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

PTOLEMY ON THE PARALLEL-POSTULATE 297
two right angles, it miist also make the other pair BFG, FGD
together greater than two right angles.
But the latter pair of angles were proved less than two
right angles : which is impossible.
Therefore the sum of the angles AFG, FGC cannot be
greater than two right angles.
(2) Similarly we can show that the sum of the two angles
AFG, FGC cannot be less than two right angles.
Z AFG + Z CGF = two right angles.
Therefore
[The fallacy here lies in the inference which I have marked
by italics. When Ptolemy says that AF, CG are no more
parallel than FB, GD, he is in effect assuming that through
any one point only one parallel can be drawn to a given straight
line, which is an equivalent for the very Postulate he is
endeavouring to prove. The alternative Postulate is known
as ' Playfair's axiom ', but it is of ancient origin, since it is
distinctly enunciated in Proclus's note on Eucl. I. 31.]
III. Post. 5 is now deduced, thus.
Suppose that the straight lines making with a transversal
angles the sum of which is less than two right angles do not
meet on the side on which those angles are.
Then, a fortiori, they will not meet on the other side on
which are the angles the sum of which is greater than two
right angles. [This is enforced by a supplementary proposition
showing that, if the lines met on that side, Eucl. I. 16
would be contradicted.]
Hence the straight lines cannot meet in either direction :
they are therefore parallel.
But in that case the angles made with the transversal are
equal to two right angles : which contradicts the assumption.
Therefore the straight lines will meet.
MENSURATION:
XVIII
HERON OF ALEXANDRIA
Controversies as to Heron's date.
The vexed question of Heron's date has perhaps called
forth as much discussion as any doubtful point in the history
of mathematics. In the early stages of the controversy much
was made of the supposed relation of Heron to Ctesibius.
The Belopoeica of Heron has, in the best manuscript, the
heading "Hponvos Krr\cn^iov BeXoirouKa, and from this, coupled
with an expression used by an anonymous Byzantine writer
of the tenth century, 6 'AcrKprjvb? Kr-qcrifiios 6 Tov'AXtgavSpim
"Hpavos KadrjyrjTrjs, Ctesibius of Ascra, the teacher of Heron
'
of Alexandria ', it was inferred that Heron was a pupil of
Ctesibius. The question then was, when did Ctesibius live ?
Martin took him to be a certain barber of that name who
lived in the time of Ptolemy Euergetes II, that is, Ptolemy VII,
called Physcon (died 117 B.C.), and who is said to have made
an improved water-organ l ;
Martin therefore placed Heron at
the beginning of the first century (say 126-50) B.C. But
Philon of Byzantium, who repeatedly mentions Ctesibius by
name, says that the first mechanicians (rex^Tai) had the
great advantage of being under kings who loved fame and
supported the arts. 2 This description applies much better
to Ptolemy II Philadelphus (285-247) and Ptolemy III Euergetes
I (247-222). It is more probable, therefore, that Ctesibius
was the mechanician Ctesibius who is mentioned by Athenaeus
as having made an elegant drinking-horn in the time of
Ptolemy Philadelphus 3 ;
a pupil then of Ctesibius would
probably belong to the end of the third and the beginning of
the second century B.C. But in truth we cannot safely conclude
that Heron was an immediate pupil of Ctesibius. The
Byzantine writer probably only inferred this from the title
1
Athenaeus, Deipno-Soph. iv. c. 75, p. 174 b-e: cf. Vitruvius, x. 9, 13.
2
Philon, Mechan. Synt., p. 50. 38, ed. Schone.
3
Athenaeus, xi. c. 97, p. 497 b-e.