## ‘On the Theory of Probabilities, and in particular on Mitchell's Problem of the Distribution of the Fixed Stars,’ pp. 521-530 in: The London, Edinburgh and Dublin Philosophical Magazine, Fourth Series, Vol. 1, No. 7 – Supplement, June, 1851.

London: Richard Taylor, 1851.

First edition, journal issue in the original printed wrappers, of Boole’s first paper on probability. “Since about 1850, Boole had become increasingly interested in the mathematical theory of probability and more especially its philosophical aspects … In 1976, in his book *Boole’s Logic and Probability*, Theodore Halperin wrote: ‘The basic relationship between Boolean algebra and the calculus of events, now commonplace in treatises on probability, was first fully understood and exploited by Boole …’ Boole was of course familiar with the classical theory of probability as enunciated for example by Laplace, but his specific interest in its development seems to have stemmed from a paper he came across late in 1850. In ‘On the Theory of Probabilities, and in particular on Mitchell's Problem of the Distribution of the Fixed Stars,’ published in the *Philosophical Magazine* in June 1851, Boole remarks: ‘My attention has lately been directed to a communication by Professor Forbes, in the *Philosophical Magazine* for December 1850, entitled ‘On the alleged evidence for a physical connexion between stars forming binary or multiple groups, deduced from the doctrine of chances.’ I have read Professor Forbes’ observations with great care and interest; and desire, both because the subject of them is important, and because it is closely related to a class of speculations in the pursuit of which I have long been engaged, to offer a few remarks which have been suggested to me by the perusal of the paper’ … This seems to have been the first mention, by any author, of the close connection, both in essence and in form, between logic and probability and indeed of the dependence of the theory of probability on an underlying mathematical theory of logic” (MacHale, pp. 239-240). Boole makes this even clearer later in his paper (p. 524): Boole writes: “Although the immediate business of the theory of probabilities is with the frequency of the occurrence of events, and although it therefore borrows some of its elements from the science of number, yet as the expression of the occurrence of those events, and also of the relations, of whatever kind, which connect them, is the office of language, the common instrument of reason, so the theory of probabilities must bear some definite relation to logic”.

In 1767, John Michell (1724? – 1793) (consistently called ‘Mitchell’ by Boole) published in the *Philosophical Transactions* ‘An Inquiry Into the Probable Parallax and Magnitude of the Fixed Stars From the Quantity of Light Which They Afford Us, and the Particular Circumstances of Their Situation,’ in which he “pointed out that the frequency of the angular separation of close pairs of stars known at that time deviated grossly from what one could expect for chance projection of stars uniformly distributed in space—there appeared to be an excessive number of close pairs—and, according to Michell: ‘The natural conclusion from hence is, that it is highly probable, and next to a certainty in general, that such double stars as appear to consist of two or more stars placed very near together, do really consist of stars placed nearly together, and under the influence of some general law … to whatever cause this may be owing, whether to their mutual gravitation, or to some other law or appointment of the Creator’” (DSB, under Michell). “On the assumption of a random distribution of the stars over the celestial sphere [with equal probability of a star being in any of the 13,131 subregions of 2° diameter], he computes the odds that, of the 230 stars comparable in brightness to the double star *β* Capricorni, two should fall within that angular distance and finds it to be 80 to 1. When more stars are taken into account, e.g. the six brightest stars of the Pleiades, the odds against such a close grouping amount to 500,000 to 1 …

“Michell’s argument was widely accepted. Not until some 80 years later was it (the argument, not the conclusion) vehemently objected to by J. D. Forbes (1850). His paper analyses Michell’s argument at great length, describing fallacies he finds and, incidentally, some errors in the probability calculations (e.g. that the *β* Capricorni odds should, on Michell’s assumptions, be 160 to 1). We shall simply state Forbes’ two main objections: (i) Michell takes the high improbability of an event’s happening, when it is one of a great many possibilities as that of the event where it is already the case. (‘The improbability, for instance, of a given deal producing a given hand at whist is so immense, that were we to assume Michell’s principle, we should be compelled to assign it as the result of an active cause with far more probability than even found by him for the physical connexion of the six star of the Pleiades.’) (ii) Michell’s assumption of a uniform probability distribution for any star and any of the subregions of the celestial sphere ‘leads to conclusions obviously at variance with the idea of a random or lawless distribution, and is therefore not the expression of that Idea.’ Forbes likens it to assuming that any face of a die has an equal chance of coming up before one knows whether the die is loaded or not” (Halperin, pp. 355-357).

“In 1851 Boole entered the controversy [in the offered paper]. As he saw it, the statement of Michell’s problem in relation to *β* Capricorni was as follows:

- Upon the hypothesis that a given number of stars have been distributed over the heavens to a law or manner whose consequences we should be altogether unable to foretell, what is the probability that a star such as
*β*Capricorni would nowhere be found? - Such a star as
*β*Capricorni having been found, what is the probability that the law or manner of distribution was not one whose consequences we should be altogether unable to foretell?

“Denoting the two probabilities by *p* and *P* respectively, Boole finds *p* to be a determined number, and finds the fallacy to lie in the identification of *p* and *P*” (Dale, p. 74). Boole finds that *p* = 159/160, agreeing with Forbes, and deduces, using a form of Bayes’ theorem, that *P* = 80/81 – the value Michell had found, but for *p* rather than *P*!

“Boole considers the problem, though without making any further specific comments, in Chapter XX of *An Investigation of the Laws of Thought* of 1854” (*ibid*., p. 75).

“Boole (1815-64) was given his first lessons in mathematics by his father, a tradesman, who also taught him to make optical instruments. Aside from his father’s help and a few years at local schools, however, Boole was self-taught in mathematics. When his father’s business declined, George had to work to support the family. From the age of 16 he taught in village schools in the West Riding of Yorkshire, and he opened his own school in Lincoln when he was 20. During scant leisure time he read mathematics journals in the Lincoln’s Mechanics Institute. There he also read Isaac Newton’s *Principia*, Pierre-Simon Laplace’s *Traité de mécanique céleste*, and Joseph-Louis Lagrange’s *Mécanique analytique**,* and began to solve advanced problems in algebra.

“Boole submitted a stream of original papers to the new *Cambridge Mathematical Journal*, beginning in 1839 with his ‘Researches on the Theory of Analytical Transformations.’ These papers were on differential equations and the algebraic problem of linear transformation, emphasizing the concept of invariance. In 1844, in an important paper in the *Philosophical Transactions of the Royal Society* for which he was awarded the Royal Society’s first gold medal for mathematics, he discussed how methods of algebra and calculus might be combined. Boole soon saw that his algebra could also be applied in logic” (Britannica).

“His principal interest soon turned to an English specialty: the ‘calculus of operations’, now called ‘differential operators’, where differentiation was represented by the letter ‘D’, higher-order differentiation by ‘D^{2}, D^{3}, . . .’, integration by ‘D^{-1}’, and so on. This tradition had developed under the influence of the algebraised calculus propounded by J. L. Lagrange, initially by some French mathematicians; but from the 1810s this algebra and related topics were prosecuted in England by Charles Babbage and John Herschel as part of the revival of research mathematics there. Boole was to become a major figure in this movement in the next generation” (Grattan-Guinness, p. 471).

“Developing novel ideas on logical method and confident in the symbolic reasoning he had derived from his mathematical investigations, he published in 1847 a pamphlet, *Mathematical Analysis of Logic*, in which he argued persuasively that logic should be allied with mathematics, not philosophy. He won the admiration of the English logician Augustus de Morgan, who published *Formal Logic* the same year. On the basis of his publications, Boole in 1849 was appointed professor of mathematics at Queen’s College, County Cork, even though he had no university degree. In 1854 he published *An Investigation into the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities*, which he regarded as a mature statement of his ideas. The next year he married Mary Everest, niece of Sir George Everest, for whom the mountain is named. The Booles had five daughters … In 1857 Boole was elected a fellow of the Royal Society. The influential *Treatise on Differential Equations* appeared in 1859 and was followed the next year by its sequel, *Treatise on the Calculus of Finite Differences*. Used as textbooks for many years, these works embody an elaboration of Boole’s more important discoveries” (Britannica).

Dale, *A History of Inverse Probability: From Thomas Bayes to Karl Pearson*, 2012. Grattan-Guinness, ‘George Boole, An Investigation of the Laws of Thought on which are Founded the Mathematical Theory of Logic and Probabilities,’ Ch. 36 in *Landmark Writings in Western Mathematics 1640-1940* (Grattan-Guinness, ed.), 2005. MacHale, *The Life and Work of George Boole. A Prelude to the Digital Age*, 2014.

8vo (225 x 145 mm). Original blue printed wrappers, uncut and unopened (upper left corner of rear wrapper torn away).

Item #2974

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Price:
$1,250.00
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