A Budget of Paradoxes Volume I Part 49

[616] There are many similar series and products. Among the more interesting are the following:

[pi] 2244668...

---- = ----------------, 2 1335577...

[pi]-3 = 1 1 1 ------ = ----- - ----- + ----- - ..., 4 234 456 678

[pi] 1 1 1 1 1 ---- = sqrt - (1 - --- + ----- - ----- + ----- - ...), 6 3 33 3^25 3^37 3^49

[pi] 1 1 1 1 ---- = 4 ( - - ----- + ----- - ----- + ...) 4 5 35^3 55^5 75^7

1 1 1 - ( --- - ------- + ------- - ...).

239 3239^3 5239^5

[617] "To a privateer, a privateer and a half."

[618] Joshua Milne (1776-1851) was actuary of the Sun Life a.s.surance Society. He wrote _A Treatise on the Valuation of Annuities and a.s.surances on Lives and Survivorships; on the Construction of tables of mortality; and on the Probabilities and Expectations of Life_, London, 1815. Upon the basis of the Carlisle bills of mortality of Dr. Heysham he reconstructed the mortality tables then in use and which were based upon the Northampton table of Dr. Price. His work revolutionized the actuarial science of the time. In later years he devoted his attention to natural history.

[619] See note 576, page 252. He also wrote the _Theory of Parallels. The proof of Euclid's axiom looked for in the properties of the equiangular spiral_ (London, 1840), which went through four editions, and the _Theory of Parallels. The proof that the three angles of a triangle are equal to two right angles looked for in the inflation of the sphere_ (London, 1853), of which there were three editions.

[620] For the latest summary, see W. B. Frankland, _Theories of Parallelism, an historical critique_, Cambridge, 1910.

[621] Joseph Louis Lagrange (1736-1813), author of the _Mecanique a.n.a.lytique_ (1788), _Theorie des functions a.n.a.lytiques_ (1797), _Traite de la resolution des equations numeriques de tous degres_ (1798), _Lecons sur le calcul des fonctions_ (1806), and many memoirs. Although born in Turin and spending twenty of his best years in Germany, he is commonly looked upon as the great leader of French mathematicians. The last twenty-seven years of his life were spent in Paris, and his remarkable productivity continued to the time of his death. His genius in the theory of numbers was probably never excelled except by Fermat. He received very high honors at the hands of Napoleon and was on the first staff of the Ecole polytechnique (1797).

[622] "I shall have to think it over again."

[623] Henry Goulburn (1784-1856) held various government posts. He was under-secretary for war and the colonies (1813), commissioner to negotiate peace with America (1814), chief secretary to the Lord Lieutenant of Ireland (1821), and several times Chancellor of the Exchequer. On the occasion mentioned by De Morgan he was standing for parliament, and was successful.

[624] On Drinkwater Bethune see note 165, page 99.

[625] Charles Henry Cooper (1808-1866) was a biographer and antiquary. He was town clerk of Cambridge (1849-1866) and wrote the _Annals of Cambridge_ (1842-1853). His _Memorials of Cambridge_ (1874) appeared after his death.

Thompson Cooper was his son, and the two collaborated in the _Athenae Cantabrigiensis_ (1858).

[626] William Yates Peel (1789-1858) was a brother of Sir Robert Peel, he whose name degenerated into the familiar t.i.tle of the London "Bobby" or "Peeler." Yates Peel was a member of parliament almost continuously from 1817 to 1852. He represented Cambridge at Westminster from 1831 to 1835.

[627] Henry John Temple, third Viscount of Palmerston (1784-1865), was member for Cambridge in 1811, 1818, 1820, 1826 (defeating Goulburn), and 1830. He failed of reelection in 1831 because of his advocacy of reform.

This must have been the time when Goulburn defeated him. He was Foreign Secretary (1827) and Secretary of State for Foreign Affairs (1830-1841, and 1846-1851). It is said of him that he "created Belgium, saved Portugal and Spain from absolutism, rescued Turkey from Russia and the highway to India from France." He was Prime Minister almost continuously from 1855 to 1865, a period covering the Indian Mutiny and the American Civil War.

[628] William Cavendish, seventh Duke of Devonshire (1808-1891). He was member for Cambridge from 1829 to 1831, but was defeated in 1831 because he had favored parliamentary reform. He became Earl of Burlington in 1834, and Duke of Devonshire in 1858. He was much interested in the promotion of railroads and in the iron and steel industries.

[629] Richard Sheepshanks (1794-1855) was a brother of John Sheepshanks the benefactor of art. (See note 314, p. 147.) He was a fellow of Trinity College, Cambridge, a fellow of the Royal Society and secretary of the Astronomical Society. Babbage (See note 469, p. 207) suspected him of advising against the government support of his calculating machine and attacked him severely in his _Exposition of 1851_, in the chapter on _The Intrigues of Science_. Babbage also showed that Sheepshanks got an astronomical instrument of French make through the custom house by having Troughton's (See note 332, page 152) name engraved on it. Sheepshanks admitted this second charge, but wrote a _Letter in Reply to the Calumnies of Mr. Babbage_, which was published in 1854. He had a highly controversial nature.

[630] See note 469, page 207. The work referred to is _Pa.s.sages from the Life of a Philosopher_, London, 1864.

[631] Drinkwater Bethune. See note 165, page 99.

[632] Simeon-Denis Poisson (1781-1840) was professor of calculus and mechanics at the Ecole polytechnique. He was made a baron by Napoleon, and was raised to the peerage in 1837. His chief works are the _Traite de mecanque_ (1811) and the _Traite mathematique de la chaleur_ (1835).

[633] "As to M. Poisson, I really wish I had a thousandth part of his mathematical knowledge that I might prove my system to the incredulous."

[634] This list includes most of the works of Antoine-Louis-Guenard Demonville. There was also the _Nouveau systeme du monde ... et hypotheses conformes aux experiences sur les vents, sur la lumiere et sur le fluide electro-magnetique_, Paris, 1830.

[635] Paris, 1835.

[636] Paris, 1833.

[637] The second part appeared in 1837. There were also editions in 1850 and 1852, and one edition appeared without date.

[638] Paris, 1842.

[639] Pa.r.s.ey also wrote _The Art of Miniature Painting on Ivory_ (1831), _Perspective Rectified_ (1836), and _The Science of Vision_ (1840), the third being a revision of the second.

[640] William Ritchie (1790-1837) was a physicist who had studied at Paris under Biot and Gay-Lussac. He contributed several papers on electricity, heat, and elasticity, and was looked upon as a good experimenter. Besides the geometry he wrote the _Principles of the Differential and Integral Calculus_ (1836).

[641] Alfred Day (1810-1849) was a man who was about fifty years ahead of his time in his attempt to get at the logical foundations of geometry. It is true that he laid himself open to criticism, but his work was by no means bad. He also wrote _A Treatise on Harmony_ (1849, second edition 1885), _The Rotation of the Pendulum_ (1851), and several works on Greek and Latin Grammar.

[642] Walter Forman wrote a number of controversial tracts. His first seems to have been _A plan for improving the Revenue without adding to the burdens of the people_, a letter to Canning in 1813. He also wrote _A New Theory of the Tides_ (1822). His _Letter to Lord John Russell, on Lord Brougham's most extraordinary conduct; and another to Sir J. Herschel, on the application of Kepler's third law_ appeared in 1832.

[643] Lord John Russell (1792-1878) first Earl Russell, was one of the strongest supporters of the reform measures of the early Victorian period.

He became prime minister in 1847, and again in 1865.

[644] Lauder seems never to have written anything else.

[645] See note 22, page 40.

[646] The names of Alphonso Cano de Molina, Yvon, and Robert Sara have no standing in the history of the subject beyond what would be inferred from De Morgan's remark.

[647] Claude Mydorge (1585-1647), an intimate friend of Descartes, was a dilletante in mathematics who read much but accomplished little. His _Recreations mathematiques_ is his chief work. Boncompagni published the "Problemes de Mydorge" in his _Bulletino_.

[648] Claude Hardy was born towards the end of the 16th century and died at Paris in 1678. In 1625 he edited the _Data Euclidis_, publishing the Greek text with a Latin translation. He was a friend of Mydorge and Descartes, but an opponent of Fermat.

[649] That is, in the _Bibliotheca Realis_ of Martin Lipen, or Lipenius (1630-1692), which appeared in six folio volumes, at Frankfort, 1675-1685.

[650] See note 29, page 43.

[651] Balda.s.sare Boncompagni (1821-1894) was the greatest general collector of mathematical works that ever lived, possibly excepting Libri. His magnificent library was dispersed at his death. His _Bulletino_ (1868-1887) is one of the greatest source books on the history of mathematics that we have. He also edited the works of Leonardo of Pisa.

[652] He seems to have attracted no attention since De Morgan's search, for he is not mentioned in recent bibliographies.

[653] Joseph-Louis Vincens de Mouleon de Causans was born about the beginning of the l8th century. He was a Knight of Malta, colonel in the infantry, prince of Conti, and governor of the princ.i.p.ality of Orange. His works on geometry are the _Prospectus apologetique pour la quadrature du cercle_ (1753), and _La vraie geometrie transcendante_ (1754).

[654] See note 119, page 80.

[655] See note 120, page 81.

[656] Lieut. William Samuel Stratford (1791-1853), was in active service during the Napoleonic wars but retired from the army in 1815. He was first secretary of the Astronomical Society (1820) and became superintendent of the Nautical Almanac in 1831. With Francis Baily he compiled a star catalogue, and wrote on Halley's (1835-1836) and Encke's (1838) comets.

[657] See Sir J. Herschel's _Astronomy_, p. 369.--A. De M.

[658] Captain Ross had just stuck a bit of bra.s.s there.--A. De M.