black 9 ^! urns w end by be+ ! same & equal to ! ratio ( ! sum ( all ! :ite balls to ! sum ( all ! black balls conta9$ 9 ! urns4 ,?us by ?is regul> mode ( *ange ! primitive irregul>ity ( ^! ratios disappe>s ev5tually 9 ord] to make room = ! mo/ simple ord]4 ,n[ if among ^! urns "o int]calate new ones 9 :i* ! ratio ( ! sum ( ! :ite balls to ! sum ( ! black balls :i* !y conta9 diff]s f ! prec$+1 cont9u+ indef9itely 9 ! totality ( ! urns ! draw+s :i* we h j indicat$1 ! simple ord] e/abli%$ 9 ! old urns w be at fir/ di/urb$1 & ! ratios ( ! :ite balls to ! black balls w become irregul>2 b little by little ?is irregul>ity w disappe> 9 ord] to make room = a new ord]1 :i* w f9ally be t ( ! equality ( ! ratios ( ! :ite balls to ! black balls conta9$ 9 ! urns4 ,we may apply ^! results to all ! comb9ations ( nature 9 :i* ! con/ant =ces by :i* _! elem5ts are animat$ e/abli% regul> modes ( action1 suit$ to br+ ab\t 9 ! v he>t ( *aos sy/ems gov]n$ by admirable laws4 ,! ph5om5a :i* seem ! mo/ dep5d5t ^u haz>d pres5t1 !n1 :5 multipli$ a t5d5cy to approa* )\t ceas+ fix$ ratios1 9 su* a mann] t if we conceive on all sides ( ea* ( ^! ratios an int]val z small z desir$1 ! probability t ! mean result ( ! obs]vations falls )9 ?is int]val w end by diff]+ f c]ta9ty only by a quantity great] ?an an assignable magnitude4 ,?us by ! calculations ( probabilities appli$ to a great numb] ( obs]vations we may recognize ! exi/5ce ( ^! ratios4 ,b be=e seek+ ! causes x is necess>y1 9 ord] n to be l$ into va9 speculations1 to assure ourselves t !y are indicat$ by a probability :i* does n p]mit u to reg>d !m z anomalies due to haz>d4 ,! !ory ( discrim9ant functions gives a v simple expression = ?is probability1 :i* is obta9$ by integrat+ ! product ( ! diff]5tial ( ! quantity ( :i* ! result d$uc$ f a great numb] ( obs]vations v>ies f ! tru? by a con/ant less ?an unity1 dep5d5t ^u ! nature ( ! problem1 & rais$ to a p[] ^: expon5t is ! ratio ( ! squ>e ( ?is v>iation to ! numb] ( obs]vations4 ,! integral tak5 betwe5 ! limits giv5 & divid$ by ! same integral1 appli$ to a positive & negative inf9ity1 w express ! probability t ! v>iation f ! tru? is compris$ betwe5 ^! limits4 ,su* is ! g5]al law ( ! probability ( results indicat$ by a great numb] ( obs]vations4 ,,*apt] ,,ix4 ,,! ,,application ,,( ,,! ,,calculus ,,( ,,probabilities ,,to ,,natural ,,philosophy4 ,! ph5om5a ( nature are mo/ (t5 envelop$ by s _m /range circum/ances1 & s great a numb] ( di/urb+ causes mix _! influ5ce1 t x is v difficult to recognize !m4 ,we may arrive at !m only by multiply+ ! obs]vations or ! exp]i5ces1 s t ! /range effects f9ally de/roy reciprocally ea* o!r1 ! mean results putt+ 9 evid5ce ^? ph5om5a & _! div]s elem5ts4 ,! m num]\s ! numb] ( obs]vations & ! less !y v>y among !mselves ! m _! results approa* ! tru?4 ,we fulfil ?is la/ condition by ! *oice ( ! me?ods ( obs]vations1 by ! precision ( ! in/rum5ts1 & by ! c>e :i* we take to obs]ve closely2 !n we det]m9e by ! !ory ( probabilities ! mo/ advantage\s mean results or ^? :i* give ! lea/ value ( ! error4 ,b t is n suffici5t2 x is fur!r necess>y to appreciate ! probability t ! errors ( ^! results are compris$ 9 ! giv5 limits2 & )\t ?is we h only an imp]fect k ( ! degree ( exactitude obta9$4 ,=mul suitable to ^! matt]s are !n true improvem5ts ( ! me?od ( sci5ces1 & x is inde$ important to add !m to ?is me?od4 ,! analysis :i* !y require is ! mo/ delicate & ! mo/ difficult ( ! !ory ( probabilities2 x is "o ( ! pr9cipal objects ( ! "w :i* ,i h publi%$ ^u ?is !ory1 & 9 :i* ,i h arriv$ at =mul ( ?is k9d :i* h ! rem>kable advantage ( be+ indep5d5t ( ! law ( ! probability ( errors & ( includ+ only ! quantities giv5 by ! obs]vations !mselves & _! expressions4 ,ea* obs]vation has = an analytic expression a function ( ! elem5ts :i* we wi% to det]m9e2 & if ^! elem5ts are ne>ly kn[n1 ?is function becomes a l9e> function ( _! corrections4 ,9 equat+ x to ! obs]vation itself !re is =m$ an equation ( condition4 ,if we h a great numb] ( simil> equations1 we comb9e !m 9 su* a mann] z to obta9 z _m f9al equations z !re are elem5ts ^: corrections we det]m9e !n by resolv+ ^! equations4 ,b :at is ! mo/ advantage\s mann] ( comb9+ equations ( condition 9 ord] to obta9 f9al equations8 ,:at is ! law ( ! probabilities ( errors ( :i* ! elem5ts are /ill susceptible t we draw f !m8 ,?is is made cle> to u by ! !ory ( probabilities4 ,! =mation ( a f9al equation by means ( ! equation ( condition am\nts to multiply+ ea* "o ( ^! by an indet]m9ate factor & by unit+ ! products2 x is necess>y to *oose ! sy/em ( factors :i* gives ! smalle/ opportunity = error4 ,b x is app>5t t if we multiply ! possible errors ( an elem5t by _! respective probabilities1 ! mo/ advantage\s sy/em w be t 9 :i* ! sum ( ^! products all1 tak51 positively is a m9imum2 = a positive or a negative error ought to be consid]$ z a loss4 ,=m+1 !n1 ?is sum ( products1 ! condition ( ! m9imum w det]m9e ! sy/em ( factors :i* x is exp$i5t to adopt1 or ! mo/ advantage\s sy/em4 ,we f9d ?us t ?is sy/em is t ( ! coeffici5ts ( ! elem5ts 9 ea* equation ( condition2 s t we =m a fir/ f9al equation by multiply+ respectively ea* equation ( condition by its coeffici5t ( ! fir/ elem5t & by unit+ all ^! equations ?us multipli$4 ,we =m a second f9al equation by employ+ 9 ! same mann] ! coeffici5ts ( ! second elem5t1 & s on4 ,9 ?is mann] ! elem5ts & ! laws ( ! ph5om5a obta9$ 9 ! collection ( a great numb] ( obs]vations are develop$ ) ! mo/ evid5ce4 ,! probability ( ! errors :i* ea* elem5t /ill leaves to be fe>$ is proportional to ! numb] ^: hyp]bolic log>i?m is unity rais$ to a p[] equal to ! squ>e ( ! error tak5 z a m9us quantity & multipli$ by a con/ant coeffici5t :i* may be consid]$ z ! modulus ( ! probability ( ! errors2 because1 ! error rema9+ ! same1 its probability decreases ) rapidity :5 ! =m] increases2 s t ! elem5t obta9$ weighs1 if ,i may ?us speak t[>d ! tru?1 z mu* m z ?is modulus is great]4 ,i w\ld call = ?is reason ?is modulus ! weight ( ! elem5t or ( ! result4 ,?is weight is ! greate/ possible 9 ! sy/em ( factors--! mo/ advantage\s2 x is ?is :i* gives to ?is sy/em sup]iority ov] o!rs4 ,by a rem>kable analogy ( ?is weight ) ^? ( bodies comp>$ at _! common c5tre ( gravity x results t if ! same elem5t is giv5 by div]s sy/ems1 compos$ ea* ( a great numb] ( obs]vations1 ! mo/ advantage\s1 ! mean result ( _! totality is ! sum ( ! products ( ea* p>tial result by its weight4 ,moreov]1 ! total weight ( ! results ( ! div]s sy/ems is ! sum ( _! p>tial weights2 s t ! probability ( ! errors ( ! mean result ( _! totality is proportional to ! numb] :i* has unity = an hyp]bolic log>i?m rais$ to a p[] equal to ! squ>e ( ! error tak5 z m9us & multipli$ by ! sum ( ! weights4 ,ea* weight dep5ds 9 tru? ^u ! law ( ! probability ( error ( ea* sy/em1 & almo/ always ?is law is unkn[n2 b happily ,i h be5 able to elim9ate ! factor :i* conta9s x by means ( ! sum ( ! squ>es ( ! v>iations ( ! obs]vations 9 ?is sy/em f _! mean result4 ,x w\ld !n be desirable 9 ord] to complete our k ( ! results obta9$ by ! totality ( a great numb] ( obs]vations t we write by ! side ( ea* result ! weight :i* corresponds to x2 analysis furni%es = ?is object bo? g5]al & simple me?ods4 ,:5 we h ?us obta9$ ! expon5tial :i* repres5ts ! law ( ! probability ( errors1 we %all h ! probability t ! error ( ! result is includ$ )9 giv5 limits by tak+ )9 ! limits ! integral ( ! product ( ?is expon5tial by ! diff]5tial ( ! error & multiply+ x by ! squ>e root ( ! weight ( ! result divid$ by ! circumf]5ce ^: diamet] is unity4 ,h5ce x foll[s t = ! same probability ! errors ( ! results are reciprocal to ! squ>e roots ( _! weights1 :i* s]ves to comp>e _! respective precision4 ,9 ord] to apply ?is me?od ) success x is necess>y to v>y ! circum/ances ( ! obs]vations or ! exp]i5ces 9 su* a mann] z to avoid ! con/ant causes ( error4 ,x is necess>y t ! obs]vations %\ld be num]\s1 & t !y %\ld be s mu* ! m s z !re are m elem5ts to det]m9e2 = ! weight ( ! mean result increases z ! numb] ( obs]vations divid$ by ! numb] ( ! elem5ts4 ,x is /ill necess>y t ! elem5ts foll[ 9 ^! obs]vations a diff]5t c\rse2 = if ! c\rse ( ! two elem5ts 7 exactly ! same1 :i* w\ld r5d] _! coeffici5ts proportional 9 equation ( conditions1 ^! elem5ts w\ld =m only a s+le unkn[n quantity & x w\ld be impossible to di/+ui% !m by ^! obs]vations4 ,f9ally x is necess>y t ! obs]vations %\ld be precise2 ?is condition1 ! fir/ ( all1 increases greatly ! weight ( ! result ! expression ( :i* has = a divisor ! sum ( ! squ>es ( ! deviations ( ! obs]vations f ?is result4 ,) ^! precautions we %all be able to make use ( ! prec$+ me?od & measure ! degree ( confid5ce :i* ! results d$uc$ f a great numb] ( obs]vations m]it4 ,! rule :i* we h j giv5 to conclude equations ( condition1 f9al equations1 am\nt to r5d]+ a m9imum ! sum ( ! squ>es ( ! errors ( obs]vations2 = ea* equation ( condition becomes exact by sub/itut+ 9 x ! obs]vation plus its error2 & if we draw f x ! expression ( ?is error1 x is easy to see t ! condition ( ! m9imum ( ! sum ( ! squ>es ( ^! expressions gives ! rule 9 "q4 ,?is rule is ! m precise z ! obs]vations are m num]\s2 b ev5 9 ! case :]e _! numb] is small x appe>s natural to employ ! same rule :i* 9 all cases (f]s a simple means ( obta9+ )\t grop+ ! corrections :i* we seek to det]m9e4 ,x s]ves fur!r to comp>e ! precision ( ! div]s a/ronomical tables ( ! same />4 ,^! tables may always be suppos$ z r$uc$ to ! same =m1 & !n !y diff] only by ! epo*s1 ! mean movem5ts & ! coeffici5ts ( ! argum5ts2 = if "o ( !m conta9s a coeffici5t :i* is n f\nd 9 ! o!rs1 x is cle> t ?is am\nts to suppos+ z]o 9 !m z ! coeffici5t ( ?is argum5t4 ,if n[ we rectify ^! tables by ! totality ( ! good obs]vations1 !y w\ld satisfy ! condition t ! sum ( ! squ>es ( ! errors %\ld be a m9imum2 ! tables :i*1 comp>$ to a consid]able numb] ( obs]vations1 approa* ne>e/ ?is condition m]it !n ! pref]5ce4 ,x is pr9cipally 9 a/ronomy t ! me?od expla9$ above may be employ$ ) advantage4 ,! a/ronomical tables owe ! truly a/oni%+ exactitude :i* !y h atta9$ to ! precision ( obs]vations & ( !ories1 & to ! use ( equations ( conditions :i* cause to concur a great numb] ( excell5t obs]vations 9 ! correction ( ! same elem5t4 ,b x rema9s to det]m9e ! probability ( ! errors t ?is correction leaves /ill to be fe>$2 & ! me?od :i* ,i h j expla9$ enables u to recognize ! probability ( ^! errors4 ,9 ord] to give "s int]e/+ applications ( x ,i h pr(it$ by ! imm5se "w :i* ,m4 ,b\v>d has j f9i%$ on ! movem5ts ( ,jupit] & ,saturn1 ( :i* he has =m$ v precise tables4 ,he has discuss$ ) ! greate/ c>e ! oppositions & quadratures ( ^! two planets obs]v$ by ,bradley & by ! a/ronom]s :o h foll[$ him d[n to ! la/ ye>s2 he has conclud$ ! corrections ( ! elem5ts ( _! movem5t & _! masses comp>$ to t ( ! sun tak5 z unity4 ,8 calculations give him ! mass ( ,saturn equal to ! #ceab? "p ( t ( ! sun4 ,apply+ to !m my =mul ( probability1 ,i f9d t x is a bet ( #aa1jjj aga9/ "o t ! error ( ?is result is n #a_/ajj ( its value1 or t :i* am\nts to almo/ ! same--t aft] a c5tury ( new obs]vations add$ to ! prec$+ ones1 & exam9$ 9 ! same mann]1 ! new result w n diff] by #a_/ajj f t ( ,m4 ,b\v>d4 ,?is wise a/ronom] f9ds aga9 ! mass ( ,jupit] equal to ! #ajga? "p ( ! sun2 & my me?od ( probability gives a bet ( #a1jjj1jjj to "o t ?is result is n #a_/ajj 9 error4 ,?is me?od may be employ$ aga9 ) success 9 geodetic op]ations4 ,we det]m9e ! l5g? ( ! great arc on ! surface ( ! e>? by triangulation1 :i* dep5ds ^u a base measur$ ) exactitude4 ,b :atev] precision may be br\ght to ! measure ( ! angles1 ! inevitable errors c1 by accumulat+1 cause ! value ( ! arc conclud$ f a great numb] ( triangles to deviate appreciably f ! tru?4 ,we recognize ?is value1 !n1 only imp]fectly unless ! probability t its error is compris$ )9 giv5 limits c be assign$4 ,! error ( a geodetic result is a function ( ! errors ( ! angles ( ea* triangle4 ,i h giv5 9 ! "w cit$ g5]al =mul 9 ord] to obta9 ! probability ( ! values ( "o or ( sev]al l9e> functions ( a great numb] ( p>tial errors ( :i* we "k ! law ( probability2 we may !n by means ( ^! =mul det]m9e ! probability t ! error ( a geodetic result is conta9$ )9 ! assign$ limits1 :atev] may be ! law ( ! probability ( p>tial errors4 ,x is moreov] m necess>y to r5d] ourselves indep5d5t ( ! law1 s9ce ! mo/ simple laws !mselves are always inf9itely less probable1 see+ ! inf9ite numb] ( ^? :i* may exi/ 9 nature4 ,b ! unkn[n law ( p>tial errors introduces into ! =mul an indet]m9ant :i* does n p]mit ( r$uc+ !m to numb]s unless we are able to elim9ate x4 ,we h se5 t 9 a/ronomical que/ions1 :]e ea* obs]vation furni%es an equation ( condition = obta9+ ! elem5ts1 we elim9ate ?is det]m9ant by means ( ! sum ( ! squ>es ( ! rema9d]s :5 ! mo/ probable values ( ! elem5ts h be5 sub/itut$ 9 ea* equation4 ,geodetic que/ions n (f]+ simil> equations1 x is necess>y to seek ano!r means ( elim9ation4 ,! quantity by :i* ! sum ( ! angles ( ea* obs]v$ triangle surpasses two "r angles plus ! sph]ical excess furni%es ?is means4 ,?us we replace by ! sum ( ! squ>es ( ^! quantities ! sum ( ! squ>es ( ! rema9d]s ( ! equations ( condition2 & we may assign 9 numb]s ! probability t ! error ( ! f9al result ( a s]ies ( geodetic op]ations w n exce$ a giv5 quantity4 ,b :at is ! mo/ advantage\s mann] ( divid+ among ! ?ree angles ( ea* triangle ! obs]v$ sum ( _! errors8 ,! analysis ( probabilities r5d]s x app>5t t ea* angle ought to be dim9i%$ by a ?ird ( ?is sum1 provid$ t ! weight ( a geodetic result be ! greate/ possible1 :i* r5d]s ! same error less probable4 ,!re is !n a great advantage 9 obs]v+ ! ?ree angles ( ea* triangle & ( correct+ !m z we h j said4 ,simple common s5se indicates ?is advantage2 b ! calculation ( probabilities alone is able to appreciate x & to r5d] app>5t t by ?is correction x becomes ! greate/ possible4 ,9 ord] to assure oneself ( ! exactitude ( ! value ( a great arc :i* re/s ^u a base measur$ at "o ( its extremities "o measures a second base t[>d ! o!r extremity2 & "o concludes f "o ( ^! bases ! l5g? ( ! o!r4 ,if ?is l5g? v>ies v little f ! obs]vation1 !re is all reason to believe t ! *a9 ( triangles :i* unites ^! bases is v ne>ly exact & likewise ! value ( ! l>ge arc :i* results f x4 ,"o corrects1 !n1 ?is value by modify+ ! angles ( ! triangles 9 su* a mann] t ! base is calculat$ accord+ to ! bases measur$4 ,b ?is may be done 9 an inf9ity ( ways1 among :i* is pref]r$ t ( :i* ! geodetic result has ! greate/ weight1 inasmu* z ! same error becomes less probable4 ,! analysis ( probabilities gives =mul = obta9+ directly ! mo/ advantage\s correction :i* results f ! measurem5ts ( ! sev]al bases & ! laws ( probability :i* ! multiplicity ( ! bases makes--laws :i* become v rapidly decreas+ by ?is multiplicity4 ,g5]ally ! errors ( ! results d$uc$ f a great numb] ( obs]vations are ! l9e> functions ( ! p>tial errors ( ea* obs]vation4 ,! coeffici5ts ( ^! functions dep5d ^u ! nature ( ! problem & ^u ! process foll[$ 9 ord] to obta9 ! results4 ,! mo/ advantage\s process is evid5tly t 9 :i* ! same error 9 ! results is less probable ?an accord+ to any o!r process4 ,! application ( ! calculus ( probabilities to natural philosophy consi/s1 !n1 9 det]m9+ analytically ! probability ( ! values ( ^! functions & 9 *oos+ _! indet]m9ant coeffici5ts 9 su* a mann] t ! law ( ?is probability %\ld be mo/ rapidly desc5d+4 ,elim9at+1 !n1 f ! =mul by ! data ( ! "q ! factor :i* is introduc$ by ! almo/ always unkn[n law ( ! probability ( p>tial errors1 we may be able to evaluate num]ically ! probability t ! errors ( ! results d n exce$ a giv5 quantity4 ,we %all ?us h all t may be desir$ t\*+ ! results d$uc$ f a great numb] ( obs]vations4 ,v approximate results may be obta9$ by o!r consid]ations4 ,suppose1 = example1 t "o has a ?\s& & "o obs]vations ( ! same quantity2 ! ari?metical mean ( all ^! obs]vations is ! result giv5 by ! mo/ advantage\s me?od4 ,b "o w\ld be able to *oose ! result accord+ to ! condition t ! sum ( ! v>iations f ea* p>tial value all tak5 positively %\ld be a m9imum4 ,x appe>s inde$ natural to reg>d z v approximate ! result :i* satisfies ?is condition4 ,x is easy to see t if "o disposes ! values giv5 by ! obs]vations accord+ to ! ord] ( magnitude1 ! value :i* w occupy ! mean w fulfil ! prec$+ condition1 & calculus r5d]s x app>5t t 9 ! case ( an inf9ite numb] ( obs]vations x w\ld co9cide ) ! tru?2 b ! result giv5 by ! mo/ advantage\s me?od is /ill pref]able4 ,we see by t :i* prec$es t ! !ory ( probabilities leaves no?+ arbitr>y 9 ! mann] ( di/ribut+ ! errors ( ! obs]vations2 x gives = ?is di/ribution ! mo/ advantage\s =mul :i* dim9i%es z mu* z possible ! errors to be fe>$ 9 ! results4 ,! consid]ation ( probabilities c s]ve to di/+ui% ! small irregul>ities ( ! cele/ial movem5ts envelop$ 9 ! errors ( obs]vations1 & to repass to ! cause ( ! anomalies obs]v$ 9 ^! movem5ts4 ,9 comp>+ all ! obs]vations x 0 ,ti*o-,brah :o recogniz$ ! necessity ( apply+ to ! moon an equation ( "t diff]5t f t :i* _h be5 appli$ to ! sun & to ! planets4 ,x 0 simil>ly ! totality ( a great numb] ( obs]vations :i* made ,may] recognize t ! coeffici5t ( ! inequality ( ! precession ought to be dim9i%$ a little = ! moon4 ,b s9ce ?is dim9ution1 al?\gh confirm$ & ev5 augm5t$ by ,mason1 did n appe> to result f univ]sal gravitation1 ! majority ( a/ronom]s neglect x 9 _! calculations4 ,hav+ submitt$ to ! calculation ( probabilities a consid]able numb] ( lun> obs]vations *os5 = ?is purpose & :i* ,m4 ,b\v>d cons5t$ to exam9e at my reque/1 x appe>$ to me to be indicat$ ) s /rong a probability t ,i believ$ ! cause ( x ought to be inve/igat$4 ,i soon saw t x w\ld be only ! ellipticity ( ! t]re/rial sph]oid1 neglect$ up to t "t 9 ! !ory ( ! lun> movem5t z be+ able to produce only imp]ceptible t]ms4 ,i conclud$ t ^! t]ms became p]ceptible by ! successive integrations ( diff]5tial equations4 ,i det]m9$ !n ^? t]ms by a p>ticul> analysis1 & ,i discov]$ fir/ ! inequality ( ! lun> movem5t 9 latitude :i* is proportional to ! s9e ( ! longitude ( ! moon1 :i* no a/ronom] be=e _h suspect$4 ,i recogniz$ !n by means ( ?is inequality t ano!r exi/s 9 ! lun> movem5t 9 longitude :i* produces ! dim9ution obs]v$ by ,may] 9 ! equation ( ! precession applicable to ! moon4 ,! quantity ( ?is dim9ution & ! coeffici5t ( ! prec$+ inequality 9 latitude are v appropriate to fix ! oblat5ess ( ! e>?4 ,hav+ communicat$ my rese>*es to ,m4 ,burg1 :o 0 occupi$ at t "t 9 p]fect+ ! tables ( ! moon by ! comp>ison ( all ! good obs]vations1 ,i reque/$ him to det]m9e ) a p>ticul> c>e ^! two quantities4 ,by a v rem>kable agreem5t ! values :i* he has f\nd give to ! e>? ! same oblat5ess1 #a_/cje1 :i* diff]s little f ! mean d]iv$ f ! measurem5ts ( ! degrees ( ! m]idian & ! p5dulum2 b ^? reg>d$ f ! po9t ( view ( ! influ5ce ( ! errors ( ! obs]vations & ( ! p]turb+ causes 9 ^! measurem5ts1 did n appe> to me exactly det]m9$ by ^! lun> inequalities4 ,x 0 aga9 by ! consid]ation ( probabilities t ,i recogniz$ ! cause ( ! secul> equation ( ! moon4 ,! mod]n obs]vations ( ?is /> comp>$ to ! anci5t eclipses _h indicat$ to a/ronom]s an accel]ation 9 ! lun> movem5t2 b ! geometricians1 & p>ticul>ly ,lagrange1 hav+ va9ly s\ght 9 ! p]turbations :i* ?is movem5t exp]i5c$ ! t]ms ^u :i* ?is accel]ation dep5ds1 reject x4 ,an att5tive exam9ation ( ! anci5t & mod]n obs]vations & ( ! int]m$i>y eclipses obs]v$ by ! ,arabians conv9c$ me t x 0 indicat$ ) a great probability4 ,i took up aga9 !n f ?is po9t ( view ! lun> !ory1 & ,i recogniz$ t ! secul> equation ( ! moon is due to ! action ( ! sun ^u ?is satellite1 comb9$ ) ! secul> v>iation ( ! ecc5tricity ( ! t]re/rial orb2 ?is br\ght me to ! discov]y ( ! secul> equations ( ! movem5ts ( ! nodes & ( ! p]igees ( ! lun> orbit1 :i* equations _h n be5 ev5 suspect$ by a/ronom]s4 ,! v rem>kable agreem5t ( ?is !ory ) all ! anci5t & mod]n obs]vations has br\ght x to a v high degree ( evid5ce4 ,! calculus ( probabilities has l$ me simil>ly to ! cause ( ! great irregul>ities ( ,jupit] & ,saturn4 ,comp>+ mod]n obs]vations ) anci5t1 ,halley f\nd an accel]ation 9 ! movem5t ( ,jupit] & a ret>dation 9 t ( ,saturn4 ,9 ord] to conciliate ! obs]vations he r$uc$ ! movem5ts to two secul> equations ( contr>y signs & increas+ z ! squ>es ( ! times pass$ s9ce #agjj4 ,eul] & ,lagrange submitt$ to analysis ! alt]ations :i* ! mutual attraction ( ^! two planets ought to produce 9 ^! movem5ts4 ,!y f\nd 9 do+ ?is ! secul> equations2 b _! results 7 s diff]5t t "o ( ! two at lea/ ought to be errone\s4 ,i det]m9$ !n to take up aga9 ?is important problem ( cele/ial me*anics1 & ,i recogniz$ ! inv>iability ( ! mean planet>y movem5ts1 :i* nullifi$ ! secul> equations introduc$ by ,halley 9 ! tables ( ,jupit] & ,saturn4 ,?us !re rema91 9 ord] to expla9 ! great irregul>ity ( ^! planets1 only ! attractions ( ! comets to :i* _m a/ronom]s _h effective rec\rse1 or ! exi/5ce ( an irregul>ity ov] a long p]iod produc$ 9 ! movem5ts ( ! two planets by _! reciprocal action & affect$ by contr>y signs = ea* ( !m4 ,a !orem :i* ,i f\nd 9 reg>d to ! inequalities ( ?is k9d r5d]$ ?is inequality v probable4 ,accord+ to ?is !orem1 if ! movem5t ( ,jupit] is accel]at$1 t ( ,saturn is ret>d$1 :i* has already con=m$ to :at ,halley _h notic$2 moreov]1 ! accel]ation ( ,jupit] result+ f ! same !orem is to ! ret>dation ( ,saturn v ne>ly 9 ! ratio ( ! secul> equations propos$ by ,halley4 ,consid]+ ! mean movem5ts ( ,jupit] & ,saturn ,i 0 enabl$ easily to recognize t two times t ( ,jupit] diff]$ only by a v small quantity f five times t ( ,saturn4 ,! p]iod ( an irregul>ity :i* w\ld h = an argum5t ?is diff]5ce w\ld be ab\t n9e c5turies4 ,inde$ its coeffici5t w\ld be ( ! ord] ( ! cubes ( ! ecc5tricities ( ! orbits2 b ,i knew t by virtue ( successive integrations x acquir$ = divisor ! squ>e ( ! v small multipli] ( ! "t 9 ! argum5t ( ?is inequality :i* is able to give x a great value2 ! exi/5ce ( ?is inequality appe>$ to me !n v probable4 ,! foll[+ obs]vation increas$ !n its probability4 ,suppos+ its argum5t z]o t[>d ! epo* ( ! obs]vations ( ,ti*o-,brah1 ,i saw t ,halley ought to h f\nd by ! comp>ison ( mod]n ) anci5t obs]vations ! alt]ations :i* he _h indicat$2 :ile ! comp>ison ( ! mod]n obs]vations among !mselves ought to (f] contr>y alt]ations simil> to ^? :i* ,lamb]t _h conclud$ f ?is comp>ison4 ,i did n !n hesitate at all to und]take ?is long & t$i\s calculation necess>y to assure myself ( ?is inequality4 ,x 0 entirely confirm$ by ! result ( ?is calculation1 :i* moreov] made me recognize a great numb] ( o!r inequalities ( :i* ! totality has incl9$ ! tables ( ,jupit] & ,saturn to ! precision ( ! same obs]vations4 ,x 0 aga9 by means ( ! calculus ( probabilities t ,i recogniz$ ! rem>kable law ( ! mean movem5ts ( ! ?ree fir/ satellites ( ,jupit]1 accord+ to :i* ! mean longitude ( ! fir/ m9us ?ree times t ( ! second plus two times t ( ! ?ird is rigor\sly equal to ! half-circumf]5ce4 ,! approximation ) :i* ! mean movem5ts ( ^! />s satisfy ?is law s9ce _! discov]y indicates its exi/5ce ) an extreme probability4 ,i s\ght !n ! cause ( x 9 _! mutual action4 ,! se>*+ exam9ation ( ?is action conv9c$ me t x 0 suffici5t if 9 ! beg9n+ ! ratios ( _! mean movem5ts _h approa*$ ?is law )9 c]ta9 limits1 because _! mutual action _h e/abli%$ & ma9ta9$ x rigor\sly4 ,?us ^! ?ree bodies w balance "o ano!r et]nally 9 space accord+ to ! prec$+ law unless /range causes1 su* z comets1 %\ld *ange sudd5ly _! movem5ts ab\t ,jupit]4 ,accord+ly x is se5 h[ necess>y x is to be att5tive to ! indications ( nature :5 !y are ! result ( a great numb] ( obs]vations1 al?\gh 9 o!r respects !y may be inexplicable by kn[n means4 ,! extreme difficulty ( problems relative to ! sy/em ( ! _w has =c$ geometricians to recur to ! approximation :i* always leaves room = ! fe> t ! quantities neglect$ may h an appreciable influ5ce4 ,:5 !y h be5 w>n$ ( ?is influ5ce by ! obs]vations1 !y h recurr$ to _! analysis2 9 rectify+ x !y h always f\nd ! cause ( ! anomalies obs]v$2 !y h det]m9$ ! laws & (t5 !y h anticipat$ ! obs]vations 9 discov]+ ! inequalities :i* x _h n yet indicat$4 ,?us "o may say t nature itself has concurr$ 9 ! analytical p]fection ( ! !ories bas$ ^u ! pr9ciple ( univ]sal gravity2 & ?is is to my m9d "o ( ! /ronge/ pro(s ( ! tru? ( ?is admirable pr9ciple4 ,9 ! cases :i* ,i h j consid]$ ! analytical solution ( ! "q has *ang$ ! probability ( ! causes into c]ta9ty4 ,b mo/ (t5 ?is solution is impossible & x rema9s only to augm5t m & m ?is probability4 ,9 ! mid/ ( num]\s & incalculable modifications :i* ! action ( ! causes receives !n f /range circum/ances ^! causes cons]ve always ) ! effects obs]v$ ! prop] ratios to make !m recognizable & to v]ify _! exi/5ce4 ,det]m9+ ^! ratios & comp>+ !m ) a great numb] ( obs]vations if "o f9ds t !y con/antly satisfy x1 ! probability ( ! causes may increase to ! po9t ( equall+ t ( facts 9 reg>d to :i* !re is no d\bt4 ,! inve/igation ( ^! ratios ( causes to _! effects is n less useful 9 natural philosophy ?an ! direct solution ( problems :e!r x be to v]ify ! reality ( ^! causes or to det]m9e ! laws f _! effects2 s9ce x may be employ$ 9 a great numb] ( que/ions ^: direct solution is n possible1 x replaces x 9 ! mo/ advantage\s mann]4 ,i %all discuss "h ! application :i* ,i h made ( x to "o ( ! mo/ int]e/+ ph5om5a ( nature1 ! fl[ & ! ebb ( ! sea4 ,pl9y has giv5 ( ?is ph5om5on a description rem>kable = its exactitude1 & 9 x "o sees t ! anci5ts _h obs]v$ t ! tides ( ea* mon? are greate/ t[>d ! syzygies & smalle/ t[>d ! quadratures2 t !y are high] 9 ! p]igees ?an 9 ! apogees ( ! moon1 & high] 9 ! equ9oxes ?an 9 ! sol/ices4 ,!y conclud$ f ?is t ?is ph5om5on is due to ! action ( ! sun & moon ^u ! sea4 ,9 ! preface ( 8 "w ,de ,/ella ,m>tis ,kepl] admits a t5d5cy ( ! wat]s ( ! sea t[>d ! moon2 b1 ignorant ( ! law ( ?is t5d5cy1 he 0 able to give on ?is subject only a probable idea4 ,newton conv]t$ into c]ta9ty ! probability ( ?is idea by atta*+ x to 8 great pr9ciple ( univ]sal gravity4 ,he gave ! exact expression ( ! attractive =ces :i* produc$ ! flood & ! ebb ( ! sea2 & 9 ord] to det]m9e ! effects he suppos$ t ! sea takes at ea* in/ant ! position ( equilibrium :i* is agreeable to ^! =ces4 ,he expla9$ 9 ?is mann] ! pr9cipal ph5om5a ( ! tides2 b x foll[$ f ?is !ory t 9 our ports ! two tides ( ! same day w\ld be v unequal if ! sun & ! moon %\ld h a great decl9ation4 ,at ,bre/1 = example1 ! ev5+ tide w\ld be 9 ! syzygies ( ! sol/ices ab\t eight times great] ?an ! morn+ tide1 :i* is c]ta9ly contr>y to ! obs]vations :i* prove t ^! two tides are v ne>ly equal4 ,?is result f ! ,newtonian !ory might hold to ! supposition t ! sea is agreeable at ea* in/ant to a position ( equilibrium1 a supposition :i* is n at all admissible4 ,b ! inve/igation ( ! true figure ( ! sea pres5ts great difficulties4 ,aid$ by ! discov]ies :i* ! geometricians _h j made 9 ! !ory ( ! movem5t ( fluids & 9 ! calculus ( p>tial diff]5ces1 ,i und]took ?is inve/igation1 & ,i gave ! diff]5tial equations ( ! movem5t ( ! sea by suppos+ t x cov]s ! entire e>?4 ,9 draw+ ?us ne> to nature ,i _h ! satisfaction ( see+ t my results approa*$ ! obs]vations1 especially 9 reg>d to ! little diff]5ce :i* exi/s 9 our ports betwe5 ! two tides ( ! sol/itial syzygies ( ! same day4 ,i f\nd t !y w\ld be equal if ! sea _h ev]y:]e ! same dep?2 ,i f\nd fur!r t 9 giv+ to ?is dep? conv5i5t values "o 0 able to augm5t ! height ( ! tides 9 a port con=mably to ! obs]vations4 ,b ^! inve/igations1 9 spite ( _! g5]ality1 did n satisfy at all ! great diff]5ces :i* ev5 adjac5t ports pres5t 9 ?is reg>d & :i* prove ! influ5ce ( local circum/ances4 ,! impossibility ( kn[+ ^! circum/ances & ! irregul>ity ( ! bas9 ( ! seas & t ( integrat+ ! equations ( p>tial diff]5ces :i* are relative has compell$ me to make up ! defici5cy by ! me?od ,i h indicat$ above4 ,i !n endeavor$ to det]m9e ! greate/ ratios possible among ! =ces :i* affect all ! molecules ( ! sea1 & _! effects obs]vable 9 our ports4 ,= ?is ,i made use ( ! foll[+ pr9ciple1 :i* may be appli$ to _m o!r ph5om5a4 ,! /ate ( ! sy/em ( a body 9 :i* ! primitive conditions ( ! movem5t h disappe>$ by ! resi/ances :i* ?is movem5t meets is p]iodic z ! =ces :i* animate x4 ,comb9+ ?is pr9ciple ) t ( ! coexi/5ce ( v small oscillations1 ,i h f\nd an expression ( ! height ( ! tides ^: arbitr>ies conta9 ! effect ( local circum/ances ( ea* port & are r$uc$ to ! smalle/ numb] possible2 x is only necess>y to comp>e x to a great numb] ( obs]vations4 ,^u ! invitation ( ! ,academy ( ,sci5ces1 obs]vations 7 made at ! beg9n+ ( ! la/ c5tury at ,bre/ ^u ! tides1 :i* 7 cont9u$ dur+ six consecutive ye>s4 ,! situation ( ?is port is v favorable to ?is sort ( obs]vations2 x communicates ) ! sea by a canal :i* empties into a va/ road/ead at ! f> end ( :i* ! port has be5 con/ruct$4 ,! irregul>ities ( ! sea ext5d ?us only to a small degree into ! port1 j z ! oscillations :i* ! irregul> movem5t ( a vessel produces 9 a b>omet] are dim9i%$ by a ?rottl+ made 9 ! tube ( ?is in/rum5t4 ,moreov]1 ! tides be+ consid]able at ,bre/1 ! accid5tal v>iations caus$ by ! w9ds are only feeble2 likewise we notice 9 ! obs]vations ( ^! tides1 h[ev] little we multiply !m1 a great regul>ity :i* induc$ me to propose to ! gov]nm5t to ord] 9 ?is port a new s]ies ( obs]vations ( ! tides1 cont9u$ dur+ a p]iod ( ! movem5t ( ! nodes ( ! lun> orbit4 ,?is has be5 done4 ,! obs]vations began ,june #a1 #ahjf2 & s9ce ?is "t !y h be5 made e day )\t int]ruption4 ,i am indebt$ to ! indefatigable zeal ( ,m4 ,b\v>d1 = all t int]e/s a/ronomy1 ! imm5se calculations :i* ! comp>ison ( my analysis ) ! obs]vations has dem&$4 ,!re h be5 us$ ab\t six ?\s& obs]vations1 made dur+ ! ye> #ahjg & ! fifte5 ye>s foll[+4 ,x results f ?is comp>ison t my =mul repres5t ) a rem>kable precision all ! v>ieties ( ! tides relative to ! digression ( ! moon1 f ! sun1 to ! decl9ation ( ^! />s1 to _! di/ances f ! e>?1 & to ! laws ( v>iation at ! maximum & m9imum ( ea* ( ^! elem5ts4 ,!re results f ?is accord a probability t ! fl[ & ! ebb ( ! sea is due to ! attraction ( ! sun & moon1 s approa*+ c]ta9ty t x ought to leave room = no reasonable d\bt4 ,x *anges into c]ta9ty :5 we consid] t ?is attraction is d]iv$ f ! law ( univ]sal gravity demon/rat$ by all ! cele/ial ph5om5a4 ,! action ( ! moon ^u ! sea is m ?an d\ble t ( ! sun4 ,newton & 8 successors 9 ! developm5t ( ?is action h paid att5tion only to ! t]ms divid$ by ! cube ( ! di/ance f ! moon to ! e>?1 judg+ t ! effects due to ! foll[+ t]ms ought to be inappreciable4 ,b ! calculation ( probabilities makes x cle> to u t ! smalle/ effects ( regul> causes may manife/ !mselves 9 ! results ( a great numb] ( obs]vations arrang$ 9 ! ord] mo/ suitable to indicate !m4 ,?is calculation aga9 det]m9es _! probability & up to :at po9t x is necess>y to multiply ! obs]vations to make x v great4 ,apply+ x to ! num]\s obs]vations discuss$ by ,m4 ,b\v>d ,i recogniz$ t at ,bre/ ! action ( ! moon ^u ! sea is great] 9 ! full moons ?an 9 ! new moons1 & great] :5 ! moon is au/ral ?an :5 x is boreal--ph5om5a :i* c result only f ! t]ms ( ! lun> action divid$ by ! f\r? p[] ( ! di/ance f ! moon to ! e>?4 ,to arrive at ! ocean ! action ( ! sun & ! moon trav]ses ! atmosph]e1 :i* ought consequ5tly to feel its influ5ce & to be subject$ to movem5ts simil> to ^? ( ! sea4 ,^! movem5ts produce 9 ! b>omet] p]iodic oscillations4 ,analysis has made x cle> to me t !y are inappreciable 9 our climates4 ,b z local circum/ances increase consid]ably ! tides 9 our ports1 ,i h inquir$ aga9 if simil> circum/ances h made appreciable ^! oscillations ( ! b>omet]4 ,= ?is ,i h made use ( ! meteorological obs]vations :i* h be5 made e day = _m ye>s at ! royal obs]vatory4 ,! heights ( ! b>omet] & ( ! !rmomet] are obs]v$ !re at n9e o'clock 9 ! morn+1 at noon1 at ?ree o'clock 9 ! aft]noon1 & at elev5 o'clock 9 ! ev5+4 ,m4 ,b\v>d has inde$ wi%$ to take up ! consid]ation ( obs]vations ( ! eight ye>s elaps$ f ,octob] #a1 #ahae1 to ,octob] #a1 #ahbc1 on ! regi/]s4 ,9 dispos+ ! obs]vations 9 ! mann] mo/ suitable to indicate ! lun> atmosph]ic flood at ,p>is1 ,i f9d only "o eighte5? ( a millimet] = ! ext5t ( ! correspond+ oscillation ( ! b>omet]4 ,x is ?is especially :i* has made u feel ! necessity ( a me?od = det]m9+ ! probability ( a result1 & )\t ?is me?od "o is =c$ to pres5t z ! laws ( nature ! results ( irregul> causes :i* has (t5 happ5$ 9 meteorology4 ,?is me?od appli$ to ! prec$+ result %[s ! unc]ta9ty ( x 9 spite ( ! great numb] ( obs]vations employ$1 :i* x w\ld be necess>y to increase t5fold 9 ord] to obta9 a result suffici5tly probable4 ,! pr9ciple :i* s]ves z a basis = my !ory ( ! tides may be ext5d$ to all ! effects ( haz>d to :i* v>iable causes are jo9$ accord+ to regul> laws4 ,! action ( ^! causes produces 9 ! mean results ( a great numb] ( effects v>ieties :i* foll[ ! same laws & :i* "o may recognize by ! analysis ( probabilities4 ,9 ! measure :i* ^! effects are multipli$ ^? v>ieties are manife/$ ) an ev]-increas+ probability1 :i* w\ld approa* c]ta9ty if ! numb] ( ! effects ( ! results %\ld become inf9ite4 ,?is !orem is analog\s to t :i* ,i h already develop$ ^u ! action ( con/ant causes4 ,e "t1 !n1 t a cause ^: progress is regul> c h influ5ce ^u a k9d ( ev5ts1 we may seek to discov] its influ5ce by multiply+ ! obs]vations & arrang+ !m 9 ! mo/ suitable ord] to indicate x4 ,:5 ?is influ5ce appe>s to manife/ itself ! analysis ( probabilities det]m9es ! probability ( its exi/5ce & t ( its int5sity2 ?us ! v>iation ( ! temp]ature f day to night modify+ ! pressure ( ! atmosph]e & consequ5tly ! height ( ! b>omet]1 x is natural to ?9k t ! multipli$ obs]vations ( ^! heights ought to %[ ! influ5ce ( ! sol> heat4 ,inde$ !re has long be5 recogniz$ at ! equator1 :]e ?is influ5ce appe>s to be greate/1 a small diurnal v>iation 9 ! height ( ! b>omet] ( :i* ! maximum occurs ab\t n9e o'clock 9 ! morn+ & ! m9imum ab\t ?ree o'clock 9 ! aft]noon4 ,a second maximum occurs ab\t elev5 o'clock 9 ! ev5+ & a second m9imum ab\t f\r o'clock 9 ! morn+4 ,! oscillations ( ! night are less ?an ^? ( ! day1 ! ext5t ( :i* is ab\t two millimet]s4 ,! incon/ancy ( our climate has n tak5 ?is v>iation f our obs]v]s1 al?\gh x may be less appreciable ?an 9 ! tropics4 ,m4 ,ramond has recogniz$ & det]m9$ x at ,cl]mont1 ! *ief place ( ! di/rict ( ,puy-de-,dme1 by a s]ies ( precise obs]vations made dur+ sev]al ye>s2 he has ev5 f\nd t x is small] 9 ! mon?s ( w9t] ?an 9 o!r mon?s4 ,! num]\s obs]vations :i* ,i h discuss$ 9 ord] to e/imate ! influ5ce ( attractions ( ! sun & ! moon ^u ! b>ometric heights at ,p>is h s]v$ me 9 det]m9+ _! diurnal v>iation4 ,comp>+ ! heights at n9e o'clock 9 ! morn+ ) ^? ( ! same days at ?ree o'clock 9 ! aft]noon1 ?is v>iation is manife/$ ) s mu* evid5ce t its mean value ea* mon? has be5 con/antly positive = ea* ( ! sev5ty-two mon?s f ,janu>y #a1 #ahag1 to ,janu>y #a1 #ahbc2 its mean value 9 ^! sev5ty-two mon?s has be5 almo/ 4#h ( a millimet]1 a little less ?an at ,cl]mont & mu* less ?an at ! equator4 ,i h recogniz$ t ! mean result ( ! diurnal v>iations ( ! b>omet] f #i o'clock ,a4,m4 to #c ,p4,m4 has be5 only 4#edbh millimet] 9 ! ?ree mon?s ( ,novemb]1 ,decemb]1 ,janu>y1 & t x has ris5 to #a4jefc millimet]s 9 ! ?ree foll[+ mon?s1 :i* co9cides ) ! obs]vations ( ,m4 ,ramond4 ,! o!r mon?s (f] no?+ simil>4 ,9 ord] to apply to ^! ph5om5a ! calculation ( ^! probabilities1 ,i comm5c$ by det]m9+ ! law ( ! probability ( ! anomalies ( ! diurnal v>iation due to haz>d4 ,apply+ x !n to ! obs]vations ( ?is ph5om5on1 ,i f\nd t x 0 a bet ( m ?an #cjj1jjj aga9/ "o t a regul> cause produc$ x4 ,i d n seek to det]m9e ?is cause2 ,i cont5t myself ) /at+ its exi/5ce4 ,! p]iod ( ! diurnal v>iation regulat$ by ! sol> day indicates evid5tly t ?is v>iation is due to ! action ( ! sun4 ,! extreme smallness ( ! attractive action ( ! sun ^u ! atmosph]e is prov$ by ! smallness ( ! effects due to ! unit$ attractions ( ! sun & ! moon4 ,x is !n by ! action ( its heat t ! sun produces ! diurnal v>iation ( ! b>omet]2 b x is impossible to subject to calculus ! effects ( its action on ! height ( ! b>omet] & ^u ! w9ds4 ,! diurnal v>iation ( ! magnetic ne$le is c]ta9ly a result ( ! action ( ! sun4 ,b does ?is /> act "h z 9 ! diurnal v>iation ( ! b>omet] by its heat or by its influ5ce ^u electricity & ^u magnetism1 or f9ally by ! union ( ^! influ5ces8 ,a long s]ies ( obs]vations made 9 diff]5t c\ntries w enable u to appreh5d ?is4 ,"o ( ! mo/ rem>kable ph5om5a ( ! sy/em ( ! _w is t ( all ! movem5ts ( rotation & ( revolution ( ! planets & ! satellites 9 ! s5se ( ! rotation ( ! sun & ab\t 9 ! same plane ( its equator4 ,a ph5om5on s rem>kable is n ! effect ( haz>d3 x indicates a g5]al cause :i* has det]m9$ all its movem5ts4 ,9 ord] to obta9 ! probability ) :i* ?is cause is indicat$ we %all obs]ve t ! planet>y sy/em1 su* z we "k x to-day1 is compos$ ( elev5 planets & ( eighte5 satellites at lea/1 if we attribute ) ,h]s*el six satellites to ! planet ,uranus4 ,! movem5ts ( ! rotation ( ! sun1 ( six planets1 ( ! moon1 ( ! satellites ( ,jupit]1 ( ! r+ ( ,saturn1 & ( "o ( its satellites h be5 recogniz$4 ,^! movem5ts =m ) ^? ( revolution a totality ( =ty-?ree movem5ts direct$ 9 ! same s5se2 b "o f9ds by ! analysis ( probabilities t x is a bet ( m ?an #djjjjjjjjjjjj aga9/ "o t ?is disposition is n ! result ( haz>d2 ?is =ms a probability inde$ sup]ior to t ( hi/orical ev5ts 9 reg>d to :i* no d\bt exi/s4 ,we ought !n to believe at lea/ ) equal confid5ce t a primitive cause has direct$ ! planet>y movem5ts1 especially if we consid] t ! incl9ation ( ! greate/ numb] ( ^! movem5ts at ! sol> equator is v small4 ,ano!r equally rem>kable ph5om5on ( ! sol> sy/em is ! small degree ( ! ecc5tricity ( ! orbs ( ! planets & ! satellites1 :ile ^? ( ! comets are v elongat$1 ! orbs ( ! sy/em n (f]+ any int]m$iate %ades betwe5 a great & a small ecc5tricity4 ,we are aga9 =c$ to recognize "h ! effect ( a regul> cause2 *ance has c]ta9ly n giv5 an almo/ circul> =m to ! orbits ( all ! planets & _! satellites2 x is !n t ! cause :i* has det]m9$ ! movem5ts ( ^! bodies has r5d]$ !m almo/ circul>4 ,x is necess>y1 aga91 t ! great ecc5tricities ( ! orbits ( ! comets %\ld result f ! exi/5ce ( ?is cause )\t its hav+ influ5c$ ! direction ( _! movem5ts2 = x is f\nd t !re are almo/ z _m retrograde comets z direct comets1 & t ! mean incl9ation ( all _! orbits to ! ecliptic approa*es v ne>ly half a "r angle1 z x ought to be if ! bodies _h be5 ?r[n at haz>d4 ,:atev] may be ! nature ( ! cause 9 "q1 s9ce x has produc$ or direct$ ! movem5t ( ! planets1 x is necess>y t x %\ld h embrac$ all ! bodies & consid]$ all ! di/ances :i* sep>ate !m1 x c h be5 only a fluid ( an imm5se ext5sion4 ,!re=e 9 ord] to h giv5 !m 9 ! same s5se an almo/ circul> movem5t ab\t ! sun x is necess>y t ?is fluid %\ld h surr\nd$ ?is /> z an atmosph]e4 ,! consid]ation ( ! planet>y movem5ts leads u !n to ?9k t by virtue ( an excessive heat ! atmosph]e ( ! sun 0 orig9ally ext5d$ beyond ! orbits ( all ! planets1 & t x has contract$ gradually to its pres5t limits4 ,9 ! primitive /ate :]e we imag9e ! sun x resembl$ ! nebul t ! telescope %[s u compos$ ( a nucleus m or less brilliant surr\nd$ by a nebula :i*1 cond5s+ at ! surface1 ought to trans=m x "s day into a />4 ,if "o conceives by analogy all ! />s =m$ 9 ?is mann]1 "o c imag9e _! ant]ior /ate ( nebulosity itself prec$$ by o!r />s 9 :i* ! nebul\s matt] 0 m & m diffuse1 ! nucleus be+ less & less lum9\s & d5se4 ,go+ back1 !n1 z f> z possible1 "o w\ld arrive at a nebulosity s diffuse t "o w\ld be able sc>cely to suspect its exi/5ce4 ,su* is inde$ ! fir/ /ate ( ! nebul :i* ,h]s*el obs]v$ ) p>ticul> c>e by means ( 8 p[]ful telescopes1 & 9 :i* he has foll[$ ! progress ( cond5sation1 n 9 a s+le "o1 ^! /ages n becom+ appreciable to u except aft] c5turies1 b 9 _! totality1 j ab\t z "o c 9 a va/ =e/ foll[ ! increase ( ! trees by ! individuals ( ! div]s ages :i* ! =e/ conta9s4 ,he has obs]v$ f ! beg9n+ nebul\s matt] spread out 9 div]s masses 9 ! diff]5t p>ts ( ! heav5s1 ( :i* x occupies a great ext5t4 ,he has se5 9 "s ( ^! masses ?is matt] slightly cond5s$ ab\t "o or sev]al fa9tly lum9\s nebul4 ,9 ! o!r nebul ^! nuclei %9e1 moreov]1 9 proportion to ! nebulosity :i* surr\nds !m4 ,! atmosph]es ( ea* nucleus becom+ sep>at$ by an ult]ior cond5sation1 !re result ! multifold nebul =m$ ( brilliant nuclei v adjac5t & surr\nd$ ea* by an atmosph]e2 sometimes ! nebul\s matt]1 by cond5s+ 9 a uni=m mann]1 has produc$ ! nebul :i* are call$ planet>y4 ,f9ally a great] degree ( cond5sation trans=ms all ^! nebul into />s4 ,! nebul class$ accord+ to ?is philosophic view indicate ) an extreme probability _! future trans=mation into />s & ! ant]ior /ate ( nebulosity ( exi/+ />s4 ,! foll[+ consid]ations come to ! aid ( pro(s drawn f ^! analogies4 ,= a long "t ! p>ticul> disposition ( c]ta9 />s visible to ! nak$ eye has /ruck ! att5tion ( philosophical obs]v]s4 ,mit*el has already rem>k$ h[ improbable x is t ! />s ( ! ,pleiades1 = example1 %\ld h be5 conf9$ 9 ! n>r[ space :i* conta9 !m by ! *ances ( haz>d alone1 & he has conclud$ f ?is t ?is gr\p ( />s & ! simil> gr\ps t ! heav5 pres5ts u are ! results ( a primitive cause or ( a g5]al law ( nature4 ,^! gr\ps are a necess>y result ( ! cond5sation ( ! nebul at sev]al nuclei2 x is app>5t t ! nebul\s matt] be+ attract$ cont9u\sly by ! div]s nuclei1 !y ought to =m 9 "t a gr\p ( />s equal to t ( ! ,pleiades4 ,! cond5sation ( ! nebul at two nuclei =ms simil>ly v adjac5t />s1 revolv+ ! "o ab\t ! o!r1 equal to ^? ^: respective movem5ts ,h]s*el has already consid]$4 ,su* are1 fur!r1 ! #fa/ ( ! ,swan & its foll[+ "o 9 :i* ,bessel has j recogniz$ p>ticul> movem5ts s consid]able & s little diff]5t t ! proximity ( ^! />s to "o ano!r & _! movem5t ab\t ! common c5tre ( gravity ought to leave no d\bt4 ,?us "o desc5ds by degrees f ! cond5sation ( nebul\s matt] to ! consid]ation ( ! sun surr\nd$ =m]ly by a va/ atmosph]e1 a consid]ation to :i* "o repasses1 z has be5 se51 by ! exam9ation ( ! ph5om5a ( ! sol> sy/em4 ,a case s rem>kable gives to ! exi/5ce ( ?is ant]ior /ate ( ! sun a probability /rongly approa*+ c]ta9ty4 ,b h[ has ! sol> atmosph]e det]m9$ ! movem5ts ( rotation & revolution ( ! planets & ! satellites8 ,if ^! bodies _h p5etrat$ deeply ! atmosph]e its resi/ance w\ld h caus$ !m to fall ^u ! sun2 "o is !n l$ to believe ) mu* probability t ! planets h be5 =m$ at ! successive limits ( ! sol> atmosph]e :i*1 contract+ by ! cold1 ought to h ab&on$ 9 ! plane ( its equator zones ( vapors :i* ! mutual attraction ( _! molecules has *ang$ into div]s sph]oids4 ,! satellites h be5 simil>ly =m$ by ! atmosph]es ( _! respective planets4 ,i h develop$ at l5g? 9 my ,exposition ( ! ,sy/em ( ! ,_w ?is hypo!sis1 :i* appe>s to me to satisfy all ! ph5om5a :i* ?is sy/em pres5ts u4 ,i %all cont5t myself "h ) consid]+ t ! angul> velocity ( rotation ( ! sun & ! planets be+ accel]at$ by ! successive cond5sation ( _! atmosph]es at _! surfaces1 x ought to surpass ! angul> velocity ( revolution ( ! ne>e/ bodies :i* revolve ab\t !m4 ,obs]vation has inde$ confirm$ ?is ) reg>d to ! planets & satellites1 & ev5 9 ratio to ! r+ ( ,saturn1 ! duration ( ^: revolution is 4#dch days1 :ile ! duration ( ! rotation ( ,saturn is 4#dbg days4 ,9 ?is hypo!sis ! comets are /rang]s to ! planet>y sy/em4 ,9 atta*+ _! =mation to t ( ! nebul !y may be reg>d$ z small nebul at ! nuclei1 w&]+ f sy/ems to sol> sy/ems1 & =m$ by ! cond5sation ( ! nebul\s matt] spread out 9 su* great pr(usion 9 ! univ]se4 ,! comets w\ld be ?us1 9 relation to our sy/em1 z ! a]olites are relatively to ! ,e>?1 to :i* !y w\ld appe> /rang]s4 ,:5 ^! />s become visible to u !y (f] s p]fect resemblance to ! nebul t !y are (t5 conf\nd$ ) !m2 & x is only by _! movem5t1 or by ! k ( all ! nebul conf9$ to t "p ( ! heav5s :]e !y appe>1 t we succe$ 9 di/+ui%+ !m4 ,?is supposition expla9s 9 a happy mann] ! great ext5sion :i* ! heads & tails ( comets take 9 ! measure t !y approa* ! sun1 & ! extreme r>ity ( ^! tails :i*1 9 spite ( _! imm5se dep?1 d n weak5 at all appreciably ! light ( ! />s :i* we look across4 ,:5 ! little nebul come into t "p ( space :]e ! attraction ( ! sun is pr$om9ant1 & :i* we %all call ! sph]e ( activity ( ?is />1 x =ces !m to describe elliptic or hyp]bolic orbits4 ,b _! spe$ be+ equally possible 9 all directions !y ought to move indiff]5tly 9 all ! s5ses & "u all incl9ations ( ! elliptic1 :i* is con=mable to t :i* has be5 obs]v$4 ,! great ecc5tricity ( ! comet>y orbits results aga9 f ! prec$+ hypo!sis4 ,inde$ if ^! orbits are elliptical !y are v elongat$1 s9ce _! great axes are at lea/ equal to ! radius ( ! sph]e ( activity ( ! sun4 ,b ^! orbits may be hyp]bolic2 & if ! axes ( ^! hyp]bol are n v l>ge 9 proportion to ! mean di/ance f ! sun to ! e>?1 ! movem5t ( ! comets :i* describe !m w appe> s5sibly hyp]bolic4 ,h[ev]1 ( ! hundr$ comets ( :i* we already h ! elem5ts1 n "o has appe>$ c]ta9ly to move 9 an hyp]bola2 x is necess>y1 !n1 t ! *ances :i* give an appreciable hyp]bola %\ld be extremely r>e 9 proportion to ! contr>y *ances4 ,! comets are s small t1 9 ord] to become visible1 _! p]ihelion di/ance ought to be inconsid]able4 ,up to ! pres5t ?is di/ance has surpass$ only twice ! diamet] ( ! t]re/rial orbit1 & mo/ (t5 x has be5 bel[ ! radius ( ?is orbit4 ,x is conceiv$ t1 9 ord] to approa* s ne> ! sun1 _! spe$ at ! mom5t ( _! entrance into its sph]e ( activity ought to h a magnitude & a direction conf9$ )9 n>r[ limits4 ,9 det]m9+ by ! analysis ( probabilities ! ratio ( ! *ances :i*1 9 ^! limits1 give an appreciable hyp]bola1 to ! *ances :i* give an orbit :i* may be conf\nd$ ) a p>abola1 ,i h f\nd t x is a bet ( at lea/ #fjjj aga9/ "o t a nebula :i* p5etrates into ! activity ( ! sun 9 su* a mann] z to be obs]v$ w describe ei!r a v elongat$ ellipse or an hyp]bola4 ,by ! magnitude ( its axis1 ! latt] w be appreciably conf\nd$ ) a p>abola 9 ! "p :i* is obs]v$2 x is !n n surpris+ t1 up to ?is "t1 hyp]bolic movem5ts h n be5 recogniz$4 ,! attraction ( ! planets1 &1 p]haps fur!r1 ! resi/ance ( ! e!real c5tres1 ought to h *ang$ _m comet>y orbits 9 ! ellipses ^: great axis is less ?an ! radius ( ! sph]e ( activity ( ! sun1 :i* augm5ts ! *ances ( ! elliptical orbits4 ,we may believe t ?is *ange has tak5 place ) ! comet ( #agei1 & ) ! comet ^: duration is only twelve hundr$ days1 & :i* w reappe> )\t ceas+ 9 ?is %ort int]val1 unless ! evaporation :i* x meets at ea* ( its returns to ! p]ihelion ends by r5d]+ x invisible4 ,we are able fur!r1 by ! analysis ( probabilities1 to v]ify ! exi/5ce or ! influ5ce ( c]ta9 causes ^: action is believ$ to exi/ ^u organiz$ be+s4 ,( all ! in/rum5ts t we are able to employ 9 ord] to recognize ! imp]ceptible ag5ts ( nature ! mo/ s5sitive are ! n]ves1 especially :5 p>ticul> causes increase _! s5sibility4 ,x is by _! aid t ! feeble electricity :i* ! contact ( two het]og5e\s metals develops has be5 discov]$2 ?is has op5$ a va/ field to ! rese>*es ( physici/s & *emi/s4 ,! s+ul> ph5om5a :i* results f extreme s5sibility ( ! n]ves 9 "s individuals h giv5 bir? to div]s op9ions ab\t ! exi/5ce ( a new ag5t :i* has be5 nam$ animal magnetism1 ab\t ! action on ord9>y magnetism1 & ab\t ! influ5ce ( ! sun & moon 9 "s n]v\s affections1 & f9ally1 ab\t ! impressions :i* ! proximity ( metals or ( runn+ wat] makes felt4 ,x is natural to ?9k t ! action ( ^! causes is v feeble1 & t x may be easily di/urb$ by accid5tal circum/ances2 ?us because 9 "s cases x is n manife/$ at all its exi/5ce ought n to be d5i$4 ,we are s f> f recogniz+ all ! ag5ts ( nature & _! div]s modes ( action t x w\ld be unphilosophical to d5y ! ph5om5a solely because !y are inexplicable 9 ! pres5t /ate ( our k4 ,b we ought to exam9e !m ) an att5tion z mu* ! m scrupul\s z x appe>s ! m difficult to admit !m2 & x is "h t ! calculation ( probabilities becomes indisp5sable 9 det]m9+ to j :at po9t x is necess>y to multiply ! obs]vations or ! exp]i5ces 9 ord] to obta9 9 favor ( ! ag5ts :i* !y indicate1 a probability sup]ior to ! reasons :i* c be obta9$ else:]e = n admitt+ !m4 ,! calculation ( probabilities c make appreciable ! advantages & ! inconv5i5ces ( ! me?ods employ$ 9 ! speculative sci5ces4 ,?us 9 ord] to recognize ! be/ ( ! treatm5ts 9 use 9 ! heal+ ( a malady1 x is suffici5t to te/ ea* ( !m on an equal numb] ( pati5ts1 mak+ all ! conditions exactly simil>2 ! sup]iority ( ! mo/ advantage\s treatm5t w manife/ itself m & m 9 ! measure t ! numb] is increas$2 & ! calculation w make app>5t ! correspond+ probability ( its advantage & ! ratio accord+ to :i* x is sup]ior to ! o!rs4 ,,*apt] ,x4 ,,application ,,( ,,! ,,calculus ,,( ,,probabilities ,,to ,,! ,,moral ,,sci5ces4 ,we h j se5 ! advantages ( ! analysis ( probabilities 9 ! inve/igation ( ! laws ( natural ph5om5a ^: causes are unkn[n or s complicat$ t _! results _c be submitt$ to calculus4 ,?is is ! case ( ne>ly all subjects ( ! moral sci5ces4 ,s _m un=ese5 causes1 ei!r hidd5 or inappreciable1 influ5ce human in/itutions t x is impossible to judge priori ! results4 ,! s]ies ( ev5ts :i* "t br+s ab\t develops ^! results & indicates ! means ( rem$y+ ^? t are h>mful4 ,wise laws h (t5 be5 made 9 ?is reg>d2 b because we _h neglect$ to cons]ve ! motives _m h be5 abrogat$ z useless1 & ! fact t vexati\s exp]i5ces h made ! ne$ felt anew ought to h re/abli%$ !m4 ,x is v important to keep 9 ea* bran* ( ! public adm9i/ration an exact regi/] ( ! results :i* ! v>i\s means us$ h produc$1 & :i* are s _m exp]i5ces made on a l>ge scale by gov]nm5ts4 ,let u apply to ! political & moral sci5ces ! me?od f\nd$ ^u obs]vation & ^u calculus1 ! me?od :i* has s]v$ u s well 9 ! natural sci5ces4 ,let u n (f] 9 ! lea/ a useless & (t5 dang]\s resi/ance to ! inevitable effects ( ! progress ( k2 b let u *ange only ) an extreme circumspection our in/itutions & ! usages to :i* we h already s long con=m$4 ,we %\ld "k well by ! exp]i5ce ( ! pa/ ! difficulties :i* !y pres5t2 b we are ignorant ( ! ext5t ( ! evils :i* _! *ange c produce4 ,9 ?is ignorance ! !ory ( probability directs u to avoid all *ange2 especially is x necess>y to avoid ! sudd5 *anges :i* 9 ! moral _w z well z 9 ! physical _w nev] op]ate )\t a great loss ( vital =ce4 ,already ! calculus ( probabilities has be5 appli$ ) success to sev]al subjects ( ! moral sci5ces4 ,i %all pres5t "h ! pr9cipal results4 ,,*apt] ,,xi4 ,,conc]n+ ,,! ,,probabilities ,,( ,,te/imonies4 ,! majority ( our op9ions be+ f\nd$ on ! probability ( pro(s x is inde$ important to submit x to calculus4 ,?+s x is true (t5 become impossible by ! difficulty ( appreciat+ ! v]acity ( witnesses & by ! great numb] ( circum/ances :i* accompany ! de$s !y atte/2 b "o is able 9 sev]al cases to resolve ! problems :i* h mu* analogy ) ! que/ions :i* are propos$ & ^: solutions may be reg>d$ z suitable approximations to guide & to def5d u aga9/ ! errors & ! dang]s ( false reason+ to :i* we are expos$4 ,an approximation ( ?is k9d1 :5 x is well made1 is always pref]able to ! mo/ speci\s reason+s4 ,let u try !n to give "s g5]al rules = obta9+ x4 ,a s+le numb] has be5 drawn f an urn :i* conta9s a ?\s& ( !m4 ,a witness to ?is draw+ ann\nces t numb] #gi is drawn2 "o asks ! probability ( draw+ ?is numb]4 ,let u suppose t exp]i5ce has made kn[n t ?is witness deceives "o "t 9 t51 s t ! probability ( 8 te/imony is #a_/aj4 ,"h ! ev5t obs]v$ is ! witness atte/+ t numb] #gi is drawn4 ,?is ev5t may result f ! two foll[+ hypo!ses1 namely3 t ! witness utt]s ! tru? or t he deceives4 ,foll[+ ! pr9ciple t has be5 exp\nd$ on ! probability ( causes drawn f ev5ts obs]v$ x is necess>y fir/ to det]m9e priori ! probability ( ! ev5t 9 ea* hypo!sis4 ,9 ! fir/1 ! probability t ! witness w ann\nce numb] #gi is ! probability itself ( ! draw+ ( ?is numb]1 t is to say1 #a_/ajjj4 ,x is necess>y to multiply x by ! probability #f_/aj ( ! v]acity ( ! witness2 "o w h !n #i_/ajjjj = ! probability ( ! ev5t obs]v$ 9 ?is hypo!sis4 ,if ! witness deceives1 numb] #gi is n drawn1 & ! probability ( ?is case is #iii_/ajjj4 ,b to ann\nce ! draw+ ( ?is numb] ! witness has to *oose x among ! #iii numb]s n drawn2 & z he is suppos$ to h no motive ( pref]5ce = ! ones r ?an ! o!rs1 ! probability t he w *oose numb] #gi is #a_/iii2 multiply+1 !n1 ?is probability by ! prec$+ "o1 we %all h #a_/ajjj = ! probability t ! witness w ann\nce numb] #gi 9 ! second hypo!sis4 ,x is necess>y aga9 to multiply ?is probability by #a_/aj ( ! hypo!sis itself1 :i* gives #a_/ajjjj = ! probability ( ! ev5t relative to ?is hypo!sis4 ,n[ if we =m a fraction ^: num]ator is ! probability relative to ! fir/ hypo!sis1 & ^: d5om9ator is ! sum ( ! probabilities relative to ! two hypo!ses1 we %all h1 by ! six? pr9ciple1 ! probability ( ! fir/ hypo!sis1 & ?is probability w be #i_/aj2 t is to say1 ! v]acity itself ( ! witness4 ,?is is likewise ! probability ( ! draw+ ( numb] #gi4 ,! probability ( ! falsehood ( ! witness & ( ! failure ( draw+ ?is numb] is #a_/aj4 ,if ! witness1 wi%+ to deceive1 has "s int]e/ 9 *oos+ numb] #gi among ! numb]s n drawn1--if he judges1 = example1 t hav+ plac$ ^u ?is numb] a consid]able /ake1 ! ann\ncem5t ( its draw+ w increase 8 cr$it1 ! probability t he w *oose ?is numb] w no long] be z at fir/1 #a_/iii1 x w !n be 1 1 etc41 accord+ to ! int]e/ t he w h 9 ann\nc+ its draw+4 ,suppos+ x to be #a_/i1 x w be necess>y to multiply by ?is fraction ! probability #iii_/ajjj 9 ord] to get 9 ! hypo!sis ( ! falsehood ! probability ( ! ev5t obs]v$1 :i* x is necess>y /ill to multiply by #a_/aj1 :i* gives #aaa_/ajjjj = ! probability ( ! ev5t 9 ! second hypo!sis4 ,!n ! probability ( ! fir/ hypo!sis1 or ( ! draw+ ( numb] #gi1 is r$uc$ by ! prec$+ rule to #i_/abj4 ,x is !n v mu* decreas$ by ! consid]ation ( ! int]e/ :i* ! witness may h 9 ann\nc+ ! draw+ ( numb] #gi4 ,9 tru? ?is same int]e/ increases ! probability #i_/aj t ! witness w speak ! tru? if numb] #gi is drawn4 ,b ?is probability _c exce$ unity or #aj_/aj2 ?us ! probability ( ! draw+ ( numb] #gi w n surpass #aj_/aba4 ,common s5se tells u t ?is int]e/ ought to inspire di/ru/1 b calculus appreciates ! influ5ce ( x4 ,! probability priori ( ! numb] ann\nc$ by ! witness is unity divid$ by ! numb] ( ! numb]s 9 ! urn2 x is *ang$ by virtue ( ! pro( into ! v]acity itself ( ! witness2 x may !n be decreas$ by ! pro(4 ,if1 = example1 ! urn conta9s only two numb]s1 :i* gives = ! probability priori ( ! draw+ ( numb] #a1 & if ! v]acity ( a witness :o ann\nces x is #d_/aj1 ?is draw+ becomes less probable4 ,inde$ x is app>5t1 s9ce ! witness has !n m incl9ation t[>ds a falsehood ?an t[>ds ! tru?1 t 8 te/imony ought to decrease ! probability ( ! fact atte/$ e "t t ?is probability equals or surpasses 4 ,b if !re are ?ree numb]s 9 ! urn ! probability priori ( ! draw+ ( numb] #a is increas$ by ! affirmation ( a witness ^: v]acity surpasses 4 ,suppose n[ t ! urn conta9s #iii black balls & "o :ite ball1 & t "o ball hav+ be5 drawn a witness ( ! draw+ ann\nces t ?is ball is :ite4 ,! probability ( ! ev5t obs]v$1 det]m9$ priori 9 ! fir/ hypo!sis1 w be "h1 z 9 ! prec$+ "q1 equal to #i_/ajjjj4 ,b 9 ! hypo!sis :]e ! witness deceives1 ! :ite ball is n drawn & ! probability ( ?is case is #iii_/ajjj4 ,x is necess>y to multiply x by ! probability #a_/aj ( ! falsehood1 :i* gives #iii_/ajjjj = ! probability ( ! ev5t obs]v$ relative to ! second hypo!sis4 ,?is probability 0 only #a_/ajjjj 9 ! prec$+ "q2 ?is great diff]5ce results f ?is--t a black ball hav+ be5 drawn ! witness :o wi%es to deceive has no *oice at all to make among ! #iii balls n drawn 9 ord] to ann\nce ! draw+ ( a :ite ball4 ,n[ if "o =ms two fractions ^: num]ators are ! probabilities relative to ea* hypo!sis1 & ^: common d5om9ator is ! sum ( ^! probabilities1 "o w h #i_/ajjh = ! probability ( ! fir/ hypo!sis & ( ! draw+ ( a :ite ball1 & #iii_/ajjh = ! probability ( ! second hypo!sis & ( ! draw+ ( a black ball4 ,?is la/ probability /rongly approa*es c]ta9ty2 x w\ld approa* x mu* ne>] & w\ld become #iiiiii_/ajjjjjh if ! urn conta9$ a million balls ( :i* "o 0 :ite1 ! draw+ ( a :ite ball becom+ !n mu* m extraord9>y4 ,we see ?us h[ ! probability ( ! falsehood increases 9 ! measure t ! de$ becomes m extraord9>y4 ,we h suppos$ up to ?is "t t ! witness 0 n mi/ak5 at all2 b if "o admits1 h[ev]1 ! *ance ( 8 error ! extraord9>y incid5t becomes m improbable4 ,!n 9 place ( ! two hypo!ses "o w h ! f\r foll[+ ones1 namely3 t ( ! witness n deceiv+ & n be+ mi/ak5 at all2 t ( ! witness n deceiv+ at all & be+ mi/ak52 ! hypo!sis ( ! witness deceiv+ & n be+ mi/ak5 at all2 f9ally1 t ( ! witness deceiv+ & be+ mi/ak54 ,det]m9+ priori 9 ea* ( ^! hypo!ses ! probability ( ! ev5t obs]v$1 we f9d by ! six? pr9ciple ! probability t ! fact atte/$ is false equal to a fraction ^: num]ator is ! numb] ( black balls 9 ! urn multipli$ by ! sum ( ! probabilities t ! witness does n deceive at all & is mi/ak51 or t he deceives & is n mi/ak51 & ^: d5om9ator is ?is num]ator augm5t$ by ! sum ( ! probabilities t ! witness does n deceive at all & is n mi/ak5 at all1 or t he deceives & is mi/ak5 at ! same "t4 ,we see by ?is t if ! numb] ( black balls 9 ! urn is v great1 :i* r5d]s ! draw+ ( ! :ite ball extraord9>y1 ! probability t ! fact atte/$ is n true approa*es mo/ ne>ly to c]ta9ty4 ,apply+ ?is conclusion to all extraord9>y de$s x results f x t ! probability ( ! error or ( ! falsehood ( ! witness becomes z mu* great] z ! fact atte/$ is m extraord9>y4 ,"s au?ors h advanc$ ! contr>y on ?is basis t ! view ( an extraord9>y fact be+ p]fectly simil> to t ( an ord9>y fact ! same motives ought to lead u to give ! witness ! same cr$5ce :5 he affirms ! "o or ! o!r ( ^! facts4 ,simple common s5se rejects su* a /range ass]tion2 b ! calculus ( probabilities1 :ile confirm+ ! f9d+s ( common s5se1 appreciates ! greate/ improbability ( te/imonies 9 reg>d to extraord9>y facts4 ,^! au?ors insi/ & suppose two witnesses equally wor?y ( belief1 ( :om ! fir/ atte/s t he saw an individual dead fifte5 days ago :om ! second witness affirms to h se5 ye/]day full ( life4 ,! "o or ! o!r ( ^! facts (f]s no improbability4 ,! res]vation ( ! individual is a result ( _! comb9ation2 b ! te/imonies d n br+ u at all directly to ?is result1 al?\gh ! cr$5ce :i* is due ^! te/imonies ought n to be decreas$ by ! fact t ! result ( _! comb9ation is extraord9>y4 ,b if ! conclusion :i* results f ! comb9ation ( ! te/imonies 0 impossible "o ( !m w\ld be necess>ily false2 b an impossible conclusion is ! limit ( extraord9>y conclusions1 z error is ! limit ( improbable conclusions2 ! value ( ! te/imonies :i* becomes z]o 9 ! case ( an impossible conclusion ought !n to be v mu* decreas$ 9 t ( an extraord9>y conclusion4 ,?is is inde$ confirm$ by ! calculus ( probabilities4 ,9 ord] to make x pla9 let u consid] two urns1 ,a & ,b1 ( :i* ! fir/ conta9s a million :ite balls & ! second a million black balls4 ,"o draws f "o ( ^! urns a ball1 :i* he puts back into ! o!r urn1 f :i* "o !n draws a ball4 ,two witnesses1 ! "o ( ! fir/ draw+1 ! o!r ( ! second1 atte/ t ! ball :i* !y h se5 drawn is :ite )\t indicat+ ! urn f :i* x has be5 drawn4 ,ea* te/imony tak5 alone is n improbable2 & x is easy to see t ! probability ( ! fact atte/$ is ! v]acity itself ( ! witness4 ,b x foll[s f ! comb9ation ( ! te/imonies t a :ite ball has be5 extract$ f ! urn ,a at ! fir/ draw1 & t !n plac$ 9 ! urn ,b x has reappe>$ at ! second draw1 :i* is v extraord9>y2 = ?is second urn1 conta9+ !n "o :ite ball among a million black balls1 ! probability ( draw+ ! :ite ball is #a_/ajjjjja4 ,9 ord] to det]m9e ! dim9ution :i* results 9 ! probability ( ! ?+ ann\nc$ by ! two witnesses we %all notice t ! ev5t obs]v$ is "h ! affirmation by ea* ( !m t ! ball :i* he has se5 extract$ is :ite4 ,let u repres5t by #i_/aj ! probability t he ann\nces ! tru?1 :i* c occur 9 ! pres5t case :5 ! witness does n deceive & is n mi/ak5 at all1 & :5 he deceives & is mi/ak5 at ! same "t4 ,"o may =m ! f\r foll[+ hypo!ses3 #a/4 ,! fir/ & second witness speak ! tru?4 ,!n a :ite ball has at fir/ be5 drawn f ! urn ,a1 & ! probability ( ?is ev5t is 1 s9ce ! ball drawn at ! fir/ draw may h be5 drawn ei!r f ! "o or ! o!r urn4 ,consequ5tly ! ball drawn1 plac$ 9 ! urn ,b1 has reappe>$ at ! second draw2 ! probability ( ?is ev5t is #a_/ajjjjja1 ! probability ( ! fact ann\nc$ is !n #a_/bjjjjjb4 ,multiply+ x by ! product ( ! probabilities #i_/aj & #i_/aj t ! witnesses speak ! tru? "o w h #ha_/bjjjjjbjj = ! probability ( ! ev5t obs]v$ 9 ?is fir/ hypo!sis4 #b;d4 ,! fir/ witness speaks ! tru? & ! second does n1 :e!r he deceives & is n mi/ak5 or he does n deceive & is mi/ak54 ,!n a :ite ball has be5 drawn f ! urn ,a at ! fir/ draw1 & ! probability ( ?is ev5t is 4 ,!n ?is ball hav+ be5 plac$ 9 ! urn ,b a black ball has be5 drawn f x3 ! probability ( su* draw+ is #ajjjjjj_/ajjjjja2 "o has !n #ajjjjjj_/bjjjjjb = ! probability ( ! comp\nd ev5t4 ,multiply+ x by ! product ( ! two probabilities #i_/aj & #a_/aj t ! fir/ witness speaks ! tru? & t ! second does n1 "o w h #ijjjjjj_/bjjjjjbjj = ! probability = ! ev5t obs]v$ 9 ! second hypo!sis4 #c;d4 ,! fir/ witness does n speak ! tru? & ! second ann\nces x4 ,!n a black ball has be5 drawn f ! urn ,b at ! fir/ draw+1 & aft] hav+ be5 plac$ 9 ! urn ,a a :ite ball has be5 drawn f ?is urn4 ,! probability ( ! fir/ ( ^! ev5ts is & t ( ! second is #ajjjjjj_/ajjjjja2 ! probability ( ! comp\nd ev5t is !n #ajjjjjj_/bjjjjjb4 ,multiply+ x by ! product ( ! probabilities #a_/aj & #i_/aj t ! fir/ witness does n speak ! tru? & t ! second ann\nces x1 "o w h #ijjjjjj_/bjjjjjbjj = ! probability ( ! ev5t obs]v$ relative to ?is hypo!sis4 #d?4 ,f9ally1 nei!r ( ! witnesses speaks ! tru?4 ,!n a black ball has be5 drawn f ! urn ,b at ! fir/ draw2 !n hav+ be5 plac$ 9 ! urn ,a x has reappe>$ at ! second draw+3 ! probability ( ?is comp\nd ev5t is #a_/bjjjjjb4 ,multiply+ x by ! product ( ! probabilities #a_/aj & #a_/aj t ea* witness does n speak ! tru? "o w h #a_/bjjjjjbjj = ! probability ( ! ev5t obs]v$ 9 ?is hypo!sis4 ,n[ 9 ord] to obta9 ! probability ( ! ?+ ann\nc$ by ! two witnesses1 namely1 t a :ite ball has be5 drawn at ea* draw1 x is necess>y to divide ! probability correspond+ to ! fir/ hypo!sis by ! sum ( ! probabilities relative to ! f\r hypo!ses2 & !n "o has = ?is probability #ha_/ahjjjjhb1 an extremely small fraction4 ,if ! two witnesses affirm ! fir/1 t a :ite ball has be5 drawn f "o ( ! two urns ,a & ,b2 ! second t a :ite ball has be5 likewise drawn f "o ( ! two urns ,a & ,b1 q simil> to ! fir/ ones1 ! probability ( ! ?+ ann\nc$ by ! two witnesses w be ! product ( ! probabilities ( _! te/imonies1 or #ha_/ajj2 x w !n be at lea/ a hundr$ & eighty ?\s& times great] ?an ! prec$+ "o4 ,"o sees by ?is h[ mu*1 9 ! fir/ case1 ! reappe>ance at ! second draw ( ! :ite ball drawn at ! fir/ draw1 ! extraord9>y conclusion ( ! two te/imonies decreases ! value ( x4 ,we w\ld give no cr$5ce to ! te/imony ( a man :o %\ld atte/ to u t 9 ?r[+ a hundr$ dice into ! air !y _h all fall5 on ! same face4 ,if we _h ourselves be5 spectators ( ?is ev5t we %\ld believe our own eyes only aft] hav+ c>efully exam9$ all ! circum/ances1 & aft] hav+ br\ght 9 ! te/imonies ( o!r eyes 9 ord] to be q sure t !re _h be5 nei!r halluc9ation nor deception4 ,b aft] ?is exam9ation we %\ld n hesitate to admit x 9 spite ( its extreme improbability2 & no "o w\ld be tempt$1 9 ord] to expla9 x1 to recur to a d5ial ( ! laws ( vision4 ,we ought to conclude f x t ! probability ( ! con/ancy ( ! laws ( nature is = u great] ?an ?is1 t ! ev5t 9 "q has n tak5 place at all--a probability great] ?an t ( ! majority ( hi/orical facts :i* we reg>d z inconte/able4 ,"o may judge by ?is ! imm5se weight ( te/imonies necess>y to admit a susp5sion ( natural laws1 & h[ improp] x w\ld be to apply to ?is case ! ord9>y rules ( criticism4 ,all ^? :o )\t (f]+ ?is imm5sity ( te/imonies support ?is :5 mak+ recitals ( ev5ts contr>y to ^? laws1 decrease r ?an augm5t ! belief :i* !y wi% to inspire2 = !n ^? recitals r5d] v probable ! error or ! falsehood ( _! au?ors4 ,b t :i* dim9i%es ! belief ( educat$ m5 increases (t5 t ( ! un$ucat$1 always gre$y = ! wond]ful4 ,!re are ?+s s extraord9>y t no?+ c balance _! improbability4 ,b ?is1 by ! effect ( a dom9ant op9ion1 c be weak5$ to ! po9t ( appe>+ inf]ior to ! probability ( ! te/imonies2 & :5 ?is op9ion *anges an absurd /atem5t admitt$ unanim\sly 9 ! c5tury :i* has giv5 x bir? (f]s to ! foll[+ c5turies only a new pro( ( ! extreme influ5ce ( ! g5]al op9ion ^u ! m enlight5$ m9ds4 ,two great m5 ( ! c5tury ( ,l\is ,,xiv4--,rac9e & ,pascal--are /rik+ examples ( ?is4 ,x is pa9ful to see ) :at complaisance ,rac9e1 ?is admirable pa9t] ( ! human he>t & ! mo/ p]fect poet t has ev] liv$1 reports z miracul\s ! recov]y ( ,mlle4 ,p]ri]1 a niece ( ,pascal & a day pupil at ! mona/]y ( ,port-,royal2 x is pa9ful to read ! reasons by :i* ,pascal seeks to prove t ?is miracle %\ld be necess>y to religion 9 ord] to ju/ify ! doctr9e ( ! monks ( ?is abbey1 at t "t p]secut$ by ! ,jesuits4 ,! "y ,p]ri] _h be5 afflict$ = ?ree ye>s & a half by a la*rymal fi/ula2 %e t\*$ h] afflict$ eye ) a relic :i* 0 pret5d$ to be "o ( ! ?orns ( ! cr[n ( ! ,savi\r & %e _h fai? 9 in/ant recov]y4 ,"s days aft]w>d ! physicians & ! surgeons atte/ ! recov]y1 & !y decl>e t nature & ! rem$ies h _h no "p 9 x4 ,?is ev5t1 :i* took place 9 #afef1 made a great s5sation1 & all ,p>is ru%$1 says ,rac9e1 to ,port-,royal4 ,! cr[d increas$ f day to day1 & ,god himself seem$ to take pleasure 9 au?oriz+ ! devotion ( ! p by ! numb] ( miracles :i* 7 p]=m$ 9 ?is *ur*4 ,at ?is "t miracles & sorc]y did n yet appe> improbable1 & "o did n hesitate at all to attribute to !m ! s+ul>ities ( nature :i* c\ld n be expla9$ o!rwise4 ,?is mann] ( view+ extraord9>y results is f\nd 9 ! mo/ rem>kable works ( ! c5tury ( ,l\is ,,xiv42 ev5 9 ! ,essay on ! ,human ,und]/&+ by ! philosoph] ,locke1 :o says1 9 speak+ ( ! degree ( ass5t3 ,?\gh ! common exp]i5ce & ! ord9>y c\rse ( ?+s h ju/ly a mighty influ5ce on ! m9ds ( m51 to make !m give or refuse cr$it to any?+ propos$ to _! belief2 yet !re is "o case1 :]e9 ! /rang5ess ( ! fact less5s n ! ass5t to a fair te/imony ( x4 ,= :]e su* sup]natural ev5ts are suitable to ends aim$ at by him :o has ! p[] to *ange ! c\rse ( nature1 !re1 "u su* circum/ances1 !y may be ! fitt] to procure belief1 by h[ mu* ! m !y are beyond or contr>y to ord9>y obs]vation4 ,! true pr9ciples ( ! probability ( te/imonies hav+ be5 ?us misund]/ood by philosoph]s to :om reason is pr9cipally indebt$ = its progress1 ,i h ?\ght x necess>y to pres5t at l5g? ! results ( calculus ^u ?is important subject4 ,!re comes up naturally at ?is po9t ! discussion ( a fam\s argum5t ( ,pascal1 t ,craig1 an ,engli% ma!matician1 has produc$ "u a geometric =m4 ,witnesses decl>e t !y h x f ,div9ity t 9 con=m+ to a c]ta9 ?+ "o w enjoy n "o or two b an inf9ity ( happy lives4 ,h[ev] feeble ! probability ( ! pro(s may be1 provid$ t x be n inf9itely small1 x is cle> t ! advantage ( ^? :o con=m to ! prescrib$ ?+ is inf9ite s9ce x is ! product ( ?is probability & an inf9ite good2 "o ought n to hesitate !n to procure = oneself ?is advantage4 ,?is argum5t is bas$ ^u ! inf9ite numb] ( happy lives promis$ 9 ! "n ( ! ,div9ity by ! witnesses2 x is necess>y !n to prescribe !m1 precisely because !y exagg]ate _! promises beyond all limits1 a consequ5ce :i* is repugnant to good s5se4 ,also calculus tea*es u t ?is exagg]ation itself enfeebles ! probability ( _! te/imony to ! po9t ( r5d]+ x inf9itely small or z]o4 ,inde$ ?is case is simil> to t ( a witness :o %\ld ann\nce ! draw+ ( ! highe/ numb] f an urn fill$ ) a great numb] ( numb]s1 "o ( :i* has be5 drawn & :o w\ld h a great int]e/ 9 ann\nc+ ! draw+ ( ?is numb]4 ,"o has already se5 h[ mu* ?is int]e/ enfeebles 8 te/imony4 ,9 evaluat+ only at ! probability t if ! witness deceives he w *oose ! l>ge/ numb]1 calculus gives ! probability ( 8 ann\ncem5t z small] ?an a fraction ^: num]ator is unity & ^: d5om9ator is unity plus ! half ( ! product ( ! numb] ( ! numb]s by ! probability ( falsehood consid]$ priori or indep5d5tly ( ! ann\ncem5t4 ,9 ord] to comp>e ?is case to t ( ! argum5t ( ,pascal x is suffici5t to repres5t by ! numb]s 9 ! urn all ! possible numb]s ( happy lives :i* ! numb] ( ^! numb]s r5d]s inf9ite2 & to obs]ve t if ! witnesses deceive !y h ! greate/ int]e/1 9 ord] to accr$it _! falsehood1 9 promis+ an et]nity ( happ9ess4 ,! expression ( ! probability ( _! te/imony becomes !n inf9itely small4 ,multiply+ x by ! inf9ite numb] ( happy lives promis$1 inf9ity w\ld disappe> f ! product :i* expresses ! advantage resultant f ?is promise :i* de/roys ! argum5t ( ,pascal4 ,let u consid] n[ ! probability ( ! totality ( sev]al te/imonies ^u an e/abli%$ fact4 ,9 ord] to fix our ideas let u suppose t ! fact be ! draw+ ( a numb] f an urn :i* conta9s a hundr$ ( !m1 & ( :i* "o s+le numb] has be5 drawn4 ,two witnesses ( ?is draw+ ann\nce t numb] #b has be5 drawn1 & "o asks = ! resultant probability ( ! totality ( ^! te/imonies4 ,"o may =m ^! two hypo!ses3 ! witnesses speak ! tru?2 ! witnesses deceive4 ,9 ! fir/ hypo!sis ! numb] #b is drawn & ! probability ( ?is ev5t is #a_/ajj4 ,x is necess>y to multiply x by ! product ( ! v]acities ( ! witnesses1 v]acities :i* we w suppose to be #i_/aj & #g_/aj3 "o w h !n #fc_/ajjjj = ! probability ( ! ev5t obs]v$ 9 ?is hypo!sis4 ,9 ! second1 ! numb] #b is n drawn & ! probability ( ?is ev5t is #ii_/ajj4 ,b ! agreem5t ( ! witnesses requires !n t 9 seek+ to deceive !y bo? *oose ! numb] #b f ! #ii numb]s n drawn3 ! probability ( ?is *oice if ! witnesses d n h a secret agreem5t is ! product ( ! fraction #a_/ii by itself2 x becomes necess>y !n to multiply ^! two probabilities toge!r1 & by ! product ( ! probabilities #a_/aj & #c_/aj t ! witnesses deceive2 "o w h ?us #a_/ccjjjj = ! probability ( ! ev5t obs]v$ 9 ! second hypo!sis4 ,n[ "o w h ! probability ( ! fact atte/$ or ( ! draw+ ( numb] #b 9 divid+ ! probability relative to ! fir/ hypo!sis by ! sum ( ! probabilities relative to ! two hypo!ses2 ?is probability w be !n #bjgi_/bjhj1 & ! probability ( ! failure to draw ?is numb] & ( ! falsehood ( ! witnesses w be #a_/bjhj4 ,if ! urn %\ld conta9 only ! numb]s #a & #b "o w\ld f9d 9 ! same mann] #ba_/bb = ! probability ( ! draw+ ( numb] #b1 & consequ5tly #a_/bb = ! probability ( ! falsehood ( ! witnesses1 a probability at lea/ n9ety-f\r times l>g] ?an ! prec$+ "o4 ,"o sees by ?is h[ mu* ! probability ( ! falsehood ( ! witnesses dim9i%es :5 ! fact :i* !y atte/ is less probable 9 itself4 ,inde$ "o conceives t !n ! accord ( ! witnesses1 :5 !y deceive1 becomes m difficult1 at lea/ :5 !y d n h a secret agreem5t1 :i* we d n suppose "h at all4 ,9 ! prec$+ case :]e ! urn conta9$ only two numb]s ! priori probability ( ! fact atte/$ is 1 ! resultant probability ( ! te/imonies is ! product ( ! v]acities ( ! witnesses divid$ by ?is product add$ to t ( ! respective probabilities ( _! falsehood4 ,x n[ rema9s = u to consid] ! influ5ce ( "t ^u ! probability ( facts transmitt$ by a traditional *a9 ( witnesses4 ,x is cle> t ?is probability ought to dim9i% 9 proportion z ! *a9 is prolong$4 ,if ! fact has no probability itself1 su* z ! draw+ ( a numb] f an urn :i* conta9s an inf9ity ( !m1 t :i* x acquires by ! te/imonies decreases accord+ to ! cont9u$ product ( ! v]acity ( ! witnesses4 ,if ! fact has a probability 9 itself2 if1 = example1 ?is fact is ! draw+ ( ! numb] #b f an urn :i* conta9s an inf9ity ( !m1 & ( :i* x is c]ta9 t "o has drawn a s+le numb]2 t :i* ! traditional *a9 adds to ?is probability decreases1 foll[+ a cont9u$ product ( :i* ! fir/ factor is ! ratio ( ! numb] ( numb]s 9 ! urn less "o to ! same numb]1 & ( :i* ea* o!r factor is ! v]acity ( ea* witness dim9i%$ by ! ratio ( ! probability ( 8 falsehood to ! numb] ( ! numb]s 9 ! urn less "o2 s t ! limit ( ! probability ( ! fact is t ( ?is fact consid]$ priori1 or indep5d5tly ( ! te/imonies1 a probability equal to unity divid$ by ! numb] ( ! numb]s 9 ! urn4 ,! action ( "t enfeebles !n1 )\t ceas+1 ! probability ( hi/orical facts j z x *anges ! mo/ durable monum5ts4 ,"o c inde$ dim9i% x by multiply+ & cons]v+ ! te/imonies & ! monum5ts :i* support !m4 ,pr9t+ (f]s = ?is purpose a great means1 un=tunately unkn[n to ! anci5ts4 ,9 spite ( ! inf9ite advantages :i* x procures ! physical & moral revolutions by :i* ! surface ( ?is globe w always be agitat$ w end1 9 conjunction ) ! inevitable effect ( "t1 by r5d]+ d\btful aft] ?\s&s ( ye>s ! hi/orical facts reg>d$ to-day z ! mo/ c]ta94 ,craig has tri$ to submit to calculus ! gradual enfeebl+ ( ! pro(s ( ! ,*ri/ian religion2 suppos+ t ! _w ought to end at ! epo* :5 x w cease to be probable1 he f9ds t ?is ought to take place #aded ye>s aft] ! "t :5 he writes4 ,b 8 analysis is z faulty z 8 hypo!sis ^u ! duration ( ! moon is biz>re4 ,,*apt] ,,xii4 ,,conc]n+ ,,! ,,selections ,,& ,,! ,,decisions ,,( ,,assemblies4 ,! probability ( ! decisions ( an assembly dep5ds ^u ! plurality ( votes1 ! intellig5ce & ! imp>tiality ( ! memb]s :o compose x4 ,s _m passions & p>ticul> int]e/s s (t5 add _! influ5ce t x is impossible to submit ?is probability to calculus4 ,!re are1 h[ev]1 "s g5]al results dictat$ by simple common s5se & confirm$ by calculus4 ,if1 = example1 ! assembly is poorly in=m$ ab\t ! subject submitt$ to its decision1 if ?is subject requires delicate consid]ations1 or if ! tru? on ?is po9t is contr>y to e/abli%$ prejudices1 s t x w\ld be a bet ( m ?an "o aga9/ "o t ea* vot] w err2 !n ! decision ( ! majority w be probably wrong1 & ! fe> ( x w be ! bett] bas$ z ! assembly is m num]\s4 ,x is important !n1 9 public affairs1 t assemblies %\ld h to pass ^u subjects )9 rea* ( ! greate/ numb]2 x is important = !m t in=mation be g5]ally diffus$ & t good works f\nd$ ^u reason & exp]i5ce %\ld enlight5 ^? :o are call$ to decide ! lot ( _! fell[s or to gov]n !m1 & %\ld =ew>n !m aga9/ false ideas & ! prejudices ( ignorance4 ,s*ol>s h _h frequ5t occasion to rem>k t fir/ conceptions (t5 deceive & t ! tru? is n always probable4 ,x is difficult to und]/& & to def9e ! desire ( an assembly 9 ! mid/ ( a v>iety ( op9ions ( its memb]s4 ,let u attempt to give "s rules 9 reg>d to ?is matt] by consid]+ ! two mo/ ord9>y cases3 ! election among sev]al c&idates1 & t among sev]al propositions relative to ! same subject4 ,:5 an assembly has to *oose among sev]al c&idates :o pres5t !mselves = "o or = sev]al places ( ! same k9d1 t :i* appe>s simple/ is to h ea* vot] write ^u a ticket ! names ( all ! c&idates accord+ to ! ord] ( m]it t he attributes to !m4 ,suppos+ t he classifies !m 9 good fai?1 ! inspection ( ^! tickets w give ! results ( ! elections 9 su* a mann] t ! c&idates may be comp>$ among !mselves2 s t new elections c give no?+ m 9 ?is reg>d4 ,x is a "q n[ to conclude ! ord] ( pref]5ce :i* ! tickets e/abli% among ! c&idates4 ,let u imag9e t "o gives to ea* vot] an urn :i* conta9s an inf9ity ( balls by means ( :i* he is able to %ade all ! degrees ( m]it ( ! c&idates2 let u conceive aga9 t he draws f 8 urn a numb] ( balls proportional to ! m]it ( ea* c&idate1 & let u suppose ?is numb] writt5 ^u a ticket at ! side ( ! "n ( ! c&idate4 ,x is cle> t by mak+ a sum ( all ! numb]s relative to ea* c&idate ^u ea* ticket1 t "o ( all ! c&idates :o %all h ! l>ge/ sum w be ! c&idate :om ! assembly pref]s2 & t 9 g5]al ! ord] ( pref]5ce ( ! c&idates w be t ( ! sums relative to ea* ( !m4 ,b ! tickets d n m>k at all ! numb] ( balls :i* ea* vot] gives to ! c&idates2 !y indicate solely t ! fir/ has m ( !m ?an ! second1 ! second m ?an ! ?ird1 & s on4 ,9 suppos+ !n at fir/ ^u a giv5 ticket a c]ta9 numb] ( balls all ! comb9ations ( ! inf]ior numb]s :i* fulfil ! prec$+ conditions are equally admissible2 & "o w h ! numb] ( balls relative to ea* c&idate by mak+ a sum ( all ! numb]s :i* ea* comb9ation gives him & divid+ x by ! entire numb] ( comb9ations4 ,a v simple analysis %[s t ! numb]s :i* mu/ be writt5 ^u ea* ticket at ! side ( ! la/ "n1 ( ! "o be=e ! la/1 etc41 are proportional to ! t]ms ( ! ari?metical progression #a1 #b1 #c1 etc4 ,writ+ !n ?us ^u ea* ticket ! t]ms ( ?is progression1 & add+ ! t]ms relative to ea* c&idate ^u ^! tickets1 ! div]s sums w indicate by _! magnitude ! ord] ( _! pref]5ce :i* ought to be e/abli%$ among ! c&idates4 ,su* is ! mode ( election :i* ,! ,!ory ( ,probabilities indicates4 ,)\t d\bt x w\ld be bett] if ea* vot] %\ld write ^u 8 ticket ! names ( ! c&idates 9 ! ord] ( m]it :i* he attributes to !m4 ,b p>ticul> int]e/s & _m /range consid]ations ( m]it w\ld affect ?is ord] & place sometimes 9 ! la/ rank ! c&idate mo/ =midable to t "o :om "o pref]s1 :i* gives too great an advantage to ! c&idates ( m$iocre m]it4 ,likewise exp]i5ce has caus$ ! ab&onm5t ( ?is mode ( election 9 ! societies :i* _h adopt$ x4 ,! election by ! absolute majority ( ! suffrages unites to ! c]ta9ty ( n admitt+ any "o ( ! c&idates :om ?is majority rejects1 ! advantage ( express+ mo/ (t5 ! desire ( ! assembly4 ,x always co9cides ) ! prec$+ mode :5 !re are only two c&idates4 ,inde$ x exposes an assembly to ! inconv5i5ce ( r5d]+ elections int]m9able4 ,b exp]i5ce has %[n t ?is inconv5i5ce is nil1 & t ! g5]al desire to put an end to elections soon unites ! majority ( ! suffrages ^u "o ( ! c&idates4 ,! *oice among sev]al propositions relative to ! same object ought to be subject$1 seem+ly1 to ! same rules z ! election among sev]al c&idates4 ,b !re exi/s betwe5 ! two cases ?is diff]5ce1 namely1 t ! m]it ( a c&idate does n exclude t ( 8 competitors2 b if x is necess>y to *oose among propositions :i* are contr>y1 ! tru? ( ! "o excludes ! tru? ( ! o!rs4 ,let u see h[ "o ought !n to view ?is "q4 ,let u give to ea* vot] an urn :i* conta9s an inf9ite numb] ( balls1 & let u suppose t he di/ributes !m ^u ! div]s propositions accord+ to ! respective probabilities :i* he attributes to !m4 ,x is cle> t ! total numb] ( balls express+ c]ta9ty1 & ! vot] be+ by ! hypo!sis assur$ t "o ( ! propositions ought to be true1 he w di/ribute ?is numb] at l5g? ^u ! propositions4 ,! problem is r$uc$ !n to ?is1 namely1 to det]m9e ! comb9ations 9 :i* ! balls w be di/ribut$ 9 su* a mann] t !re may be m ( !m ^u ! fir/ proposition ( ! ticket ?an ^u ! second1 m ^u ! second ?an ^u ! ?ird1 etc42 to make ! sums ( all ! numb]s ( balls relative to ea* proposition 9 ! div]s comb9ations1 & to divide ?is sum by ! numb] ( comb9ations2 ! quoti5ts w be ! numb]s ( balls t "o ought to attribute to ! propositions ^u a c]ta9 ticket4 ,"o f9ds by analysis t 9 go+ f ! la/ proposition ^! quoti5ts are among !mselves z ! foll[+ quantities3 fir/1 unity divid$ by ! numb] ( propositions2 second1 ! prec$+ quantity1 augm5t$ by unity1 divid$ by ! numb] ( propositions less "o2 ?ird1 ?is second quantity1 augm5t$ by unity1 divid$ by ! numb] ( propositions less two1 & s on = ! o!rs4 ,"o w write !n ^u ea* ticket ^! quantities at ! side ( ! correspond+ propositions1 & add+ ! relative quantities to ea* proposition ^u ! div]s tickets ! sums w indicate by _! magnitude ! ord] ( pref]5ce :i* ! assembly gives to ^! propositions4 ,let u speak a ^w ab\t ! mann] ( r5ew+ assemblies :i* %\ld *ange 9 totality 9 a def9ite numb] ( ye>s4 ,ought ! r5ewal to be made at "o "t1 or is x advantage\s to divide x among ^! ye>s8 ,accord+ to ! la/ me?od ! assembly w\ld be =m$ "u ! influ5ce ( ! div]s op9ions dom9ant dur+ ! "t ( its r5ewal2 ! op9ion :i* obta9$ !n w\ld be probably ! mean ( all ^! op9ions4 ,! assembly w\ld receive ?us at ! "t ! same advantage t is giv5 to x by ! ext5sion ( ! elections ( its memb]s to all p>ts ( ! t]ritory :i* x repres5ts4 ,n[ if "o consid]s :at exp]i5ce has only too cle>ly taught1 namely1 t elections are always direct$ 9 ! greate/ degree by dom9ant op9ions1 "o w feel h[ useful x is to temp] ^! op9ions1 ! ones by ! o!rs1 by means ( a p>tial r5ewal4 ,,*apt] ,,xiii4 ,,conc]n+ ,,! ,,probability ,,( ,,! ,,judgm5ts ,,( ,,tribunals4 ,analysis confirms :at simple common s5se tea*es u1 namely1 ! correctness ( judgm5ts is z mu* m probable z ! judges are m num]\s & m enlight5$4 ,x is important !n t tribunals ( appeal %\ld fulfil ^! two conditions4 ,! tribunals ( ! fir/ in/ance /&+ 9 clos] relation to ^? am5able (f] to ! high] tribunal ! advantage ( a fir/ judgm5t already probable1 & ) :i* ! latt] (t5 agree1 be x 9 compromis+ or 9 desi/+ f _! claims4 ,b if ! unc]ta9ty ( ! matt] 9 litigation & its importance det]m9e a litigant to h rec\rse to ! tribunal ( appeals1 he ought to f9d 9 a great] probability ( obta9+ an equitable judgm5t great] security = 8 =tune & ! comp5sation = ! tr\ble & exp5se :i* a new proc$ure entails4 ,x is ?is :i* _h no place 9 ! in/itution ( ! reciprocal appeal ( ! tribunals ( ! di/rict1 an in/itution !reby v prejudicial to ! int]e/ ( ! citiz5s4 ,x w\ld be p]haps prop] & con=mable to ! calculus ( probabilities to dem& a majority ( at lea/ two votes 9 a tribunal ( appeal 9 ord] to invalidate ! s5t5ce ( ! l[] tribunal4 ,"o w\ld obta9 ?is result if ! tribunal ( appeal be+ compos$ ( an ev5 numb] ( judges ! s5t5ce %\ld /& 9 ! case ( ! equality ( votes4 ,i %all consid] p>ticul>ly ! judgm5ts 9 crim9al matt]s4 ,9 ord] to condemn an accus$ x is necess>y )\t d\bt t ! judges %\ld h ! /ronge/ pro(s ( 8 (f5ce4 ,b a moral pro( is nev] m ?an a probability2 & exp]i5ce has only too cle>ly %[n ! errors ( :i* crim9al judgm5ts1 ev5 ^? :i* appe> to be ! mo/ j1 are /ill susceptible4 ,! impossibility ( am5d+ ^! errors is ! /ronge/ argum5t ( ! philosoph]s :o h wi%$ to proscribe ! p5alty ( dea?4 ,we %\ld !n be oblig$ to ab/a9 f judg+ if x 7 necess>y = u to await ma!matical evid5ce4 ,b ! judgm5t is requir$ by ! dang] :i* w\ld result f ! impunity ( ! crime4 ,?is judgm5t r$uces itself1 if ,i am n mi/ak51 to ! solution ( ! foll[+ "q3 ,has ! pro( ( ! (f5ce ( ! accus$ ! high degree ( probability necess>y s t ! citiz5s w\ld h less reason to d\bt ! errors ( ! tribunals1 if he is innoc5t & condemn$1 ?an !y w\ld h to fe> 8 new crimes & ^? ( ! un=tunate ones :o w\ld be embold5$ by ! example ( 8 impunity if he 7 guilty & acquitt$8 ,! solution ( ?is "q dep5ds ^u sev]al elem5ts v difficult to asc]ta94 ,su* is ! em95ce ( dang] :i* w\ld ?reat5 society if ! crim9al accus$ %\ld rema9 unpuni%$4 ,sometimes ?is dang] is s great t ! magi/rate sees himself con/ra9$ to waive =ms wisely e/abli%$ = ! protection ( innoc5ce4 ,b t :i* r5d]s almo/ always ?is "q insoluble is ! impossibility ( appreciat+ exactly ! probability ( ! (f5ce & ( fix+ t :i* is necess>y = ! condemnation ( ! accus$4 ,ea* judge 9 ?is respect is =c$ to rely ^u 8 own judgm5t4 ,he =ms 8 op9ion by comp>+ ! div]s te/imonies & ! circum/ances by :i* ! (f5ce is accompani$1 to ! results ( 8 reflections & 8 exp]i5ces1 & 9 ?is respect a long habitude ( int]rogat+ & judg+ accus$ p]sons gives great advantage 9 asc]ta9+ ! tru? 9 ! mid/ ( indices (t5 contradictory4 ,! prec$+ "q dep5ds aga9 ^u ! c>e tak5 9 ! inve/igation ( ! (f5ce2 = "o dem&s naturally mu* /rong] pro(s = impos+ ! dea? p5alty ?an = inflict+ a det5tion ( "s mon?s4 ,x is a reason = proportion+ ! c>e to ! (f5ce1 great c>e tak5 ) an unimportant case inevitably cle>+ _m guilty ones4 ,a law :i* gives to ! judges p[] ( mod]at+ ! c>e 9 ! case ( att5uat+ circum/ances is !n con=mable at ! same "t to pr9ciples ( humanity t[>ds ! culprit1 & to ! int]e/ ( society4 ,! product ( ! probability ( ! (f5ce by its gravity be+ ! measure ( ! dang] to :i* ! acquittal ( ! accus$ c expose society1 "o w\ld ?9k t ! c>e tak5 ought to dep5d ^u ?is probability4 ,?is is done indirectly 9 ! tribunals :]e "o reta9s = "s "t ! accus$ aga9/ :om !re are v /rong pro(s1 b insuffici5t to condemn him2 9 ! hope ( acquir+ new light "o does n place him imm$iately 9 ! mid/ ( 8 fell[ citiz5s1 :o w\ld n see him aga9 )\t great al>m4 ,b ! arbitr>9ess ( ?is measure & ! abuse :i* "o c make ( x h caus$ its rejection 9 ! c\ntries :]e "o atta*es ! greate/ price to individual lib]ty4 ,n[ :at is ! probability t ! decision ( a tribunal :i* c condemn only by a giv5 majority w be j1 t is to say1 con=m to ! true solution ( ! "q propos$ above8 ,?is important problem well solv$ w give ! means ( comp>+ among !mselves ! diff]5t tribunals4 ,! majority ( a s+le vote 9 a num]\s tribunal indicates t ! affair 9 "q is v d\btful2 ! condemnation ( ! accus$ w\ld be !n contr>y to ! pr9ciples ( humanity1 protectors ( innoc5ce4 ,! unanimity ( ! judges w\ld give v /rong probability ( a j decision2 b 9 ab/a9+ f x too _m guilty ones w\ld be acquitt$4 ,x is necess>y1 !n1 ei!r to limit ! numb] ( judges1 if "o wi%es t !y %\ld be unanim\s1 or increase ! majority necess>y = a condemnation1 :5 ! tribunal becomes m num]\s4 ,i %all attempt to apply calculus to ?is subject1 be+ p]suad$ t x is always ! be/ guide :5 "o bases x ^u ! data :i* common s5se sugge/s to u4 ,! probability t ! op9ion ( ea* judge is j ent]s z ! pr9cipal elem5t into ?is calculation4 ,if 9 a tribunal ( a ?\s& & "o judges1 five hundr$ & "o are ( "o op9ion1 & five hundr$ are ( ! contr>y op9ion1 x is app>5t t ! probability ( ! op9ion ( ea* judge surpasses v little 2 = suppos+ x obvi\sly v l>ge a s+le vote ( diff]5ce w\ld be an improbable ev5t4 ,b if ! judges are unanim\s1 ?is indicates 9 ! pro(s t degree ( /r5g? :i* entails conviction2 ! probability ( ! op9ion ( ea* judge is !n v ne> unity or c]ta9ty1 provid$ t ! passions or ! ord9>y prejudices d n affect at ! same "t all ! judges4 ,outside ( ^! cases ! ratio ( ! votes = or aga9/ ! accus$ ought alone to det]m9e ?is probability4 ,i suppose ?us t x c v>y f to unity1 b t x _c be bel[ 4 ,if t 7 n ! case ! decision ( ! tribunal w\ld be z insignificant z *ance2 x has value only 9 s f> z ! op9ion ( ! judge has a great] t5d5cy to tru? ?an to error4 ,x is ?us by ! ratio ( ! numb]s ( votes favorable1 & contr>y to ! accus$1 t ,i det]m9e ! probability ( ?is op9ion4 ,^! data suffice to asc]ta9 ! g5]al expression ( ! probability t ! decision ( a tribunal judg+ by a kn[n majority is j4 ,9 ! tribunals :]e ( eight judges five votes w\ld be necess>y = ! condemnation ( an accus$1 ! probability ( ! error to be fe>$ 9 ! ju/ice ( ! decision w\ld surpass 4 ,if ! tribunal %\ld be r$uc$ to six memb]s :o are able to condemn only by a plurality ( f\r votes1 ! probability ( ! error to be fe>$ w\ld be bel[ 4 ,!re w\ld be !n = ! accus$ an advantage 9 ?is r$uction ( ! tribunal4 ,9 bo? cases ! majority requir$ is ! same & is equal to two4 ,?us ! majority rema9+ con/ant1 ! probability ( error increases ) ! numb] ( judges2 ?is is g5]al :atev] may be ! majority requir$1 provid$ t x rema9s ! same4 ,tak+1 !n1 = ! rule ! ari?metical ratio1 ! accus$ f9ds himself 9 a position less & less advantage\s 9 ! measure t ! tribunal becomes m num]\s4 ,"o might believe t 9 a tribunal :]e "o might dem& a majority ( twelve votes1 :atev] ! numb] ( ! judges 01 ! votes ( ! m9ority1 neutraliz+ an equal numb] ( votes ( ! majority1 ! twelve rema9+ votes w\ld repres5t ! unanimity ( a jury ( twelve memb]s1 requir$ 9 ,engl& = ! condemnation ( an accus$2 b "o w\ld be greatly mi/ak54 ,common s5se %[s t !re is a diff]5ce betwe5 ! decision ( a tribunal ( two hundr$ & twelve judges1 ( :i* "o hundr$ & twelve condemn ! accus$1 :ile "o hundr$ acquit him1 & t ( a tribunal ( twelve judges unanim\s = condemnation4 ,9 ! fir/ case ! hundr$ votes favorable to ! accus$ w>rant 9 ?9k+ t ! pro(s are f> f atta9+ ! degree ( /r5g? :i* entails conviction2 9 ! second case1 ! unanimity ( ! judges leads to ! belief t !y h atta9$ ?is degree4 ,b simple common s5se does n suffice at all to appreciate ! extreme diff]5ce ( ! probability ( error 9 ! two cases4 ,x is necess>y !n to recur to calculus1 & "o f9ds ne>ly "o fif? = ! probability ( error 9 ! fir/ case1 & only #a_/haib = ?is probability 9 ! second case1 a probability :i* is n "o ?\s&? ( ! fir/4 ,x is a confirmation ( ! pr9ciple t ! ari?metical ratio is unfavorable to ! accus$ :5 ! numb] ( judges increases4 ,on ! contr>y1 if "o takes = a rule ! geometrical ratio1 ! probability ( ! error ( ! decision dim9i%es :5 ! numb] ( judges increases4 ,= example1 9 ! tribunals :i* c condemn only by a plurality ( two ?irds ( ! votes1 ! probability ( ! error to be fe>$ is ne>ly "o f\r? if ! numb] ( ! judges is six2 x is bel[ #a_/g if ?is numb] is increas$ to twelve4 ,?us "o ought to be gov]n$ nei!r by ! ari?metical ratio nor by ! geometrical ratio if "o wi%es t ! probability ( error %\ld nev] be above nor bel[ a giv5 fraction4 ,b :at fraction ought to be det]m9$ ^u8 ,x is "h t ! arbitr>9ess beg9s & ! tribunals (f] 9 ?is reg>d ! greate/ v>iety4 ,9 ! special tribunals :]e five ( ! eight votes suffice = ! condemnation ( ! accus$1 ! probability ( ! error to be fe>$ 9 reg>d to ju/ice ( ! judgm5t is #fe_/bef1 or m ?an 4 ,! magnitude ( ?is fraction is dreadful2 b t :i* ought to reassure u a little is ! consid]ation t mo/ frequ5tly ! judge :o acquits an accus$ does n reg>d him z innoc5t2 he pron\nces solely t x is n atta9$ by pro(s suffici5t = condemnation4 ,"o is especially reassur$ by ! pity :i* nature has plac$ 9 ! he>t ( man & :i* disposes ! m9d to see only ) reluctance a culprit 9 ! accus$ submitt$ to 8 judgm5t4 ,?is s5tim5t1 m active 9 ^? :o h n ! habitude ( crim9al judgm5ts1 comp5sates = ! inconv5i5ces atta*$ to ! inexp]i5ce ( ! jurors4 ,9 a jury ( twelve memb]s1 if ! plurality dem&$ = ! condemnation is eight ( twelve votes1 ! probability ( ! error to be fe>$ #ajic_/haib1 or a little m ?an "o eigh?1 x is almo/ #a_/bb if ?is plurality consi/s ( n9e votes4 ,9 ! case ( unanimity ! probability ( ! error to be fe>$ is #a_/haib1 t is to say1 m ?an a ?\s& times less ?an 9 our juries4 ,?is supposes t ! unanimity results only f pro(s favorable or contr>y to ! accus$2 b motives t are entirely /range1 ought (t5times to concur 9 produc+ x1 :5 x is impos$ ^u ! jury z a necess>y condition ( its judgm5t4 ,!n its decisions dep5d+ ^u ! temp]am5t1 ! "*1 ! habits ( ! jurors1 & ! circum/ances 9 :i* !y are plac$1 !y are sometimes contr>y to ! decisions :i* ! majority ( ! jury w\ld h made if !y _h li/5$ only to ! pro(s2 ?is seems to me to be a great fault ( ?is mann] ( judg+4 ,! probability ( ! decision is too feeble 9 our juries1 & ,i ?9k t 9 ord] to give a suffici5t gu>antee to innoc5ce1 "o ought to dem& at lea/ a plurality ( n9e votes 9 twelve4 ,,*apt] ,,xiv4 ,,conc]n+ ,,tables ,,( ,,mortality1 ,,& ,,( ,,mean ,,durations ,,( ,,life1 ,,( ,,m>riages1 ,,& ,,( ,,associations4 ,! mann] ( prep>+ tables ( mortality is v simple4 ,"o takes 9 ! civil regi/]s a great numb] ( individuals ^: bir? & dea? are indicat$4 ,"o det]m9es h[ _m ( ^! individuals h di$ 9 ! fir/ ye> ( _! age1 h[ _m 9 ! second ye>1 & s on4 ,x is conclud$ f ^! ! numb] ( individuals liv+ at ! comm5cem5t ( ea* ye>1 & ?is numb] is writt5 9 ! table at ! side ( t :i* indicates ! ye>4 ,?us "o writes at ! side ( z]o ! numb] ( bir?s2 at ! side ( ! ye> #a ! numb] ( infants :o h atta9$ "o ye>2 at ! side ( ! ye> #b ! numb] ( infants :o h atta9$ two ye>s1 & s on = ! re/4 ,b s9ce 9 ! fir/ two ye>s ( life ! mortality is v great1 x is necess>y = ! sake ( great] exactitude to indicate 9 ?is fir/ age ! numb] ( survivors at ! end ( ea* half ye>4 ,if we divide ! sum ( ! ye>s ( ! life ( all ! individuals inscrib$ 9 a table ( mortality by ! numb] ( ^! individuals we %all h ! mean duration ( life :i* corresponds to ?is table4 ,= ?is1 we w multiply by a half ye> ! numb] ( dea?s 9 ! fir/ ye>1 a numb] equal to ! diff]5ce ( ! numb]s ( individuals inscrib$ at ! side ( ! ye>s #j & #a4 ,_! mortality be+ di/ribut$ ov] ! entire ye> ! mean duration ( _! life is only a half ye>4 ,we w multiply by a ye> & a half ! numb] ( dea?s 9 ! second ye>2 by two ye>s & a half ! numb] ( dea?s 9 ! ?ird ye>2 & s on4 ,! sum ( ^! products divid$ by ! numb] ( bir?s w be ! mean duration ( life4 ,x is easy to conclude f ?is t we w obta9 ?is duration1 by mak+ ! sum ( ! numb]s inscrib$ 9 ! table at ! side ( ea* ye>1 divid+ x by ! numb] ( bir?s & subtract+ "o half f ! quoti5t1 ! ye> be+ tak5 z unity4 ,! mean duration ( life t rema9s1 />t+ f any age1 is det]m9$ 9 ! same mann]1 work+ ^u ! numb] ( individuals :o h arriv$ at ?is age1 z has j be5 done ) ! numb] ( bir?s4 ,b x is n at ! mom5t ( bir? t ! mean duration ( life is ! greate/2 x is :5 "o has escap$ ! dang]s ( infancy & x is !n ab\t =ty-?ree ye>s4 ,! probability ( arriv+ at a c]ta9 age1 />t+ f a giv5 age is equal to ! ratio ( ! two numb]s ( individuals indicat$ 9 ! table at ^! two ages4 ,! precision ( ^! results dem&s t = ! =mation ( tables we %\ld employ a v great numb] ( bir?s4 ,analysis gives !n v simple =mul = appreciat+ ! probability t ! numb]s indicat$ 9 ^! tables w v>y f ! tru? only )9 n>r[ limits4 ,we see by ^! =mul t ! int]val ( ! limits dim9i%es & t ! probability increases 9 proportion z we take into consid]ation m bir?s2 s t ! tables w\ld repres5t exactly ! true law ( mortality if ! numb] ( bir?s employ$ 7 inf9ite4 ,a table ( mortality is !n a table ( ! probability ( human life4 ,! ratio ( ! individuals inscrib$ at ! side ( ea* ye> to ! numb] ( bir?s is ! probability t a new bir? w atta9 ?is ye>4 ,z we e/imate ! value ( hope by mak+ a sum ( ! products ( ea* b5efit hop$ =1 by ! probability ( obta9+ x1 s we c equally evaluate ! mean duration ( life by add+ ! products ( ea* ye> by half ! sum ( ! probabilities ( atta9+ ! comm5cem5t & ! end ( x1 :i* leads to ! result f\nd above4 ,b ?is mann] ( view+ ! mean duration ( life has ! advantage ( %[+ t 9 a /ation>y population1 t is to say1 su* t ! numb] ( bir?s equals t ( dea?s1 ! mean duration ( life is ! ratio itself ( ! population to ! annual bir?s2 = ! population be+ suppos$ /ation>y1 ! numb] ( individuals ( an age compris$ betwe5 two consecutive ye>s ( ! table is equal to ! numb] ( annual bir?s1 multipli$ by half ! sum ( ! probabilities ( atta9+ ^! ye>s2 ! sum ( all ^! products w be !n ! entire population4 ,n[ x is easy to see t ?is sum1 divid$ by ! numb] ( annual bir?s1 co9cides ) ! mean duration ( life z we h j def9$ x4 ,x is easy by means ( a table ( mortality to =m ! correspond+ table ( ! population suppos$ to be /ation>y4 ,= ?is we take ! ari?metical means ( ! numb]s ( ! table ( mortality correspond+ to ! ages z]o & "o ye>1 "o & two ye>s1 two & ?ree ye>s1 etc4 ,! sum ( all ^! means is ! entire population2 x is writt5 at ! side ( ! age z]o4 ,!re is subtract$ f ?is sum ! fir/ mean & ! rema9d] is ! numb] ( individuals ( "o ye> & upw>ds2 x is writt5 at ! side ( ! ye> #a4 ,!re is subtract$ f ?is fir/ rema9d] ! second mean2 ?is second rema9d] is ! numb] ( individuals ( two ye>s & upw>ds2 x is writt5 at ! side ( ! ye> #b1 & s on4 ,s _m v>iable causes influ5ce mortality t ! tables :i* repres5t x ought to be *ang$ accord+ to place & "t4 ,! div]s /ates ( life (f] 9 ?is reg>d appreciable diff]5ces relative to ! fatigues & ! dang]s insep>able f ea* /ate & ( :i* x is indisp5sable to keep acc\nt 9 ! calculations f\nd$ ^u ! duration ( life4 ,b ^! diff]5ces h n be5 suffici5tly obs]v$4 ,"s day !y w be & !n w be kn[n :at sacrifice ( life ea* pr(ession dem&s & "o w pr(it by ?is k to dim9i% ! dang]s4 ,! great] or less salubrity ( ! soil1 its elevation1 its temp]ature1 ! cu/oms ( ! inhabitants1 & ! op]ations ( gov]nm5ts h a consid]able influ5ce ^u mortality4 ,b x is always necess>y to prec$e ! inve/igation ( ! cause ( ! diff]5ces obs]v$ by t ( ! probability ) :i* ?is cause is indicat$4 ,?us ! ratio ( ! population to annual bir?s1 :i* "o has se5 rais$ 9 ,france to tw5ty-eight & "o ?ird1 is n equal to tw5ty-five 9 ! anci5t du*y ( ,milan4 ,^! ratios1 bo? e/abli%$ ^u a great numb] ( bir?s1 d n p]mit ( call+ into "q ! exi/5ce among ! ,milanese ( a special cause ( mortality1 :i* x is ( mom5t = ! gov]nm5t ( our c\ntry to inve/igate & remove4 ,! ratio ( ! population to ! bir?s w\ld increase aga9 if we c\ld dim9i% & remove c]ta9 dang]\s & widely spread maladies4 ,?is has happily be5 done = ! smallpox1 at fir/ by ! inoculation ( ?is disease1 !n 9 a mann] mu* m advantage\s1 by ! inoculation ( vacc9e1 ! ine/imable discov]y ( ,j5n]1 :o has !reby become "o ( ! greate/ b5efactors ( humanity4 ,! smallpox has ?is 9 p>ticul>1 namely1 t ! same individual is n twice affect$ by x1 or at lea/ su* cases are s r>e t !y may be ab/ract$ f ! calculation4 ,?is malady1 f :i* few escap$ be=e ! discov]y ( vacc9e1 is (t5 fatal & causes ! dea? ( "o sev5? ( ^? :om x attacks4 ,sometimes x is mild1 & exp]i5ce has taught t x c be giv5 ?is latt] "* by inoculat+ x ^u heal?y p]sons1 prep>$ = x by a prop] diet & 9 a favorable season4 ,!n ! ratio ( ! individuals :o die to ! inoculat$ ones is n "o ?ree hundr$?4 ,?is great advantage ( inoculation1 jo9$ to ^? ( n alt]+ ! appe>ance & ( pres]v+ f ! griev\s consequ5ces :i* ! natural smallpox (t5 br+s1 caus$ x to be adopt$ by a great numb] ( p]sons4 ,! practice 0 /rongly recomm5d$1 b x 0 /rongly combat$1 z is ne>ly always ! case 9 ?+s subject to inconv5i5ce4 ,9 ! mid/ ( ?is dispute ,daniel ,b]n\lli propos$ to submit to ! calculus ( probabilities ! influ5ce ( inoculation ^u ! mean duration ( life4 ,s9ce precise data ( ! mortality produc$ by ! smallpox at ! v>i\s ages ( life 7 lack+1 he suppos$ t ! dang] ( hav+ ?is malady & t ( dy+ ( x are ! same at e age4 ,by means ( ^! suppositions he succe$$ by a delicate analysis 9 conv]t+ an ord9>y table ( mortality into t :i* w\ld be us$ if smallpox did n exi/1 or if x caus$ ! dea? ( only a v small numb] ( ^? affect$1 & he concludes f x t inoculation w\ld augm5t by ?ree ye>s at lea/ ! mean duration ( life1 :i* appe>$ to him beyond d\bt ! advantage ( ?is op]ation4 ,d',alemb]t attack$ ! analysis ( ,b]n\lli3 at fir/ 9 reg>d to ! unc]ta9ty ( 8 two hypo!ses1 !n 9 reg>d to its insuffici5cy 9 ?is1 t no comp>ison 0 made ( ! imm$iate dang]1 al?\gh v small1 ( dy+ ( inoculation1 to ! v great b v remote dang] ( succumb+ to natural smallpox4 ,?is consid]ation1 :i* disappe>s :5 "o consid]s a great numb] ( individuals1 is = ?is reason immat]ial = gov]nm5ts & ! advantages ( inoculation = !m /ill rema92 b x is ( great weight = ! fa!r ( a family :o mu/ fe>1 9 hav+ 8 *ildr5 inoculat$1 to see t "o p]i% :om he holds mo/ de> & to be ! cause ( x4 ,_m p>5ts 7 re/ra9$ by ?is fe>1 :i* ! discov]y ( vacc9e has happily dissipat$4 ,by "o ( ^? my/]ies :i* nature (f]s to u s frequ5tly1 vacc9e is a prev5tive ( smallpox j z c]ta9 z v>iol> virus1 & !re is no dang] at all2 x does n expose to any malady & dem&s only v little c>e4 ,!re=e ! practice ( x has spread quickly2 & to r5d] x univ]sal x rema9s only to ov]come ! natural in]tia ( ! p1 aga9/ :i* x is necess>y to /rive cont9ually1 ev5 :5 x is a "q ( _! de>e/ int]e/s4 ,! simple/ means ( calculat+ ! advantage :i* ! ext9ction ( a malady w\ld produce consi/s 9 det]m9+ by obs]vation ! numb] ( individuals ( a giv5 age :o die ( x ea* ye> & subtract+ ?is numb] f ! numb] ( dea?s at ! same age4 ,! ratio ( ! diff]5ce to ! total numb] ( individuals ( ! giv5 age w\ld be ! probability ( dy+ 9 ! ye> at ?is age if ! malady did n exi/4 ,mak+1 !n1 a sum ( ^! probabilities f bir? up to any giv5 age1 & subtract+ ?is sum f unity1 ! rema9d] w be ! probability ( liv+ to t age correspond+ to ! ext9ction ( ! malady4 ,! s]ies ( ^! probabilities w be ! table ( mortality relative to ?is hypo!sis1 & we may conclude f x1 by :at prec$es1 ! mean duration ( life4 ,x is ?us t ,duvil>d has f\nd t ! increase ( ! mean duration ( life1 due to inoculation ) vacc9e1 is ?ree ye>s at ! lea/4 ,an increase s consid]able w\ld produce a v great increase 9 ! population if ! latt]1 = o!r reasons1 7 n re/ra9$ by ! relative dim9ution ( subsi/5ces4 ,x is pr9cipally by ! lack ( subsi/5ces t ! progressive m>* ( ! population is arre/$4 ,9 all k9ds ( animals & vegetables1 nature t5ds )\t ceas+ to augm5t ! numb] ( individuals until !y are on a level ( ! means ( subsi/5ce4 ,9 ! human race moral causes h a great influ5ce ^u ! population4 ,if easy cle>+s ( ! =e/ c furni% an abundant n\ri%m5t = new g5]ations1 ! c]ta9ty ( be+ able to support a num]\s family enc\rages m>riages & r5d]s !m m productive4 ,^u ! same soil ! population & ! bir?s ought to increase at ! same "t simultane\sly 9 geometric progression4 ,b :5 cle>+s become m difficult & m r>e !n ! increase ( population dim9i%es2 x approa*es cont9ually ! v>iable /ate ( subsi/5ces1 mak+ oscillations ab\t x j z a p5dulum ^: p]iodicity is ret>d$ by *ang+ ! po9t ( susp5sion1 oscillates ab\t ?is po9t by virtue ( its own weight4 ,x is difficult to evaluate ! maximum increase ( ! population2 x appe>s aft] obs]vations t 9 favorable circum/ances ! population ( ! human race w\ld be d\bl$ e fifte5 ye>s4 ,we e/imate t 9 ,nor? ,am]ica ! p]iod ( ?is d\bl+ is tw5ty-two ye>s4 ,9 ?is /ate ( ?+s1 ! population1 bir?s1 m>riages1 mortality1 all increase accord+ to ! same geometric progression ( :i* we h ! con/ant ratio ( consecutive t]ms by ! obs]vation ( annual bir?s at two epo*s4 ,by means ( a table ( mortality repres5t+ ! probabilities ( human life1 we may det]m9e ! duration ( m>riages4 ,suppos+ 9 ord] to simplify ! matt] t ! mortality is ! same = ! two sexes1 we %all obta9 ! probability t ! m>riage w subsi/ "o ye>1 or two1 or ?ree1 etc41 by =m+ a s]ies ( fractions ^: common d5om9ator is ! product ( ! two numb]s ( ! table correspond+ to ! ages ( ! consorts1 & ^: num]ators are ! successive products ( ! numb]s correspond+ to ^! ages augm5t$ by "o1 by two1 by ?ree1 etc41 ye>s4 ,! sum ( ^! fractions augm5t$ by "o half w be ! mean duration ( m>riage1 ! ye> be+ tak5 z unity4 ,x is easy to ext5d ! same rule to ! mean duration ( an association =m$ ( ?ree or ( a great] numb] ( individuals4 ,,*apt] ,,xv4 ,,conc]n+ ,,! ,,b5efits ,,( ,,in/itutions ,,:i* ,,dep5d ,,^u ,,! ,,probability ,,( ,,ev5ts4 ,let u recall "h :at has be5 said 9 speak+ ( hope4 ,x has be5 se5 t 9 ord] to obta9 ! advantage :i* results f sev]al simple ev5ts1 ( :i* ! ones produce a b5efit & ! o!rs a loss1 x is necess>y to add ! products ( ! probability ( ea* favorable ev5t by ! b5efit :i* x procures1 & subtract f _! sum t ( ! products ( ! probability ( ea* unfavorable ev5t by ! loss :i* is atta*$ to x4 ,b :atev] may be ! advantage express$ by ! diff]5ce ( ^! sums1 a s+le ev5t compos$ ( ^! simple ev5ts does n gu>antee aga9/ ! fe> ( exp]i5c+ a loss4 ,"o imag9es t ?is fe> ought to decrease :5 "o multiplies ! comp\nd ev5t4 ,! analysis ( probabilities leads to ?is g5]al !orem4 ,by ! repetition ( an advantage\s ev5t1 simple or comp\nd1 ! real b5efit becomes m & m probable & increases )\t ceas+2 x becomes c]ta9 9 ! hypo!sis ( an inf9ite numb] ( repetitions2 & divid+ x by ?is numb] ! quoti5t or ! mean b5efit ( ea* ev5t is ! ma!matical hope itself or ! advantage relative to ! ev5t4 ,x is ! same ) a loss :i* becomes c]ta9 9 ! long run1 h[ev] small ! disadvantage ( ! ev5t may be4 ,?is !orem ^u b5efits & losses is analog\s to ^? :i* we h already giv5 ^u ! ratios :i* are indicat$ by ! indef9ite repetition ( ev5ts simple or comp\nd2 &1 l !m1 x proves t regul>ity ends by e/abli%+ itself ev5 9 ! ?+s :i* are mo/ subord9at$ to t :i* we "n haz>d4 ,:5 ! ev5ts are 9 great numb]1 analysis gives ano!r v simple expression ( ! probability t ! b5efit w be compris$ )9 det]m9$ limits4 ,?is is ! expression :i* ent]s aga9 into ! g5]al law ( probability giv5 above 9 speak+ ( ! probabilities :i* result f ! indef9ite multiplication ( ev5ts4 ,! /ability ( in/itutions :i* are bas$ ^u probabilities dep5ds ^u ! tru? ( ! prec$+ !orem4 ,b 9 ord] t x may be appli$ to !m x is necess>y t ^? in/itutions %\ld multiply ^! advantage\s ev5ts = ! sake ( num]\s ?+s4 ,!re h be5 bas$ ^u ! probabilities ( human life div]s in/itutions1 su* z life annuities & tont9es4 ,! mo/ g5]al & ! mo/ simple me?od ( calculat+ ! b5efits & ! exp5ses ( ^! in/itutions consi/s 9 r$uc+ ^! to actual am\nts4 ,! annual int]e/ ( unity is t :i* is call$ ! rate ( int]e/4 ,at ! end ( ea* ye> an am\nt acquires = a factor unity plus ! rate ( int]e/2 x increases !n accord+ to a geometrical progression ( :i* ?is factor is ! ratio4 ,?us 9 ! c\rse ( "t x becomes imm5se4 ,if1 = example1 ! rate ( int]e/ is #a_/bj or five p] c5t1 ! capital d\bles v ne>ly 9 f\rte5 ye>s1 quadruples 9 tw5ty-n9e ye>s1 & 9 less ?an ?ree c5turies x becomes two million times l>g]4 ,an increase s prodigi\s has giv5 bir? to ! idea ( mak+ use ( x 9 ord] to pay (f ! public debt4 ,"o =ms = ?is purpose a s9k+ fund to :i* is devot$ an annual fund employ$ = ! r$emption ( public bills & )\t ceas+ increas$ by ! int]e/ ( ! bills r$eem$4 ,x is cle> t 9 ! long run ?is fund w absorb a great "p ( ! national debt4 ,if1 :5 ! ne$s ( ! ,/ate make a loan necess>y1 a "p ( ?is loan is devot$ to ! increas+ ( ! annual s9k+ fund1 ! v>iation ( public bills w be less2 ! confid5ce ( ! l5d]s & ! probability ( retir+ )\t loss ( capital loan$ :5 "o desires w be augm5t$ & w r5d] ! conditions ( ! loan less on]\s4 ,favorable exp]i5ces h fully confirm$ ^! advantages4 ,b ! fidelity 9 engagem5ts & ! /ability1 s necess>y to ! success ( su* in/itutions1 c be gu>ante$ only by a gov]nm5t 9 :i* ! legislative p[] is divid$ among sev]al indep5d5t p[]s4 ,! confid5ce :i* ! necess>y cop]ation ( ^! p[]s inspires1 d\bles ! /r5g? ( ! ,/ate1 & ! sov]eign himself ga9s !n 9 legal p[] m ?an he loses 9 arbitr>y p[]4 ,x results f t :i* prec$es t ! actual capital equival5t to a sum :i* is to be paid only aft] a c]ta9 numb] ( ye>s is equal to ?is sum multipli$ by ! probability t x w be paid at t "t & divid$ by unity augm5t$ by ! rate ( int]e/ & rais$ to a p[] express$ by ! numb] ( ^! ye>s4 ,x is easy to apply ?is pr9ciple to life annuities ^u "o or sev]al p]sons1 & to sav+s banks1 & to assurance societies ( any nature4 ,suppose t "o proposes to =m a table ( life annuities accord+ to a giv5 table ( mortality4 ,a life annuity payable at ! end ( five ye>s1 = example1 & r$uc$ to an actual am\nt is1 by ?is pr9ciple1 equal to ! product ( ! two foll[+ quantities1 namely1 ! annuity divid$ by ! fif? p[] ( unity augm5t$ by ! rate ( int]e/ & ! probability ( pay+ x4 ,?is probability is ! inv]se ratio ( ! numb] ( individuals inscrib$ 9 ! table opposite to ! age ( t "o :o settles ! annuity to ! numb] inscrib$ opposite to ?is age augm5t$ by five ye>s4 ,=m+1 !n1 a s]ies ( fractions ^: d5om9ators are ! products ( ! numb] ( p]sons indicat$ 9 ! table ( mortality z liv+ at ! age ( t "o :o settles ! annuity1 by ! successive p[]s ( unity augm5t$ by ! rate ( int]e/1 & ^: num]ators are ! products ( ! annuity by ! numb] ( p]sons liv+ at ! same age augm5t$ successively by "o ye>1 by two ye>s1 etc41 ! sum ( ^! fractions w be ! am\nt requir$ = ! life annuity at t age4 ,let u suppose t a p]son wi%es by means ( a life annuity to assure to 8 heirs an am\nt payable at ! end ( ! ye> ( 8 dea?4 ,9 ord] to det]m9e ! value ( ?is annuity1 "o may imag9e t ! p]son borr[s 9 life at a bank ?is capital & t he places x at p]petual int]e/ 9 ! same bank4 ,x is cle> t ?is same capital w be due by ! bank to 8 heirs at ! end ( ! ye> ( 8 dea?2 b he w h paid ea* ye> only ! excess ( ! life int]e/ ov] ! p]petual int]e/4 ,! table ( life annuities w !n %[ t :i* ! p]son ought to pay annually to ! bank 9 ord] to assure ?is capital aft] 8 dea?4 ,m>itime assurance1 t aga9/ fire & /orms1 & g5]ally all ! in/itutions ( ?is k9d1 are comput$ on ! same pr9ciples4 ,a m]*ant hav+ vessels at sea wi%es to assure _! value & t ( _! c>goes aga9/ ! dang]s t !y may run2 9 ord] to d ?is1 he gives a sum to a company :i* becomes responsible to him = ! e/imat$ value ( 8 c>goes & 8 vessels4 ,! ratio ( ?is value to ! sum :i* ought to be giv5 = ! price ( ! assurance dep5ds ^u ! dang]s to :i* ! vessels are expos$ & c be appreciat$ only by num]\s obs]vations ^u ! fate ( vessels :i* h sail$ f port = ! same de/9ation4 ,if ! p]sons assur$ %\ld give to ! assurance company only ! sum indicat$ by ! calculus ( probabilities1 ?is company w\ld n be able to provide = ! exp5ses ( its in/itution2 x is necess>y !n t !y %\ld pay a sum mu* great] ?an ! co/ ( su* insurance4 ,:at !n is _! advantage8 ,x is "h t ! consid]ation ( ! moral disadvantage atta*$ to an unc]ta9ty becomes necess>y4 ,"o conceives t ! faire/ game becomes1 z has already be5 se51 disadvantage\s1 because ! play] ex*anges a c]ta9 /ake = an unc]ta9 b5efit2 assurance by :i* "o ex*anges ! unc]ta9 = ! c]ta9 ought to be advantage\s4 ,x is inde$ ?is :i* results f ! rule :i* we h giv5 above = det]m9+ moral hope & by :i* "o sees moreov] h[ f> ! sacrifice may ext5d :i* ought to be made to ! assurance company by res]v+ always a moral advantage4 ,?is company c !n 9 procur+ ?is advantage itself make a great b5efit1 if ! numb] ( ! assur$ p]sons is v l>ge1 a condition necess>y to its cont9u$ exi/5ce4 ,!n its b5efits become c]ta9 & ! ma!matical & moral hopes co9cide2 = analysis leads to ?is g5]al !orem1 namely1 t if ! expectations are v num]\s ! two hopes approa* ea* o!r )\t ceas+ & end by co9cid+ 9 ! case ( an inf9ite numb]4 ,we h said 9 speak+ ( ma!matical & moral hopes t !re is a moral advantage 9 di/ribut+ ! risks ( a b5efit :i* "o expects ov] sev]al ( its p>ts4 ,?us 9 ord] to s5d a sum ( money to a di/ant "p x is mu* bett] to s5d x on sev]al vessels ?an to expose x on "o4 ,?is "o does by means ( mutual assurances4 ,if two p]sons1 ea* hav+ ! same sum ^u two diff]5t vessels :i* h sail$ f ! same port to ! same de/9ation1 agree to divide equally all ! money :i* may arrive1 x is cle> t by ?is agreem5t ea* ( !m divides equally betwe5 ! two vessels ! sum :i* he expects4 ,inde$ ?is k9d ( assurance always leaves unc]ta9ty z to ! loss :i* "o may fe>4 ,b ?is unc]ta9ty dim9i%es 9 proportion z ! numb] ( policy-hold]s increases2 ! moral advantage increases m & m & ends by co9cid+ ) ! ma!matical advantage1 its natural limit4 ,?is r5d]s ! association ( mutual assurances :5 x is v num]\s m advantage\s to ! assur$ ones ?an ! companies ( assurance :i*1 9 proportion to ! b5efit t !y give1 give a moral advantage always inf]ior to ! ma!matical advantage4 ,b ! surveillance ( _! adm9i/ration c balance ! advantage ( ! mutual assurances4 ,all ^! results are1 z has already be5 se51 indep5d5t ( ! law :i* expresses ! moral advantage4 ,"o may look ^u a free p z a great association ^: memb]s secure mutually _! prop]ties by support+ proportionally ! *>ges ( ?is gu>anty4 ,! conf$]ation ( sev]al peoples w\ld give to !m advantages analog\s to ^? :i* ea* individual enjoys 9 ! society4 ,a congress ( _! repres5tatives w\ld discuss objects ( a utility common to all & )\t d\bt ! sy/em ( weights1 measures1 & moneys propos$ by ! ,fr5* sci5ti/s w\ld be adopt$ 9 ?is congress z "o ( ! ?+s mo/ useful to comm]cial relations4 ,among ! in/itutions f\nd$ ^u ! probabilities ( human life ! bett] ones are ^? 9 :i*1 by means ( a light sacrifice ( 8 rev5ue1 "o assures 8 exi/5ce & t ( 8 family = a "t :5 "o ought to fe> to be unable to satisfy _! ne$s4 ,z f> z games are immoral1 s f> ^! in/itutions are advantage\s to cu/oms by favor+ ! /ronge/ b5ts ( our nature4 ,! gov]nm5t ought !n to enc\rage !m & respect !m 9 ! vicissitudes ( public =tune2 s9ce ! hopes :i* !y pres5t look t[>d a di/ant future1 !y are able to prosp] only :5 %elt]$ f all inquietude dur+ _! exi/5ce4 ,x is an advantage t ! in/itution ( a repres5tative gov]nm5t assures !m4 ,let u say a ^w ab\t loans4 ,x is cle> t 9 ord] to borr[ p]petually x is necess>y to pay ea* ye> ! product ( ! capital by ! rate ( int]e/4 ,b "o may wi% to dis*>ge ?is pr9cipal 9 equal paym5ts made dur+ a def9ite numb] ( ye>s1 paym5ts :i* are call$ annuities & ^: value is obta9$ 9 ?is mann]4 ,ea* annuity 9 ord] to be r$uc$ at ! actual mom5t ought to be divid$ by a p[] ( unity augm5t$ by ! rate ( int]e/ equal to ! numb] ( ye>s aft] :i* ?is annuity ought to be paid4 ,=m+ !n a geometric progression ^: fir/ t]m is ! annuity divid$ by unity augm5t$ by ! rate ( int]e/1 & ^: la/ t]m is ?is annuity divid$ by ! same quantity rais$ to a p[] equal to ! numb] ( ye>s dur+ :i* ! paym5t %\ld h be5 made1 ! sum ( ?is progression w be equival5t to ! capital borr[$1 :i* w det]m9e ! value ( ! annuity4 ,a s9k+ fund is at bottom only a means ( conv]t+ into annuities a p]petual r5t ) ! sole diff]5ce t 9 ! case ( a loan by annuities ! int]e/ is suppos$ con/ant1 :ile ! int]e/ ( funds acquir$ by ! s9k+ fund is v>iable4 ,if x 7 ! same 9 bo? cases1 ! annuity correspond+ to ! funds acquir$ w\ld be =m$ by ^! funds & f ?is annuity ! ,/ate contributes annually to ! s9k+ fund4 ,if "o wi%es to make a life loan x w be obs]v$ t ! tables ( life annuities give ! capital requir$ to con/itute a life annuity at any age1 a simple proportion w give ! r5t :i* "o ought to pay to ! individual f :om ! capital is borr[$4 ,f ^! pr9ciples all ! possible k9ds ( loans may be calculat$4 ,! pr9ciples :i* we h j exp\nd$ conc]n+ ! b5efits & ! losses ( in/itutions may s]ve to det]m9e ! mean result ( any numb] ( obs]vations already made1 :5 "o wi%es to reg>d ! deviations ( ! results correspond+ to div]s obs]vations4 ,let u designate by x ! correction ( ! lea/ result & by x augm5t$ successively by q1 q1 q1 etc41 ! corrections ( ! foll[+ results4 ,let u "n e1 e1 e1 etc41 ! errors ( ! obs]vations ^: law ( probability we w suppose kn[n4 ,ea* obs]vation be+ a function ( ! result1 x is easy to see t by suppos+ ! correction x ( ?is result to be v small1 ! error e ( ! fir/ obs]vation w be equal to ! product ( x by a det]m9$ coeffici5t4 ,likewise ! error e ( ! second obs]vation w be ! product ( ! sum q plus x1 by a det]m9$ coeffici5t1 & s on4 ,! probability ( ! error e be+ giv5 by a kn[n function1 x w be express$ by ! same function ( ! fir/ ( ! prec$+ products4 ,! probability ( e w be express$ by ! same function ( ! second ( ^! products1 & s on ( ! o!rs4 ,! probability ( ! simultane\s exi/5ce ( ! errors e1 e1 e1 etc41 w be !n proportional to ! product ( ^! div]s functions1 a product :i* w be a function ( x4 ,?is be+ grant$1 if "o conceives a curve ^: abscissa is x1 & ^: correspond+ ord9ate is ?is product1 ?is curve w repres5t ! probability ( ! div]s values ( x1 ^: limits w be det]m9$ by ! limits ( ! errors e1 e1 e1 etc4 ,n[ let u designate by ,x ! abscissa :i* x is necess>y to *oose2 ,x dim9i%$ by x w be ! error :i* w\ld be committ$ if ! abscissa x 7 ! true correction4 ,?is error1 multipli$ by ! probability ( x or by ! correspond+ ord9ate ( ! curve1 w be ! product ( ! loss by its probability1 reg>d+1 z "o %\ld1 ?is error z a loss atta*$ to ! *oice ,x4 ,multiply+ ?is product by ! diff]5tial ( x ! integral tak5 f ! fir/ extremity ( ! curve to ,x w be ! disadvantage ( ,x result+ f ! values ( x inf]ior to ,x4 ,= ! values ( x sup]ior to ,x1 x less ,x w\ld be ! error ( ,x if x 7 ! true correction2 ! integral ( ! product ( x by ! correspond+ ord9ate ( ! curve & by ! diff]5tial ( x w be !n ! disadvantage ( ,x result+ f ! values x sup]ior to x1 ?is integral be+ tak5 f x equal to ,x up to ! la/ extremity ( ! curve4 ,add+ ?is disadvantage to ! prec$+ "o1 ! sum w be ! disadvantage atta*$ to ! *oice ( ,x4 ,?is *oice ought to be det]m9$ by ! condition t ?is disadvantage be a m9imum2 & a v simple calculation %[s t = ?is1 ,x ought to be ! abscissa ^: ord9ate divides ! curve into two equal p>ts1 s t x is ?us probable t ! true value ( x falls on nei!r ! "o side nor ! o!r ( ,x4 ,celebrat$ geometricians h *os5 = ,x ! mo/ probable value ( x & consequ5tly t :i* corresponds to ! l>ge/ ord9ate ( ! curve2 b ! prec$+ value appe>s to me evid5tly t :i* ! !ory ( probability indicates4 ,,*apt] ,,xvi4 ,,conc]n+ ,,illusions ,,9 ,,! ,,e/imation ,,( ,,probabilities4 ,! m9d has its illusions z ! s5se ( sight2 & 9 ! same mann] t ! s5se ( feel+ corrects ! latt]1 reflection & calculation correct ! =m]4 ,probability bas$ ^u a daily exp]i5ce1 or exagg]at$ by fe> & by hope1 /rikes u m ?an a sup]ior probability b x is only a simple result ( calculus4 ,?us we d n fe> 9 return = small advantages to expose our life to dang]s mu* less improbable ?an ! draw+ ( a qu9t 9 ! lott]y ( ,france2 & yet no "o w\ld wi% to procure = himself ! same advantages ) ! c]ta9ty ( los+ 8 life if ?is qu9t %\ld be drawn4 ,our passions1 our prejudices1 & dom9at+ op9ions1 by exagg]at+ ! probabilities :i* are favorable to !m & by att5uat+ ! contr>y probabilities1 are ! abundant s\rces ( dang]\s illusions4 ,pres5t evils & ! cause :i* produc$ !m effect u mu* m ?an ! remembrance ( evils produc$ by ! contr>y cause2 !y prev5t u f appreciat+ ) ju/ice ! inconv5i5ces ( ! ones & ! o!rs1 & ! probability ( ! prop] means to gu>d ourselves aga9/ !m4 ,x is ?is :i* leads alt]nately to despotism & to an>*y ! p :o are driv5 f ! /ate ( repose to :i* !y nev] return except aft] long & cruel agitations4 ,?is vivid impression :i* we receive f ! pres5ce ( ev5ts1 & :i* all[s u sc>cely to rem>k ! contr>y ev5ts obs]v$ by o!rs1 is a pr9cipal cause ( error aga9/ :i* "o _c suffici5tly gu>d himself4 ,x is pr9cipally at games ( *ance t a multitude ( illusions support hope & su/a9 x aga9/ unfavorable *ances4 ,! majority ( ^? :o play at lott]ies d n "k h[ _m *ances are to _! advantage1 h[ _m are contr>y to !m4 ,!y see only ! possibility by a small /ake ( ga9+ a consid]able sum1 & ! projects :i* _! imag9ation br+s =?1 exagg]ate to _! eyes ! probability ( obta9+ x2 ! poor man especially1 excit$ by ! desire ( a bett] fate1 risks at play 8 necessities by cl++ to ! mo/ unfavorable comb9ations :i* promise him a great b5efit4 ,all w\ld be )\t d\bt surpris$ by ! imm5se numb] ( /akes lo/ if !y c\ld "k ( !m2 b "o takes c>e on ! contr>y to give to ! w9n+s a great publicity1 :i* becomes a new cause ( excitem5t = ?is fun]eal play4 ,:5 a numb] 9 ! lott]y ( ,france has n be5 drawn = a long "t ! cr[d is eag] to cov] x ) /akes4 ,!y judge s9ce ! numb] has n be5 drawn = a long "t t x ought at ! next draw+ to be drawn 9 pref]5ce to o!rs4 ,s common an error appe>s to me to re/ ^u an illusion by :i* "o is c>ri$ back involunt>ily to ! orig9 ( ev5ts4 ,x is1 = example1 v improbable t at ! play ( heads & tails "o w ?r[ heads t5 times 9 succession4 ,?is improbability :i* /rikes u inde$ :5 x has happ5$ n9e times1 leads u to believe t at ! t5? ?r[ tails w be ?r[n4 ,b ! pa/ indicat+ 9 ! co9 a great] prop5sity = heads ?an = tails r5d]s ! fir/ ( ! ev5ts m probable ?an ! second2 x increases z "o has se5 ! probability ( ?r[+ heads at ! foll[+ ?r[4 ,a simil> illusion p]suades _m p t "o c c]ta9ly w9 9 a lott]y by plac+ ea* "t ^u ! same numb]1 until x is drawn1 a /ake ^: product surpasses ! sum ( all ! /akes4 ,b ev5 :5 simil> speculations w\ld n (t5 be /opp$ by ! impossibility ( su/a9+ !m !y w\ld n dim9i% ! ma!matical disadvantage ( speculators & !y w\ld increase _! moral disadvantage1 s9ce at ea* draw+ !y w\ld risk a v l>ge "p ( _! =tune4 ,i h se5 m51 ard5tly desir\s ( hav+ a son1 :o c\ld le>n only ) anxiety ( ! bir?s ( boys 9 ! mon? :5 !y expect$ to become fa!rs4 ,imag9+ t ! ratio ( ^! bir?s to ^? ( girls ought to be ! same at ! end ( ea* mon?1 !y judg$ t ! boys already born w\ld r5d] m probable ! bir?s next ( girls4 ,?us ! extraction ( a :ite ball f an urn :i* conta9s a limit$ numb] ( :ite balls & ( black balls increases ! probability ( extract+ a black ball at ! foll[+ draw+4 ,b ?is ceases to take place :5 ! numb] ( balls 9 ! urn is unlimit$1 z "o mu/ suppose 9 ord] to comp>e ?is case ) t ( bir?s4 ,if1 9 ! c\rse ( a mon?1 !re 7 born _m m boys ?an girls1 "o might suspect t t[>d ! "t ( _! conception a g5]al cause _h favor$ mascul9e conception1 :i* w\ld r5d] m probable ! bir? next ( a boy4 ,! irregul> ev5ts ( nature are n exactly comp>able to ! draw+ ( ! numb]s ( a lott]y 9 :i* all ! numb]s are mix$ at ea* draw+ 9 su* a mann] z to r5d] ! *ances ( _! draw+ p]fectly equal4 ,! frequ5cy ( "o ( ^! ev5ts seems to indicate a cause slightly favor+ x1 :i* increases ! probability ( its next return1 & its repetition prolong$ = a long "t1 su* z a long s]ies ( ra9y days1 may develop unkn[n causes = its *ange2 s t at ea* expect$ ev5t we are n1 z at ea* draw+ ( a lott]y1 l$ back to ! same /ate ( indecision 9 reg>d to :at ought to happ54 ,b 9 proportion z ! obs]vation ( ^! ev5ts is multipli$1 ! comp>ison ( _! results ) ^? ( lott]ies becomes m exact4 ,by an illusion contr>y to ! prec$+ ones "o seeks 9 ! pa/ draw+s ( ! lott]y ( ,france ! numb]s mo/ (t5 drawn1 9 ord] to =m comb9ations ^u :i* "o ?9ks to place ! /ake to advantage4 ,b :5 ! mann] 9 :i* ! mix+ ( ! numb]s 9 ?is lott]y is consid]$1 ! pa/ ought to h no influ5ce ^u ! future4 ,! v frequ5t draw+s ( a numb] are only ! anomalies ( *ance2 ,i h submitt$ sev]al ( !m to calculation & h con/antly f\nd t !y are includ$ )9 ! limits :i* ! supposition ( an equal possibility ( ! draw+ ( all ! numb]s all[s u to admit )\t improbability4 ,9 a long s]ies ( ev5ts ( ! same k9d ! s+le *ances ( haz>d ought sometimes to (f] ! s+ul> ve9s ( good luck or bad luck :i* ! majority ( play]s d n fail to attribute to a k9d ( fatality4 ,x happ5s (t5 9 games :i* dep5d at ! same "t ^u haz>d & ^u ! compet5cy ( ! play]s1 t t "o :o loses1 tr\bl$ by 8 loss1 seeks to repair x by haz>d\s ?r[s :i* he w\ld %un 9 ano!r situation2 ?us he aggravates 8 own ill luck & prolongs its duration4 ,x is !n t prud5ce becomes necess>y & t x is ( importance to conv9ce oneself t ! moral disadvantage atta*$ to unfavorable *ances is increas$ by ! ill luck itself4 ,! op9ion t man has long be5 plac$ 9 ! c5tre ( ! univ]se1 consid]+ himself ! special object ( ! c>es ( nature1 leads ea* individual to make himself ! c5tre ( a m or less ext5d$ sph]e & to believe t haz>d has pref]5ce = him4 ,su/a9$ by ?is belief1 play]s (t5 risk consid]able sums at games :5 !y "k t ! *ances are unfavorable4 ,9 ! conduct ( life a simil> op9ion may sometimes h advantages2 b mo/ (t5 x leads to disa/r\s ent]prises4 ,"h z ev]y:]e illusions are dang]\s & tru? alone is g5]ally useful4 ,"o ( ! great advantages ( ! calculus ( probabilities is to tea* u to di/ru/ fir/ op9ions4 ,z we recognize t !y (t5 deceive :5 !y may be submitt$ to calculus1 we ought to conclude t 9 o!r matt]s confid5ce %\ld be giv5 only aft] extreme circumspection4 ,let u prove ?is by example4 ,an urn conta9s f\r balls1 black & :ite1 b :i* are n all ( ! same color4 ,"o ( ^! balls has be5 drawn ^: color is :ite & :i* has be5 put back 9 ! urn 9 ord] to proce$ aga9 to simil> draw+s4 ,"o dem&s ! probability ( extract+ only black balls 9 ! f\r foll[+ draw+s4 ,if ! :ite & black 7 9 equal numb] ?is probability w\ld be ! f\r? p[] ( ! probability ( extract+ a black ball at ea* draw+2 x w\ld be !n #a_/af4 ,b ! extraction ( a :ite ball at ! fir/ draw+ indicates a sup]iority 9 ! numb] ( :ite balls 9 ! urn2 = if "o supposes 9 ! urn ?ree :ite balls & "o black ! probability ( extract+ a :ite ball is 2 x is #b_/d if "o supposes two :ite balls & two black2 f9ally x is r$uc$ to if "o supposes ?ree black balls & "o :ite4 ,foll[+ ! pr9ciple ( ! probability ( causes drawn f ev5ts ! probabilities ( ^! ?ree suppositions are among !mselves z ! quantities 1 #b_/d1 2 !y are consequ5tly equal to #c_/f1 #b_/f1 4 ,x is ?us a bet ( #e aga9/ #a t ! numb] ( black balls is inf]ior1 or at ! mo/ equal1 to t ( ! :ite4 ,x seems !n t aft] ! extraction ( a :ite ball at ! fir/ draw+1 ! probability ( extract+ successively f\r black balls ought to be less ?an 9 ! case ( ! equality ( ! colors or small] ?an "o sixte5?4 ,h[ev]1 x is n1 & x is f\nd by a v simple calculation t ?is probability is great] ?an "o f\rte5?4 ,inde$ x w\ld be ! f\r? p[] ( 1 ( #b_/d1 & ( 9 ! fir/1 ! second1 & ! ?ird ( ! prec$+ suppositions conc]n+ ! colors ( ! balls 9 ! urn4 ,multiply+ respectively ea* p[] by ! probability ( ! correspond+ supposition1 or by #c_/f1 #b_/f1 & 1 ! sum ( ! products w be ! probability ( extract+ successively f\r black balls4 ,"o has ?us = ?is probability #bi_/chd1 a fraction great] ?an #a_/ad4 ,?is p>adox is expla9$ by consid]+ t ! indication ( ! sup]iority ( :ite balls ov] ! black ones at ! fir/ draw+ does n exclude at all ! sup]iority ( ! black balls ov] ! :ite ones1 a sup]iority :i* excludes ! supposition ( ! equality ( ! colors4 ,b ?is sup]iority1 ?\gh b slightly probable1 ought to r5d] ! probability ( draw+ successively a giv5 numb] ( black balls great] ?an 9 ?is supposition if ! numb] is consid]able2 & "o has j se5 t ?is comm5ces :5 ! giv5 numb] is equal to f\r4 ,let u consid] aga9 an urn :i* conta9s sev]al :ite & black balls4 ,let u suppose at fir/ t !re is only "o :ite ball & "o black4 ,x is !n an ev5 bet t a :ite ball w be extract$ 9 "o draw+4 ,b x seems = ! equality ( ! bet t "o :o bets on extract+ ! :ite ball ought to h two draw+s if ! urn conta9s two black & "o :ite1 ?ree draw+s if x conta9s ?ree black & "o :ite1 & s on2 x is suppos$ t aft] ea* draw+ ! extract$ ball is plac$ aga9 9 ! urn4 ,we are conv9c$ easily t ?is fir/ idea is errone\s4 ,inde$ 9 ! case ( two black & "o :ite ball1 ! probability ( extract+ two black 9 two draw+s is ! second p[] ( or #d_/i2 b ?is probability add$ to t ( draw+ a :ite ball 9 two draw+s is c]ta9ty or unity1 s9ce x is c]ta9 t two black balls or at lea/ "o :ite ball ought to be drawn2 ! probability 9 ?is la/ case is !n #e_/i1 a fraction great] ?an 4 ,!re w\ld /ill be a great] advantage 9 ! bet ( draw+ "o :ite ball 9 five draws :5 ! urn conta9s five black & "o :ite ball2 ?is bet is ev5 advantage\s 9 f\r draw+s2 x returns !n to t ( ?r[+ six 9 f\r ?r[s ) a s+le die4 ,! ,*evali] de ,m]1 :o caus$ ! inv5tion ( ! calculus ( probabilities by enc\rag+ 8 fri5d ,pascal1 ! great geometrician1 to occupy himself ) x1 said to him t he _h f\nd error 9 ! numb]s by ?is ratio4 ,if we und]take to make six ) "o die !re is an advantage 9 und]tak+ x 9 f\r ?r[s1 z #fga to #fbe4 ,if we und]take to make two sixes ) two dice1 !re is a disadvantage 9 und]tak+ 9 #bd ?r[s4 ,at lea/ #bd is to #cf1 ! numb] ( ! faces ( ! two dice1 z #d is to #f1 ! numb] ( faces ( "o die4 ,?is 01 wrote ,pascal to ,f]mat1 8 great sc&al :i* caus$ him to say boldly t ! propositions 7 n con/ant & t ari?metic 0 dem5t$4444 ,he has a v good m9d1 b he is n a geometrician1 :i* is1 z y "k1 a great fault4 ,! ,*evali] de ,m]1 deceiv$ by a false analogy1 ?\ght t 9 ! case ( ! equality ( bets ! numb] ( ?r[s ought to increase 9 proportion to ! numb] ( all ! *ances possible1 :i* is n exact1 b :i* approa*es exactness z ?is numb] becomes l>g]4 ,"o has endeavor$ to expla9 ! sup]iority ( ! bir?s ( boys ov] ^? ( girls by ! g5]al desire ( fa!rs to h a son :o w\ld p]petuate ! "n4 ,?us by imag9+ an urn fill$ ) an inf9ity ( :ite & black balls 9 equal numb]1 & suppos+ a great numb] ( p]sons ea* ( :om draws a ball f ?is urn & cont9ues ) ! int5tion ( /opp+ :5 he %all h extract$ a :ite ball1 "o has believ$ t ?is int5tion ought to r5d] ! numb] ( :ite balls extract$ sup]ior to t ( ! black ones4 ,inde$ ?is int5tion gives necess>ily aft] all ! draw+s a numb] ( :ite balls equal to t ( p]sons1 & x is possible t ^! draw+s w\ld nev] lead a black ball4 ,b x is easy to see t ?is fir/ notion is only an illusion2 = if "o conceives t 9 ! fir/ draw+ all ! p]sons draw at once a ball f ! urn1 x is evid5t t _! int5tion c h no influ5ce ^u ! color ( ! balls :i* ought to appe> at ?is draw+4 ,its unique effect w be to exclude f ! second draw+ ! p]sons :o %all h drawn a :ite "o at ! fir/4 ,x is likewise app>5t t ! int5tion ( ! p]sons :o %all take "p 9 ! new draw+ w h no influ5ce ^u ! color ( ! balls :i* %all be drawn1 & t x w be ! same at ! foll[+ draw+s4 ,?is int5tion w h no influ5ce !n ^u ! color ( ! balls extract$ 9 ! totality ( draw+s2 x w1 h[ev]1 cause m or few] to p>ticipate at ea* draw+4 ,! ratio ( ! :ite balls extract$ to ! black ones w diff] ?us v little f unity4 ,x foll[s t ! numb] ( p]sons be+ suppos$ v l>ge1 if obs]vation gives betwe5 ! colors extract$ a ratio :i* diff]s s5sibly f unity1 x is v probable t ! same diff]5ce is f\nd betwe5 unity & ! ratio ( ! :ite balls to ! black conta9$ 9 ! urn4 ,i c\nt aga9 among illusions ! application :i* ,liebnitz & ,daniel ,b]n\lli h made ( ! calculus ( probabilities to ! summation ( s]ies4 ,if "o r$uces ! fraction ^: num]ator is unity & ^: d5om9ator is unity plus a v>iable1 9 a s]ies prescrib$ by ! ratio to ! p[]s ( ?is v>iable1 x is easy to see t 9 suppos+ ! v>iable equal to unity ! fraction becomes 1 & ! s]ies becomes plus "o1 m9us "o1 plus "o1 m9us "o1 etc4 ,9 add+ ! fir/ two t]ms1 ! second two1 & s on1 ! s]ies is trans=m$ into ano!r ( :i* ea* t]m is z]o4 ,gr&i1 an ,italian ,jesuit1 conclud$ f ?is ! possibility ( ! creation2 because ! s]ies be+ always 1 he saw ?is fraction spr+ f an inf9ity ( z]os or f no?+4 ,x 0 ?us t ,liebnitz believ$ he saw ! image ( creation 9 8 b9>y ari?metic :]e he employ$ only ! two *>act]s1 unity & z]o4 ,he imag9$1 s9ce ,god c be repres5t$ by unity & no?+ by z]o1 t ! ,supreme ,be+ _h drawn f no?+ all be+s1 z unity ) z]o expresses all ! numb]s 9 ?is sy/em ( ari?metic4 ,?is idea 0 s pleas+ to ,liebnitz t he communicat$ x to ! ,jesuit ,grimaldi1 presid5t ( ! tribunal ( ma!matics 9 ,*9a1 9 ! hope t ?is emblem ( creation w\ld conv]t to ,*ri/ianity ! emp]or !re :o p>ticul>ly lov$ ! sci5ces4 ,i report ?is incid5t only to %[ to :at ext5t ! prejudices ( infancy c mislead ! greate/ m54 ,liebnitz1 always l$ by a s+ul> & v loose metaphysics1 consid]$ t ! s]ies plus "o1 m9us "o1 plus "o1 etc41 becomes unity or z]o accord+ z "o /ops at a numb] ( t]ms odd or ev52 & z 9 inf9ity !re is no reason to pref] ! ev5 numb] to ! odd1 "o ought foll[+ ! rules ( probability1 to take ! half ( ! results relative to ^! two k9ds ( numb]s1 & :i* are z]o & unity1 :i* gives = ! value ( ! s]ies4 ,daniel ,b]n\lli has s9ce ext5d$ ?is reason+ to ! summation ( s]ies =m$ f p]iodic t]ms4 ,b all ^! s]ies h no values prop]ly speak+2 !y get !m only 9 ! case :]e _! t]ms are multipli$ by ! successive p[]s ( a v>iable less ?an unity4 ,!n ^! s]ies are always conv]g5t1 h[ev] small "o supposes ! diff]5ce ( ! v>iable f unity2 & x is easy to demon/rate t ! values assign$ by ,b]n\lli1 by virtue ( ! rule ( probabilities1 are ! same values ( ! g5]ative fraction ( ! s]ies1 :5 "o supposes 9 ^! fractions ! v>iable equal to unity4 ,^! values are aga9 ! limits :i* ! s]ies approa* m & m1 9 proportion z ! v>iable approa*es unity4 ,b :5 ! v>iable is exactly equal to unity ! s]ies cease to be conv]g5t2 !y h values only z f> z "o arre/s !m4 ,! rem>kable ratio ( ?is application ( ! calculus ( probabilities ) ! limits ( ! values ( p]iodic s]ies supposes t ! t]ms ( ^! s]ies are multipli$ by all ! consecutive p[]s ( ! v>iable4 ,b ?is s]ies may result f ! developm5t ( an inf9ity ( diff]5t fractions 9 :i* ?is did n occur4 ,?us ! s]ies plus "o1 m9us "o1 plus "o1 etc41 may spr+ f ! developm5t ( a fraction ^: num]ator is unity plus ! v>iable1 & ^: d5om9ator is ?is num]ator augm5t$ by ! squ>e ( ! v>iable4 ,suppos+ ! v>iable equal to unity1 ?is developm5t *anges1 9 ! s]ies propos$1 & ! g5]ative fraction becomes equal to 2 ! rules ( probabilities w\ld give !n a false result1 :i* proves h[ dang]\s x w\ld be to employ simil> reason+1 especially 9 ! ma!matical sci5ces1 :i* ought to be especially di/+ui%$ by ! rigor ( _! op]ations4 ,we are l$ naturally to believe t ! ord] accord+ to :i* we see ?+s r5ew$ ^u ! e>? has exi/$ f all times & w cont9ue always4 ,inde$ if ! pres5t /ate ( ! univ]se 7 exactly simil> to ! ant]ior /ate :i* has produc$ x1 x w\ld give bir? 9 its turn to a simil> /ate2 ! succession ( ^! /ates w\ld !n be et]nal4 ,i h f\nd by ! application ( analysis to ! law ( univ]sal gravity t ! movem5t ( rotation & ( revolution ( ! planets & satellites1 & ! position ( ! orbits & ( _! equators are subject$ only to p]iodic inequalities4 ,9 comp>+ ) anci5t eclipses ! !ory ( ! secul> equation ( ! moon ,i h f\nd t s9ce ,hipp>*us ! duration ( ! day has n v>i$ by ! hundr$? ( a second1 & t ! mean temp]ature ( ! e>? has n dim9i%$ ! "o-hundr$? ( a degree4 ,?us ! /ability ( actual ord] appe>s e/abli%$ at ! same "t by !ory & by obs]vations4 ,b ?is ord] is effect$ by div]s causes :i* an att5tive exam9ation reveals1 & :i* x is impossible to submit to calculus4 ,! actions ( ! ocean1 ( ! atmosph]e1 & ( meteors1 ( e>?quakes1 & ! eruptions ( volcanoes1 agitate cont9ually ! surface ( ! e>? & ought to effect 9 ! long run great *anges4 ,! temp]ature ( climates1 ! volume ( ! atmosph]e1 & ! proportion ( ! gases :i* con/itute x1 may v>y 9 an inappreciable mann]4 ,! in/rum5ts & ! means suitable to det]m9e ^! v>iations be+ new1 obs]vation has be5 unable up to ?is "t to tea* u any?+ 9 ?is reg>d4 ,b x is h>dly probable t ! causes :i* absorb & r5ew ! gases con/itut+ ! air ma9ta9 exactly _! respective proportions4 ,a long s]ies ( c5turies w %[ ! alt]ations :i* are exp]i5c$ by all ^! elem5ts s ess5tial to ! cons]vation ( organiz$ be+s4 ,al?\gh hi/orical monum5ts d n g back to a v great antiquity !y (f] u nev]!less suffici5tly great *anges :i* h come ab\t by ! sl[ & cont9u$ action ( natural ag5ts4 ,se>*+ 9 ! b[els ( ! e>? "o discov]s num]\s dbris ( =m] nature1 entirely diff]5t f ! pres5t4 ,moreov]1 if ! entire e>? 0 9 ! beg9n+ fluid1 z ev]y?+ appe>s to indicate1 "o imag9es t 9 pass+ f t /ate to ! "o :i* x has n[1 its surface ought to h exp]i5c$ prodigi\s *anges4 ,! heav5s itself 9 spite ( ! ord] ( its movem5ts1 is n un*angeable4 ,! resi/ance ( light & ( o!r e!real fluids1 & ! attraction ( ! />s ought1 aft] a great numb] ( c5turies1 to alt] consid]ably ! planet>y movem5ts4 ,! v>iations already obs]v$ 9 ! />s & 9 ! =m ( ! nebul give u a pres5tim5t ( ^? :i* "t w develop 9 ! sy/em ( ^! great bodies4 ,"o may repres5t ! successive /ates ( ! univ]se by a curve1 ( :i* "t w\ld be ! abscissa & ( :i* ! ord9ates are ! div]s /ates4 ,sc>cely kn[+ an elem5t ( ?is curve we are f> f be+ able to g back to its orig92 & if 9 ord] to satisfy ! imag9ation1 always re/less f our ignorance ( ! cause ( ! ph5om5a :i* int]e/ x1 "o v5tures "s conjectures x is wise to pres5t !m only ) extreme res]ve4 ,!re exi/s 9 ! e/imation ( probabilities a k9d ( illusions1 :i* dep5d+ especially ^u ! laws ( ! intellectual organization dem&s1 9 ord] to secure oneself aga9/ !m1 a pr(\nd exam9ation ( ^! laws4 ,! desire to p5etrate into ! future & ! ratios ( "s rem>kable ev5ts1 to ! pr$ictions ( a/rolog]s1 ( div9]s & soo?say]s1 to pres5tim5ts & dreams1 to ! numb]s & ! days reput$ lucky or unlucky1 h giv5 bir? to a multitude ( prejudices /ill v widespread4 ,"o does n reflect ^u ! great numb] ( non-co9cid5ces :i* h made no impression or :i* are unkn[n4 ,h[ev]1 x is necess>y to be acqua9t$ ) !m 9 ord] to appreciate ! probability ( ! causes to :i* ! co9cid5ces are attribut$4 ,?is k w\ld confirm )\t d\bt t :i* reason tells u 9 reg>d to ^! prejudices4 ,?us ! philosoph] ( antiquity to :om is %[n 9 a temple1 9 ord] to exalt ! p[] ( ! god :o is ador$ !re1 ! ex veto ( all ^? :o aft] hav+ invok$ x 7 sav$ f %ipwreck1 pres5ts an incid5t consonant ) ! calculus ( probabilities1 obs]v+ t he does n see inscrib$ ! names ( ^? :o1 9 spite ( ?is invocation1 h p]i%$4 ,cic]o has refut$ all ^! prejudices ) mu* reason & eloqu5ce 9 8 ,treatise on ,div9ation1 :i* he ends by a passage :i* ,i %all cite2 = "o loves to f9d aga9 among ! anci5ts ! ?und]bolts ( reason1 :i*1 aft] hav+ dissipat$ all ! prejudices by its light1 %all become ! sole f\ndation ( human in/itutions4 ,x is necess>y1 says ! ,roman orator1 to reject div9ation by dreams & all simil> prejudices4 ,widespread sup]/ition has subjugat$ ! majority ( m9ds & has tak5 possession ( ! feebl5ess ( m54 ,x is ?is we h exp\nd$ 9 our books ^u ! nature ( ! gods & especially 9 ?is "w1 p]suad$ t we %all r5d] a s]vice to o!rs & to ourselves if we succe$ 9 de/roy+ sup]/ition4 ,h[ev] 7& ,i desire especially 9 ?is reg>d my ?\ght be well compreh5d$71 9 de/roy+ sup]/ition ,i am f> f wi%+ to di/urb religion4 ,wisdom enjo9s u to ma9ta9 ! in/itutions & ! c]emonies ( our ance/ors1 t\*+ ! cult ( ! gods4 ,moreov]1 ! beauty ( ! univ]se & ! ord] ( cele/ial ?+s =ce u to recognize "s sup]ior nature :i* ought to be rem>k$ & admir$ by ! human race4 ,b z f> z x is prop] to propagate religion1 :i* is jo9$ to ! k ( nature1 s f> x is necess>y to "w t[>d ! extirpation ( sup]/ition1 = x torm5ts "o1 importunes "o1 & pursues "o cont9ually & 9 all places4 ,if "o consult a div9] or a soo?say]1 if "o immolates a victim1 if "o reg>ds ! flight ( a bird1 if "o enc\nt]s a ,*aldean or an aruspex1 if x light5s1 if x ?und]s1 if ! ?und]bolt /rikes1 f9ally1 if !re is born or is manife/$ a k9d ( prodigy1 ?+s "o ( :i* ought (t5 to happ51 !n sup]/ition dom9ates & leaves no repose4 ,sleep itself1 ?is refuge ( mortals 9 _! tr\bles & _! labors1 becomes by x a new s\rce ( inquietude & fe>4 ,all ^! prejudices & ! t]rors :i* !y inspire are connect$ ) physiological causes :i* cont9ue sometimes to op]ate /rongly aft] reason has disabus$ u ( !m4 ,b ! repetition ( acts contr>y to ^! prejudices c always de/roy !m4 ,,*apt] ,,xvii4 ,,conc]n+ ,,! ,,v>i\s ,,means ,,( ,,approa*+ ,,c]ta9ty4 ,induction1 analogy1 hypo!ses f\nd$ ^u facts & rectifi$ cont9ually by new obs]vations1 a happy tact giv5 by nature & /r5g!n$ by num]\s comp>isons ( its indications ) exp]i5ce1 su* are ! pr9cipal means = arriv+ at tru?4 ,if "o consid]s a s]ies ( objects ( ! same nature "o p]ceives among !m & 9 _! *anges ratios :i* manife/ !mselves m & m 9 proportion z ! s]ies is prolong$1 & :i*1 ext5d+ & g5]aliz+ cont9ually1 lead f9ally to ! pr9ciple f :i* !y 7 d]iv$4 ,b ^! ratios are envelop$ by s _m /range circum/ances t x requires great sagacity to dis5tangle !m & to recur to ?is pr9ciple3 x is 9 ?is t ! true g5ius ( sci5ces consi/s4 ,analysis & natural philosophy owe _! mo/ important discov]ies to ?is fruitful means1 :i* is call$ induction4 ,newton 0 indebt$ to x = 8 !orem ( ! b9omial & ! pr9ciple ( univ]sal gravity4 ,x is difficult to appreciate ! probability ( ! results ( induction1 :i* is bas$ ^u ?is t ! simple/ ratios are ! mo/ common2 ?is is v]ifi$ 9 ! =mul ( analysis & is f\nd aga9 9 natural ph5om5a1 9 cry/allization1 & 9 *emical comb9ations4 ,?is simplicity ( ratios w n appe> a/oni%+ if we consid] t all ! effects ( nature are only ma!matical results ( a small numb] ( immutable laws4 ,yet induction1 9 lead+ to ! discov]y ( ! g5]al pr9ciples ( ! sci5ces1 does n suffice to e/abli% !m absolutely4 ,x is always necess>y to confirm !m by demon/rations or by decisive exp]i5ces2 = ! hi/ory ( ! sci5ces %[s u t induction has sometimes l$ to inexact results4 ,i %all cite1 = example1 a !orem ( ,f]mat 9 reg>d to prime numb]s4 ,?is great geometrician1 :o _h m$itat$ pr(\ndly ^u ?is !orem1 s\ght a =mula :i*1 conta9+ only prime numb]s1 gave directly a prime numb] great] ?an any o!r numb] assignable4 ,induction l$ him to ?9k t two1 rais$ to a p[] :i* 0 itself a p[] ( two1 =m$ ) unity a prime numb]4 ,?us1 two rais$ to ! squ>e plus "o1 =ms ! prime numb] five2 two rais$ to ! second p[] ( two1 or sixte51 =ms ) "o ! prime numb] sev5te54 ,he f\nd t ?is 0 /ill true = ! eigh? & ! sixte5? p[] ( two augm5t$ by unity2 & ?is induction1 bas$ ^u sev]al ari?metical consid]ations1 caus$ him to reg>d ?is result z g5]al4 ,h[ev]1 he av[$ t he _h n demon/rat$ x4 ,inde$1 ,eul] recogniz$ t ?is does n hold = ! ?irty-second p[] ( two1 :i*1 augm5t$ by unity1 gives #d1bid1ifg1big1 a numb] divisible by #fda4 ,we judge by induction t if v>i\s ev5ts1 movem5ts1 = example1 appe> con/antly & h be5 long connect$ by a simple ratio1 !y w cont9ue to be subject$ to x2 & we conclude f ?is1 by ! !ory ( probabilities1 t ?is ratio is due1 n to haz>d1 b to a regul> cause4 ,?us ! equality ( ! movem5ts ( ! rotation & ! revolution ( ! moon2 t ( ! movem5ts ( ! nodes ( ! orbit & ( ! lun> equator1 & ! co9cid5ce ( ^! nodes2 ! s+ul> ratio ( ! movem5ts ( ! fir/ ?ree satellites ( ,jupit]1 accord+ to :i* ! mean longitude ( ! fir/ satellite1 less ?ree times t ( ! second1 plus two times t ( ! ?ird1 is equal to two "r angles2 ! equality ( ! int]val ( ! tides to t ( ! passage ( ! moon to ! m]idian2 ! return ( ! greate/ tides ) ! syzygies1 & ( ! smalle/ ) ! quadratures2 all ^! ?+s1 :i* h be5 ma9ta9$ s9ce !y 7 fir/ obs]v$1 indicate ) an extreme probability1 ! exi/5ce ( con/ant causes :i* geometricians h happily succe$$ 9 atta*+ to ! law ( univ]sal gravity1 & ! k ( :i* r5d]s c]ta9 ! p]petuity ( ^! ratios4 ,! *ancellor ,bacon1 ! eloqu5t promot] ( ! true philosophical me?od1 has made a v /range misuse ( induction 9 ord] to prove ! immobility ( ! e>?4 ,he reasons ?us 9 ! ,novum ,organum1 8 f9e/ "w3 ,! movem5t ( ! />s f ! ori5t to ! occid5t increases 9 swiftness1 9 proportion to _! di/ance f ! e>?4 ,?is movem5t is swifte/ ) ! />s2 x slack5s a little ) ,saturn1 a little m ) ,jupit]1 & s on to ! moon & ! highe/ comets4 ,x is /ill p]ceptible 9 ! atmosph]e1 especially betwe5 ! tropics1 on acc\nt ( ! great circles :i* ! molecules ( ! air describe !re2 f9ally1 x is almo/ inappreciable ) ! ocean2 x is !n nil = ! e>?4 ,b ?is induction proves only t ,saturn1 & ! />s :i* are inf]ior to x1 h _! own movem5ts1 contr>y to ! real or app>5t movem5t :i* sweeps ! :ole cele/ial sph]e f ! ori5t to ! occid5t1 & t ^! movem5ts appe> sl[] ) ! m remote />s1 :i* is con=mable to ! laws ( optics4 ,bacon ought to h be5 /ruck by ! inconceivable swiftness :i* ! />s require 9 ord] to accompli% _! diurnal revolution1 if ! e>? is immovable1 & by ! extreme simplicity ) :i* its rotation expla9s h[ bodies s di/ant1 ! ones f ! o!rs1 z ! />s1 ! sun1 ! planets1 & ! moon1 all seem subject$ to ?is revolution4 ,z to ! ocean & to ! atmosph]e1 he ought n to comp>e _! movem5t ) t ( ! />s :i* are deta*$ f ! e>?2 b s9ce ! air & ! sea make "p ( ! t]re/rial globe1 !y ought to p>ticipate 9 its movem5t or 9 its repose4 ,x is s+ul> t ,bacon1 c>ri$ to great prospects by 8 g5ius1 0 n won ov] by ! maje/ic idea :i* ! ,cop]nican sy/em ( ! univ]se (f]s4 ,he 0 able1 h[ev]1 to f9d 9 favor ( t sy/em1 /rong analogies 9 ! discov]ies ( ,galileo1 :i* 7 cont9u$ by him4 ,he has giv5 = ! se>* aft] tru? ! precept1 b n ! example4 ,b by insi/+1 ) all ! =ce ( reason & ( eloqu5ce1 ^u ! necessity ( ab&on+ ! insignificant subtleties ( ! s*ool1 9 ord] to apply oneself to obs]vations & to exp]i5ces1 & by indicat+ ! true me?od ( asc5d+ to ! g5]al causes ( ph5om5a1 ?is great philosoph] contribut$ to ! imm5se /rides :i* ! human m9d made 9 ! gr& c5tury 9 :i* he t]m9at$ 8 c>e]4 ,analogy is bas$ ^u ! probability1 t simil> ?+s h causes ( ! same k9d & produce ! same effects4 ,?is probability increase z ! similitude becomes m p]fect4 ,?us we judge )\t d\bt t be+s provid$ ) ! same organs1 do+ ! same ?+s1 exp]i5ce ! same s5sations1 & are mov$ by ! same desires4 ,! probability t ! animals :i* resemble u h s5sations analog\s to ours1 al?\gh a little inf]ior to t :i* is relative to individuals ( our species1 is /ill exce$+ly great2 & x has requir$ all ! influ5ce ( religi\s prejudices to make u ?9k ) "s philosoph]s t animals are m]e automatons4 ,! probability ( ! exi/5ce ( feel+ decreases 9 ! same proportion z ! similitude ( ! organs ) ours dim9i%es1 b x is always v great1 ev5 ) insects4 ,9 see+ ^? ( ! same species execute v complicat$ ?+s exactly 9 ! same mann] f g5]ation to g5]ation1 & )\t hav+ le>n$ !m1 "o is l$ to believe t !y act by a k9d ( aff9ity analog\s to t :i* br+s toge!r ! molecules ( cry/als1 b :i*1 toge!r ) ! s5sation atta*$ to all animal organization1 produces1 ) ! regul>ity ( *emical comb9ations1 comb9ations t are mu* m s+ul>2 "o might1 p]haps1 "n ?is m+l+ ( elective aff9ities & s5sations animal aff9ity4 ,al?\gh !re exi/s a great analogy betwe5 ! organization ( plants & t ( animals1 x does n seem to me suffici5t to ext5d to vegetables ! s5se ( feel+2 b no?+ au?orizes u 9 d5y+ x to !m4 ,s9ce ! sun br+s =?1 by ! b5efic5t action ( its light & ( its heat1 ! animals & plants :i* cov] ! e>?1 we judge by analogy t x produces simil> effects ^u ! o!r planets2 = x is n natural to ?9k t ! cause ^: activity we see develop$ 9 s _m ways %\ld be /]ile ^u s great a planet z ,jupit]1 :i*1 l ! t]re/rial globe1 has its days1 its nights1 & its ye>s1 & ^u :i* obs]vations indicate *anges :i* suppose v active =ces4 ,yet ?is w\ld be giv+ too great an ext5sion to analogy to conclude f x ! similitude ( ! inhabitants ( ! planets & ( ! e>?4 ,man1 made = ! temp]ature :i* he enjoys1 & = ! elem5t :i* he brea!s1 w\ld n be able1 accord+ to all appe>ance1 to live ^u ! o!r planets4 ,b ought !re n to be an inf9ity ( organization relative to ! v>i\s con/itutions ( ! globes ( ?is univ]se8 ,if ! s+le diff]5ce ( ! elem5ts & ( ! climates make s mu* v>iety 9 t]re/rial productions1 h[ mu* great] ! diff]5ce ought to be among ^? ( ! v>i\s planets & ( _! satellites6 ,! mo/ active imag9ation c =m no idea ( x2 b _! exi/5ce is v probable4 ,we are l$ by a /rong analogy to reg>d ! />s z s _m suns end[$1 l ours1 ) an attractive p[] proportional to ! mass & reciprocal to ! squ>e ( ! di/ances2 = ?is p[] be+ demon/rat$ = all ! bodies ( ! sol> sy/em1 & = _! smalle/ molecules1 x appe>s to app]ta9 to all matt]4 ,already ! movem5ts ( ! small />s1 :i* h be5 call$ d\ble1 on acc\nt ( _! be+ b9>y1 appe> to indicate x2 a c5tury at mo/ ( precise obs]vations1 by v]ify+ _! movem5ts ( revolution1 ! ones ab\t ! o!rs1 w place beyond d\bt _! reciprocal attractions4 ,! analogy :i* leads u to make ea* /> ! c5tre ( a planet>y sy/em is f> less /rong ?an ! prec$+ "o2 b x acquires probability by ! hypo!sis :i* has be5 propos$ 9 reg>d to ! =mation ( ! />s & ( ! sun2 = 9 ?is hypo!sis ea* />1 hav+ be5 l ! sun1 primitively environ$ by a va/ atmosph]e1 x is natural to attribute to ?is atmosph]e ! same effects z to ! sol> atmosph]e1 & to suppose t x has produc$1 9 cond5s+1 planets & satellites4 ,a great numb] ( discov]ies 9 ! sci5ces is due to analogy4 ,i %all cite z "o ( ! mo/ rem>kable1 ! discov]y ( atmosph]ic electricity1 to :i* "o has be5 l$ by ! analogy ( electric ph5om5a ) ! effects ( ?und]4 ,! sure/ me?od :i* c guide u 9 ! se>* = tru?1 consi/s 9 ris+ by induction f ph5om5a to laws & f laws to =ces4 ,laws are ! ratios :i* connect p>ticul> ph5om5a toge!r3 :5 !y h %[n ! g5]al pr9ciple ( ! =ces f :i* !y are d]iv$1 "o v]ifies x ei!r by direct exp]i5ces1 :5 ?is is possible1 or by exam9ation if x agrees ) kn[n ph5om5a2 & if by a rigor\s analysis we see !m proce$ f ?is pr9ciple1 ev5 9 _! small details1 & if1 moreov]1 !y are q v>i$ & v num]\s1 !n sci5ce acquires ! highe/ degree ( c]ta9ty & ( p]fection t x is able to atta94 ,su*1 a/ronomy has become by ! discov]y ( univ]sal gravity4 ,! hi/ory ( ! sci5ces %[s t ! sl[ & labori\s pa? ( induction has n always be5 t ( inv5tors4 ,! imag9ation1 impati5t to arrive at ! causes1 takes pleasure 9 creat+ hypo!ses1 & (t5 x *anges ! facts 9 ord] to adapt !m to its "w2 !n ! hypo!ses are dang]\s4 ,b :5 "o reg>ds !m only z ! means ( connect+ ! ph5om5a 9 ord] to discov] ! laws2 :51 by refus+ to attribute !m to a reality1 "o rectifies !m cont9ually by new obs]vations1 !y are able to lead to ! v]itable causes1 or at lea/ put u 9 a position to conclude f ! ph5om5a obs]v$ ^? :i* giv5 circum/ances ought to produce4 ,if we %\ld try all ! hypo!ses :i* c be =m$ 9 reg>d to ! cause ( ph5om5a we %\ld arrive1 by a process ( exclusion1 at ! true "o4 ,?is means has be5 employ$ ) success2 sometimes we h arriv$ at sev]al hypo!ses :i* expla9 equally well all ! facts kn[n1 & among :i* s*ol>s are divid$1 until decisive obs]vations h made kn[n ! true "o4 ,!n x is int]e/+1 = ! hi/ory ( ! human m9d1 to return to ^! hypo!ses1 to see h[ !y succe$ 9 expla9+ a great numb] ( facts1 & to inve/igate ! *anges :i* !y ought to und]go 9 ord] to agree ) ! hi/ory ( nature4 ,x is ?us t ! sy/em ( ,ptolemy1 :i* is only ! realization ( cele/ial appe>ances1 is trans=m$ into ! hypo!sis ( ! movem5t ( ! planets ab\t ! sun1 by r5d]+ equal & p>allel to ! sol> orbit ! circles & ! epicycles :i* he causes to be describ$ annually1 & ! magnitude ( :i* he leaves undet]m9$4 ,x suffices1 !n1 9 ord] to *ange ?is hypo!sis into ! true sy/em ( ! _w1 to transport ! app>5t movem5t ( ! sun 9 a s5se contr>y to ! e>?4 ,x is almo/ always impossible to submit to calculus ! probability ( ! results obta9$ by ^! v>i\s means2 ?is is true likewise = hi/orical facts4 ,b ! totality ( ! ph5om5a expla9$1 or ( ! te/imonies1 is sometimes su* t )\t be+ able to appreciate ! probability we _c reasonably p]mit ourselves any d\bt 9 reg>d to !m4 ,9 ! o!r cases x is prud5t to admit !m only ) great res]ve4 ,,*apt] ,,xviii4 ,,hi/orical ,,notice ,,conc]n+ ,,! ,,calculus ,,( ,,probabilities4 ,long ago 7 det]m9$1 9 ! simple/ games1 ! ratios ( ! *ances :i* are favorable or unfavorable to ! play]s2 ! /akes & ! bets 7 regulat$ accord+ to ^! ratios4 ,b no "o be=e ,pascal & ,f]mat _h giv5 ! pr9ciples & ! me?ods = submitt+ ?is subject to calculus1 & no "o _h solv$ ! r complicat$ que/ions ( ?is k9d4 ,x is1 !n1 to ^! two great geometricians t we mu/ ref] ! fir/ elem5ts ( ! sci5ce ( probabilities1 ! discov]y ( :i* c be rank$ among ! rem>kable ?+s :i* h r5d]$ illu/ri\s ! sev5te5? c5tury--! c5tury :i* has done ! greate/ honor to ! human m9d4 ,! pr9cipal problem :i* !y solv$ by diff]5t me?ods1 consi/s1 z we h se51 9 di/ribut+ equitably ! /ake among ! play]s1 :o are suppos$ to be equally skilful & :o agree to /op ! game be=e x is f9i%$1 ! condition ( play be+ t1 9 ord] to w9 ! game1 "o mu/ ga9 a giv5 numb] ( po9ts diff]5t = ea* ( ! play]s4 ,x is cle> t ! di/ribution %\ld be made proportionally to ! respective probabilities ( ! play]s ( w9n+ ?is game1 ! probabilities dep5d+ ^u ! numb]s ( po9ts :i* are /ill lack+4 ,! me?od ( ,pascal is v +5i\s1 & is at bottom only ! equation ( p>tial diff]5ces ( ?is problem appli$ 9 det]m9+ ! successive probabilities ( ! play]s1 by go+ f ! smalle/ numb]s to ! foll[+ ones4 ,?is me?od is limit$ to ! case ( two play]s2 t ( ,f]mat1 bas$ ^u comb9ations1 applies to any numb] ( play]s4 ,pascal believ$ at fir/ t x 01 l 8 own1 re/rict$ to two play]s2 ?is br\ght ab\t betwe5 !m a discussion1 at ! conclusion ( :i* ,pascal recogniz$ ! g5]ality ( ! me?od ( ,f]mat4 ,huyg5s unit$ ! div]s problems :i* _h already be5 solv$ & add$ new ones 9 a little treatise1 ! fir/ t has appe>$ on ?is subject & :i* has ! title ,de ,ratioc9iis 9 ludo ale4 ,sev]al geometricians h occupi$ !mselves ) ! subject s9ce3 ,hudde1 ! great p5sion>y1 ,witt 9 ,holl&1 & ,halley 9 ,engl&1 appli$ calculus to ! probabilities ( human life1 & ,halley publi%$ 9 ?is field ! fir/ table ( mortality4 ,ab\t ! same "t ,jacques ,b]n\lli propos$ to geometricians v>i\s problems ( probability1 ( :i* he aft]w>ds gave solutions4 ,f9ally he compos$ 8 beautiful "w entitl$ ,ars conject&i1 :i* appe>$ sev5 ye>s aft] 8 dea?1 :i* occurr$ 9 #agjf4 ,! sci5ce ( probabilities is m pr(\ndly inve/igat$ 9 ?is "w ?an 9 t ( ,huyg5s4 ,! au?or gives a g5]al !ory ( comb9ations & s]ies1 & applies x to sev]al difficult que/ions conc]n+ haz>ds4 ,?is "w is /ill rem>kable on acc\nt ( ! ju/ice & ! clev]ness ( view1 ! employm5t ( ! =mula ( ! b9omial 9 ?is k9d ( que/ions1 & by ! demon/ration ( ?is !orem1 namely1 t 9 multiply+ indef9itely ! obs]vations & ! exp]i5ces1 ! ratio ( ! ev5ts ( diff]5t natures approa*es t ( _! respective probabilities 9 ! limits ^: int]val becomes m & m n>r[ 9 proportion z !y are multipli$1 & become less ?an any assignable quantity4 ,?is !orem is v useful = obta9+ by obs]vations ! laws & ! causes ( ph5om5a4 ,b]n\lli atta*es1 ) reason1 a great importance to 8 demon/ration1 ^u :i* he has said to h m$itat$ = tw5ty ye>s4 ,9 ! int]val1 f ! dea? ( ,jacques ,b]n\lli to ! publication ( 8 "w1 ,montmort & ,moivre produc$ two treatises ^u ! calculus ( probabilities4 ,t ( ,montmort has ! title ,essai sur les ,jeux de has>d2 x conta9s num]\s applications ( ?is calculus to v>i\s games4 ,! au?or has add$ 9 ! second edition "s lett]s 9 :i* ,nicolas ,b]n\lli gives ! +5i\s solutions ( sev]al difficult problems4 ,! treatise ( ,moivre1 lat] ?an t ( ,montmort1 appe>$ at fir/ 9 ! ,transactions philosophiques ( ! ye> #agaa4 ,!n ! au?or publi%$ x sep>ately1 & he has improv$ x successively 9 ?ree editions4 ,?is "w is pr9cipally bas$ ^u ! =mula ( ! b9omial & ! problems :i* x conta9s h1 l _! solutions1 a gr& g5]ality4 ,b its di/+ui%+ feature is ! !ory ( recurr5t s]ies & _! use 9 ?is subject4 ,?is !ory is ! integration ( l9e> equations ( f9ite diff]5ces ) con/ant coeffici5ts1 :i* ,moivre made 9 a v happy mann]4 ,9 8 "w1 ,moivre has tak5 up aga9 ! !ory ( ,jacques ,b]n\lli 9 reg>d to ! probability ( results det]m9$ by a great numb] ( obs]vations4 ,he does n cont5t himself ) %[+1 z ,b]n\lli does1 t ! ratio ( ! ev5ts :i* ought to occur approa*es )\t ceas+ t ( _! respective probabilities2 b he gives besides an elegant & simple expression ( ! probability t ! diff]5ce ( ^! two ratios is conta9$ )9 ! giv5 limits4 ,= ?is purpose he det]m9es ! ratio ( ! greate/ t]m ( ! developm5t ( a v high p[] ( ! b9omial to ! sum ( all its t]ms1 & ! hyp]bolic log>i?m ( ! excess ( ?is t]m above ! t]ms adjac5t to x4 ,! greate/ t]m be+ !n ! product ( a consid]able numb] ( factors1 8 num]ical calculus becomes impracticable4 ,9 ord] to obta9 x by a conv]g5t approximation1 ,moivre makes use ( a !orem ( ,/irl+ 9 reg>d to ! mean t]m ( ! b9omial rais$ to a high p[]1 a rem>kable !orem1 especially 9 ?is1 t x introduces ! squ>e root ( ! ratio ( ! circumf]5ce to ! radius 9 an expression :i* seem+ly ought to be irrelevant to ?is transc5d5t4 ,moreov]1 ,moivre 0 greatly /ruck by ?is result1 :i* ,/irl+ _h d$uc$ f ! expression ( ! circumf]5ce 9 inf9ite products2 ,wallis _h arriv$ at ?is expression by a s+ul> analysis :i* conta9s ! g]m ( ! v curi\s & useful !ory ( def9ite integrals4 ,_m s*ol>s1 among :om "o ought to "n ,dep>cieux1 ,k]sseboom1 ,w>g5t91 ,dupr de ,sa9t-,maure1 ,simpson1 ,sussmil*1 ,messne1 ,moheau1 ,price1 ,bailey1 & ,duvill>d1 h collect$ a great am\nt ( precise data 9 reg>d to population1 bir?s1 m>riages1 & mortality4 ,!y h giv5 =mul & tables relative to life annuities1 tont9es1 assurances1 etc4 ,b 9 ?is %ort notice ,i c only indicate ^! useful works 9 ord] to adh]e to orig9al ideas4 ,( ?is numb] special m5tion is due to ! ma!matical & moral hopes & to ! +5i\s pr9ciple :i* ,daniel ,b]n\lli has giv5 = submitt+ ! latt] to analysis4 ,su* is aga9 ! happy application :i* he has made ( ! calculus ( probabilities to inoculation4 ,"o ought especially to include1 9 ! numb] ( ^! orig9al ideas1 direct consid]ation ( ! possibility ( ev5ts drawn f ev5ts obs]v$4 ,jacques ,b]n\lli & ,moivre suppos$ ^! possibilities kn[n1 & !y s\ght ! probability t ! result ( future exp]i5ces w m & m ne>ly repres5t !m4 ,bayes1 9 ! ,transactions philosophiques ( ! ye> #agfc1 s\ght directly ! probability t ! possibilities indicat$ by pa/ exp]i5ces are compris$ )9 giv5 limits2 & he has arriv$ at ?is 9 a ref9$ & v +5i\s mann]1 al?\gh a little p]plex+4 ,?is subject is connect$ ) ! !ory ( ! probability ( causes & future ev5ts1 conclud$ f ev5ts obs]v$4 ,"s ye>s lat] ,i exp\nd$ ! pr9ciples ( ?is !ory ) a rem>k z to ! influ5ce ( ! inequalities :i* may exi/ among ! *ances :i* are suppos$ to be equal4 ,al?\gh x is n kn[n :i* ( ! simple ev5ts ^! inequalities favor1 nev]!less ?is ignorance itself (t5 increases ! probability ( comp\nd ev5ts4 ,9 g5]aliz+ analysis & ! problems conc]n+ probabilities1 ,i 0 l$ to ! calculus ( p>tial f9ite diff]5ces1 :i* ,lagrange has s9ce treat$ by a v simple me?od1 elegant applications ( :i* he has us$ 9 ?is k9d ( problems4 ,! !ory ( g5]ative functions :i* ,i publi%$ ab\t ! same "t includes ^! subjects among ^? x embraces1 & is adapt$ ( itself & ) ! greate/ g5]ality to ! mo/ difficult que/ions ( probability4 ,x det]m9es aga91 by v conv]g5t approximations1 ! values ( ! functions compos$ ( a great numb] ( t]ms & factors2 & 9 %[+ t ! squ>e root ( ! ratio ( ! circumf]5ce to ! radius ent]s mo/ frequ5tly into ^! values1 x %[s t an inf9ity ( o!r transc5d5ts may be introduc$4 ,te/imonies1 votes1 & ! decisions ( electoral & delib]ative assemblies1 & ! judgm5ts ( tribunals1 h be5 submitt$ likewise to ! calculus ( probabilities4 ,s _m passions1 div]s int]e/s1 & circum/ances complicate ! que/ions relative to ! subjects1 t !y are almo/ always insoluble4 ,b ! solution ( v simple problems :i* h a great analogy ) !m1 may (t5 %$ ^u difficult & important que/ions great light1 :i* ! surety ( calculus r5d]s always pref]able to ! mo/ speci\s reason+s4 ,"o ( ! mo/ int]e/+ applications ( ! calculus ( probabilities conc]ns ! mean values :i* mu/ be *os5 among ! results ( obs]vations4 ,_m geometricians h /udi$ ! subject1 & ,lagrange has publi%$ 9 ! ,mmoires de ,tur9 a beautiful me?od = det]m9+ ^! mean values :5 ! law ( ! errors ( ! obs]vations is kn[n4 ,i h giv5 = ! same purpose a me?od bas$ ^u a s+ul> contrivance :i* may be employ$ ) advantage 9 o!r que/ions ( analysis2 & ?is1 by p]mitt+ indef9ite ext5sion 9 ! :ole c\rse ( a long calculation ( ! functions :i* ought to be limit$ by ! nature ( ! problem1 indicates ! modifications :i* ea* t]m ( ! f9al result ought to receive by virtue ( ^! limitations4 ,x has already be5 se5 t ea* obs]vation furni%es an equation ( condition ( ! fir/ degree1 :i* may always be dispos$ ( 9 su* a mann] t all its t]ms be 9 ! fir/ memb]1 ! second be+ z]o4 ,! use ( ^! equations is "o ( ! pr9cipal causes ( ! great precision ( our a/ronomical tables1 because an imm5se numb] ( excell5t obs]vations has ?us be5 made to concur 9 det]m9+ _! elem5ts4 ,:5 !re is only "o elem5t to be det]m9$ ,ctes prescrib$ t ! equations ( condition %\ld be prep>$ 9 su* a mann] t ! coeffici5t ( ! unkn[n elem5t be positive 9 ea* ( !m2 & t all ^! equations %\ld be add$ 9 ord] to =m a f9al equation1 :5ce is d]iv$ ! value ( ?is elem5t4 ,! rule ( ,ctes 0 foll[$ by all calculators1 b s9ce he fail$ to det]m9e sev]al elem5ts1 !re 0 no fix$ rule = comb9+ ! equations ( condition 9 su* a mann] z to obta9 ! necess>y f9al equations2 b "o *ose = ea* elem5t ! obs]vations mo/ suitable to det]m9e x4 ,x 0 9 ord] to obviate ^! grop+s t ,leg5dre & ,gauss conclud$ to add ! squ>es ( ! fir/ memb]s ( ! equations ( condition1 & to r5d] ! sum a m9imum1 by v>y+ ea* unkn[n elem5t2 by ?is means is obta9$ directly z _m f9al equations z !re are elem5ts4 ,b d ! values det]m9$ by ^! equations m]it ! pref]5ce ov] all ^? :i* may be obta9$ by o!r means8 ,?is "q1 ! calculus ( probabilities alone 0 able to answ]4 ,i appli$ x1 !n1 to ?is subject1 & obta9$ by a delicate analysis a rule :i* includes ! prec$+ me?od1 & :i* adds to ! advantage ( giv+1 by a regul> process1 ! desir$ elem5ts t ( obta9+ !m ) ! greate/ %[ ( evid5ce f ! totality ( obs]vations1 & ( det]m9+ ! values :i* leave only ! smalle/ possible errors to be fe>$4 ,h[ev]1 we h only an imp]fect k ( ! results obta9$1 z long z ! law ( ! errors ( :i* !y are susceptible is unkn[n2 we mu/ be able to assign ! probability t ^! errors are conta9$ )9 giv5 limits1 :i* am\nts to det]m9+ t :i* ,i h call$ ! weight ( a result4 ,analysis leads to g5]al & simple =mul = ?is purpose4 ,i h appli$ ?is analysis to ! results ( geodetic obs]vations4 ,! g5]al problem consi/s 9 det]m9+ ! probabilities t ! values ( "o or ( sev]al l9e> functions ( ! errors ( a v great numb] ( obs]vations are conta9$ )9 any limits4 ,! law ( ! possibility ( ! errors ( obs]vations introduces into ! expressions ( ^! probabilities a con/ant1 ^: value seems to require ! k ( ?is law1 :i* is almo/ always unkn[n4 ,happily ?is con/ant c be det]m9$ f ! obs]vations4 ,9 ! inve/igation ( a/ronomical elem5ts x is giv5 by ! sum ( ! squ>es ( ! diff]5ces betwe5 ea* obs]vation & ! calculat$ "o4 ,! errors equally probable be+ proportional to ! squ>e root ( ?is sum1 "o c1 by ! comp>ison ( ^! squ>es1 appreciate ! relative exactitude ( ! diff]5t tables ( ! same />4 ,9 geodetic op]ations ^! squ>es are replac$ by ! squ>es ( ! errors ( ! sums obs]v$ ( ! ?ree angles ( ea* triangle4 ,! comp>ison ( ! squ>es ( ^! errors w enable u to judge ( ! relative precision ( ! in/rum5ts ) :i* ! angles h be5 measur$4 ,by ?is comp>ison is se5 ! advantage ( ! repeat+ circle ov] ! in/rum5ts :i* x has replac$ 9 geodesy4 ,!re (t5 exi/s 9 ! obs]vations _m s\rces ( errors3 ?us ! positions ( ! />s be+ det]m9$ by means ( ! m]idian telescope & ( ! circle1 bo? susceptible ( errors ^: law ( probability ought n to be suppos$ ! same1 ! elem5ts t are d$uc$ f ^! positions are affect$ by ^! errors4 ,! equations ( condition1 :i* are made to obta9 ^! elem5ts1 conta9 ! errors ( ea* in/rum5t & !y h v>i\s coeffici5ts4 ,! mo/ advantage\s sy/em ( factors by :i* ^! equations ought to be multipli$ respectively1 9 ord] to obta91 by ! union ( ! products1 z _m f9al equations z !re are elem5ts to be det]m9$1 is no long] t ( ! coeffici5ts ( ! elem5ts 9 ea* equation ( condition4 ,! analysis :i* ,i h us$ leads easily1 :atev] ! numb] ( ! s\rces ( error may be1 to ! sy/em ( factors :i* gives ! mo/ advantage\s results1 or ^? 9 :i* ! same error is less probable ?an 9 any o!r sy/em4 ,! same analysis det]m9es ! laws ( probability ( ! errors ( ^! results4 ,^! =mul conta9 z _m unkn[n con/ants z !re are s\rces ( error1 & !y dep5d ^u ! laws ( probability ( ^! errors4 ,x has be5 se5 t1 9 ! case ( a s+le s\rce1 ?is con/ant c be det]m9$ by =m+ ! sum ( ! squ>es ( ! residuals ( ea* equation ( condition1 :5 ! values f\nd = ^! elem5ts h be5 sub/itut$4 ,a simil> process g5]ally gives values ( ^! con/ants1 :atev] _! numb] may be1 :i* completes ! application ( ! calculus ( probabilities to ! results ( obs]vations4 ,i ought to make "h an important rem>k4 ,! small unc]ta9ty t ! obs]vations1 :5 !y are n num]\s1 leave 9 reg>d to ! values ( ! con/ants ( :i* ,i h j spok51 r5d]s a little unc]ta9 ! probabilities det]m9$ by analysis4 ,b x almo/ always suffices to "k if ! probability1 t ! errors ( ! results obta9$ are compris$ )9 n>r[ limits1 approa*es closely to unity2 & :5 x is n1 x suffices to "k up to :at po9t ! obs]vations %\ld be multipli$1 9 ord] to obta9 a probability su* t no reasonable d\bt rema9s 9 reg>d to ! correctness ( ! results4 ,! analytic =mul ( probabilities satisfy p]fectly ?is requirem5t2 & 9 ?is connection !y may be view$ z ! necess>y complem5t ( ! sci5ces1 bas$ ^u a totality ( obs]vations susceptible ( error4 ,!y are likewise indisp5sable 9 solv+ a great numb] ( problems 9 ! natural & moral sci5ces4 ,! regul> causes ( ph5om5a are mo/ frequ5tly ei!r unkn[n1 or too complicat$ to be submitt$ to calculus2 aga91 _! action is (t5 di/urb$ by accid5tal & irregul> causes2 b its impression always rema9s 9 ! ev5ts produc$ by all ^! causes1 & x leads to modifications :i* only a long s]ies ( obs]vations c det]m9e4 ,! analysis ( probabilities develops ^! modifications2 x assigns ! probability ( _! causes & x indicates ! means ( cont9ually increas+ ?is probability4 ,?us 9 ! mid/ ( ! irregul> causes :i* di/urb ! atmosph]e1 ! p]iodic *anges ( sol> heat1 f day to night1 & f w9t] to summ]1 produce 9 ! pressure ( ?is great fluid mass & 9 ! correspond+ height ( ! b>omet]1 ! diurnal & annual oscillations2 & num]\s b>ometric obs]vations h reveal$ ! =m] ) a probability at lea/ equal to t ( ! facts :i* we reg>d z c]ta94 ,?us x is aga9 t ! s]ies ( hi/orical ev5ts %[s u ! con/ant action ( ! great pr9ciples ( e?ics 9 ! mid/ ( ! passions & ! v>i\s int]e/s :i* di/urb societies 9 e way4 ,x is rem>kable t a sci5ce1 :i* comm5c$ ) ! consid]ation ( games ( *ance1 %\ld be elevat$ to ! rank ( ! mo/ important subjects ( human k4 ,i h collect$ all ^! me?ods 9 my ,?orie analytique des ,probabilits1 9 :i* ,i h propos$ to exp\nd 9 ! mo/ g5]al mann] ! pr9ciples & ! analysis ( ! calculus ( probabilities1 likewise ! solutions ( ! mo/ int]e/+ & mo/ difficult problems :i* calculus pres5ts4 ,x is se5 9 ?is essay t ! !ory ( probabilities is at bottom only common s5se r$uc$ to calculus2 x makes u appreciate ) exactitude t :i* exact m9ds feel by a sort ( in/9ct )\t be+ able (ttimes to give a reason = x4 ,x leaves no arbitr>9ess 9 ! *oice ( op9ions & sides to be tak52 & by its use c always be det]m9$ ! mo/ advantage\s *oice4 ,!reby x supplem5ts mo/ happily ! ignorance & ! weakness ( ! human m9d4 ,if we consid] ! analytical me?ods to :i* ?is !ory has giv5 bir?2 ! tru? ( ! pr9ciples :i* s]ve z a basis2 ! f9e & delicate logic :i* _! employm5t 9 ! solution ( problems requires2 ! e/abli%m5ts ( public utility :i* re/ ^u x2 ! ext5sion :i* x has receiv$ & :i* x c /ill receive by its application to ! mo/ important que/ions ( natural philosophy & ! moral sci5ce2 if we consid] aga9 t1 ev5 9 ! ?+s :i* _c be submitt$ to calculus1 x gives ! sure/ h9ts :i* c guide u 9 our judgm5ts1 & t x tea*es u to avoid ! illusions :i* (ttimes confuse u1 !n we %all see t !re is no sci5ce m wor?y ( our m$itations1 & t no m useful "o c\ld be incorporat$ 9 ! sy/em ( public in/ruction4