CONTENTS.
CHAPTER I. CARDAN. KEPLER. GALILEO.
1
Commentary on Dante, 1. Cardan, De Ludo Aleoe, 1. Kepler, De Stella Nova, 4. Galileo, Considerazione sopra il Giuco dei Dadi, 4; Lettere, 5.
 
CHAPTER II. PASCAL AND FERMAT.
7
Quotations from Laplace, Poisson, and Boole, 7. De Méré's Problems, 7. Problem of Points, 9. De Méré's dissatisfaction, 11. Opinion of Leibnitz, 12. Fermat's solution of the Problem of Points, 13. Roberval, 13. Pascal's error, 14. The Arithmetical Triangle, 17. Pascal's design, 20. Contemporary mathematicians, 21.
 
CHAPTER III. HUYGENS.
22
De Ratiociniis in Ludo Aleoe, 22. English translations, 23. Huygens's solution of a problem, 24; Problems proposed for solution, 25.
 
CHAPTER IV. ON COMBINATIONS.
26
William Buckley, 26. Bernardus Bauhusius and Erycius Puteanus, 27. Quotation from James Bernoulli, 28. Pascal, 29. Schooten, 30. Leibnitz, Dissertatio de Arte Combinatoria, 31; his fruitless attempts, 33. Wallis's Algebra, 34; his errors, 35.
 
CHAPTER V. MORTALITY AND LIFE INSURANCE.
37
John Graunt, 37. Van Hudden and John de Witt, 38. Sir William Petty, 39. Correspondence between Leibnitz and James Bernoulli, 40. Halley, 41; his table, 42; geometrical illustration, 43.
 
CHAPTER VI. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1670 AND 1700.
44
Caramuel's Mathesis Biceps, 44; his errors, 45, 46. Sauveur on Bassette, 46. James Bernoulli's two problems, 47. Leibnitz, 47; his error, 48. Of the Laws of Chance, ascribed to Motte, 48; really by Arbuthnot, 49; quotation from the preface, 50; error, 52; problem proposed, 53. Francis Roberts, An Arithmetical Paradox, 53. Craig's Theologioe Christianoe Principia Mathematica, 54. Credibility of Human Testimony, 55.
 
CHAPTER VII. JAMES BERNOULLI.
56
Correspondence with Leibnitz, 56; Ars Conjectandi, 57. Error of Montucla, 58. Contents of the Ars Conjectandi, 58. Problem of Points, 59. James Bernoulli's own method for problems on chances, 60; his solution of a problem on Duration of Play, 61; he points out a plausible mistake, 63; treats of Permutations and Combinations, 64; his Numbers, 65; Problem of Points, 66; his problem with a false but plausible solution, 67; his famous Theorem, 71; memoir on infinite series, 73; letter on the game of Tennis, 75. Gouraud's opinion, 77.
 
CHAPTER VIII. MONTMORT.
78
Fontenelle's Eloge, 78. Two editions of Montmort's book, 79; contents of the book, 80; De Moivre's reference to Montmort, 81; Montmort treats of Combinations and the Binomial Theorem, 82; demonstrates a formula given by De Moivre, 84; sums certain Series, 86; his researches on Pharaon, 87; Treize, 91; Bassette, 93. Problem sol ed by a lady, 95. Problem of Points, 96; Bowls, 100; Duration of Play, 101; Her, 106; Tas, 110. Letter from John Bernoulli, 113. Nicolas Bernoulli's game of chance, 116. Treize, 120. Summation of Series, 121. Waldegrave's problem, 122. Summation of Series, 125. Malebranche, 126. Pascal, 128. Sum of a series, 129. Argument by Arbuthnot and 's Gravesande on Divine Providence, 130. James Bernoulli's Theorem, 131. Montmort's views on a History of Mathematics, 132. Problems by Nicolas Bernoulli, 133. Petersburg Problem, 134.
 
CHAPTER IX. DE MOIVRE.
135
Testimony of John Bernoulli and of Newton, 135. Editions of the Doctrine of Chances, 136. De Mensura Sortis, 137. De Moivre's approximate formula, 138; his Lemma, 138; Waldegrave's problem, 139; Duration of Play, 140; Doctrine of Chances, 141; Introduction to it, 142; continued fractions, 143; De Moivre's approximate formula, 144; Duration of Play, 147; Woodcock's problem, 147; Bassette and Pharaon, 150; Numbers of Bernoulli, 151; Pharaon, 152; Treize or Rencontre, 153; Bowls, 159; Problem on Dice, 160; Waldegrave's problem, 162; Hazard, 163; Whist, 164; Piquet, 166; Duration of Play, 167; Recurring Series, 178; Cuming's problem, 182; James Bernoulli's Theorem, 183; problem on a Run of Events, 184; Miscellanea Analytica, 187; controversy with Montmort, 188; Stirling's theorem, 189; Arbuthnot's argument, 193.
 
CHAPTER X. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1700 AND 1750.
194
Nicolas Bernoulli, 194. Barbeyrac, 196. Arbuthnot's argument on Divine Providence, 197. Waldegrave's problem, 199. Browne's translation of Huygens's treatise, 199. Mairan on Odd and Even, 200. Nicole, 201. Buffon, 203. Ham, 203. Trente-et-quarante, 205. Simpson's Nature and Laws of Chance, 206; he adds something to De Moivre's results, 207; sums certain Series, 210; his Miscellaneous Tracts, 211. Problem by John Bernoulli, 212.
 
CHAPTER XI. DANIEL BERNOULLI.
213
Theory of Moral Expectation, 213; Petersburg Problem, 220; Inclination of planes of Planetary Orbits, 222; Small-pox, 224; mean duration of marriages, 229; Daniel Bernoulli's problem, 231; Births of boys and girls, 235; Errors of observations, 236.
 
CHAPTER XII. EULER.
239
Treize, 239; Mortality, 240; Annuities, 242; Pharaon, 243; Lottery, 245; Lottery, 247; notes on Lagrange, 249; Lottery, 250; Life Assurance, 256.
 
CHAPTER XIII. D'ALEMBERT.
258
Croix ou Pile, 258; Petersburg Problem, 259; Small-pox, 265; Petersburg Problem, 275; Mathematical Expectation, 276; Inoculation, 277; Croix ou Pile, 279; Petersburg Problem, 280; Inoculation, 282; refers to Laplace, 287; Petersburg Problem, 288; error in a problem, 290.
 
CHAPTER XIV. BAYES.
294
Bayes's theorem, 295; his mode of investigation, 296; area of a curve, 298. Price's example, 299. Approximations to an area, 300.
 
CHAPTER XV. LAGRANGE.
301
Theory of errors, 301; Recurring Series, 313; Problem of Points, 315; Duration of Play, 316; Annuities, 320.
 
CHAPTER XVI. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1750 AND 1780.
321
Kaestner, 321. Dodson, 322. Hoyle, 322. Clark's Laws of Chance, 323. Mallet, 325. John Bernoulli, 325. Beguelin, on a Lottery problem, 328; on the Petersburg Problem, 332. Michell, 332. John Bernoulli, 335. Lambert, 335. Mallet, 337. Emerson, 343. Buffon, on gambling, 344; on the Petersburg Problem, 345; his own problem, 347. Fuss, 349.
 
CHAPTER XVII. CONDORCET.
351
Discours Préliminaire, 351; Essai, 353; first Hypothesis, 353; second Hypothesis, 357; problem on a Run of Events, 361; election of candidates for an office, 370; problems on inverse probability, 378; Risk which may be neglected, 386; Trial by Jury, 388; advantageous Tribunals, 391; expectation, 392; Petersburg Problem, 393; evaluation of feudal rights, 395; probability of future events, 398; extraordinary facts, 400; credibility of Roman History, 406. Opinions on Condorcet's merits, 409.
 
CHAPTER XVIII. TREMBLEY.
411
Problem of Points, 412; probability of causes, 413; problem of births, 415; lottery problem, 421; small-pox, 423; duration of marriages, 426; theory of errors, 428; Her, 429.
 
CHAPTER XIX. MISCELLANEOUS INVESTIGATIONS BETWEEN THE YEARS 1780 AND 1800.
432
Prevost, 432. Borda, 432. Malfatti, 434. Bicquilley, 438. Encyclopédie Méthodique, 441. D'Anieres, 445. Waring, 446. Ancillon, 453. Prevost and Lhuilier, 453. Young, 463.
 
CHAPTER XX. LAPLACE.
464
Memoirs of 1774, 464; recurring series, 464; Duration of Play, 465; Odd and Even, 465; probability of causes, 465; theory of errors, 468; Petersburg Problem, 470; Memoir of 1773, 473; Odd and Even, 473; Problem of Points, 474; Duration of Play, 474; Inclination of Orbits of Comets, 475; Memoir of 1781, 476; Duration of Play, 476; approximation to integrals, 478; problem of births, 482; theory of errors, 484; Memoir of 1779, 484; Generating Functions, 484; Memoir of 1782, 485; Memoirs of 1783, 485; Memoir of 1809, 487; Memoir of 1810, 489; Connaissance des Tems, 490; Problem on Comets, 491; Théorie...des Probabilités, 495; editions of it, 495; dedication to Napoleon, 496; Laplace's researches in Physical Astronomy, 499; Pascal's argument, 500; illusions, 501; Bacon, 503; Livre I. 505; Generating Functions, 505; Method of approximation, 512; examples, 516; Livre II. first Chapter, 527; second Chapter 527; Odd and Even, 527; Problem of Points, 528; Fourth Supplement, 532; Waldegrave's Problem, 535; Run of Events, 539; Inclination of the Orbits of Planets, 542; election of candidates, 547; third Chapter, 548; James Bernoulli's Theorem, 548; Daniel Bernoulli's problem, 558; fourth Chapter, 560; Poisson's problem, 561; Least Squares, 571; history of this subject, 588; fifth Chapter, 589. Buffon's problem, 590; sixth Chapter, 592; a Definite Integral, 594; seventh Chapter, 598; eighth Chapter, 601; Small-pox, 601; duration of marriages, 602; ninth Chapter, 605; extension of James Bernoulli's Theorem, 607; tenth Chapter, 609; inequality, 609; eleventh Chapter, 609; first Supplement, 610; second Supplement, 611; third Supplement, 612; quotation from Poisson, 613.
 
APPENDIX.
614
John de Witt, 614. Rizzetti, 614. Kahle, 615. 's Gravesande, 616. Quotation from John Bernoulli, 616. Mendelsohn, 616. Lhuilier, 618. Waring, 618.