The Origins of the Geometric Probability in England

WDS'08 Proceedings of Contributed Papers, Part I, 7–12, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS

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 (Académie Royale des Sciences) formulated and solved two problems in which he introduced differential and integral calculus into probability theory. Buffon’s Mémoire sur le jeu de franc- carreau was presented in the Academy in April 1733 and the reports from academicians Pierre

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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, générale et particulière (1749–1778 in 36 volumes, 8 additional volumes published after his death by Lacépède). In [Buffon, 1777] the problems from Mémoire 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 “interesting 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 parquetée ou pavée de carreaux égaux, d’une figure quelconque, on jette en l’air un écu; l’un des joueurs parie que cet écu après sa chute se trouvera àfranc-carreau, c’est-à/dire, sur une seul carreau; le second parie que cet écu se trouvera sur deux carreaux, c’est-à-dire, qu’il couvrira un des joints qui les séparent; un troisième joueur parie que l’écu se trouvera sur deux joints; un quatrième parie que l’écu 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 divisépar les joints parallèles, on jette en l’air une baguette, & que l’un des joueurs parie que la baguette ne croisera aucune des parallèles du parquet, & que l’autre au contraire parie que la baguette croisera quelques-unes de ces parallèles; on demande le sort de ces deux joueurs. On peut jouer ce jeu sur un damier avec une aiguille àcoudre ou une épingle sans tête.

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 examination 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). Todhunter was also interested in the history and bibliography of science. He admired the work of professor Robert Woodhouse (1773–1827) who had written a history of the calculus of variations, ending with the eighteenth century, and decided to continue the history of this calculus during the nineteenth century. History of the Calculus of Variation was published in 1861. In 1862 he was elected a fellow of the Royal Society of London. In 1864 the financial success of his books was such that he was able to marry Louisa Anna Maria Davies, daughter of Captain George Davies of the Royal Navy. This step involved resigning of his fellowship. In 1865 the London Mathematical Society was organized under the guidance of A. De Morgan and Todhunter became a member in the first year of its existence. His second historical work, History of the Mathematical Theory of Probability appeared in the same year. In 1873 Todhunter published his History of the Mathematical Theories of Attraction consisting of two volumes of nearly 1000 pages altogether. In the same year he published The Conflict of Studies and Other Essays on Subjects Connected with Education, related to facets of undergraduate mathematical education in Cambridge University in the later part of the 19th century. In 1874 Todhunter was elected honorary fellow of St John’s College. In the summer of 1880 he began to suffer from his eyesight and from that date he slowly and gradually became weaker. In September 1883 he partially lost the use of the right arm. In January of the next year he suffered another attack and died on the 1st March 1884. Todhunter received many awards for his contributions to mathematics. He served on Council of the Royal Society of London in 1874 and in the same year he was awarded the Adams Prize for his work Researches on the Calculus of Variations.

Todhunter and geometric probability Todhunter was probably the first in England who was really concerned with geometric probability. Chapter XIV (Application of the Integral Calculus to Questions of Mean Value and Probability) of [Todhunter, 1857] is exclusively dedicated to this topic. He formulates and solves Buffon’s needle problem and proposes some of its generalizations in the section of examples. In the first edition (1857), he proposes to throw a cube (Example 4), a rod whose length is equal

9 KALOUSOVA: GEOMETRIC PROBABILITY to r-times the perpendicular distance between two consecutive lines (Example 6) and an ellipse whose major axis is less then the distance between consecutive lines3 (Example 11) at random on the plane ruled with parallel equidistant lines. In the second edition (1862), Example 4 is formulated more precisely (diagonal of the cube is less than the distance between consecutive lines) and the result is given. In Example 11 the ellipse is replaced by a closed curve which has no singular points and whose greatest diameter (in the contemporary terminology, Todhunter’s diameter is called the breadth and the greatest breadth is called the diameter) is less than the distance between consecutive lines. Two another problems are given in the third edition (1868). A large area is ruled with parallel equidistant straight lines, and also with a second set of parallel equidistant straight lines at right angles to the former set. A thin rod (Example 13) or a cube (Example 14) are thrown at random on that area now. Todhunter recalls [Todhunter, 1865] for corresponding results. In these editions, Buffon is not introduced as the author of these problems. In later editions (1880), Todhunter writes that “this problem (i.e. the needle problem) was first proposed by the celebrated naturalist Buffon, and was afterwards discussed by Laplace”. These later editions are also complemented by new (Crofton’s) results. In the third edition, the journal Mathematical Questions, with their solutions from Educational Times is recommended for “a large number of very interesting problems relating to the subject of the present Chapter”. In [Todhunter, 1865] Todhunter writes about Buffon’s problems too. He adverts to [Laplace, 1812], where Buffon’s problems are given wthout any reference to Buffon. Todhunter writes also about [Buffon, 1777], analyses its parts and corrects the wrong result. He also shows that Buffon had solved them at a much earlier date and cites [Fontenelle, 1735[.

Examples Let us recall Todhunter’s solutions of Buﬀon’s needle problem and of some its generalizations which are formulated and solved in the 5th edition of [Todhunter, 1857]. Denote by 2a the distance between two consecutive lines and 2b the length of the rod. Suppose that the centre of the rod falls on a straight line drown between two consecutive lines of the given system and meeting them in the right angles. Denote by x the distance between the centre and the nearer of two selected parallels. Now, suppose the rod to revolve round its 2Φ centre. Obviously, the chance that the rod crosses the straight line is π , where x = b cos Φ. ∆x · We may denote by a the chance that the centre of the rod falls between the distance x and 2 x + ∆x from the nearer of the two parallels. Thus the required chance will be πa Φ dx. The limits of x are 0 and b, hence the result is R

π 2b 2 2b Φ sin Φ dΦ = . πa Z0 · πa Analogously, we can solve the problem in which the rod (straight line) is replaced by an ellipse. We denote by AA′ its major axis and L its perimeter. Take two consecutive lines and choose one of them; we will estimate the chance that the ellipse crosses this line, and, by doubling this result, we get the chance that the ellipse crosses the system. Let A be at the distance x from the chosen line. Suppose the ellipse to revolve around A. Obviously the chance Φ that the ellipse crosses the chosen line is π , where Φ is the angle when the ellipse touches the ∆x chosen line. We may denote by a the chance that A falls between the distance x and x + ∆x from the chosen parallel. Thus, the required chance will be 1 1 Φ dx = (xΦ x dΦ). πa Z πa − Z

3Sylvester in [Sylvester,1890-1891] wrote: “this important step in the development of the theory is, I am informed, currently attributed to the late Mr Leslie Ellis, of the University of Cambridge.”

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When x = 0 we have Φ = π and when x = AA′ we have Φ = 0; the limits of x are 0 and AA′ , thus | | | | 0 π Φ dx = 0 x dΦ = x dΦ. Z − Zπ Z0 Because π x dΦ = L , the chance that the ellipse crosses the chosen line is 0 2 R 1 π L x dΦ = πa Z0 2πa L and the required chance is (by doubling) πa . If we consider a closed curve having no singular points, whose greatest diameter is less than a and which is symmetrical with respect to AA′ (greatest diameter) we can use the same deduction and we obtain the same result. If the closed curve is not symmetrical the chance that it crosses the chosen line is Φ1+Φ2 , where Φ and Φ denote the angles when the curve touches 2π 1 2 the chosen line on one side and on other side. Finally we have 1 π 1 π L x dΦ1 + x dΦ2 = 2πa Z0 πa Z0 2πa L and the required chance is (by doubling) πa . We recall some other examples. 1. Prove that the mean of all the radius-vectores of an ellipse, the focus being the origin, is equal to half the minor axis, when the straight lines are drown at equal angular intervals; and is equal to the half of the major axis when the straight lines are drawn so that the abscissae of their extremities increase uniformly. 2. Two arrows are sticking in a circular target: shew that the chance that their distance is greater than the radius of the target is 3√3 . 4π 3. A certain territory is bounded by two meridian circles and by two parallels of latitude which differ in longitude and latitude respectively by one degree, and is known to lie within certain limits of latitude; find the mean superficial area. 4. From any point within a closed curve straight lines are drown at equal angular intervals to the circumference: shew that the mean value of the squares on these straight lines is 1 the product of π into the area of the curve.

Followers France In France, young student of Ecole´ normale supérieure Joseph-Emile´ Barbier (1839–1889) took also lectures of Gabriel Lamé(1795–1870) at Sorbonne. Lamélectured about Buffon’s needle problem and its generalizations to circles, ellipses and polygons. Barbier noticed that in all these examples, the probability that the figure crosses one of the parallels is the same, namely L πa where L is the perimeter of the figure and a is the distance between parallels. The rod can be thought of as a limiting case of an ellipse with the major axis equal to L . In 1860, the same 2 year when he graduated from Ecole´ normale supérieure, Barbier presented in [Barbier, 1860] the general theorem that the probability that an arbitrary convex disk of diameter less than the L distance of parallel lines intersects a line is πa , where L is the length of the disk circumference and a is the distance between parallels.4

4Soit un disque convexe de forme quelconque qui ne puisse, dans aucune de ses positions sur le plan, rencontrer àla fois plusieurs lignes de division. Qu’elle sera la probabilitéde la rencontre? l l On peut prouver qu’elle est πa , étant la longueur du contour du disque, en s’appuyant sur diverses con- sidérations.

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This result is subsequently extended even for an arbitrary rigid rectiﬁable curve (closed or not). Then, he computes the mean number of intersections of a rectiﬁable curve with a system of parallels, generalizes this result and deduces formulas which are now used in stereology (stochastic inference of object properties from its sections). However, such applications of Barbier’s results were not noticed.

England In England, the development of the geometric probability (or Local probability as they called it) was closely linked to the journal Educational Times edited by College of Preceptors. The College was founded in 1846 to standardize the teaching profession. The Educational Times was launched in 1847 and its mathematical department was created two years later. The mathematical problems were published under the heading Mathematical Questions from the beginning. Because of a great interest not only among students and teachers but also among the leading mathematicians a new separate journal Mathematical Questions with their Solutions from the Educational Times was launched in 1864. In next years, speciﬁc problems concerning geometric probability were posed and solved in this journal. The best known contributors were James Joseph Sylvester (1814–1897), Thomas Archer Hirst (1830–1892), Arthur Cayley (1821–1895) and, especially, Morgan William Crofton (1826–1915). Sylvester formulated the well known four point problem: what is the probability that four points chosen at random in a planar region have a convex hull which is a quadrilateral. The problem was published in the Educational Times of 1864, question 1491, and completely solved by Blaschke in 1917. Crofton is known mainly due to the Crofton’s formulas which appeared in [Crofton, 1868]. He supposed to use them for computation of complicated integrals. His most important con- tribution to the ﬁeld of geometric probability was probably an entry of Probability in the 9th edition of Encyclopaedia Britannica (1885) with its 6th section on local, i.e. geometric, probability. In contemporary editions of Encyclopaedia Britannica, (e.g. the 15th edition, 1995) the entry Probability is completely changed and Crofton’s name is not at all mentioned.

References Barbier, J.-E.,´ Note sur problème de l’aiguille et le jeu du joint couvert, Journal de mathématiques pures et appliqués, 5, 273–287, 1860. Buffon, G.-L. Leclerc, comte de, Essai d’Arithmétique morale, Histoire naturelle, générale et particulière, Supplément, Tome IV, Paris, Imprimerie Royale, 46–168, 1777. Crofton, M.W., On the Theory of Local Probability, Applied to Straight Lines Drawn at Random in a Plane, the Metods Used Being Also Extended to the Proof of Certain New Theorems in the Integral Calculus, Phil. Trans., 158, 181–199, 1868. Fontenelle, B. le B. de, Summary of Buffon’s memoir, Histoire de l’Académie royale des Sciences en 1733, Paris, Imprimerie Royale, 43–45, 1735. Halley, E., An Estimate of the Degrees of Mortality of Mankind, drown from curious Tables of the Births and Funerals at the city of Breslaw; with an Attempt to ascertain the Price of Annuities upon Lives, Phil. Trans., VII, 596–610, 1693. Laplace, P.-S. de, Théorie analytique des probabilités, Paris, Imprimerie Royale, 1812. Newton, I., in Derek T. Whiteside (ed.) The Matematical Papers of Isaac Newton, Vol. I, University Press Cambridge, 60–62, 1967. Sylvester, J.J., On a Funicular Solution of Buffon’s “Problem of the Needle” in its Most General Form, Acta Mathematica, 14, 185–205, 1890–1891. Todhunter, I., History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange, Cambridge and London: MacMillan and Co., 1865. Todhunter, I., Treatise on the Integral Calculus and its Applications with Numerous Examples, Cambridge and London: MacMillan and Co., 1857, 1862, 1868, 1874, 1878, 1880.