Salinon

The salinon (meaning 'salt-cellar' in Greek) is a

Let A, D, E, and B be four points on a line in the plane, in that order, with AD = EB. Let O be the bisector of segment AB (and of DE). Draw semicircles above line AB with diameters AB, AD, and EB, and another semicircle below with diameter DE. A salinon is the figure bounded by these four semicircles.^[2]

Properties

Area

Archimedes introduced the salinon in his Book of Lemmas by applying Book II, Proposition 10 of Euclid's Elements. Archimedes noted that "the area of the figure bounded by the circumferences of all the semicircles [is] equal to the area of the circle on CF as diameter."^[3]

Namely, if ${\displaystyle r_{1}}$ is the radius of large enclosing semicircle, and ${\displaystyle r_{2}}$ is the radius of the small central semicircle, then the area of the salinon is:^[4]

$${\displaystyle A={\frac {1}{4}}\pi \left(r_{1}+r_{2}\right)^{2}.}$$

Arbelos

Should points D and E converge with O, it would form an

• Lune of Hippocrates

References