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

THE OPTICS OF PTOLEMY 295
but the translation into Latin (now included in the Teubner
edition of Heron, ii, 1900, pp. 316-64), which was made by
William of Moerbeke in 1269, was evidently made from the
Greek and not from the Arabic, as is shown by Graecisms in
the translation.
296 TRIGONOMETRY
I. To prove I. 28, Ptolemy takes two straight lines AB, CD,
and a transversal EFGH. We have to prove that, if the sum
A mechanical work, Tlepl poircou.
There are allusions in Simplicius 1 and elsewhere to a book
by Ptolemy of mechanical content, nepl poncou, on balancings
or turnings of the scale, in which Ptolemy maintained as
'
against Aristotle that air or water (e.g.) in their own place '
'
have no weight, and, when they are in their own place ', either
remain at rest or rotate simply, the tendency to go up or to
fall down being due to the desire of things which are not in
their own places to move to them. Ptolemy went so far as to
maintain that a bottle full of air was not only not heavier
than the same bottle empty (as Aristotle held), but actually
lighter when inflated than when empty. The same work is
apparently meant by the book on the elements ' ' mentioned
by Simplicius. 2 Suidas attributes to Ptolemy three Books of
Mechanica.
Simplicius 3 also mentions a single book, wepl o^aorao-ea)?,
On t&mension ', i. e. dimensions, in which Ptolemy tried to
1
show that the possible number of dimensions is limited to
three.
Attempt to prove the Parallel-Postulate.
Nor should we omit to notice
Ptolemy's attempt to prove
the Parallel-Postulate. Ptolemy devoted a tract to this
subject, and Proclus 4 has given us the essentials of the argument
used. Ptolemy gives, first, a proof of Eucl. I. 28, and
then an attempted proof of I. 29, from which he deduces
Postulate 5.
1
Simplicius on Arist. De caelo, p. 710. 14, Heib. (Ptoleniv, ed. Heib.,
vol. ii, p. 263).
2
3
lb., p. 20. 10 sq.
lb., p. 9. 21 sq., (Ptolemy, ed. Heib., vol. ii, p. 265).
4
Proclus on Eucl. I, pp. 362. 14 sq., 365. 7-367. 27 (Ptolemy, ed. Heib.,
vol. ii, pp. 266-70).
of the angles BFG, FGD is equal to two right angles, the
straight lines AB, CD are parallel, i.e. non-secant.
Since AFG is the supplement of BFG, and FGC of FGD, it
follows that the sum of the angles AFG, FGC is
also equal to
two right angles.
,
Now suppose, if possible, that FB, GD, making the sum of
the angles BFG FGD equal to two right angles, meet at K
y ;
then similarly FA, GC making the sum of the angles AFG,
FGC equal to two right angles must also meet, say at L.
[Ptolemy would have done better to point out that not
only are the two sums equal but the angles .themselves are
equal in pairs, i.e. AFG to FGD and FGC to BFG, and we can
therefore take the triangle KFG and apply it to FG on the other
side so that the sides FK, GK may lie along GC, FA respectively,
in which case GC, FA will meet at the point where
K falls.]
Consequently the straight lines LABK, LCDK enclose a
space :
which is impossible.
It follows that AB, CD cannot meet in either direction ;
they are therefore parallel.
II. To prove I. 29, Ptolemy takes two parallel lines AB,
CD and the transversal FG, and argues thus. It is required
to prove that
Z AFG + Z CGF = two right angles.
For, if the sum is not equal to two right angles, it must be
either (1) greater or (2) less.
(1) If it is greater, the sum of the angles on the other side,
BFG, FGD, which are the supplements of the first pair of
angles, must be less than two right angles.
But AF, CG are no more parallel than FB, GD, so that, if
FG makes one pair of angles AFG, FGC together greater than