In Moralia, the Greek biographer and philosopher Plutarch states "Chrysippus says that the number of compound propositions that can be made from only ten simple
propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing
that on the affirmative side there are compound statements, and on the negative side .)" These numbers are known as the Plutarch numbers.
can be interpreted as the number
of bracketings
on ten letters (Stanley 1997, Habsieger et al. 1998). Similarly, Plutarch's
second number is given by (Habsieger et al. 1998).
Biermann, K.-R. and Mau, J. "Überprüfung einer frühen Anwendung der Kombinatorik in der Logik." J. Symbolic Logic23,
129-132, 1958.Biggs, N. L. "The Roots of Combinatorics."
Historia Mathematica6, 109-136, 1979.Habsieger, L.; Kazarian,
M.; and Lando, S. "On the Second Number of Plutarch." Amer. Math. Monthly105,
446, 1998.Heath, T. L. A
History of Greek Mathematics, Vol. 2: From Aristarchus to Diophantus.
New York: Dover, p. 256, 1981.Kneale, W. and Kneale, M. The
Development of Logic. Oxford, England: Oxford University Press, p. 162,
1971.Neugebauer, O. A
History of Ancient Mathematical Astronomy. New York: Springer-Verlag, p. 338,
1975.Plutarch. §VIII.9 in Moralia,
Vol. 9. Cambridge, MA: Harvard University Press, p. 732, 1961.Stanley,
R. P. Enumerative
Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press,
p. 63, 1996.Stanley, R. P. "Hipparchus, Plutarch, Schröder,
and Hough." Amer. Math. Monthly104, 344-350, 1997.
compound statements, and on the negative side 
.)" These numbers are known as the Plutarch numbers.
can be interpreted as the number
of bracketings
on ten letters (Stanley 1997, Habsieger et al. 1998). Similarly, Plutarch's
second number is given by 
(Habsieger et al. 1998).
