WDS'08 Proceedings of Contributed Papers, Part I, 7–12, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS The Origins of the Geometric Probability in England A. Kalousov´a Czech Technical University in Prague, Faculty of Electrical Engineering, Prague, Czech Republic. Abstract. This article describes the origins of the geometric probability from the first intuitive examples in the 17th century (Newton, Halley). We recall the Buffon’s problems and their generalizations by Barbier, Crofton and, in particular, by Todhunter. Introduction The geometric probability is not taught at elementary and secondary schools, it is seldom taught at universities even though it is frequently used in medicine, crystallography, materials science, biology, ecology and in common life too. The main feature of geometric probability is that the random events are measured and not counted (both numbers of favourable and considered events have the cardinality of continuum). Very roughly speaking, whereas in the common probability, e.g., the chance of throwing six is calculated, in geometric probability the chance of hitting a target or its part is estimated. The origins of geometric probability Usually, Georges-Louis Leclerc, comte de Buffon (1707–1788) is considered to be the first who used geometry for computation of probability. But the first examples can be found nearly one century ago in England. The first hesitant steps It was Isaac Newton (1642–1727), who first applied geometry to probability calculation. In his manuscript [Newton, 1967] written sometimes between 1664-1666 stands: “If ye Proportion of the chances for any stake bee irrationall the interest in the stake may bee found after ye same manner. As if ye Radij ab, ac, divide ye horizontal circle bcd into two pts abec & abdc in such proportion as 2 to √5. And if a ball falling perpendicularly upon ye center a doth tumble into ye portion abec I winn (a): but if into ye other portion, I win b, my hopes is worth 2a+b√5 ”. 2+√5 Newton’s intention was to show that a chance of a simple event could be irrational and he unwittingly discovered the basic rule of geometric probability, namely measurement instead of computing. Newton’s manuscript is only a few pages long and it remained unknown to his contemporaries. The first who applied geometric probability intentionally in a published work was Edmond Halley (1656–1742). In his memoir [Halley, 1693] he deduced formulas for various annuities at first analytically and then, according to the fashion of the time, he added their geometric illustration. To France for a while The first difficult problem in geometric probability was examined in France. Young Georges- Louis Leclerc, later comte de Buffon, wanting to join the French Royal Academy of Sciences (Acad´emie Royale des Sciences) formulated and solved two problems in which he introduced differential and integral calculus into probability theory. Buffon’s M´emoire sur le jeu de franc- carreau was presented in the Academy in April 1733 and the reports from academicians Pierre 7 KALOUSOVA: GEOMETRIC PROBABILITY Louis Maupertuis (1698–1759) and Alexis Claude Clairaut (1713–1765) were commendatory. In January 1734 Buffon joined the Academy, section of mechanics. But he was interested more and more in botany. In March 1739, he transfered to section of botany and in July 1739 he became Keeper of the King’s Garden (Jardin du Roi). Therefore, Buffon is best known as naturalist and author of Histoire naturelle, g´en´erale et particuli`ere (1749–1778 in 36 volumes, 8 additional volumes published after his death by Lac´ep`ede). In [Buffon, 1777] the problems from M´emoire sur le jeu de franc-carreau were included. Buffon claims in the 23rd section that up to the present time arithmetic has been the only instrument used in estimating probabilities and he promises to show examples which would require the aid of geometry. Then he shows three types of such examples. The first is the generalization of the game franc-carreau where a round coin is thrown at random down on a large plane area divided into regular figures (squares, equilateral triangles, or regular hexagons) and the chance that it will fall clear off the bounding lines of the figure (position franc-carreau), or hit one of them, or two of them, and so on, is required.1 He solves them using only simple calculation without any need of the integral calculus. The second one is the problem which is now known as Buffon’s Needle Problem. A slender rod is thrown at random down on a large plane area ruled with equidistant parallel straight lines; one of the players bets that the rod will not cross any of the lines while the other bets that the rod will cross several of them. The odds for these two players are required.2 Buffon 2b 2b solves the problem correctly using integral calculus and shows that the odds are (1 πa ):( πa ) where 2a is the distance between the parallel lines and 2b is the length of the rod (the− inequality b a is assumed to hold). ≤ The third one is a generalization of the previous problem. The plane area on which a rod is thrown down is ruled with a second set of equidistant equally spaced parallel straight lines, oriented at right angles to the former one. Buffon merely gives the result, but it is incorrect. At the end of the 23th section, Buffon proposes to put some questions which can be “in- teresting and useful for common life”, for example: to what extent do we risk when we pass a river on a plank (less or more wide), how much should we be afraid of a lightning or of a bomb etc. The basic and new feature of these problems, namely comparison of measures (lengths), was unnoticed by Buffon’s contemporaries and a considerable attention arose only to the occurrence of π in a probabilistic formula. Pierre-Simon de Laplace (1749–1827) suggests in [Laplace, 1812] to turn the needle problem and use it to determine the value of π. He mentions both needle problems without any reference to Buffon and gives the correct solution of the problem with two systems of parallel straight lines. In the half of 19th century, the geometric probability returned to England. It was Isaac Todhunter (1820–1888), who had known Buffon’s problems and began to study them. Isaac Todhunter Life and work Isaac Todhunter was born at Rye, Sussex, 23rd November 1820, as the second son of George Todhunter, Congregationalist minister, and Mary, his wife. Father died when Isaac was 1Dans une chambre parquet´ee ou pav´ee de carreaux ´egaux, d’une figure quelconque, on jette en l’air un ´ecu; l’un des joueurs parie que cet ´ecu apr`es sa chute se trouvera `afranc-carreau, c’est-`a/dire, sur une seul carreau; le second parie que cet ´ecu se trouvera sur deux carreaux, c’est-`a-dire, qu’il couvrira un des joints qui les s´eparent; un troisi`eme joueur parie que l’´ecu se trouvera sur deux joints; un quatri`eme parie que l’´ecu se trouvera sur trois, quatre ou six joints; on demande les sorts de chacun de ces joueurs. 2Je suppose que dans une chambre, dont le parquet est simplement divis´epar les joints parall`eles, on jette en l’air une baguette, & que l’un des joueurs parie que la baguette ne croisera aucune des parall`eles du parquet, & que l’autre au contraire parie que la baguette croisera quelques-unes de ces parall`eles; on demande le sort de ces deux joueurs. On peut jouer ce jeu sur un damier avec une aiguille `acoudre ou une ´epingle sans tˆete. 8 KALOUSOVA: GEOMETRIC PROBABILITY six years old and left family in narrow circumstances. The widow with four sons decided to move to Hastings and open a girls’ school. Isaac was sent to a boys’ school in Hastings and subsequently to a new school set by J.B. Austin from London. Under this new teacher he made rapid progress. After leaving school he became an assistant school master with J.B. Austin at a school in Pecham and attended simultaneously evening classes at University College, London, where his lecturer was Augustus De Morgan (1806–1871). In 1839 he passed the matriculation exami- nation of the University of London, in 1842 he passed the B.A. examination and in 1844 he obtained the degree of M.A. with the gold medal. One of his lecturers at the University was also James Joseph Sylvester (1814–1897). In 1844 Todhunter entered St John’s College, Cambridge, where he was senior wrangler in 1848, and gained the first Smith’s prize and Burney prize. In 1849 he was elected to a fellowship, and began his life of college lecturer and private tutor. Mathematicians Peter Guthrie Tait (1831–1901) and Edward John Routh (1831–1907) were among his pupils. Todhunter was very successful as a lecturer and it was not long before he began to publish textbooks on the subjects of his lectures. Most of his textbooks passed through many editions and have been widely used in Great Britain and North America. Let us remind some of them: Treatise on the Differential Calculus and the Elements of the Integral Calculus (1852), Treatise on Analytical Statics (1853), Treatise on Integral Calculus (1857), Treatise on Algebra (1857), Treatise on Plane Coordinate Geometry (1858), Examples of Analytical Geometry of Three Dimensions (1858), Plane Trigonometry (1859), Spherical Trigonometry (1859), Theory of Equations (1861), Mechanics (1867), Researches in the Calculus of Variations (1871), Elementary Treatise on Laplace’s, Lame’s and Bessel’s Functions (1875).