WDS'09 Proceedings of Contributed Papers, Part I, 198–203, 2009.
ISBN 978-80-7378-101-9 © MATFYZPRESS
Solutions of Buffon’s problems in the 19th century
A. Kalousová
Czech Technical University in Prague, Faculty of Electrical Engineering, Prague, Czech
Republic.
Abstract. The Buffon’s needle problem is a well known problem of geometric
probability. But Buffon formulated and solved also other problems in this topic. We
recall them and their solutions by Buffon, Laplace, Todhunter, Lamé and Barbier.
Introduction
Although Georges-Louis Leclerc de Buffon was not the first one who used geometry for
computation of probability (see [Newton, 1967] and [Halley, 1693]), his problems presented
in [Buffon, 1777] became a basis of a new branch of the probability theory, the geometric
probability. In the time of Buffon, i. e. in the 18th century, the problems did not seem to
be interesting or useful but in the 19th century, many mathematicians began to study them
and proposed some generalizations. There were Pierre-Simon de Laplace (1749–1827), Gabriel
Lamé (1795–1870) in France and Isaac Todhunter (1820–1888) in England. Moreover, the
results of Joseph-Émile Barbier (1839–1889) generalize Buffon’s approach and can be considered
as a foundation of geometric statistics estimating the object properties from an incomplete
information.
Georges-Louis Leclerc, comte de Buffon, and his problems
Life and work
Georges-Louis Leclerc was born in Montbard, 7th September 1707 as the eldest son of
Benjamin-François Leclerc and Anne-Christine Marlin. Georges Blaisot (mother’s uncle and
Georges’ godfather) died in 1714, his wife three years later and the Leclerc’s family inherited a
large sum of money. Benjamin bought the manor terre de Buffon and a function in Dijon. The
family moved to Dijon and Georges entered the Jesuit Collège des Godrans. Although he was
attracted to mathematics, he met his father’s wishes and began to study law in 1723. After he
obtained the licence de droit he gave up the career of a lawyer and moved to Angers in order
to study medicine. He was also interested in botany and mathematics. Unfortunately, he had
to leave Angers due to his participation in a duel.
Georges met a young English nobleman, the Duke of Kingston, and travelled with him and
his tutor in southern France and Italy. After his mother’s death, Georges left his friends and
rejoined France. He resided in Paris and decided to be a scientist. After he won a quarrel at
law with his father concerning a heritage of his mother, he rebought terre de Buffon (sold by his
father) and began to call himself de Buffon. In Paris, Buffon made the acquaintance of notable
intellectuals (Voltaire) and also of some politicians (comte de Maurepas).
In April 1733, Buffon’s memoir Solutions de problèmes sur le jeu de franc-carreau was
presented in the French Royal Academy of Sciences (Académie Royale des Sciences) and the
reports were commendatory. In the same year his memoir Fine mécanique was presented and
in January 1734 Buffon was elected to the Academy, section of mechanics. He published a
generalization of the memoir about the game franc-carreau in 1736 and the memoir Sur les
mesures in 1738 (they are not extant). In 1739 he translated Newton’s Method of fluxions and
infinite series and wrote a preface, which covers the history of the infinitesimal calculus. But
he was more and more attracted to botany in these years. In March 1739, Buffon transferred to
the section of botany and in July 1739, when the keeper of the King’s garden (Jardin du Roi)
Charles de Cisternay du Fay died from chicken-pox, Buffon was appointed his successor. In the
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KALOUSOVA: BUFFON’S PROBLEMS
subsequent years, he converted the Cabinet d’histoire naturelle appurtenant to the garden to a
research centre and a modern museum; the garden was enlarged and enriched with many trees
and plants from around the world.
Therefore, Buffon is best known as the naturalist and author of Histoire naturelle, générale
et particulière (1749–1788 in 36 volumes; 8 additional volumes were published after his death
by Lacépède). He opened many controversial themes, for example about age and origin of the
Earth, reproduction, classification of animals and plants (he disagreed with Carl von Linné)
etc. He also constructed an iron mill in the village Buffon, which served him as a great chemical
laboratory, made experiments with Archimedes’ fire and researched trees in the forest around
his domicile in Montbard. The king ennobled him in 1772. Georges-Louis Leclerc, comte de
Buffon died in Paris, 16th April 1888.
Franc-carreau
In [Fontenelle, 1735], we can find a summary of Buffon’s memoir Solutions de problèmes
sur le jeu de franc-carreau. Buffon presents the game franc-carreau, reportedly popular at the
French court: In a room paved with squared tiles, a coin is tossed at random in the air and
one of the players bets that the coin hits only one tile (position franc-carreau) while the other
bets that it hits more tiles. It is obvious that the coin hits only one tile if and only if its centre
falls into the square inscribed in the tile and having the side of the length c − 2r where c is the
length of the tile side and r is the coin radius. Thus, the probability that the coin hits only one
tile is
(c − 2r)2
.
c2
Buffon concludes that the problem is more difficult if the coin is not round but he does not
solve it.
In [Buffon, 1777], the coin problem is generalized. The tiles may be of arbitrary shape
(Buffon examines the cases of equilateral triangle, rhombus with the angle π/3 and regular
hexagon) and we search the odds for the first player who bets that the coin misses the figure
sides, for the second player betting that only one side is hit, for the third one who bets two such
hits and for the fourth one who bets three, four or six hits.1 Buffon also looks for such a ratio
of the figure side to the coin diameter d = 2r that the game would be equal (i. e., the odds are
1 : 1).
At first, Buffon examines the odds for the first and for the second player. He finds out q
that
in the case of the square, the ratio of the figure side to the coin diameter should be 1 : (1 − 12 )
(thus the side should be approximately three
and half times longer than the coin diameter), in
√
the case of the equilateral triangle 1 :
1
2
3
p 1 (thus the side should be approximately six times
3+3
2
1
√
3
2 √
(thus the side should be
longer than the coin diameter), in the case of the rhombus 1 : 2+
2
approximately
four times longer than the coin diameter) and in the case of the regular hexagon
√
1:
1
2
1+
3
p 1 (thus the side should be approximately two times longer than the coin diameter).
2
Further, Buffon examines the odds for the third player. More precisely, he asks what should
be the ratio of the figure side to the coin diameter in order that the probability of coin hitting
at least two joints were 1/2. He obtains that the ratio of the figure side to the coin diameter
√
should be 1 :
q
1
2
in the case of the square, 1 :
1
2
1
in the case of the equilateral triangle, 1 :
1
2√
3
2
Dans 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 à franc-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.
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KALOUSOVA: BUFFON’S PROBLEMS
√
in the case of the rhombus and 1 : 12 3 in the case of the regular hexagon.
Finally, Buffon examines the odds for the fourth player and for the ratio of the figure
side to the coin diameter such that the probability of the coin hitting four (in the case of the
square and the rhombus), six (in the case of the equilateral triangle) or three (in the
q case of the
22
he obtains the following results: 1 :
7q
√
1 : 7223 for the equilateral triangle and
regular hexagon) sides is 1/2. Setting π =
incorrectly writes 1 :
and 1 :
q
√
21 3
44
q
11
7 )
for the square,
7
11
(Buffon
the rhombus
for the regular hexagon.
Needle problems
In [Fontenelle, 1735], the problem known as Buffon’s needle problem is published for the
first time. In a room the floor of which is formed by parallel planks of equal breadth, a rod of
the known length (and a negligible breadth) is tossed in the air. When does it fall onto one
plank only?2 Buffon solves it in the following way. The centre of the rod falls onto the floor at
a certain distance from the plank side. Let the rod revolve round its centre from the position
in which the rod is parallel with the plank sides to the position in which it is perpendicular to
them. The end of the rod describes a quarter of a circle; one part of the circle lies within the
plank, the other part lies out of it and the probability of the rod hitting only one plank depends
on the length of the arc included in the same plank as the circle centre.
In [Buffon, 1777], the needle problem in the well known form is presented. 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 miss the lines and the other bets that the rod will hit some
of them. The odds for these two players are required.3 Buffon solves the problem similarly to
[Fontenelle, 1735]. He makes two new parallel lines at the distance equal to the half of the rod
length (supposed to be less than or equal to the distance between the original parallels). It is
clear that if the centre of the rod falls in the area between these new parallels, the rod does not
intersect any parallel. Otherwise the rod can cross a parallel depending on the angle between
the rod and the parallels. Buffon expresses the lengths of the arc parts lying within or out of the
area between the original parallels and by means of it he computes the odds of the players. He
examines the ratio of the rod length to the distance between parallels giving the odds 1 : 1 and
concludes that the rod length should be approximately 3/4 of the distance between parallels
(the exact number is π/4).
Buffon finally generalizes the needle problem. The rod is thrown down on the floor paved
with squared tiles and he looks for the ratio of the rod length to the square side such that the
bets on hitting or missing are equal.4 He uses the same calculation as in the previous case and
computes that the rod length should be approximately a half of the square side length. But
this result is incorrect.
Pierre-Simon, marquis de Laplace
Pierre-Simon de Laplace was born in Beaumont-en-Auge, 23rd March 1749. He attended a
Benedictine priory school there and at the age of sixteen he entered the University of Caen to
study theology. However, he was more attracted to mathematics due to his enthusiastic teachers.
2
Sur un plancher qui n’est formé que de planches égales & paralleles, on jette une Baguette d’une certaine
longueur, & qu’on suppose sans largeur. Quand tombera-t-elle franchement sur une seule planche?
3
Je suppose que dans une chambre, dont le parquet est simplement divisé par 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 à coudre ou une épingle sans tête.
4
La solution de ce premier cas nous conduit aisément à celle d’un autre qui d’abord aurois paru plus difficile,
qui est déterminer le sort de ces deux joueurs dans une chambre pavée de carreaux carrés.
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KALOUSOVA: BUFFON’S PROBLEMS
After two years, he interrupted his studies and went to Paris with a letter of introduction
to d’Alembert, who immediately recognized his mathematical talent, recommended him as a
teacher at the École militaire and further directed his mathematical studies.
Laplace’s first memoir was presented in the French Royal Academy of Sciences (Académie
royale des sciences) in 1770 but he was not elected until 1773. In the following years Laplace
produced much of his astronomical work. He was also interested in mathematics, physics and
probability theory. In 1793 Laplace left Paris with his family to secure them from the Jacobin
terror. They returned after July 1794. When Napoleon’s power increased, Laplace was nominated to the post of the Minister of the Interior. He was not successful at this post and after
less than six weeks was dismissed. He entered Senat, was its chancellor, received the Legion
of Honours and became the Count of Empire. After 1814 Laplace offered his services to the
Bourbons and in 1817 he was awarded the title of marquis.
Pierre-Simon de Laplace died in Paris, 5th March 1827. He is considered one of the greatest
scientists of all times, frequently called French Newton. He was a member of all important world
academies.
His most important work about probability is [Laplace, 1812]. At the end of Chapter V of
this book, Laplace introduces a new application of the probability calculation, namely estimating
the length of curves and area of surfaces. Then he mentions needle problems without making
any reference to their author. Firstly, Laplace solves the needle problem on the plain area ruled
with equidistant parallel straight lines.5 He solved it simpler than Buffon. Let us choose two
adjacent straight lines (denote the distance between the lines by a) and erect a perpendicular in
an arbitrary point of one of them. Suppose the centre of the rod of the length 2r ≤ a falls onto
this perpendicular at the distance y of the chosen point and the rod revolves round its centre.
Denote by ϕ the angle between the rod and the perpendicular at the instant when the rod end
hits the line. Laplace deduces that the
favourable events (i. e. the rod hits the line containing
Ra
the chosen point) have the measure 0 4ϕ dy. We have to take it two times because the rod can
hit also the other line, and the considered events have the measure a · 2π. Thus the probability
of intersection is
Ra
R π2
R π2
4r
4y
dϕ
4r cos ϕ dϕ
4ϕ
dy
= 0
= 0
=
.
2 0
2aπ
aπ
aπ
aπ
Then Laplace introduces the needle problem on the plane divided by two orthogonal systems
of equidistant parallel straight lines having distances a and b, respectively, and thus forming a
rectangular grid.6 It is a generalization of the original Buffon’s problem with a = b and Laplace
solves it correctly. He chooses one of the rectangles and makes the parallels inside it at the
distance r from each rectangle side. The lines delimit the internal rectangle of sides a − 2r and
b − 2r, two lesser rectangles with sides a − 2r and r, two others with sides r and b − 2r and four
squares with sides r. If the rod centre falls into the internal rectangle, the rod cannot intersect
any side of the original rectangle. If it falls into some of the peripheral rectangles, the measure
of the favourable events can be computed as in the previous case (it is 2 · 4r(a − 2r) = 8r(a − 2r)
or 2 · 4r(b − 2r) = 8r(b − 2r). The case of the four squares is slightly more complicated. Let
us describe a quarter-circle having the centre in the vertex of the original rectangle and radius
r. If the rod centre falls into this quarter circle, the rod must intersect some line. The measure
2
= 12 π 2 r 2 . Otherwise the intersection depends on the angle
of the favorable events is 2π·πr
4
between the rod and the system of parallels. In this case, the measure of the favourable events
is 12 r 2 (12 − π 2 ). Consequently, we obtain 4 · ( 12 r 2 (12 − π 2 ) + 12 π 2 r 2 ) = 24r 2 for the four squares
5
Imaginons un plan divisé par des lignes parallèles, equidistantes de la quantité a; concevons de plus un
cylindre très-étroit dont 2r soit la longueur supposée égale ou moindre qua a. On demande la probabilité qu’en
le projetant il rencontrera une des division du plan.
6
Concevons maintenant le plan précédent divisé encore par des lignes perpendiculaires aux précédentes, et
équidistantes d’une quantité b égale ou plus grande que la longueur 2r du cylindre.
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KALOUSOVA: BUFFON’S PROBLEMS
and 8ar − 16r 2 + 8br − 16r 2 + 24r 2 = 8(a + b)r − 8r 2 for the whole rectangle. The measure of
the considered events is then ab · 2π and the intersection probability is
4(a + b)r − 4r 2
8(a + b)r − 8r 2
=
.
ab · 2π
ab · π
The appearance of these problems in a mathematical book attracted perhaps other mathematicians to study them, too. Some generalizations of needle problem have been made in
England. James Joseph Sylvester (1814–1897) wrote in [Sylvester, 1890-1891] that Leslie Ellis
from the University of Cambridge made the important step and examined the case in which the
ellipse (instead of the rod) is thrown down onto the plane with equidistant parallel straight lines
but we have not any corresponding paper. Isaac Todhunter (1820–1884) in [Todhunter, 1857]
proposes and solves Buffon’s needle problem and some generalizations (an ellipse with the major
axis less then the distance between parallels, a square with the diagonal less then the distance
between parallels and a rod with the length equal to r-times distance between parallels). In the
third edition (1868) the rod or the square is thrown down onto a rectangular grid. Todhunter
also notices that a rod is a special case of an ellipse with the minor axis equal to zero.
Todhunter was also interested in the history of science. When studying the history of
probability he found out that the original author of the needle problems was not Laplace but
Buffon. Consequently, Buffon is introduced in the further editions. In [Todhunter, 1865] Todhunter writes also about Buffon’s problems and their solutions by Buffon and Laplace. More
about Todhunter’s contribution can be found in [Kalousová, 2008].
In France Gabriel Lamé (1795–1870) lectured at Sorbonne about Buffon’s needle problem
and its generalization to circles, ellipses and polygon. His lectures were also listened by JosephÉmile Barbier, a young student of the École normale supérieure, who provided many other
generalizations.
Joseph-Émile Barbier
Joseph-Émile Barbier was born in St Hilaire-Cottes, 18th March 1839 in the family of a
soldier. He atanded the special mathematics class of the Lycée Henri IV in Paris and then the
École normale supérieure. Barbier finished his study in 1860 and in the same year he published
[Barbier, 1860], where many important generalizations of the Buffon’s needle problem appeared.
Barbier obtained the first post as the professor at a lycée in Nice but he left it soon and took
an offered post at the Observatoire de Paris. In 1865 Barbier left Paris and broke all contacts
with his colleagues because of his mental problems. His former lecturer Joseph Bertrand (1822–
1900) found him in an asylum in Charenton-St-Maurice in 1880 and encouraged him to return
to Paris and continue mathematical research. In 1880s Barbier wrote 14 mathematical articles
published (with one exception) in Comptes rendus de l’Académie des sciences. Joseph-Émile
Barbier died in St Genest, 28th January 1889.
In [Barbier, 1860] Barbier recalls the needle problem and its generalizations about which
he has learned from Lamé, at first. He indicates Laplace as the author of the needle problem.
Analogously as Todhunter, Barbier notices that the needle can be considered as the limit case
of an ellipse with the minor axis equal to zero. Then he proves a general theorem that for a
convex disk of arbitrary form that cannot in any position on the plane intersect more than one
of dividing lines, the probability of intersection is L/(πa) where L is the perimeter of the disk
and a is the distance between parallels. The proof is based on the idea that every convex figure
can be approximated with an arbitrary precision by a polygon having the sides of the same
length. For that polygon the probability of intersection is proportional to the number of sides
and, consequently, to its perimeter. So, the probability of intersection is equal for all figures
having a given perimeter. Barbier computes the probability for the simplest case, namely for a
circle. Evidently, a circle having diameter less than or equal to the distance between parallels
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KALOUSOVA: BUFFON’S PROBLEMS
intersects one of parallels if the distance between its centre and the parallel is less than or equal
to its radius. Hence, the probability that a circle intersects a system of parallels is
2πr
L
2r
=
=
,
a
πa
πa
where r is the radius of the circle, L is its perimeter and a is the distance between parallels.
Barbier makes two important observations in this part. The first one is that any curve
(open or closed) can be approximated by a polygon (open or closed); the second one is that the
probability of intersection depends only on the length of the curve, not on its shape. Consequently, it is sufficient to examine always the simplest case. These observations permit Barbier
other generalizations. He deduces certain relations from which the basic stereological formulas follow. Barbier’s results went unnoticed by his contemporaries and the formulas, simply
deducible from Barbier’s relations, were rediscovered as late as in the middle of the 20th century. More about Barbier’s article is in [Kalousová, 2009a], [Kalousová, 2009b], [Saxl, Ilucová,
Kalousová, 2008].
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