CONTROVERSIES AS TO HERON’S DATE 303 
machines used by the two for the same purpose frequently 
differ in details; e. g. in Vitruvius’s hodometer a pebble drops 
into a box at the end of each Roman mile. 1 while in Heron’s 
the distance completed is marked by a pointer. 2 It is indeed 
pointed out that the water-organ of Heron is in many respects 
more primitive than that of Vitruvius; but, as the instru 
ments 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 agrimen 
sores. 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 
h ( s + h) h + i? (i 6 ') 2 ; 
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’ (oí á.KpL^é(TTepov efrjTrjKOTes). 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 Pro 
clus, 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 
1 Vitruvius, x. 14. . 2 Heron, Dioptra, c. 34. 
3 Heron, Métrica, i. 31, p. 74. 21. 
4 Proclus, Hypotyposis, pp. 120. 9-15, 124, 7-26.