1864

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Galton, Francis (1822-1911). Autograph letter signed to Archibald Smith (1813-72). 4pp. London, November 11, 1864. 179 x 113 mm. Light oil-stain in lower margins, but very good otherwise. From Galton to Scottish lawyer and mathematician Archibald Smith, best known for his work on magnetism and the Earth's magnetic field, particularly in relation to navigation. Galton was (among many other things) a Fellow of the Royal Geographical Society; he devoted the whole of his letter to Smith to geographical subjects, beginning with a discussion of the measurement of the Dead Sea's elevation: "Sir H. James is really in error in believing that the elevation of the Dead Sea has never been determined by actual levelling. In 1843 Lieut. Symonds R.E. received the gold medal of the Geographical Society for having triangulated & levelled between the Mediterranean and Dead Sea. He did his work very carefully. You will see an account of it in Vols. 12 & 13 of the Geograph. Society's transactions, in the first in the President's address & in the second in the President's speech, when he gave the gold medal. I had thought, but I believe now I was wrong, that it had also been levelled in more recent years, but I see that Lynch only used barometers & I suppose (from you not having referred to him) that Van der Welde did the same." Galton here refers to the measurement of the level of the Dead Sea by Lieutenant John F. A. Symonds (d. 1852), the leader of the Royal Engineers' survey of Palestine undertaken in 1841. As Galton notes, this achievement earned Symonds the Royal Geographical Society's gold medal in 1843. Galton also mentions the measurement made by William F. Lynch (1801-65), leader of the U.S. Navy's 1848 expedition to Jordan and the Dead Sea. "Van der Welde" refers to Dutch cartographer Carel W. M. Van de Velde (1818-98), who published a map of Palestine in 1858; Van de Velde did not do any surveying, however, and his Dead Sea measurements were taken from Symonds. "Sir H. James" is Sir Henry James (1803-77), who served as the director-general of the British government's Ordnance Survey from 1854 to 1875. Galton next brings up the question of whether Smith would "press the pendulum matter" before the Geographical Society: "If you do not think it well to press the pendulum matter, pure & simple, there remains nothing else for our Society to take up but if you do care to press it I will move at our meeting of council on Monday that a committee should be formed consisting say of Everest Waugh Spottiswoode & one or two others, to examine the question & report to the ensuing meeting. They would then communicate with you, Sabine & the Astronomer Royal, & no time would be lost. I know Everest to be very keen about pendulum experiments being now established in many places, & I think he would greatly assist. . . ." Galton here is referring to the Royal Society's proposal, supported by Smith, that pendulum measurements of the Dead Sea elevation be made as part of the Ordnance Survey's survey of Jerusalem, then in progress. The Jerusalem survey at first lacked both the money and the trained personnel necessary to perform these measurements, but Smith's advocacy of the project evidently paid off: The Royal Society and the Royal Geographical Society each contributed £100 towards the cost of the pendulum work and the Ordnance Survey team was able to make accurate measurements of the difference between the levels of the Mediterranean and the Dead Sea (see Wilson, Ordnance Survey of Jerusalem). "Everest" is Sir George Everest (1790-1866), Surveyor-General of India from 1830 to 1843 and the namesake of Mount Everest; "Waugh" is Sir Andrew Scott Waugh (1810-78), who succeeded Everest as Surveyor-General of India and who was responsible for naming Mount Everest after his predecessor; "Spottiswoode" is mathematician and physicist William Spottiswoode (1825-83), who served on the council of the Royal Geographical Society from 1862 to 1864. Spottiswoode was the. Bookseller Inventory # 42850

Title: **Autograph letter signed to Archibald Smith**

Publication Date: **1864**

Signed: **Signed by Author(s)**

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Both from St James's Rectory Piccadilly London. 22 September and 21 June 1858
(1856)

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**Book Description **Both from St James's Rectory Piccadilly London. 22 September and 21 June 1858, 1856. Both items in good condition, lightly aged and worn. ONE: 22 September 1856. 4pp., 12mo. Bifolium. Docketted: 'Revd J. E. Kempe about Annie's XG. & Tait, new Bp of London | Sep 1856'. After discussing arrangements for meeting he turns to Tait, about to be consecrated Bishop of London. 'You ask about our new Bishop. I have reason to think it an excellent appointment. Great administrative ability, a remarkable talent for business, indefatigable industry, most conciliatory manners, a total freedom from party of any kind, considerable learning, and great piety - these are the qualities which I am assured by those who know him are united in Dr. Tait. The only doubt seems to be as to his physical strength.' He discusses his examining chaplain, likely to be 'A. P.?> Stanley, and is pleased at 'the prospect of an occupant of London House who is able, amiable & a gentleman. For some time I was trembling lest "Dean of Carlisle" should be a mis-print for "Bishop"!!' Postscript regarding a pamphlet Tait produced 'at the time of the proceedings against Ward', which 'produced a great impression'. TWO: 21 June 1858. 2pp., 12mo. He explains that on the suggested day 'we are engaged to dine with Mr & Mrs H. Smith 16 Devonshire St.' In consequence he leaves it to 'the Rev S. Smith to judge whether it will be possible for me to t{ake] late Coffee at S George's Passage at 5 and to attend a Camberwell Meeting at 6'. He has been 'tantalized by the offer of Cartwell - the Lakes - Morecambe Bay &c &c.'. Seller Inventory # 16684

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(1863)

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**Book Description **Cambridge, 1863. Three Mathematical Autograph Letters, Signed, Plus a Four Page Autograph Mathematics Manuscript Cayley, Arthur (1821-95). (1) Autograph letter signed ("A. C.") to Archibald Smith (1813-72), written in the margins of an autograph letter signed from Smith to Cayley. 1 page. London, 9 August 1863 [date of Smith's letter]. 280 x 226 mm. (2) Autograph letter signed to Smith. 3 - 1/2pp. Cambridge, 14 November 1866. 180 x 112 mm. (3) Autograph letter signed to Smith. 3pp. Cambridge, 19 October 1868. 180 x 112 mm. (4) Autograph manuscript (originally enclosed with Cayley's 19 Oct. 1868 letter). 4ff., tied with linen tape. 412 x 257 mm. N.p., n.d. [1868]. Together 4 items. Very minor marginal tears in nos. (1) and (4), otherwise very good. Exceptionally rare mathematical correspondence from Arthur Cayley, one of the founders of the British school of pure mathematics, consisting of three autograph letters, all containing mathematics, plus an extensive 4-page mathematical proof written on four extra-large legal sheets. These are the first mathematical letters or manuscripts by Cayley that we have seen on the market in over forty years. Cayley was the author of over 900 papers covering nearly every aspect of modern mathematics; his greatest contributions were his development of the algebra of matrices, his work in non-Euclidean geometry and n-dimensional geometry, and his contributions to invariant theory. A large number of mathematical constructs bear his name, including Cayley's theorem (group theory), the Cayley-Hamilton theorem (linear algebra), Cayley's formula (graph theory) and the Cayley-Klein model (hyperbolic geometry). Cayley's correspondent was Archibald Smith, who helped to found the Cambridge Mathematical Journal in 1836 and made significant contributions to the study of magnetism and the Earth's magnetic field. Both Cayley and Smith were alumni of Trinity College, Cambridge and both subsequently entered Lincoln's Inn to study law, with Smith being called to the bar in 1841 and Cayley in 1849. Cayley remained in the legal profession until 1863, at which time he left the bar to take the newly established Sadleirean professorship of pure mathematics at Cambridge. It is evident from our letters that Smith and Cayley had a cordial relationship based on their shared love for mathematics; Smith apparently was in the habit of sending Cayley mathematical problems and requesting his help in solving them. Letter (1) contains both Smith's query and Cayley's response; in it Smith asked Cayley to give him "a simple construction for the following problem. Given three points in a plane to draw the straight line such that the sum of the squares of the perpendiculars shall be a minimum." Cayley responded by writing his solution in the margins of Smith's letter: "Considering the points as material points of equal mass, then taking the origin at the C. G. it is easy to show that the required line is the principal axis of least moment. E.g. if the three points be an isosceles base [.] less than 60° required line is [parallel] to base-if greater than 60° it is [perpendicular] to base-if equal 60° or triangle is equilateral there the position is indeterminate. The construction for the principal axes depends of course on finding the axes of an elliptic. A. C." In letter (2), dated 1866, Cayley gave the solution to another of Smith's mathematical queries; "it comes out easily & prettily enough." Cayley stated the problem and solution as follows: "Given two lines A, B & a point O, to draw from O to the lines A, B two lines of [equal] length & enclosing a given [angle] . Imagine the line A rigidly connected with O and let it revolve round this point thro' the given [angle] , so as to assume the position A'. Then if P be any point in A and P' the corresponding point in A', it is clear that OP = OP' & [angle] POP' = . . ." In letter (3), dated 1868, Cayley enclosed the four-page autograph manuscript listed above under no. (4), containing "a solution, such as I have been able to obtain, of your problem, but the solution is I am afraid in a form which will not be of any use to you. May I send the problem-of course in your name-to the Educational Times; it is very likely that you will so obtain a solution of it in a more practical form; and at any rate, the problem, quà problem is an excellent one." In the manuscript Cayley stated the problem as follows: "Considering any two circles O, O' and taking M, M' the corresponding points in the line OO', let the angles MP, M'P' be respectively , - where is a variable angle; it is required to find the envelope of the line PP'." Cayley's solution covers four folio pages and includes several equations and two diagrams. Biggs et al., Graph Theory, ch. 3. Dictionary of Scientific Biography. Kline, Mathematical Thought from Ancient to Modern Times, pp. 804-9. Ewald, From Kant to Hilbert, p. 542. Seller Inventory # 42843