b. 7 September 1707, Montbard, France

d. 16 April 1788, Paris, France

Buffon
created the area of Geometric Probability
with his study of the game of Franc-carreau.
The first
notice occurs in the Histoire de
l'Académie de
Sciences in 1733. This was
partially reprinted in the
Encyclopedia of Diderot in 1752 under the word Carreau.
In *Supplement
*IV, Essai
d'Arithemétique
Morale (1777) to his *Histoire
naturelle*, the study,
probably written in 1760, speaks at length on both Franc-carreau and
the famous Needle
Problem. In this same essay he also considers the
Petersburg
problem for which see both Nicolas
Bernoulli and Daniel Bernoulli.

Considerable literature has developed on the Needle Problem. Included here is all of the experimental trials concerning this problem. One should consult N. T. Gridgeman: "Geometric Probability and the number π," Scripta Mathematica 25 (1960), pp. 183-195.

- 1812 Laplace shows how to estimate π and generalizes to a needle cast on a rectangular grid. Theorie Analytic des Probabilités, the end of Chapter 5, p. 363-369.
- 1843 Lalanne describes the problem in Un million Faits. The link is to the 4th edition of 1846 The relevant text is also quoted by Wolf immediately below.
- 1850 Rudolf Wolf provides experimental evidence. Mittheilungen der Naturforschenden Gesellschaft in Bern, 176, 85-88 and 193, 209-212.
- 1859 A. de Morgan:

Augustus de Morgan writes in A Budget of Paradoxes (2nd edition, Vol, 1, 1915, pp. 283-284) of the Needle Problem:

The
paradoxes of what is called chance,
or hazard, might themselves make a small volume. All the world
understands that there is a long run, a general average; but great part
of the world is surprised that this general average should be computed
and predicted. There are many remarkable cases of verification; and one
of themrelates to the quadrature of the circle. I give some account of
this and another. Throw a penny time after time until head
arrives, which it will do
before long: let this be called a set.
Accordingly, H is the smallest set, TH the next smallest, then TTH,
&c. For abbreviation, let a set in which seven tails
occur before head
turns up be T7H. In an immense
number of trials of sets, about half will be H; about a quarter TH;
about an eighth T2H. Buffon tried 2,048 sets; and several have followed
him. It will tend to illustrate the principle if I give all the
results; namely that many trials will with moral certainty show an
approach - and the greater the greater the number of trials - to that
average which sober reasoning predicts. In the first column is the most
likely number of the theory: the next column gives Buffon's result; the
three next are results obtained by trial by correspondents of mine. In
each case the number of trials is 2,048.

I come now to the way in which such considerations have led to a mode in which mere pitch-and-toss has given a more accurate approach to the quadrature of the circle than has been reached by some of my paradoxers. What would my friend in No. 14 have said to this? The method is as follows: Suppose a planked floor of the usual kind, with thin visble seams between the planks. Let there be a thin straight rod, or wire, not so long as the breadth of the plank. This rod, being tossed at hazard, will either fall quite clear of the seams, or will lay across one seam. Now Buffon, and after him Laplace, proved the following: That in the long run the fraction of the whole number of trials in which a seam is intersected will be the fraction which twice the length of the rod is of the circumference of the circle having the breadth of a plank for its diameter. In 1855 Mr. Ambrose Smith, of Aberdeen, made 3,204 trials with a rod three-fifths of the distance between the planks: there were 1,213 clear intersections, and 11 contacts on which it was difficult to decide. Divide these contacts equally, and we have 1,218½ to 3,204 for the ratio of 6 to 5π, presuming that the greatness of the number of trials gives something near to the final average, or result in the long run: this gives π = 3.1553. If all the 11 contacts had been treated as intersections, the result would have been π = 3.1412, exceedingly near. A pupil of mine made 600 trials with a rod of the length between seams, and got π = 3.137.

This method will hardly be believed until it has been repeated so often that "there never could have been any doubt about it."

The first experiment strongly illustrates the truth of the theory, well confirmed by practice: whatever can happen will happen if we make trials enough.

H | 1,024 | 1,061 | 1,048 | 1,017 | 1,039 |

TH | 512 | 494 | 507 | 547 | 480 |

T2H | 256 | 232 | 248 | 235 | 267 |

T3H | 128 | 137 | 99 | 118 | 126 |

T4H | 64 | 56 | 71 | 72 | 67 |

T5H | 32 | 29 | 38 | 32 | 33 |

T6H | 16 | 25 | 17 | 10 | 19 |

T7H | 8 | 8 | 9 | 9 | 10 |

T8H | 4 | 6 | 5 | 3 | 3 |

T9H | 2 | 3 | 2 | 4 | |

T10H | 1 | 1 | 1 | ||

T11H | 0 | 1 | |||

T12H | 0 | 0 | |||

T13H | 1 | 1 | 0 | ||

T14H | 0 | 0 | |||

T15H | 1 | 1 | |||

&c. | 0 | 0 | |||

2,048 | 2,048 | 2,048 | 2,048 | 2,048 |

I come now to the way in which such considerations have led to a mode in which mere pitch-and-toss has given a more accurate approach to the quadrature of the circle than has been reached by some of my paradoxers. What would my friend in No. 14 have said to this? The method is as follows: Suppose a planked floor of the usual kind, with thin visble seams between the planks. Let there be a thin straight rod, or wire, not so long as the breadth of the plank. This rod, being tossed at hazard, will either fall quite clear of the seams, or will lay across one seam. Now Buffon, and after him Laplace, proved the following: That in the long run the fraction of the whole number of trials in which a seam is intersected will be the fraction which twice the length of the rod is of the circumference of the circle having the breadth of a plank for its diameter. In 1855 Mr. Ambrose Smith, of Aberdeen, made 3,204 trials with a rod three-fifths of the distance between the planks: there were 1,213 clear intersections, and 11 contacts on which it was difficult to decide. Divide these contacts equally, and we have 1,218½ to 3,204 for the ratio of 6 to 5π, presuming that the greatness of the number of trials gives something near to the final average, or result in the long run: this gives π = 3.1553. If all the 11 contacts had been treated as intersections, the result would have been π = 3.1412, exceedingly near. A pupil of mine made 600 trials with a rod of the length between seams, and got π = 3.137.

This method will hardly be believed until it has been repeated so often that "there never could have been any doubt about it."

The first experiment strongly illustrates the truth of the theory, well confirmed by practice: whatever can happen will happen if we make trials enough.

- 1873 A. Hall, "An Experimental Determination of π," references the experiments of Captain Fox, Messenger of Mathematics, 113-114.
- 1880 G. Tissandier, Récréations Scientifiques. The Game of the Needle, pp. 139-145.
- 1884 E. Czuber, Geometrische Wahrscheinlichkeit und Mittelwerte, Sections 64-66, pp. 84-91.
- 1901 M. Lazzarini, "Un'Applicazione del Calculo della Probabilità," Periodico di Matimatica 4, pp. 140-143.