
05845503
j
2013a.00597
Pippenger, Nicholas
Two extensions of results of Archimedes.
Am. Math. Mon. 118, No. 1, 6671 (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 socalled ``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$(S)=168\sqrt{2}$. \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)