A history of Greek mathematics - Wilbourhall.org

THE LIBER ASSUMPTORUM 103
Lastly, we may mention the elegant theorem about the
area of the Salinon (presumably salt-cellar ' ') in Prop. 14.
ACB is a semicircle on AB as diameter, AD, EB are equal
lengths measured from A and B on AB. Semicircles are
drawn with AD, EB as diameters on the side towards G, and
a semicircle with DE as diameter is drawn on the other side of
AB. CF is the perpendicular to AB through 0, the centre
of the semicircles ACB, DFE. Then is the area bounded by
all the semicircles (the Salinon) equal to the circle on CF
as diameter.
The Arabians, through whom the Book of Lemmas has
reached us, attributed to Archimedes other works (1)
on the
Circle, (2) on the Heptagon in a Circle, (3) on Circles touching
one another, (4) on Parallel Lines, (5) on Triangles, (6) on
the properties of right-angled triangles, (7) a book of Data,
(8) De clepsydris : statements which we are not in a position
to check. But the author of a book on the finding of chords
in a circle, 1 Abu'l Raihan Muh. al-Biruni, quotes some alternative
proofs as coming from the first of these works.
of
(8) Formula for area of triangle.
More important, however, is the mention in this same work
Archimedes as the discoverer of two propositions hitherto
attributed to Heron, the first being the problem of finding
the perpendiculars of a triangle when the sides are given, and
the second the famous formula for the area of a triangle in
terms of the sides,
V{s(s — a)(s — b) (s — c)}.
1
See Bibliotheca mathematica, xi 3 , pp. 11-78.