  Ancient astronomers faced the problem of dealing with arcs and angles in
their observations and predictions without the help of modern trigonometry. The
usual method to deal with such problems was the Menelaus Theorem, explicitly
discussed and extensively used by Ptolemy. It is generally assumed that
Hipparchus, who in his Exegesis explicitly states that he developed a method to
calculate Simultaneous Risings and Settings, used either the Menelaus' Theorem,
or stereographic projections, even if both methods are only attested for later
periods and their availability to Hipparchus is debated. In this paper we argue
that simultaneous risings and settings could be calculated using less advanced
mathematical tools unquestionably available to Hipparchus: the theorem of
Pythagoras, the theory of proportions, and the table of chords. Without
claiming that this was indeed the method that Hipparchus used, we argue that
there is no need to postulate the availability of more sophisticated tools to
carry out the calculations which Hipparchus mentions in the Exegesis.
