INAUGURAL DISSERTATION MATHEMATICAL-JURIDICAL ON THE USE OF THE ART OF CONJECTURING IN LAW, Which BY AID OF DIVINE GRACE By the Authority and by the Command Of the most Magnificent & most Distinguished Order of Counselors of Law in accordance with the Fatherland’s University for the DOCTORATE DEGREE In both Roman and Canon law legitimately acquired On the day 14 June A.C. MDCCIX L.H.Q.S. Publicly he will defend M. NICOLAUS BERNOULLI, Basel 2 To the most distinguished & most knowledgeable man Master J. Jac. Battier, Doctor of Canon and Roman Law, at the Imperial Institute & Professor of Public Law, most meritorious, of the Academy here the Magnificent Director, my own Patron and Teacher to be revered continuously: And also TO THE MOST FAMOUS, MOST EXCELLENT, MOST CELEBRATED MAN MASTER JOH. BERNOULLI, Philosopher & Doctor of Medicine, Professor of Mathematics at Basel Both The Royal Society of Sciences of France & The Prussian Society Most distinguished Member, my own most respected Uncle This specimen in Mathematics-Law In indication of a grateful spirit, and of his higher recommendation, these things for which attention is proper he presents and dedicates NICOLAUS BERNOULLI Nicolaus, son. 3 PREFACE With the usual Examinations having been endured just previously for some months,towhichtheCandidatesofLawatBaselaresubjected,Ihavehopedthat Iwouldmakedobynomeansineptly,ifintheplaceofthefourthevidence,which still remains to be determined, by public debate, I would discuss some theme out of mathematics, produced by that divine knowledge, the study of which I have joined so far with the study of law with GOD favoring,and how from the first years I have continued with conspicuous love, with my most celebrated Uncles Jakob & Johan Bernoulli displaying a light for me in this knowledge, the first of whom, now indeed enrolled in the heavenly chorus of the blessed, to his own whom he has bequeathed (unedited thus far but shortly, as we hope, brought into light) the Treatise on the Art of Conjecture, he has made available to me the opportunity, of choosing this material, concerning the Use namely Of the Art of Conjecture in Law, which also there I undertake with pleasure, because I see, that many of the most useful investigations, particularly about absent men to be considered dead,likewise life annuities &c. occurring nearly daily in the Court of Justice,by this art are able to be decided. Thus by reason of the exposition having been explained with a few words, I approach the matter itself without further delay. It would be consequently. 4 J. N. D. N. J. C. 4. DISSERTATION ON THE USE OF THE ART OF CONJECTURE IN LAW Chapter 1 THE ART OF CONJECTURE IN GENERAL Following Cicero’s advice at de Offic. 1.11 every discipline which takes its be- ginning from another one, so that it may be rationally understood, ought to proceed from a statement of what it might be, from which <vantage point> it can be contested; indeed in the first place saying what might be the Art of Conjecture favors us. Moreover it is a very great pleasure to define this <Art>inaccordancewiththeverywordsofmyUncle,2 mymaster,p(rofessor) m(athematics) in the Tractate on the Art of Conjecture Part. IV Chapter II as far as it would be the Ars metiendi, quam fieri potest exactissime, probabilitates rerum eo fine, ut in judiciis & actionibus nostris semper eligere vel sequi pos- simus id, quod melius, satius, tutius aut consultius fuerit deprehen- sum.3 The goal of this Art, as the definition makes clear, is warranted by the ran- domuncertainanddoubtfulmatters, inwhichalthoughallpersuasivecertainty is not possible to be held, we are nonetheless able to delimit through conjec- 5. ture, howgreattheprobabilitywouldbe, suchasthisorthatmightbeorcould happen, or what probably might be about to exist, or which outcome would be 1“My dear son Marcus, you have now been studying a full year under Cratippus, and that too in Athens, and you should be fully equipped with the practical precepts and the principles of philosophy; so much at least one might expect from the preeminence not only ofyourteacherbutalsoofthecity;theformerisabletoenrichyouwithlearning,thelatter to supply you with models. Nevertheless, just as I for my own improvement have always combinedGreekandLatinstudies–andIhavedonethisnotonlyinthestudyofphilosophy but also in the practice of oratory – so I recommend that you should do the same, so that youmayhaveequalcommandofbothlanguages. AnditisinthisverydirectionthatIhave, if I mistake not, rendered a great service to our countrymen, so that not only those who areunacquaintedwithGreekliteraturebuteventheculturedconsiderthattheyhavegained much both in oratorical power and in mental training. Cicero, De Officiis. Trans. Walter Miller. Loeb Classical Library Vol. XXI,1921. 2JamesI(Jakob)1654-1705. 3Theartofmeasuring,aspreciselyaspossible,probabilitiesofthings,withthegoalthat wewouldbeablealwaystochooseorfollowinourjudgmentsandactionsthatcourse,which willhavebeendeterminedtobebetter,moresatisfactory,saferormoreadvantageous. 5 6 CHAPTER 1. THE ART OF CONJECTURE IN GENERAL more probable than another, or how much this or that might deviate from an integral certitude; I very much intend integral: for probability is the gradation of certainty and it differs as much from this <i. e. certainty> as the part from the whole. If certainty is indisputably integral & absolute, which we designate by the letter a or number 1, e.g. it might be supposed to consist of 5 probabil- ities, or rather “parts”, of which three would argue strongly for the <actual> existence or the future occurrence of an event, the remaining <two parts would argue forcefully> against <their occurrence>; that outcome is said to have a 3/5a or 3/5 probability. This therefore is called more probable than the other because it has the greater claim to certainty; although this would be only re- ferred to normally as probable in a positive degree, whose probability exceeds significantly 50% certainty. I stress significantly; for what approaches a 50% certainty is called uncertain or doubtful. Thus what has a 1/5 certainty is more probable than what has 1/10, although neither may be probable in a positive degree as my uncle taught in the aforementioned Tractate Part. IV Chapter I. This<essayisrelevantto>thegoaloftheArtofConjecturebecausewehave saidthatthereareuncertainordoubtfulmatters: Fromthesemattersmoreover, which are certain & whose truth can be apprehended easily, this is not done; whenfollowingthefirst<matter>accordingtotheRules,whichUnclewrotein ChapterII,thereoughtnottobeaplaceforconjecturein<assessing>matters in which one may comprehend overwhelming certainty. Thus if a thief upon interrogation will have responded that he sold stolen property to Sempronius,4 ajudgewouldtrythecaseineptly,whofromalookortoneofvoice,orfromthe quality of the merchandise stolen by the thief, or from other circumstances of the thief would want to conjecture concerning the probability of the assertion, when Sempronius is present, from whom it will be allowed that everything can be discovered certainly & easily. The foundation of this entire Art, upon which we ought to rely perpetually in assessing probability, in this general Rule consists, which Huygens5 demon- strates in his elegant Pamphlet de Ratiociniis in aleæ ludo,6 Propositions 1, 2 & 3 and my Uncle in his Notes to these same Propositions. Multiplicetur id quod singulis casibus evenit per numerum casuum, quibus unumquodque evenire deprehenditur, summaque productorum dividaturpersummamomniumcasuum,quotiensostenditquidprob- abilitereventurumsit,sivedenotabitvaloremexpectationisseugradum probabilitatis quæsitæ.7 The Rule is the same as that in which commonly the mean arithmetical proportional is sought among many more given quantities, and indeed with the 4Semproniusisanamecommonlyusedinthelegalliteratureforapartyinalegalaction. 5ChristiaanHuygens1629-1695. 6Thistreatisepublishedin1657. 7Thatwouldbemultipliedwhichoccursinindividualcasesbythenumberofcasesinwhich one or another <possibility> is considered to occur, and the sum of the products would be dividedbythesumofallthecases,thequotientindicateswhatprobablywilloccur,eitherit willindicatethevalueoftheexpectationorthedegreeofthesought-outprobability. 7 ruleofalligation,8 onwhichmatteritisapleasuretoofferthenotesofUnclemy master, which he has in his Notes to Proposition 3 of the Pamphlet of Huygens. 6. Perspicuum est ex calculi hujus consideratione, magnam illi inter- cedere affinitatem cum Regula Arith. Alligationis dicta, qua res di- versi pretii in data quantitate miscentur, & quæritur pretium rei mixtæ; aut potius calculum utrinque plane eundem esse. Sicut enim summa productorum ex quantitatibus singulorum miscibilium in sua respective pretia, divisa per aggregatum omnium miscibilium, exhibit pretiumquæsitum,quodsempermediumestinterpretiaextremorum: ita summa productorum ex numeris casuum in id quod quovis casu acquiritur, divisa per numerum omnium casuum, ostendit valorem expectationis, qui proinde semper intermedius erit inter maximum & minimum quod acquiri potest. Unde si iidem numeri assumantur, ibi pro quantitate miscibilium, eorumque pretiis: hic pro casibus,& eo quod quovis casu obtinetur; idem quoque numerus denotabit ibi pretium rei mixtæ,& hic expectationem. Ex. gr. si 3 canthari vini pretii 13 misceantur cum 2 cantharis pretii 8; multiplicatis 3 per 13 & 2 per 8, exurgit pretium omnium cantharorum 55, quo diviso per 5 numerum cantharorum, habetur 11 pretium unius cantari mixti: quanta quoque juxta regulam expectatio cujuspiam æstimanda est, qui 3 habuerit casus ad 13, & 2 ad 8.9 Butyetthatindividualandexcellentagreementdeservestobenoted,which thisRulehaswithit,whichissaid<tobe>ofthegreatestimportanceinregard tofindingthecenterofgravity. Forjustasthesumofthemomentsi.e. thesum oftheproductsoftheweightsintheirrespectivedistancesfromsomegivenfixed point, <that sum> divided by the sum of their weights, shows the distance of the center of gravity, i.e. of that point from which the weights are suspended in equilibrium: Thuseventhismidpoint,whichisobtainedbythepresentRule,is, as thus I would say, the center of gravity of all probabilities, which thus places 8Theso-calledRuleofMixtures. 9It is evident from a consideration of his calculation that to him there is a great affinity withtheArithmeticRuleofAlligation,bywhichproductsofadifferentpricearecombinedin thegivenquantity,&apriceforthemixedproductissought: orratherthecalculationdone both ways is essentially the same. For just as the sum of the products from the quantities of individual constituent commodities in their own prices respectively, when <that sum> is divided by the aggregate of all the constituent commodities, it exhibits the sought after price because it is always the mean among the prices of the extremes: thus the sum of the productsfromthenumberofcasesinthatwhichisacquiredbyanycase,whenit<thesum> isdividedbythenumberofallthecases,exhibitsthevalueoftheexpectation,whichwillbe alwayslikewiseintermedialbetweentheminimumandmaximumwhichisabletobeacquired. Whenceifthesesamenumbersmaybeassumed,<theyareconstant>thereinregardtothe quantity of the constituent commodities and also in their prices; they are constant here in regardtocasesand<inregardto>thatwhichisobtainedbyrandomchance;sotoothesame number will represent the price of the mixed product & this <represents> the expectation. Forexample,if3canthariofwinepricedat13aremixedwith2pricedat8;bymultiplying3 by13&2by8,thegrosspriceofthecanthariis55,dividingby5canthari,thepriceofone cantharusofmixedwineis11: Alsohowgreattheexpectationofanythingmustbeestimated closetotherule,inorderthat3willhavehadchancesupto13&2upto8. 8 CHAPTER 1. THE ART OF CONJECTURE IN GENERAL these <things> in equilibrium, so that neither of these probabilities, which fall away on both sides from this mean, takes the greater weight for itself in turn. Our legal experts intending to preserve such an equilibrium in doubtful and vexing <cases> ought to follow the mean as seems apparent from l. 3. ff. si pars her. pet. there: Prudentissime juris auctores medietatem quandam secuti sunt;10 andmoreoverinthesamed. l. 3 theyshallhavepursuedthemeanprecisely,as weshallseebelow. Thisalsopertainstothisdegreetotheproverbialexpression: Semper in obscuris, quod minimum est, sequimur.11 to l. 9 ff. de R. J. c. 30. de R. J. in 6 , a similar thing is said in l. 115 ff. c. to 45 in 6 that one ought to determine in problematic cases what is most likely to be true or what has the greater claim <to be true>. For this law of ours shows where there is even the least danger of deviating from the truth: in the middle evidently outsideof whichall otherprobabilities yieldmore towards the extremes, i.e. they incline more to those things which most rarely happen. 7. Finally this must moreover be noted which my Uncle cautions in Scholium to Proposition I of Huygens’ de ratiociniis in ale ae ludo; the word expectation; in some manner we have said that that mean which is obtained by dividing the sum of the products from the cases in it which are attained by some kind of chance,bythenumberofallcases,cannotbetakenheretomeaninitsordinary sense, in which we are said to expect or hope generally that which is the best of all, granted that for us something worse can happen, but to what extent our hopeofobtainingthebesthasbeentemperedanddiminishedbyfearofaworse outcome: sothatsomethingisalwaysrepresentedbythevaluationofthatwhich is intermediate between the best we can hope and the worst we fear. Thus he who has 3 chances to gain 13 & 2 to gain 8 cannot be said to expect 13, but 11 which is mean between 13 & 8. 10Mostprudentlyjudgeshavefollowedacertainmean. SeeChapterVIII. 11Wealwaysfollowinproblematic<cases>whatistheminimum. Chapter 2 CONCERNING HOW THE PROBABILITY OF HUMAN LIFE SPAN IS COMPUTED, OR RATHER OF A MAN OF WHATEVER AGE In the prior chapter we have considered the art of conjecture in general, there follows how we make evident the use of it through some particular examples. FirstwhatfromthisArtIhavededuced,&inthefollowingitispossibletohave utilitynotdeservingofscorn,itistheestimationofthelongevityofhumanlife; foralthoughtheendofourlifebemostuncertain,&thehourofdeathisknown to no one except GOD the highest and best the ultimate giver of our life, who is able to take from us this his own gift, at whatever time he himself shall have pleased,heisabletotakeaway;nothingremainsforusother,thanthatthrough conjecture we would determine, how many years up to this time of the lifetime of anyone at all a man would probably gain, or how much be the probability that he would exceed some given year or not &c. I see however that, there are many who will oppose in the case of the begin- 8. ning itself immediately & they will say, not only that it is impossible, that all this can be estimated according to the Art of Conjecture, for there is required the exact enumeration of all chances in which anything is able to happen, but there is no one of mortals, who could ever define the number, e.g. of diseases, asjustsomanychances,whichinvadethecountlesspartsofthehumanbodyat anyage,andtheyarevigoroustoinflictdeathuponus,andhewouldknowhow much easier this than that, plague than dropsy, dropsy than fever kills a man, so that thence concerning the future state of life and death a conjecture is able tobeformed,sinceallthisdependsuponcausesentirelyhidden&onknowledge removedfromourperception; indeedinanotherwaythematterrevealsitselfto bethesubjectindivination&games,whichfatealonegoverns,forsinceinthese the expectation is able to be determined precisely and scientifically, because we accurately & clearly perceive the number of chances, according to which infalli- blythereoughttofollowprofitorloss,&sincethesechancesmanagethemselves indifferently, & they would be equally likely to happen, at least if one should be more probable than the other, we are able to define scientifically, how much 9 10 CHAPTER 2. CONCERNING HUMAN LIFE SPAN more probable it would be.1 To this we shall respond, that for us another way here of investigating the number of chances is sufficient, which if not a priori or from reason, at least a posteriori or from an event observed many times in similar examples it will be allowedtobringtolight(forIamlessabletostrayfromtrueproportion,ifmore frequently, than more rarely I should observe <something>); since it ought to be presumed, that in so many chances each one in the future can happen & not happen, how often previously in a similar condition of the events it will have been discerned to have happened & not to have happened. Forife.g. afterhavingonceconductedanexperimentonthreehundredmen, of the same age and physical constitution, of which Titius is now, you will have observed that two hundred of them died before exactly ten years have passed, <while> the remaining prolonged life beyond, safely you will have collected enough, that it is twice as likely, that by them & by Titius that the debt to nature must be paid within ten years, rather than that it is possible for them to live beyond that limit. Hence it is manifest what must be realized about the excellence of the Art of Conjecture, for how much less those things, which are fortuitous and also uncertain, seem to be able to be comprehended within the limits of reason, so much more admirable the Art must be valued, to which lesser things are also subject, as Huygens says to Schootens2 in the preface of his pamphlet De ratiociniis in aleæ ludo. Thetruthofhisassertionrevealsitselfveryclearlythusfar,ifnowtruthfully we shall have revealed the method of raising the probability of human life by 9. means of a calculation drawn from observations made concerning the Bills of mortality, of which kind are accustomed to be distributed in Paris & London monthly or weekly. Uncle, my distinguished master, p(rofessor) m(athematics) refersinDissertationedeConversione&OppositioneEnunciationumannex31.3 from Ephemeris Erud. Gall. Anni. 1666 Num. 31.4 that it has been observed from the collection of many such bills, that out of one hundred infants born at the same time there remain surviving 64 after six years have elapsed, after XVI years have elapsed 40, after XXVI years 25. XXXVI years 16. XLVI years 10. LVI years 6. LXVI 3. LXXVI years 1. LXXXVI years 0.5 This being said, if one should be driven to estimate the lifetime of some newborninfant, thusonewillhavetoconsider: Thisnewborninfantisincluded either among those 36. who die within the first six years; or among those 24 whodiebetweenthesixthandsixteenthyear;oramongthe15whodiebetween 1ThisparagraphandthenexttwoaredrawnfromBookIV,ChapterIVoftheArs Con- jectandi. 2Franz van Schooten 1615-1660. Professor of mathematics, University of Leyden. He taught Christiaan Huygens. Huygens’ treatise on probability was appended to his Exercita- tiones Mathematicæ (1657) 3DissertationonConversionandReversionofPronouncements 4ThispublicationofJakobBernoullireferstotheJournal des S¸cavans inwhichwaspub- lishedthemortalitytablereferedtohere. 5This is John Graunt’s table as published in his Observations on the Bills of Mortality, 1662.