Georges-Louis LeClerc, Comte de Buffon
b. 7 September 1707, Montbard, France
d. 16 April 1788, Paris, France
created the area of Geometric Probability
with his study of the game of Franc-carreau.
notice occurs in the Histoire de
Sciences in 1733. This was
partially reprinted in the
Encyclopedia of Diderot in 1752 under the word Carreau.
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
problem for which see both Nicolas
Bernoulli and Daniel Bernoulli.
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.
- 1812 Laplace shows how to
estimate π and generalizes to a needle cast on a rectangular
grid. Theorie Analytic
Probabilités, the end of
Chapter 5, p.
- 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
- 1850 Rudolf Wolf
provides experimental evidence. Mittheilungen
Gesellschaft in Bern, 176,
85-88 and 193, 209-212.
- 1859 A. de Morgan:
de Morgan writes in A Budget of
Vol, 1, 1915, pp. 283-284) of the Needle Problem:
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
greatness of the number of trials gives something near to the final
average, or result in the long run: this gives π =
all the 11 contacts had been treated as intersections, the result would
have been π = 3.1412, exceedingly near. A pupil of mine
600 trials with a rod of the length between seams, and
got π =
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
- 1873 A. Hall, "An
Experimental Determination of π,"
references the experiments of Captain Fox, Messenger of Mathematics,
- 1880 G. Tissandier,
Récréations Scientifiques. The
Game of the Needle, pp.
- 1884 E. Czuber, Geometrische
Wahrscheinlichkeit und Mittelwerte, Sections
64-66, pp. 84-91.
- 1901 M. Lazzarini, "Un'Applicazione del
Matimatica 4, pp. 140-143.