A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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.
(8) Formula for area of triangle.
More important, however, is the mention in this same work
of 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,
1
V{s(s — a)(s — b) (s — c)}.
See Bibliotheca mathematica, xi 3 , pp. 11-78.
104 ERATOSTHENES
Long as the present chapter is, it is nevertheless the most
appropriate place for Eratosthenes of Cyrene. It was to him
that Archimedes dedicated The Method, and the Cattle-Problem
purports, by its heading, to have been sent through him to
the mathematicians of Alexandria. It is evident from the
preface to The Method that Archimedes thought highly of his
mathematical ability.
He was, indeed, recognized by his contemporaries
as a man of great distinction in all branches of
knowledge, though in each subject he just fell short of the
highest place. On the latter ground he was called Beta, and
another nickname applied to him, Pentathlos, has the same
implication, representing as it does an all-round athlete who
was not the first runner or wrestler but took the second prize
in these contests as well as in others. He was very little
younger than Archimedes ; the date of his birth was probably
284 b.c. or thereabouts. He was a pupil of the philosopher
Ariston of Chios, the grammarian Lysanias of Cyrene, and
the poet Callimachus ;
he is said also to have been a pupil of
Zeno the Stoic, and he may have come under the influence of
Arcesilaus at Athens, where he spent a considerable time.
Invited, when about 40 years of age, by Ptolemy Euergetes
to be tutor to his son (Philopator), he became librarian at
Alexandria ; his obligation to Ptolemy he recognized by the
column which he erected with a graceful epigram inscribed on
it.
This is the epigram, with which we are already acquainted
(vol. i, p. 260), relating to the solutions, discovered up to date,
of the problem of the duplication of the cube, and commending
his own method by means of an appliance called fxeaoXafiov,
itself represented in bronze on the column.
Eratosthenes wrote a book with the title IlXaTcoviKos, and,
whether it was a sort of commentary on the Timaeus of
Plato, or a dialogue in which the principal part was played by
Plato, it evidently dealt with the fundamental notions of
mathematics in connexion with Plato's philosophy. It was
naturally one of the important sources of Theon of Smyrna's
work on the mathematical matters which it was necessary for
the student of Plato to know
;
and Theon cites the work
twice by name. It seems to have begun with the famous
problem of Delos, telling the story quoted by Theon how the
god required, as a means of stopping a plague, that the altar