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2013a.00597 Pippenger, Nicholas Two extensions of results of Archimedes. Am. Math. Mon. 118, No. 1, 66-71 (2011). 2011 Mathematical Association of America (MAA), Washington, DC EN G40 method of Archimedes Cavalieri's principle intersection of cylinders hoof
• doi:10.4169/amer.math.monthly.118.01.066
• Let us consider the following bodies: { indent=4mm \item{--} $X$ is a filled right circular cylinder of radius 1 about the $x$-axis. \item{--} $Y$ is a similar cylinder about the $y$-axis. \item{--} $Z$ is a similar cylinder about the $z$-axis. \item{--} $R=X\cap Y$ and $S=X\cap Y\cap Z$. \item{--} $U$ is the filled right isosceles triangular cylinder with side 1 given by the inequalities $z\geq 0$, $z\leq y$ and $y\leq 1$. \item{--} $H=U\cap Z$ (the so-called hoof''). \item{--} $V$ is the filled right isosceles triangular cylinder with side 1 given by the inequalities $z\geq 0$, $z\leq x$ and $x\leq 1$. \item{--} $J=U\cap V\cap Z$. } In this paper the author uses Cavalieri's principle to compute rigorously the following volumes: { indent=6mm \item {(1)} vol$(R)=16/3$. \item {(2)} vol$(S)=16-8\sqrt{2}$. \item {(3)} vol$(H)=2/3$. \item {(4)} vol$(J)=(2-\sqrt{2})/3$. } Bodies $R$ and $H$ both appear in Archimedes' method where, obviously, different techniques are used to compute their volume. The volume of $S$ and $H$ are presented here as generalizations of the previous ones using Cavalieri's principle. Antonio M. Oller (Zaragoza)