260 
TRIGONOMETRY 
by g-Q-Q from -|§f or ■~- E , 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 infor 
mation about Greek trigonometry in its highest development 
in the &'phaerica of Menelaus. 
The date of Menelaus of Alexandria is roughly indi 
cated 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 
orhe 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 
{n pay par eta) 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 absurduin, 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 
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 ’ 
(tTapdSogos) 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 permixtorum. Light is thrown on this 
last title by one al-Chazim who (about a.d. 1121) wrote a 
1 Pappus, vi, pp. 600-2. 
2 Proclus on Eucl. I, pp. 345. 14-846. 11. 
8 Pappus, iv, p. 270. 25.