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. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...
s. It was first introduced in the ''
Book of Lemmas
The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositio ...
'', a work attributed to
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
.
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 center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
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
is the radius of large enclosing semicircle, and
is the radius of the small central semicircle, then the area of the salinon is:
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 (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
along the
''y''-axis.
[
]
See also
* Lune of Hippocrates
In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex p ...
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 (french: link=no, Centre national de la recherche scientifique, CNRS) is the French state research organisation and is the largest fundamental science agency in Europe.
In 2016, it employed 31,63 ...
Piecewise-circular curves
Archimedes