A history of Greek mathematics - Wilbourhall.org

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260 TRIGONOMETRY
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, Heron's
figure). There is little doubt
that it is to Hipparchus's work that Heron refers, though the
author is not mentioned.
While for our knowledge of Hipparchus's trigonometry we
have to rely for the most part upon what we can infer from
Ptolemy, we fortunately possess an original source of information
about Greek trigonometry in its highest development
in the Sphaerica of Menelaus.
The date of Menelaus of Alexandria is roughly indicated
by the fact that Ptolemy quotes an observation of
his made in the first year of Trajan's reign (a.d. 98). He
was therefore- a contemporary of Plutarch, who in fact
represents him as being present at the dialogue De facie in
orbe lunae, where (chap. 17) Lucius apologizes to Menelaus 'the
mathematician ' for questioning the fundamental proposition
in optics that the angles of incidence and reflection are equal.
He wrote a variety of treatises other than the Sphaerica.
We have seen that Theon mentions his work on Chords in a
Circle in six Books. Pappus says that he wrote a treatise
(Trpayixareia) on the setting (or perhaps only rising) of
different arcs of the zodiac. 1 Proclus quotes an alternative
proof by him of Eucl. I. 25, which is direct instead of by
reductio ad absurdum, 2, and he would seem to have avoided
the latter kind of proof throughout.
Again, Pappus, speaking
of the many complicated curves discovered by Demetrius of
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Alexandria (in his " Linear considerations ") and by Philon
of Tyana as the result of interweaving plectoids and other
surfaces of all kinds ', says that one curve in particular was
investigated by Menelaus and called by him ' paradoxical
(irapd8o£os) 3 ;
the nature of this curve can only be conjectured
(see below).
But Arabian tradition refers to other works by Menelaus,
(l) Elements of Geometry, edited by Thabit b. Qurra, in three
Books, (2) a Book on triangles, and (3) a work the title of
which is translated by Wenrich de cognitione quantitatis
discretae corporum permixtomm. Light is thrown on this
last title by one al-Chazini who (about A.D. 1121) wrote a
1
Pappus, vi, pp. 600-2.
2
3
Proclus on Eucl. I, pp. 345. 14-346. 11.
Pappus, iv, p. 270. 25.