A history of Greek mathematics - Wilbourhall.org

;
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
a semicircle, the formula in the last case being
less than
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
Vitruvius, x. 14.
2
Heron, Dioptra, c. 34.
1
3
Heron, Metrica, i. 31, p. 74. 21.
4
Proclus, Hypotyposis, pp. 120. 9-15, 124. 7-26.