A correction of the historiographical record on the probability integral
aa r X i v : . [ m a t h . HO ] O c t A correction of the historiographical record on theprobability integral
Fausto Di Biase ∗ ,Dipartimento di Economia, Università “G. D’Annunzio” di Chieti-Pescara,Viale Pindaro, 42, I-65127, Pescara, Italyemail: [email protected] October 22, 2019
Abstract
We correct a common (but mistaken) attribution of the evaluation of the probabilityintegral, usually attributed to Poisson, Gauss, or Laplace. The probability integral is the following one: Z R e − x dx = √ π (E)Some modern authors attribute the probability integral (E) to Poisson [Gnedenko, 1978, p.132]. However, Gauss himself (in an important book published in 1809, to which we shallreturn momentarily) attributes the result to Laplace. The work of Laplace was publishedin 1774 [Laplace, 1774, pp.35-36]. In 1812 Laplace returned to this calculation and gavea different proof [Laplace, 1812, pp.93–96]. Even modern authors attribute the result toLaplace. For example, S. Stigler, in his study of Laplace’s work, writes“Since the proof includes what seems to be the earliest evaluation of the definiteintegral R ∞ √ πσ exp (cid:16) − z σ (cid:17) dz = 1 Laplace may have been the first to integratethe normal density” [Stigler, 1986, p.360].Currently a consensus points to Laplace as the author of the evaluation of (E). We will showthat attribution of (E) to Laplace given by Gauss is mistaken. ∗ Supported in part by INdAM-GNAMPA and by Fondi di Ricerca di Ateneo Università “G. D’Annunzio” MSC: 01A85; 01A50 The correction
Firstly, observe that Laplace [Laplace, 1774, p.36] proves (E) in the form (E.0) Z [ln(1 /x ] − / dx = √ π (E.0)Indeed, the substitution x = e − t shows that (E.0) is equivalent to (E).At the beginning of his proof, Laplace gives a generic attribution to Euler, and writes“ Voir le Calcul intégral de M. Euler” [Laplace, 1774, p.35].As a matter of fact, in [Euler, 1772, p. 111], published in 1772,
Euler had alreadyproved (E.0). Moreover, in his previous work [Euler, 1738], where he introduced the fac-torial of non integer numbers, now known as the Gamma function, Euler had proved that Z [ln(1 /x )] / dx = 12 √ π (E.0’)which is again equivalent to (E). Mathematicians who were contemporaries of Gauss were very well aware of the fact that (E.0)is equivalent to (E). Indeed, inside a printed copy of Gauss’s book [Gauss, 1809], located inthe library of the ETH Zürich, a handwritten footnote by Barnaba Oriani, a contemporaryof Gauss, written in Latin, states that the evaluation in (E) is really due to Euler, not toLaplace. This handwritten footnote is located on page 212, where Gauss mentions this resultas theorema elegans primo ab ill. Laplace inventum (The elegant theorem first proved byillustrious Laplace). Here follows a translation of Oriani’s footnote.
The elegant theorem, attributed to the illustrious La Place, was in realityfirst found by Leonard Euler. As a matter of fact, Euler had proved it first, inComment. Acad. Petrop. (vol 16) that − Z q ln x dx extended from x = 1 to x = 0 is equal to π , i.e., half of the length of a circumference of radius one. Hence,putting x = e − t one has − dx √ ln x = 2 e − t dt . Hence the integral R e − t from t = 0 to t = ∞ is = √ π and therefore the same integral from t = −∞ to t = + ∞ willbe = √ π . Barnaba Oriani (1752-1832) was a mathematician and astronomer. He was a friend ofGiuseppe Piazzi (1746,1826), the astronomer who on January 1, 1801, from his observatoryin Palermo, discovered the planetoid Ceres, located between the orbits of Mars and Jupiter,and collected the data of its position until February 11, when “because of sickness, it was Indeed, the Gamma function may be written as Γ( z ) = R [ln(1 /t )] z − dt and therefore the expressionin (E.0’) is equal to Γ(3 / which is equal to Γ(1 /
2) = √ π/ . All of this had been proved by Euler, althoughin a different notation. TheoriaMotus Corporum Coelestium in sectionibus conicis solem ambientium (Theory of the Motionof Celestial bodies which revolve about the Sun in Conic Sections). [Herrmann, 1984, p.26].This is the book where Gauss attributes the calculation in (E) to Laplace.
References [Euler, 1738] Euler, L. (1738). De progressionibus transcendentibus seu quarum terminigenerales algebraice dari nequeunt.
Commentarii academiae scientiarum Petropolitanae ,5:36–57. [E19] http://eulerarchive.maa.org/pages/E019.html .[Euler, 1772] Euler, L. (1772). Evolutio formulae integralis R x f − dx ( lx ) mn integratione avalore x = 0 ad x = 1 extensa. Novi Commentarii Academiae Scientiarum ImperialisPetropolitanae , 16:91–139. [E421] http://eulerarchive.maa.org/pages/E421.html .[Gauss, 1809] Gauss, C. (1809).
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Theory of Probability . MIR Publishers.[Herrmann, 1984] Herrmann, D. (1984).
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Mémoires de l’Académie royale des sciences de Paris (Sa-vants étrangers) , VI:621–656. Reprinted in Œuvres complètes, 8:27–65, 1891 https://gallica.bnf.fr/ark:/12148/bpt6k77596b/f4.image .[Laplace, 1812] Laplace, P.-S. (1812).
Théorie analytique de probabilités . Courcier. https://gallica.bnf.fr/ark:/12148/btv1b8625611h/f7.image .[Stigler, 1986] Stigler, S. M. (1986). Laplace’s 1774 memoir on inverse probability.
Statist.Sci. , 1(3):359–363. https://projecteuclid.org/euclid.ss/1177013620https://projecteuclid.org/euclid.ss/1177013620