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

CONTROVERSIES AS TO HERON'S DATE 303
machines used by the two for the same purpose frequently
differ in details ; e. g. in Vitru vius's hodometer a pebble drops
into a box at the end of each Roman mile. 1
the distance completed is marked by a pointer. 2
while in Heron's
It is indeed
pointed out that the water-organ of Heron is in many respects
more primitive than that of Vitruvius ; but, as the instruments
are altogether different, this can scarcely be said to
prove anything.
On the other hand, there are points of contact between
certain propositions of Heron and of the Roman agrimensores.
Columella, about a.d. 62, gave certain measurements of
plane figures which agree with the formulae used by Heron,
notably those for the equilateral triangle, the regular hexagon
(in this case not only the formula but the actual figures agree
with Heron's) and the segment of a circle which is less than
a semicircle, the formula in the last case being
where s is the chord and h the height of the segment.
Here
there might seem to be dependence, one way or the other
but the possibility is not excluded that the two writers may
merely have drawn from a common source ; for Heron, in
giving the formula for the area of the segment of a circle,
states that it was the formula used by ' the more accurate
investigators' (ol ccKpL^icrrepou e ^77777 /core?). 3
We have, lastly, to consider the relation between Ptolemy
and Heron. If Heron lived about 100 B.C., he was 200 years
earlier than Ptolemy (a.d. 100—178). The argument used to
prove that Ptolemy came some time after Heron is based on
a passage of Proclus where Ptolemy is said to have remarked
on the untrustworthiness of the method in vogue among the
'
more ancient ' writers of measuring the apparent diameter of
the sun by means of water-clocks. 4 Hipparchus, says Proclus,
used his dioptra for the purpose, and Ptolemy followed
him. Proclus proceeds :
'
Let us then set out here not only the observations of
the ancients but also the construction of the dioptra of
304 HERON OF ALEXANDRIA
Hipparchus. And first we will show how we can measure an
interval of time by means of the regular efflux of water,
a procedure which was explained by Heron the mechanician
in his treatise on water-clocks.'
Theon of Alexandria has a passage to a similar effect. 1
first says that the most ancient mathematicians contrived
a vessel which would let water flow out uniformly through a
small aperture at the bottom, and then adds at the end, almost
in the same words as Proclus uses, that Heron showed how
this is managed in the first book of his work on waterclocks.
Theon's account is from Pappus's Commentary on
the Syntaxis, and this is also Proclus's source, as is shown by
the fact that Proclus gives a drawing of the water-clock
which appears to have been lost in Theon's transcription from
Pappus, but which Pappus must have reproduced from the
work of Heron. Tittel infers that Heron must have ranked
as one of the more ancient ' ' writers as compared with
Ptolemy. But this again does not seem to be a necessary
inference. No doubt Heron's work was a convenient place to
refer to for a description of a water-clock, but it does not
necessarily follow that Ptolemy was referring to Heron's
clock rather than some earlier form of the same instrument.
An entirely different conclusion from that of Tittel is
reached in the article ' Ptolemaios and Heron ' already alluded
to. 2 The arguments are shortly these. (1) Ptolemy says in
his Geography (c. 3) that his predecessors had only been able
to measure the distance between two places (as an arc of a
great circle on the earth's circumference) in the case where
the two places are on the same meridian. He claims that he
himself invented a way of doing this even in the case where
the two places are neither on the same meridian nor on the
same parallel circle, provided that the heights of the pole at
the two places respectively, and the angle between the great
circle passing through both and the meridian circle through
one of the places, are known. Now Heron in his Dioptra
deals with the problem of measuring the distance between
two places by means of the dioptra, and takes as an example
He
1
2
Vitruvius, x. 14.
Heron, Dioptra, c. 34.
3
Heron, Metrica, i. 31, p. 74. 21.
4
Proclus, Hypotyposis, pp. 120. 9-15, 124. 7-26.
1
Theon, Comm. on the Syntaxis, Basel, 1538, pp. 261 sq. (quoted in
Proclus, Hypotyposis, ed. Manitius, pp. 309-11).
2
Hammer-Jensen, op. cit.