Thomas Knight - Mathematician

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Thomas Knight of Papcastle
Mathematician

Thomas Knight and Isabella Walker

American Mathematical Monthly vol. 112, Feb. 2005

Further News from Alex Craik

I have discovered that he enrolled in medical courses at Edinburgh University during sessions 1796-7 and 1797-8. In both sessions he enrolled in "Anat. & Chir" and "Chem". These are anatomy and surgery, and chemistry for medics.
It was fairly unusual to repeat courses, though a few did. This suggests to me that either his studies were interrupted or that he was otherwise engaged. (He got married in 1797 in Edinburgh as you informed me.) I sent copies of his two signatures in the Matriculation lists to the Royal Society of London, who have confirmed that they are very similar to Knight's signature (see below) in their archive papers.
The Edinburgh connection is interesting to me because it could explain his mathematical interests IF he was then in contact then with Prof. John Playfair. Though he did not enroll in Playfair's courses, he could have attended informally.

Alex Craik (August 2007)

In search of Thomas Knight

by Alex D. D. Craik¹ and Gloria Edwards²

1. School of Mathematics & Statistics, University of St Andrews, St Andrews, Fife KY16 9SS

2. Secretary, Kirkgate Museum Group, High Moor, Hill St., Cockermouth, Cumbria CA13 0AU

Introduction
Thomas Knight is a forgotten figure, absent from biographical dictionaries and barely mentioned in works on the history of mathematics. Nevertheless, during 1809-18, he wrote a considerable number of papers on mathematics and its applications. In Cambridge, the “continental” calculus was not introduced into the Mathematical Tripos examinations until 1817, the year of Knight’s last published paper; and we are not aware of any contact between Knight and the Cambridge reformers. But Knight’s writings show that he was familiar with works of French analysis, then little-read in Britain, and that he was an accomplished practitioner of both analysis and geometry, with a particular liking for infinite series. His name has not been found among published lists of alumni of Cambridge, Oxford, or the Scottish and Irish universities. Neither did he become a Fellow of the Royal Society (though a relative, the horticulturalist Thomas Andrew Knight, did so). Niccolò Guicciardini [29] is one of very few recent writers to mention Knight’s papers, but he gives no details of Knight’s life.

The only biographical clue in Knight’s papers is his address of Papcastle, Cumberland. Then a visit to the British Library uncovered an early publication by Knight: a pamphlet [1] printed at Cockermouth, just a mile from the village of Papcastle, in which the author “T.K.” criticises Laplace’s theory of capillary action. Seeking more biographical information, the first-named author contacted the second, who is a genealogical researcher and Secretary of the Kirkgate Museum Group at Cockermouth. Our findings on Knight’s life are described in the third section of this paper; but first we describe his mathematical output.

Knight’s mathematical writings
Knight published fourteen journal articles, and he wrote some other works besides. His early pamphlet [1] on Laplace’s theory of capillary action is along similar lines to two of his papers [4], [6] in Nicholson’s Journal. Though Knight did not dispute Laplace’s conclusions, he considered Laplace’s reasoning to be loose and incorrect, and he claimed that his own modification of an earlier theory of James Jurin was more convincing. In the pamphlet, he expresses himself forcibly and at times amusingly. He claims that, according to Laplace’s argument:

If we suppose a column of water suspended in a capillary tube to become rigid... we may take away the tube, and the column will stand as high as ever.... I could as soon believe that a man might hang himself without a rope, which is perhaps not unworthy of consideration, in the present scarcity of hemp. [1, pp. 11-12]

A third article in Nicholson’s Journal concerned objections to a “new principle” in Legendre’s Éléments de géométrie [5], [31]. The latter principle recurred as a subject of controversy fifteen years later, following publication of the English translation of Legendre’s work: see Craik [26]. At issue was the validity of dimensional reasoning in geometry. Then, John Leslie of Edinburgh independently reiterated Knight’s incorrect argument that lines and angles should not be treated differently.

Knight published four short articles in Leybourn’s Mathematical Repository (not listed in the Royal Society Catalogue of Scientific Papers, 1800-1900) [2], [3], [10], [11]. These all concern series expansions. Some are rather trivial, such as his rederivations of exponential and trigonometric series. But others are more difficult, such as the trigonometric expansion of functions f(y) where y = (A + ccosx)m and two short letters about his expansion of “multinomials”, the topic which he examined at length in his first Phil. Trans. paper [7].

Seven papers by Knight appeared in the Philosophical Transactions of the Royal Society of London. This is a large number for a time when mathematics was not much esteemed by the Society. The only other mathematician to publish as many in this journal during the first three decades of the nineteenth century was the much better-known James Ivory: see Craik [27].

Knight’s paper [7] is essentially a reworking of L.F.A. Arbogast’s Calcul des Dérivations of 1800 [21], which concerns the series expansion of functions of one or more polynomials. This is perhaps Knight’s most ambitious work, which he believed to give improvements on Arbogast’s difficult work: see Craik [28]. But Knight’s complicated and at times idiosyncratic notation must have repelled his readers, and this substantial paper was neglected.
Knight’s longest paper, “Of the attraction of such solids as are terminated by planes; and of solids of greatest attraction” was published in Phil. Trans. in 1812 [8]. In this, he sets out to generalise work of John Playfair, who had calculated the gravitational attraction of parallelopipeds and of isosceles pyramids on a rectangular base. Knight’s “general problem” was: “any solid, regular or irregular, terminated by plane surfaces, being given, to find, both in quantity and direction, its action, on a point, given in position, either within or without it” [8, p. 247]. He supposes that such a body may be regarded as made up of an assemblage of pyramids and prisms with polygonal bases, and he finds the gravitational attraction of many such shapes. He then considers “the attraction of certain solids not terminated by planes” [8, pp. 269-283], such as those enclosed by a regular polygon moving perpendicular to its plane and varying its shape and size according to some given law. Several specific examples are solved. The paper concludes with a section on “solids of greatest attraction” in which he finds the shape of the solid, consisting of a given quantity of matter, that exerts the greatest gravitational force at a point on its surface. In this, he cites earlier results on special cases by Playfair and Silvabelle, and the variational method of Euler. Though the problems he solves are very artificial, of no real physical interest, they display his mastery of techniques of double integration in various coordinate systems. Though he used Newtonian “dots”

rather than differentials

, along with Leibnizian integral signs, this was clearly no impediment to him.

In the same volume of Phil. Trans. he also published “Of the penetration of a hemisphere by an indefinite number of equal and similar cylinders” [9]. This is a curious work, employing both geometrical insight and skill in integration, but of no conceivable practical importance. It stems from “well known theorems of Viviani and Bossut respecting certain portions of the surface and solidity of a hemisphere,” which are a single case of Knight’s generalised problem. This is “To pierce a hemisphere perpendicularly on the plane of its base, with any number of equal and similar cylinders; of such a kind, that, if we take away from the hemisphere those portions of the cylinders that are within it, the remaining part shall admit of an exact cubature: and if we take away, from the surface of the hemisphere, those portions cut out by the cylinders, the remaining surface shall admit an exact quadrature” [9, p. 310]. Following his rather elaborate geometrical construction, for each specified number of cylinders, a curve is obtained within the base of the hemisphere that outlines the form of the base of each desired cylinder. Calculating the volume V and curved surface area A of what remains of the hemisphere after taking away the cylinders, Knight finds that V = 8r³/9 and A = 4r², which are respectively commensurable with a cube and square of side r, where r is the radius of the hemisphere. Interestingly, these results do not depend on the number of cylinders chosen, although the shapes of the bases do. Only when two cylinders are chosen are their bases circular; but, for any number, an equation for the perimeter of the base can always be found algebraically, at least in principle.

In 1816-17, Knight published four more papers in Phil. Trans. [12], [13], [14], [15]. Two of these concern a “new demonstration of the binomial theorem” and the others are on construction of logarithms and the method of finite differences. His brief note [15] is a punctilious admission that his demonstration of the binomial theorem had been anticipated in William Spence’s “ingenious Essay on Logarithmic Transcendents, a work published in 1809, but which I have been so unfortunate as never to have seen till within the last fortnight ... The same may be said of the first proposition ... on the construction of logarithms.” In his papers on logarithms and finite differences, Knight again shows his familiarity with French sources, citing Delambre’s Preface to a paper by Borda, the [never fully published] “great French Tables,” and works by Lagrange, Prony, and Lacroix.

The Royal Society holds among its “Archived Papers” two unpublished manuscripts by Knight, again on power series, dated 1817 and 1818. The first of these [16] is a short paper showing how to express

and

as series of terms in powers of

.

One might wonder what use such series might be: but Knight observes that “Everyone knows that the expressions for sin.nA, cos.nA depend on the solution of this problem.” (This follows on setting x = expiA, but Knight does not say so.) In contrast, the second manuscript [17] is thirteen large double-sided folio pages long. It concerns the expression of various integrals as series of terms, mainly using repeated integration by parts. Along the way, he cites results of Johann Bernoulli, Waring, Frisi, Bossut and Lacroix, which he dismisses as “in almost every case ... inconvenient and useless.” Striving for greater generality, he establishes his “Proposition 1: Supposing

to be known, it is required to express

in an infinite number of ways.” (Interestingly, he has now switched from Newtonian “dot” notation to differentials.) He gives examples from William Spence’s 1809 work [35] on “logarithmic transcendents” and goes on to consider “a much more extensive class of functions than those treated by Mr Spence.” These are multiple integrals of the forms

, , etc. (r = 1, 2, 3, ...),

and, still more generally,

(any positive integers r, m, n).

The latter lead to complications and an elaborate notation which would have tested the ability of the typesetters, had the paper been recommended for publication. No doubt Knight was disappointed that it was merely “noted” and never published.

Knight’s papers were quickly forgotten. In part, this was due to a lack of true originality, to his use of rather cumbersome notations, and to the apparent lack of utility (or absence of explanation of utility) of much of his work. But a more cogent reason was the sad lack of interest in analytical mathematics in England at this time. Most of the fellows of the Royal Society were hostile to the presentation of complicated calculations which few could understand. Knight lacked the power base of a university appointment; and his sending manuscripts from Cumberland was a far less effective way of making a mark that appearing in person in London to mingle with his mathematical peers. Yet Knight deserves to be remembered as one of very few English mathematicians of his day who was able and willing to engage with continental analytical mathematics, and to read the impressive but neglected analytical work of William Spence of Greenock.

Biography

A search of the genealogical databases IGI (International Genealogical Index for Cumberland) and Family Search (of the Church of Latter Day Saints) was fruitful, and led to more information in Parish Registers for Wolverley (Worcestershire) 1775, and the Public Record Office (Wills). The records, however, are not entirely consistent. We also made contact with two descendants of Thomas Knight.

Thomas Knight was born on 25th August and baptised on 21st November 1775 at Wolverley, Worcestershire. He was the second son of John Knight (1740-1795) and Henrietta Cunyngham of Lea Castle, Wolverley, near Kidderminster. John Knight’s father was Edward Knight (1699-1780), an ironfounder of Wolverley. Thomas Knight married Isabella Walker (b. about 1777, d. 1830) and, between 1798 and 1820, they had fourteen children, eight sons and six daughters. Though IGI records one fewer daughter, the cover of Thomas Knight’s family bible, preserved by the family, confirms the fourteen offspring. Among them were triplets, one son and two daughters, whose births were reported in the Cumberland Paquet for 25th December 1804. Thomas Knight died at Papcastle, in the parish of Bridekirk, on 16th August 1853, just short of his 78th birthday. The announcement of his death in the Gentleman’s Magazine for October 1853 (p. 427) describes him as “aged 79 ... of Papcastle and Henley Hall, Shropshire.” Adding further confusion, his gravestone in Bridekirk Churchyard wrongly records his age as seventy-four.

It seems likely that Thomas Knight was educated fairly near his childhood home at Wolverley, but we have found no school record for him. It is possible that the family employed a private tutor. There are more family papers in the Worcestershire and Shropshire Archives which we have not seen, and which may shed further light on Knight’s education. Along with a description of his family coat of arms, Huddleston & Boumphrey’s Cumberland Families and Heraldry [30, p. 192] mentions that Thomas settled first at Keswick and then at Papcastle. The Keswick connection is confirmed as the birthplace in 1798 of Thomas and Isabella’s first child, also named Thomas. This Thomas was admitted to Trinity College, Cambridge in 1816 and later to the Middle Temple in London, where he pursued a legal career [36]

Thomas Knight’s children were all baptised into the Anglican faith. Accordingly, though Knight apparently did not attend an English university, he was not excluded from doing so by his religious beliefs (as Catholics, Quakers and other dissenters effectively were), and some other family members did so. As well as Thomas’s eldest son Thomas jr., who studied at Cambridge, an uncle James Knight matriculated at Christchurch College, Oxford in 1763. So too, in 1774, did Samuel Johns Knight of Henley Hall, who adopted the name Knight by royal licence in 1813 and who became a fellow of All Souls College and an Anglican vicar.

Based on the profits of the iron foundry, the extended Knight family had become rich and owned several large properties (Beesley [22]). Among them was Henley Hall, near Bitterley in Shropshire, which was eventually inherited in 1852 by John Knight (b. 1803), the third son of “our” Thomas. But many of Thomas’s family seem to have lived there before this date, as genealogical records give this as the address of all of the Knight children.

A gazetteer of 1829 [33, p. 305] describes Papcastle as “an irregularly built village, seated upon a lofty eminence ... 11/2 mile WNW of Cockermouth. Many of the houses are neat and commodious, and are occupied by gentry.” A still earlier gazetteer of 1807 describes it as a “large and handsome village ... pleasantly situated on the high N. bank of the Derwent ... well-built commodious dwellings of respectable families owning and occupying the lands ... uncommonly rich pasturage, and the cheese made in Papcastle is much admired. This is one of the pleasantest villages in the N. of England; some say in all England...” [23]. An 1816 volume on Cumberland mentions that “Thomas Knight, Esq. is building a handsome mansion for his residence, on some ground lately purchased within or adjoining to the site of the Roman station; many antiquities have been discovered in sinking the foundations” [32, p. 37]; and “Many coins and other Roman Antiquities have been lately found near the station at Papcastle, in digging the foundations of a house for Thomas Knight, Esquire” [32, p. clxxxix].

An 1832 map of Cockermouth by John Wood (see Figure 1) gives an inset of Papcastle in which the major proprietors are named. The property of Thomas Knight, a large mansion house surrounded by extensive wooded parks, is partly shown on the east side of the village. Like Lea Castle, the house no longer exists, and the archaeological site, believed to be the Roman town of Derventio (part of which is now known as “The Mount”), has been built on. Stone from the Roman remains was used to construct Cockermouth Castle in the 13th century. Some of the Roman finds are in Tullie House Museum, Carlisle and in Senhouse Roman Museum, Maryport. A recent archaeological dig was conducted by Channel 4’s “Time Team” and more excavations are currently underway.

A photograph of Knight’s former home, part of the catalogue of its sale in 1957, is shown in Figure 2. It was sold, along with other properties in the area, to pay taxes arising from the death of the owner, Lord Egremont of Cockermouth Castle. The house was demolished around 1967.

It is probably no coincidence that Thomas Knight’s mathematical activity ceased shortly after he moved to Papcastle: the management of the estate, and his already large and growing family, must have been major distractions. He must also have been discouraged by the Royal Society’s decision not to publish his manuscript papers of 1817-18. Thomas Knight lived into old age, and at least three of his children died before him. Robert (1804-1834) died quite young; the lawyer Thomas died in 1850; and Captain James Knight (1807-1836) “fell gloriously... heading the storm party against the enemy’s entrenchments on the heights of San Sebastian” in Spain [18]. In his will, Thomas Knight left all his estate, apart from a small bequest to a servant, to two of his daughters, Henrietta Mansfield and Maria Knight, the former a widow and the latter unmarried [20].

Knight and Cumbrian mathematics

Cumberland and its vicinity, though geographically isolated, was no intellectual desert in the early 1800s. Knight was not cut off from scientific or mathematical stimulus, if he sought it out. There were a number of able schoolteachers nearby. Long-established schools at the port of Whitehaven catered for the navigational training of sailors, as well as teaching book-keeping and the usual grammar-school syllabus. St Bees grammar school and the Lowther school for “the sons of gentlemen” were particularly praised [34]. Among the private schools, especially noteworthy was that of the Quaker John Slee and his son and successor, Thomas. John Slee, described as a "profound mathematician," was born at Mungrisedale, and died in 1828 at Tirril. Their school at Tirril in Westmorland was "visited by many gentlemen from the Universities, who, during the vacations, come here to receive instruction in mathematics" [33, p. 50]. From about 1830, it became the custom for private tutors at Cambridge to take their charges to rural summer retreats to prepare them for the demanding Tripos examinations. But long before then, Cambridge students had sought out private tutors in the Lake District and the Yorkshire Dales. The most remarkable of these was John Dawson of Sedbergh (then just over the Yorkshire border but now in Cumbria), who between 1781 and 1800 trained no fewer than eleven Cambridge senior wranglers and doubtless many lower wranglers besides (Warwick [37, p. 62], DNB [25]). Dawson (1734-1820) was a medical doctor and a self-taught mathematician with no direct Cambridge affiliation. In his day, Cambridge University tried to discourage students from using private tutors; but by the 1830s a private tutor had become almost indispensable for prospective wranglers.

Others with direct connections between the county and Cambridge were the mathematical divine Isaac Milner (1750-1820), who was Dean of Carlisle as well as President of Queens’ College, and several Anglican parish rectors whose livings were in the gift of colleges of the University. Christopher Wordsworth, the Master of Trinity College Cambridge from 1820 to 1841, was born in Cockermouth: and so too were his brother, the acclaimed poet William Wordsworth, and their sister Dorothy.

Several local boys attained mathematical and scientific distinction. Fearon Fallows was born in Cockermouth in 1788, and was third wrangler at Cambridge in 1813, behind John Herschel and George Peacock. Fallows was soon appointed Astronomer Royal at the Cape of Good Hope. Knight moved to Papcastle shortly after Fallows had left for Cape Town and there is no evidence that the two ever met. Another able mathematician from Cumberland whom Knight probably did not meet was Solomon Atkinson, son of a poor farm worker from Lazonby, who became senior wrangler in 1821.

The greatest scientist to be born near Cockermouth was certainly the meteorologist and chemist John Dalton (1766-1844), who pioneered the atomic theory. Dalton was the son of a handloom weaver from Eaglesfield, and one of the Quaker community. He taught mathematics for a time at Cockermouth and then at Kendal, before moving to Manchester in 1793, some time before Knight moved to Papcastle. We have found an interesting letter from Dalton to Knight, now in the Worcestershire Record Office [19], which shows that Knight was on friendly terms with John Dalton. Dated April 3rd 1822, this begins by referring to a visit that Dalton made to Papcastle in the previous summer, when he:

mentioned my intention of communicating some particulars respecting Mr. Buchan’s telescope, and also sending some of the numbers of the Annals of Philosophy with Mr. Herapath’s new mechanical notions. I have not hitherto done either, and you may justly surmise that I have been neglectful of my promise.

Dalton goes on to describe observing Jupiter, Saturn and four of Saturn’s satellites with Mr. Buchan, who “resides two miles from me: if your atmosphere does not often favour you for day views, much less does ours.” It seems, then, that Knight too was an amateur astronomer with a telescope at Papcastle.
Regarding Herapath,
Mr. Herapath’s communications in the Annals have been so numerous that scarcely a month has elapsed without something pro or con having appeared. Some notice of them has been taken in Tilloch’s magazine, which I think you occasionally see. Mr. H. has so blended his mechanical reasoning with abstract physical notions that I am afraid you would hardly think yourself requited by wading through the whole.
With kind remembrance to Mrs. Knight and the young ladies, I remain yours truly,
John Dalton

The “Mr Buchan” referred to is certainly not the Scottish meteorologist Alexander Buchan (born 1829), and we have not established his identity. John Herapath (1790-1868) was a private teacher of mathematics in Devon. He was interested in steam-driven machines and early theories of thermodynamics; and, like Charles Babbage and John Herschel, he published on functional equations. His quarrelsome nature involved him in many disputes and damaged his career. Disillusioned with science, Herapath turned to editing the world’s first railway magazine (Brush [24]).

The picture that emerges of Thomas Knight is of a well-to-do English country gentleman with an amateur interest in science and mathematics, a large estate and a large family. During the second decade of the nineteenth century, his mathematical attainments compared favourably with those of any of his compatriots, although his published work had little lasting impact. He was remarkably well-read in continental mathematical works; he tried to extend the work of the French analysts, especially Arbogast, and that of the Scot William Spence; and he was particularly interested in series expansions. His scientific interests led to papers on the theories of capillarity and gravitational attraction, and he seems to have possessed and used an astronomical telescope. How and where he learned his mathematics is unknown to us. There may well be more material about Knight in archives that we have not seen, and we wonder whether a portrait may have survived. We hope that this attempt to “put him on the map” may lead others to find more information: if so, we should like to hear about it.

Acknowledgments: We are grateful to librarians of the Worcestershire and Shropshire Record Offices, the Public Record Office, The British Library, The Royal Society, and St Andrews University Library for their assistance. We also acknowledge with thanks the permission granted to us by a descendant of Thomas Knight, Lt. Cdr. F. R. Jerram, to quote from two letters deposited in the Worcestershire Record Office, and we thank him and Miss R.M. Jerram for providing further family information. We are grateful to Mrs D. Jackson of Papcastle for supplying the photograph of Figure 2. The permission of the Royal Society to quote brief passages from Knight’s two unpublished manuscripts is also gratefully acknowledged.

References
(a) Publications of Thomas Knight:
1. “T.K.” [Thomas Knight], An Examination of M. La Place’s Theory of Capillary Action. 36pp. + errata sheet and 1 plate of figures, T. Bailey for J. Deboffe, Cockermouth, 1809.
2. On the expansion of certain functions. Leybourn’s Math. Repository 2 (1809), 64-67.
3. Two letters, on the expansion of any functions of a multinomial. Leybourn’s Math. Repository 2 (1809), 67-70.

4. On the theory of capillary attraction. A Journal of Natural Philosophy, Chemistry and the Arts (Nicholson’s Journal), New Ser. 27 (1810), 126-132.

5. Remarks on a new principle introduced by Legendre in his ‘Elements of Geometry’. A Journal of Natural Philosophy, Chemistry and the Arts (Nicholson’s Journal), New Ser. 27 (1810), 285-287.

6. Remarks on La Place’s theory of capillary action. A Journal of Natural Philosophy, Chemistry and the Arts (Nicholson’s Journal), New Ser. 28 (1811), 155-156.

7. On the expansion of any functions of multinomials. Phil. Trans. Roy. Soc. London 101 (1811), 49-88.

8. Of the attraction of such solids as are terminated by planes; and of solids of greatest attraction. Phil. Trans. Roy. Soc. London 102 (1812), 247-309.

9. Of the penetration of a hemisphere by an indefinite number of equal and similar cylinders. Phil. Trans. Roy. Soc. London 102 (1812), 310-313.
10. On the expansion of the formula fccosx)m. Leybourn’s Math. Repository 3 (1814), 32-34.
11. On the sine and cosine of the multiple arc. Leybourn’s Math. Repository 3 (1814), 34-37.

12. A new demonstration of the binomial theorem. Phil. Trans. Roy. Soc. London 106 (1816), 331-334.

13. Of the construction of logarithmic tables. Phil. Trans. Roy. Soc. London 107 (1817), 217-233.

14. Two general propositions in the method of differences. Phil. Trans. Roy. Soc. London 107 (1817), 234-244.

15. Note respecting the demonstration of the binomial theorem inserted in the last volume of the Philosophical Transactions. Phil. Trans. Roy. Soc. London 107 (1817), 245-251.

(b) Manuscripts:

16. Royal Society Archived Papers Series AP.8(ii).10 (28 October 1817), Fol. 1. Extract to express

and

by series arranged according to the powers of

.

17. Royal Society Archived Papers Series AP.8(ii).11, Fol. ff 13, “Noted. Read 26 February 1818.” A general series for

.

18. Worcestershire Record Office, BA10831(i) 1: Letters relating to Knight family of Papcastle. Letter from Brig. Gen. C. Chichester, May 9, 1836.

19. Worcestershire Record Office, BA 10831(ii) 1: John Dalton to Thos. Knight of Papcastle April 3, 1822.

20. Public Record Office, PROB 11/2182: Will of Thomas Knight.

(c) Other references:

21. Louis F. A. Arbogast, Du calcul des dérivations. Levrault, Strasbourg, 1800.

22. Pauline Beesley, A Brief History of the Knight Family. 1958 (Copy in Shropshire Archives, Shrewsbury).

23. T. Brown (of Sanquhar) The Union Gazetteer of Great Britain and Ireland 1807. Also quoted in the Cockermouth Free Press, 1912.

24. Stephen G. Brush, art. John Herapath. Dictionary of Scientific Biography (ed. Charles C. Gillispie) 6, 291-293. New York: Scribners, 1973.

25. “J. W. C--k”, art. John Dawson. Dictionary of National Biography 14, ed. Leslie Stephen, London: Smith, Elder & Co. 1888.

26. Alex D.D. Craik, Geometry versus analysis in early 19th century Scotland: William Wallace, John Leslie and Thomas Carlyle. Historia Mathematica 27 (2000), 133-163.

27. Alex D.D. Craik, James Ivory, mathematician: “the most unlucky person that ever existed”. Notes & Records of the Royal Society 54 (2000), 223-247.

28. Alex D.D. Craik, Prehistory of Faà di Bruno’s formula. American Mathematical Monthly, to appear.

29. Niccolò Guicciardini, The Development of Newtonian Calculus in Britain 1700-1800. Cambridge: Cambridge Univ. Press, 1989.

30. C. Roy Huddleston & R.S. Boumphrey, Cumberland Families and Heraldry. Trans. Cumberland and Westmorland Archaeological and Antiquarian Society [CWAAS], Extra Series 23 (1978).

31. Adrien-Marie Legendre, Éléments de géométrie, avec des notes. Paris: Firmin Didot, 1794; 8th ed. 1809.

32. Daniel and Samuel Lysons, Magna Britannia, 6 vols. 1806 -22, vol. 4: Cumberland (1816). London: T. Cadell & W. Davies. Also quoted in E. Birley, ‘Roman Papcastle’, Trans. CWAAS, 63 (1963) New ser. art. VI, 103ff.

33. William Parson & William White, History, Directory and Gazetteer of the counties of Cumberland and Westmorland.... Leeds: for W. White & Co. by E. Baines & Son, 1829. Also reprinted by Michael Moon, The Scolar Press, Ilkley, Yorks, 1976.

34. F. J. G. Robinson & Peter J. Wallis, Some early mathematical schools in Whitehaven. Trans. CWAAS, New ser. 75 (1975) 262-274.

35. William Spence, An Essay on the Theory of the Various Orders of Logarithmic Transcendents; with an Inquiry into Their Applications to the Integral Calculus and the Summation of Series. London: John Murray, Edinburgh: Arch. Constable & Co., 1809.

36. John and J.A. Venn, Alumni Cantabrigienses ... Part II (1752-1900), 6 vols. Cambridge: Cambridge University Press, 1954.

37. Andrew Warwick, Masters of Theory: Cambridge and the Rise of Mathematical Physics. Chicago: The University of Chicago Press, 2003.

CAPTIONS

Figure 1: Map of Papcastle, inset from John Wood’s map of Cockermouth, 1832, showing Thomas Knight’s property.

Figure 2: Photograph of the now-demolished house in Papcastle built by Thomas Knight, and then known as “The Mount” (from the particulars of its sale in 1957, courtesy of Mrs. D. Jackson).

.