The salinon (meaning 'salt-cellar' in Greek) is a geometrical figure that consists of four

semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, radians, or a half-turn). It only has one line of symmetr ...

s. It was first introduced in the '' Book of Lemmas'', a work attributed to

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

Construction

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

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

s ''AB'', ''AD'', and ''EB'', and another semicircle below with diameter ''DE''. A salinon is the figure bounded by these four semicircles.

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 sequal to the area of the circle on CF as diameter." Namely, if

r_1

is the radius of large enclosing semicircle, and

r_2

is the radius of the small central semicircle, then the area of the salinon is:

A=\frac\pi\left(r_1+r_2\right)^2.

Arbelos

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

arbelos In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that conta ...

, another one of Archimedes' creations, with

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

along the ''y''-axis.

References

{{Reflist

External links

L’arbelos. Partie II
by Hamza Khelif a
www.images.math.cnrs.fr
of

CNRS The French National Centre for Scientific Research (, , CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 staff, including 11,137 tenured researchers, 13,415 eng ...