A letter of Robert Leslie Ellis to William Walton on probability

Abstract

The paper discusses the background to and provides a transcription of a letter from Robert Leslie Ellis (1817–59) to William Walton (1813–1901) of 1849 on probability theory.

Although today still a largely forgotten figure, the polymath Robert Leslie Ellis (1817–59) was an important member of the Cambridge mathematical community in the mid-nineteenth century.¹ As a pupil of William Hopkins, student of George Peacock, First Wrangler in 1840, editor of the Cambridge Mathematical Journal and frequent contributor to the important British mathematical journals of the time, Ellis was much admired by his Victorian contemporaries. George Boole, in an 1857 prize-essay about applications of probability theory, wrote that ‘[t]here’ is no living mathematician for whose intellectual character I entertain a more sincere respect than I do for that of Mr. Ellis’ (Boole 1857 [1952], 350). A decade later, Francis Galton expressed a similar sentiment when describing Ellis as one ‘whose name is familiar to generations of Cambridge men as a prodigy of universal genius’ (Galton 1869 [1892], 18).

Robert Leslie Ellis and William Whewell: foundations of probability theory

During his tragically short life, Ellis wrote five papers and one note on probability theory (1844a [1863], 1844b [1863], 1844c, 1850a, 1850b, 1854 [1863]). ² As the Reverend Harvey Goodwin noted in his biographical memoir accompanying The Mathematical and Other Writings of Robert Leslie Ellis, Ellis’ papers on probability represented ‘as well as possible his special taste [...] with regard to mathematics’ (Goodwin 1863, xxix). Given his ‘speculative mind’, Ellis, who always talked about probability ‘with great pleasure’ and as a subject in which he was ‘thoroughly at home’, was naturally drawn to questions of foundations (Goodwin 1863, xxix, xxxv).

Ellis’ contributions to mathematical probability theory, written in the 1840s to 1850s, roughly consisted of two parts: firstly, an early sketch of a frequentist theory of probability and, secondly, a simplification and extension to any number of unknowns of Pierre-Simon Laplace's demonstration of the method of least squares. ³

Ellis’ ideas on what he himself saw as ‘the application to natural philosophy of the doctrine of probabilities’ (Ellis to Whewell, 8 April 1840, quoted in Zabell 1999, 135) arose from a dissatisfaction with the Laplacean definition of the subject matter of probability as a ‘quantity of belief’—a view which held sway in Britain ever since Abraham de Moivre's Doctrine of Chances (1738) and Augustus De Morgan's defense in his Essay on Probabilities (1838). Ellis was one of the very first to object to the infamous ‘principle of indifference’—which says that in so far as probability depends partly on ignorance and partly on knowledge, in the case of completely ignorance equal prior probabilities are to be assigned to each possible event, hypothesis, etc.—underlying the classical calculation of so-called ‘direct’ and ‘inverse’ probabilities. As Ellis wrote: ‘Mere ignorance is no ground for any inference whatever. Ex nihilo nihil’ (Ellis 1850a, 325).

Ellis first spoke of his frequentist scruples about the classical ‘subjective’ view in February 1842, but his diaries and letters reveal that he sketched them out as early as April 1840. On 8 April of that year Ellis wrote to his future brother-in-law, the Knightbridge Professor of Moral Philosophy William Whewell (1794–1866), saying that he wanted to ‘attempt to point out the impossibility of a strict numerical estimate of the force of belief [...]’ (Ellis quoted in Zabell 1999, 135, f 5). Ellis reflected on Whewell's response in a diary entry of 9 April 1840: ‘I wrote to Whewell, stating my notion of writing a little essay on probabilities, and asking if his work on the Philosophy of induction would interfere with it. I had a very civil answer today – saying he was glad to hear of my intention and wished me to persevere, as “he was sure I would throw light on it” [...]’ (Ellis quoted in Zabell 1999, 135, f 5). Some two years later, on 10 January 1842, Ellis once again turned to Whewell, asking his approval of a now finished essay that would reconcile probability theory with Whewell's philosophy and, thereby, place Ellis and Whewell ‘in opposition to received opinions and to the authority of great writers’ (Ellis quoted in Kilinc 2000, 257).

Whewell apparently approved of Ellis’ ‘On the foundations of the theory of probabilities’, for he communicated it to the Cambridge Philosophical Society, where Ellis would present it on 14 February 1842. The essay was subsequently published in the eighth volume of the Society's Transactions, in the year 1844.

Whewell's philosophy of science

Not unlike that of other early Victorians, Whewell's outlook was completely at odds with the alleged ‘atheism’ and ‘rationalism’ of Continental probabilists like Laplace and Condorcet. The early Victorian discussion about probability theory evolved around natural theology—the attempt to ground religion in science which received a huge boost with the establishment of the Bridgewater Treatises in the 1830s (see Richards 1997). Whewell, in his Astronomy and general physics considered with reference to natural theology (1833) (the first of the Bridgewater Treatises), upheld the view that rather than being established and proved through abstract deductive reasoning, knowledge of God's creation was obtained by a trial-and-error process of careful inductions. At the same time, Whewell noted that ‘[i]t is no easy matter, if it be possible, to analyze the process of thought by which laws of nature have thus been discovered. [...]’ (Whewell 1833, 304). During the next decade, Whewell devoted himself to this very attempt: the History of the inductive sciences (1837) and the Philosophy of the inductive sciences (1840) made a monumental effort to define the ‘dynamic personal knowing of natural theology in the rational context of philosophy’ (Richards 1997, 63). Whewell put forward a ‘reformed Baconian inductivism’ (Snyder 2006, 21) in which induction is said to consist of a combination of observed facts provided by the world and ‘Fundamental Ideas’ provided by the mind of the individual scientist. ⁴ The view of inductive science formulated in the History and Philosophy very much reflected the religious values for scientific investigation found in the Astronomy. Thus, for example:

The effort and struggle by which he [the scientist] endeavors to extend his view, makes him feel that there is a region of truth not included in his present [...] knowledge; the very imperfection of the light in which he works his way, suggests to him that there must be a source of clearer illumination at a distance from him.

(Whewell 1837, 334)

The Idea is disclosed but not fully revealed, imparted but not transfused, by the use we make of it in science. When we have taken from the foundation so much as serves our purpose, there still remains behind a deep well of truth, which we have not exhausted, and which we may easily believe to be inexhaustible.

(Whewell 1840, 73)

Given that the process of drawing black and white balls from an urn on the basis of which probability theory mathematized scientific rationality stood in stark contrast with his own emphasis on personal knowing, Whewell showed little interest in the theory. As becomes clear from their correspondence, Whewell gratefully left the task of writing on probability theory from the perspective of his own philosophico-theological outlook to the younger Ellis.

‘On the foundations of the theory of probabilities’ (1844): Ellis’ Whewellian probability theory

Ellis opened his 1844 article ‘On the foundations of the theory of probabilities’ with the claim that probability theory ‘is at once a metaphysical and a mathematical science. The mathematical part of it has been fully developed, while, [...] its metaphysical tendencies have not received much attention. This is the more remarkable, as they are in direct opposition to the views of the nature of knowledge, generally adopted at present’ (Ellis 1844a [1863], 1). What must have struck the audience at the Cambridge Philosophical Society was not so much Ellis’ acceptance of the mathematical part of probability theory as a completed whole, but also his suggestion that the Whewellian view in light of which its foundational part was to be updated had already won the day.

Ellis’ project was to reconcile probability theory with Whewell's view of knowledge by reinterpreting its foundations in such a way that it would ‘cease to be [...] in opposition to a philosophy of science which recognizes ideal elements of knowledge, and which makes the process of induction depend on them’ (Ellis 1844a [1863], 11). His main arguments attacked the foundations both of ‘direct’ and ‘inverse’ probability.

As to ‘direct’ probability, Ellis argued that all calculations are founded on a principle (‘when an event is expected rather than another, we believe it will occur more frequently on the long run’) and that this principle is itself founded on a fundamental axiom (‘that on the long run, the action of fortuitous causes disappears’). Importantly, where Ellis considered the principle to be ‘true a priori’, he held the axiom to be an ‘a priori truth, supplied by the mind itself' (Ellis 1844a [1863], 3). Thus, for example, even though he believed that the ‘law of large numbers’ amounted to a law of nature, because mathematical demonstrations cannot establish laws of nature, Ellis denied that it was to be considered as a ‘résultat du calcul’. As he rhetorically asked: ‘Are we prepared to admit, that our confidence in the regularity of nature is merely a corollary from Bernouilli's theorem?’ (Ellis 1844a [1863], 1).

As to ‘inverse’ probability, Ellis argued that in so far as the calculations are founded on the ‘mental phenomenon of expectation’, which is itself dependent upon a more fundamental belief in the ‘regularity of nature’, all ‘estimates furnished by [the] theory a posteriori of the force of inductive results are illusory’ (Ellis 1844a [1863], 5, 11). Given that it was the mind of the scientists that was ‘ever endeavouring to introduce order and regularity among the objects of its perceptions’, Ellis’ point was that inverse calculations were, firstly, not adequate to, and, secondly, premised on, the way scientists actually think (Ellis 1844a [1863], 5).

Thus, firstly, Ellis noted that, if a certain event has been observed to occur on m occasions, ‘there is a presumption that it will recur on the next occasion’, a presumption estimated by (m + 1)/(m + 2) (Ellis 1844a [1863], 7). But, Ellis asked, ‘[w]hat shall constitute a “next occasion”?, and ‘[w]hat degree of similarity in the new event to those which have preceded it, entitles it to be considered a recurrence of the same event?’ (Ellis 1844a [1863], 7). Ellis illustrated his point by means of the following example:

Ten vessels sail up a river. All have flags. The presumption that the next vessel will have a flag is 11/12. Let us suppose the ten vessels to be Indiamen. Is the passing up of any vessel whatever, from a wherry to a man of war, to be considered as constituting a “next occasion”? Or will an Indiaman only satisfy the conditions of the question? It is clear that in the latter case, the presumption that the next Indiaman would have a flag is much stronger, than that, as in the former case, the next vessel of any kind would have one. Yet the theory gives 11/12 as the presumption in both cases.

(Ellis 1844a [1863], 7)

Secondly, Ellis used the same example to show that the mind of the scientist was of great importance in assigning equal probabilities a priori to events:

Take the case of a vessel sailing up a river. The vessel has a flag. What was the a priori probability of this? Before any answer can by possibility be given to the enquiry, we must know (1) what circumstances the person who makes it rejects as irrelevant [...]; (2) what circumstances constitute in his mind the ‘trial’ [...] (3) What idea he forms to himself of a flag.

(Ellis 1844a [1863], 9)

More generally, Ellis found the ‘assertion [that] 3/4 is the probability that any observed event had on a priori probability greater than 1/2, or that three out of four observed events had such an a priori probability’ (Ellis 1844a [1863], 9) completely lacking in precision. Before proposing a detailed frequency explanation, Ellis asked: ‘a priori probability to what mind? In relation to what way of looking at them?’ (Ellis 1844a [1863], 9).

Taken together, Ellis not only reinterpreted the foundations of classical probability theory such that they were reconciled with Whewell's outlook, but, by doing so, also protected this outlook from the mathematization of induction by the probability calculus. Both in his 1844 and 1854 articles, Ellis repeated the belief that the foundations of probability theory were not known empirically or mathematically, but through personal insight leading to what Augustus De Morgan, referring to Whewell, at one point called ‘latent axioms’: ⁵

For myself, after giving a painful degree of attention to the point, I have been unable to sever the judgment that one event is more likely to happen than another, or that it is to be expected in preference to it, from the belief that on the long run it will occur more frequently.

(Ellis 1844a [1863], 3)

The principle may be [...] thus stated: “On a long run of similar trials, every possible event tends ultimately to recur in a definite ratio of frequency”. Our conviction of the truth of this proposition is, I think, intuitive, – the word being used [...] with reference to the intuitions of a mind, which has fully and clearly apprehended the subject before it, and to which therefore to have arrived at the truth and to perceive that it has done so are inseparable elements of the same act of thought.

(Ellis 1854 [1863], 49)

Ellis, Whewell and Forbes: scientific application of probability

Ellis’ position vis-à-vis the importance of probabilistic methods for physical science was twofold. On the one hand, Ellis counts as one of the most perceptive writers on the method of least squares—Gauss’ first proof of which was based on Laplace's inverse probability. ⁶ On the other hand, Ellis argued that the application of probabilistic methods, for example, to astronomy—as suggested by Gauss and Laplace—was nonsense, epistemologically speaking. As to this latter position, Ellis joined forces with Whewell and their philosophical companion James D Forbes.

Whewell, Herschel and the graphical method

In his Philosophy of 1840, Whewell had referred to ‘certain mathematical methods [that] may be employed to facilitate and give accuracy to the determination of the formula by which the observations are connected into laws' (Whewell 1840, 395). The methods that Whewell described were the ‘method of curves’, the ‘method of means’, the ‘method of residues’, and the ‘method of least squares’. Whewell considered the graphical ‘method of curves’ to be especially relevant for scientific purposes. Firstly, the method provided a coup d’oeil from which the scientist could recognize an order in the data: a graph allowed the mind to ‘see’ an idea in a set of data and arrive at a scientific law by means of a mental act that superimposed an ordering on the observed facts. Secondly, the method allowed the scientist to obtain laws from imperfect observations: by plotting numerical data it was possible to average the random errors ‘by drawing a curve among the data points so that the tendency of the law could be seen in spite of random errors’ (Hankins 2006, 618). Whewell gave two examples in which a graph of observational data gave mean values in cases where only such values could reveal any regularity: that of his own study of tides, in which case ‘a curve among plotted data points helped erase the random fluctuations caused by wind, barometric pressure [etc.]’ (Hankins 2006, 618), and of John Herschel's famous use of a graphical method to determine the orbits of double stars in 1833.

Despite underlying differences of opinion about the nature of induction—that is, on the epistemology of science, with Herschel criticizing Whewell's ‘peculiar and a priori point of view’—with regard to its methods, Whewell and Herschel agreed that in so far as induction could not be mathematized, there could be no method of invention: ‘[o]nce you lay out the data and connect the dots [...] you either see the law or you do not’ (Hankins 2006, 622).

Forbes and double stars

In 1833, the same year that he published his graphical method for determining the orbits of double stars, Herschel also published a Treatise on astronomy, which he republished some fifteen years later as Outlines of astronomy (1848). In this popular book, Herschel devoted much space to double stars, making the point that before double stars had been directly observed to rotate around one another, the best evidence for the existence of a physical connection between them came from probability theory. Following John Michell, who had made the same argument, with much theological fanfare, as early as in 1767, Herschel wrote that there were simply too many stars very close together for positioning to be an epiphenomenon of random distribution (see, for example, McCormmach 2012, chapter 6). It was this ‘disarmingly simple statement’ (Richards 1997, 65) that led a number of major figures to take up the question of how much could be established by probabilistic argument.

The first response came from the Scottish natural philosopher James D Forbes at Edinburgh—a former pupil of Whewell and a friend of Ellis (see Gower 1981). ⁷ Soon after the publication of Herschel's 1848 book, Forbes wrote a letter to the editor of the Philosophical Magazine and Journal in which he put forward

a doubt as to the legitimacy of certain reasonings with respect to the evidence for the physical connexion of binary or multiple stars arising from the mere fact of their juxtaposition, as stated and applied by some of the most eminent writers on sidereal astronomy.

(Forbes 1849, 132)

Both in his letter of July 1849, and in his lengthy article of December 1850 for the Philosophical Magazine, Forbes objected to Michell and Herschel—the ‘eminent writers’ referred to—on several grounds (Forbes 1849, 1850; see also Hankins 2006, 628–629). Firstly, following up on Whewell's position in the 1830s debate on the theological implications of probability theory, Forbes objected to Michell's argument that the observed distribution of stars was evidence of design (see Richards 1997, 55–57). He warned, instead, that all attempts to ground a proof of design on a priori and mathematical reasonings were to be approached with much caution. Secondly, Forbes targeted Herschel's (‘Laplacean’) assumption that doubts in the human mind could be used ‘to assign numerical values to the antecedent [that is ‘prior’] probability of any given arrangement or grouping [...]’ (Forbes 1849, 132). He argued that even though the frequent close proximity of two stars could give rise to an induction that might suggest a cause, this cause could not be proven from probability. Thirdly, and relatedly, Forbes wrote that Herschel had ‘confounded the measure of hypothetical antecedent probability of a given result with a probability in the nature of things, or an actual probability, and [had] used the measure of the former [...] for the measure of the latter’ (Forbes 1850, 416).

Taken together, defending ‘a Whewellian model of induction against the rigid mechanical overtones of a probabilistic one’ (Richards 1997, 65), Forbes emphasized that knowledge was grounded in personal insight, which, as such, lay beyond mathematical calculation.

Forbes wrote his article ‘On the alleged evidence for a physical connexion between stars forming binary or multiple groups, deduced from the doctrine of chances’ in May 1850, with the intention of reading it at the meeting of the British Association for the Advancement of Science (BAAS) in August 1850. Due to unknown circumstances that prevented its completion, Forbes read what he had written in May to the Physical Section of the BAAS. Meanwhile, in July 1850, there appeared in the Edinburgh Review an article by Herschel in which he defended himself against Forbes’ objection. Before publishing the article, in December 1850, Forbes consulted several friends, to wit William Thomson, Philip Kelland, George Terrot, George Biddel Airy, and Ellis (see Shairp et al. 1873, 473–483). As he wrote in his 1849 letter to the Philosophical Magazine:

I should probably have hesitated to oppose my solitary opinion to that entertained by the eminent writers whom I am about to quote [that is Michell and Herschel], had I not found it to be entirely supported by the eminent authority of [some] friends to whom I separately proposed it.

(Forbes 1849, 132)

Ellis’ letters to Forbes: double stars and inverse probability

From Ellis, Forbes received three letters, dated 3 September 1850, 8 October 1850, and 10 October 1850, in which Ellis commented on the proof-sheets. The general point of these letters was that Ellis was in full agreement with Forbes’ ‘Whewellian’ standpoint on Herschel's application of probability theory. As Ellis wrote, for example:

Between ourselves I am beginning to think the great Sir John Herschel is rather a charlatan: honorably distinguished no doubt [...] but neither clear nor deep.

(Ellis to Forbes, 20 September 1850, quoted in Richards 1997, 66)

I am greatly pleased with your argument in general; it expresses what I have long thought. The foundation of all the confusion is the notion that the numerical expression of a chance expresses the force of expectation, whereas it only expresses the proportion of frequency with which such and such an event occurs on the long run. From this notion that chances express something mental or subjective, is derived the assumption that the force of belief touching past events admits of numerical evaluation as well as the force of expectation touching future. [...] All this [is] folly [...].

(Ellis to Forbes, 8 October 1850, quoted in Shairp et al. 1873, 480, 481)

Avec des chiffres on peut tout démontrer, ought to be the motto of most of the philosophical applications of the theory of probabilities – which in its own nature and according to the plain of it, is only a development of the theory of combinations. To attempt to constitute it into the philosophy of science, is, in effect, to destroy the philosophy of science altogether.

(Ellis to Forbes, 3 September 1850, quoted in Shairp et al. 1873, 480)

In his three letters to Forbes, Ellis also expressed his views on the more specific topic of the (inverse) application of probability theory and Michell's and Herschel's use of it to arrive at a physical statement about double stars. Ellis’ message was that the philosophy of science had to be protected from any such mechanical calculations of the ‘probabilitarians’ (Ellis to Forbes, 10 October 1850, quoted in Shairp et al. 1873, 482). On the one hand, as Ellis explained his argument against inverse probability:

Everything which exists is a priori infinitely improbable, as that this sheet of paper should be the precise size that it is of all the infinite variety of possible sizes, &c. Consequently the improbability that the actual angular distance between Aldebaran and Arcturus should be within certain limits may [...] be made to exceed as much as we please the improbability that there should be 91 cases of stars lying within 4″ of each other. If, therefore, in the case of double stars we are bound to believe in physical connection because of the a priori improbability of what is observed to exist, I do not see how the wit of man can escape from the conclusion that we are bound also to believe in a physical connection between Arcturus and Aldebaran. If it be said that the cases are not parallel except quoad the numerical calculation, I answer that this is undoubtedly true; but then the remark admits that the numerical calculation is in itself no ground for inference in either case. And so, as they used to say in the schools, cadit quaestio.

(Ellis to Forbes, 3 September 1850, quoted in Shairp et al. 1873, 479–480)

On the other hand, as Ellis explained the underlying idea of his own account of explaining (the causes of) phenomena:

For everything which exists there is a definite reason why it is what it is and as it is; the only question being what analogy exists among the causes of analogous phenomena; in other words, what general propositions can be affirmed about them, or, which is again the same thing, what law or laws they fulfil. However the stars had been fixed in the visible heavens, each of them must have been fixed there by an adequate cause, fixing it just where it is. [...] The question is, Do the phenomena suggest to us the idea that the causes which placed the stars as they are, are connected with certain regions of the sky rather than with others?

(Ellis to Forbes, 10 October 1850, quoted in Shairp et al. 1873, 481–482)

Where for classical probabilists, such as Herschel around the time of Outlines of astronomy, probability theory was a calculus that could be used as a scientific method, for Ellis it was a mere branch of combinatorics not adequate to the way people, including scientists, actually think. According to Ellis, everything that was expressed in the theorems of probability theory was not known mathematically, but intuitively—‘the word being used’, he explained in an 1854 paper,

with reference to the intuitions of a mind, which has fully and clearly apprehended the subject before it, and to which therefore to have arrived at the truth and to perceive that it has done so are inseparable elements of the same act of thought. [...] Man in relation to the universe is not spectator ab extra, but in some sort a part of that which he contemplates [...] It is only when in thought we remove the action of disturbing causes to an indefinite distance that we can conceive the absolute verification of any a priori law. Only on the horizon of our mental prospect earth and sky, the fact and the idea, are seen to meet. (Ellis 1854 [1863], 606)

Ellis, Walton and probability theory at Cambridge

One of Ellis’ most frequent correspondents was William Walton (1813–1901), known to many Cambridge residents as ‘Old Father Time’ because of his flowing white beard and Scotch bonnet and plaid (see Venn 1954, 338). ⁸ Walton—admitted as pensioner to Trinity College, Cambridge in January 1831, Eight Wrangler in the 1836 Tripos, and Fellow of Trinity Hall from 1868–85—was the (co-)editor of several numbers of the Cambridge Mathematical Journal in 1840 and 1844–45, to which he contributed numerous papers, and the editor of the collected writings of Duncan Farquharson Gregory and Ellis.

Somewhere in the year 1849, Ellis wrote Walton a letter in which he informed him about (the relation between) ‘equal probability a priori’ and the ‘theory of probabilities a posteriori’, apparently responding to Henry Wilbraham's (1825–83) foundational views. ⁹ The letter is of interest for, at least, three reasons—all related to the history of probability theory. Firstly, in so far as Ellis referred to James Wood's Elements of algebra, designed for the use of students in the university, first published in 1795, as containing the ‘common’ treatment of (inverse) probability, it underlines the importance of textbook knowledge in the history of mathematics. Secondly, also in light of Walton's standing as a mature mathematician at the end of the 1840s, it is somewhat remarkable to observe the almost casual way in which Ellis conveys both the basic elements and (his criticism of) the foundations of probability theory. Thirdly, and lastly, Ellis’ statement that the recognition of ‘the subjective or conventional element in the theory of probabilities [is] of the highest importance’ counts as yet another example of the fact that even at the end of the 1840s the distinction between ‘objective’ and ‘subjective’ was not so straightforward as sometimes assumed in histories of probability (cf. Daston 1994; Verburgt 2015).

What follows is a transcription of Ellis’ letter to Walton, as found in the still largely unexplored Ellis Papers which are included in the Whewell Manuscripts in the Wren Library, Trinity College, Cambridge.

[Add.Ms.c 67, Whewell MSS, R.L. Ellis papers, Wren Library, Trinity College, Cambridge]

My dear Walton,

Wood's way of treating inverse probabilities is the common one & is quite correct – on the principle assumed, namely that the different hypotheses (as to the causes) which may have given rise to the observed event, were a priori equally probable. The question then reduces itself to this – is this principle itself correct? Wilbraham says no. I should say that taken abstractedly, it is neither correct nor incorrect, being an admissible assumption & nothing more.

To understand distinctly what is meant by equal probability a priori, let us take the following case. A person is supplied continually (by some contrivance into the nature of which we need not enquire) with a succession of urns each containing two balls. Some urns which I call m urns contain two white balls, & some, which I call n urns, a white & a black. Suppose m urns & n urns are on the long run supplied in equal numbers. Suppose A draws (& replaces) a ball from each urn that comes to his hands, & passes on every urn from which he draws a white ball to B.

B now proceeds to a second drawing: in what proportion of urns does he get a white ball? The question is answered by observing that all the urns come to his hands, &, on the long run, half the urns. These two classes of urns being equally numerous, two thirds of all the urns he [damaged] are m urns & one third n urns.

From all the former he gets white balls, &, on the long run, from half the latter, so that he gets white balls from or of all the trials he makes. That is in any single trial the chance is that he draws a white ball. And this is the common result. There is nothing precarious in the process of obtaining it, except the assumption that A is supplied with equal numbers, on the long run, [damaged]. This is in the accurate sense of the word precarious being mere matter of convention. If indeed we make any special hypotheses as to the way in which he is supplied with urns then the case is changed. Failing any such hypotheses, we may suppose both kinds to come to his hands with equal frequency – or in other words that m urns & n urns are a priori equally probable.

This supposition is inconsistent with (our) supposing that of the urns in which he finds a white ball & which he passes to B, as many are as . In order to this the frequency of supply to A of urns must be twice as great as that of urns: or the probability, antecedent to the first drawing of the former class twice as great as that of the latter.

In this use, which may be more simply conceived by supposing A to supply B with urns by putting a white ball into each & then a second ball, which is as often white & as black, the chance of B's drawing a white ball is not but . It is manifestly that he draws the white ball A put in, & that drawing the other he still draws a white ball, & therefore on the whole the chance is . And this, as I need not tell you, is the case which I mentioned to Cayley. The moral so to speak which I deduced from it, is that unless the hypothesis on which we proceed is distinctly “precisé” one result is as plausible & in reality as valid as the other.

Wilbraham's doctrine as to the a priori probabilities amounts in effect to assuming that the urn [damaged] filled by pitch & top – that is that [damaged] white [damaged] when head comes up & a black one when it does not, or vice versa: a quite legitimate hypothesis – but by no means necessarily true.

The recognition of the subjective or conventional element in the theory of probabilities a posteriori is I think of the highest importance in the appreciation of the results to which it leads. I have said something about it in what I published on probabilities in our transactions, but much less than the nature of the subject required.

I do not suppose this letter will [scribbled through] interest you much, but you may send any part or the whole of it to Wilbraham, if you think fit. I have written it to inform your conscience on the subject, so as to save you the trouble of thinking it out for yourself [...].

R.L. Ellis

(p.m.: 1849) William Walton Esq., 10 Trinity Street, Cambridge

Acknowledgments

I would like to thank John Gibbins for his support and the anonymous referee for his/her valuable comments. For the material from the Ellis Papers held by Wren Library (Cambridge), I duly acknowledge the kind permission of the Master and Fellows of Trinity College, Cambridge.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by a VENI grant from the Netherlands Organization for Scientific Research (NWO) [grant number 016.Veni.185.299]. A Grattan-Guinness Archival Research Travel Grant allowed me the travels and stays needed to undertake the research for the present paper.

Notes

¹ The present paper is part of a larger research project that is to cover all aspects of Robert Leslie Ellis’ life and work.

² For discussions of Ellis’ contributions to probability theory see, for example, Dale (1999, 370–372), Kilinc (2000), Salmon (1980), and Verburgt (2013).

³ Given the content of Ellis’ letter to Walton, the background sketch in what follows is limited to the first part of Ellis’ contributions to probability theory.

⁴ Ellis, for his part, wrote the lengthy ‘General Preface’ to Bacon's Philosophical works (Ellis 1860). About this preface, Whewell remarked that ‘Mr. Ellis has [here] given a more precise view than any of his predecessors had done of the nature of Bacon's induction and of his philosophy of discovery’ (Whewell 1860, 149–150).

⁵ As De Morgan wrote to Ellis in a famous letter about the four-colour problem: ‘On looking at the question I perceived that the thing depends upon the following [...]. On looking for demonstration of this, I found nothing more simple on which to found it, and on thinking of it, it became an axiom to my mind, and I quoted it in a paper sent to Cambridge (but not yet published) as an instance of Whewell's views about latent axioms, things which at first are not even credible, but which settle down into first principles’ (Augustus De Morgan to Robert Leslie Ellis, 24 June 1854, Wren Library, R L Ellis Papers, Add.Ms.c. 67/111–12).

⁶ See footnote 2.

⁷ From the known sources, it can be seen that the Ellis–Forbes correspondence commenced as early as February 1836, when Ellis was nineteen, and Forbes twenty-seven, years old. See Shairp et al. 1873, 121.

⁸ The Ellis Papers in the Wren Library, Cambridge, contain some 81 letters from Ellis to Walton.

⁹ Henry Wilbraham, who is known for discovering and explaining the Gibbs phenomenon nearly fifty years before William Gibbs himself did, was admitted to Trinity College, Cambridge in 1841 at the age of sixteen. He finished Seventh Wrangler in the Tripos of 1846 and became a Fellow of his college in 1848 (see Venn 1854, 464). As to probability theory, Wilbraham's contributions primarily lay in his criticisms of Boole's method of treating (inverse) probability in the Laws of thought of 1854—his point being that ‘Boole does in the great number of questions relating to chances solvable by his method [...] tacitly assume certain conditions expressible by algebraical equations, over and above the conditions expressed by the data of the problem, and to show how these assumed conditions may be algebraically expressed’ (Wilbraham 1854, 465).