64 A History ofMathematics
Fig. 5Heron’s slot machine.Heron in hisAutomataandBalancings,...or by using water to tell the time, as Heron in hisHydria, which appears to
have affinities with the science of sundials.^4 (Pappus, in Fauvel and Gray 5.A.2)The combination of quite classical geometry and detailed machine construction makes Heron
interesting, unusual, and hard to classify. Predictably, his machines attract considerable interest
on the Internet, notably the earliest description of a ‘slot machine’; however unmathematical this
may be, it is worth including to illustrate the variety of (some) mathematicians’ interests:
Sacrificial vessel which flows only when money is introduced (see Fig. 5)
If into certain sacrificial vessels a coin of five drachms be thrown, water shall flow out and surround them. Let ABCD
be a sacrificial vessel or treasure chest, having an opening in its mouth, A; and in the chest let there be a vessel, FGHK,
containing water, and a small box, L, from which a pipe, LM, conducts out of the chest. Near the vessel place a vertical
rod, NX, about which turns a lever, OP, widening at O into the plate R parallel to the bottom of the vessel, while at
the extremity P is suspended a lid, S, which fits into the box L, so that no water can flow through the tube LM: this lid,
however, must be heavier than the plate H, but lighter than the plate and coin combined. When the coin is thrown
through the mouth A, it will fall upon the plate H and, preponderating, it will turn the beam OP, and raise the lid of
the box so that the water will flow: but if the coin falls off, the lid will descend and close the box so that the discharge
ceases. (Heron 1851, section 21, to be found at http://www.history.rochester. edu/steam/hero)His geometry—the ‘Metrics’—is both inside and outside the mainstream Greek tradition, giving
rough rules for how to compute combined with Euclidean proofs on occasion. The most famous
example, although not a typical one, is what has become known as ‘Heron’s formula’ for computing
the area of a triangle given its sides. The formula (see Appendix A) is very unusual in Greek
mathematics in that it requires you to multiply four lengths and take the square root. While you
could think of the product of three lengths as a volume, the product of four has no meaning
in Greek terms—and Omar Khayyam was still dismissing such ideas a thousand years later. (In
hisAlgebra; see Fauvel and Gray 6.A.3, p. 226.) All the same, the formula became widespread
and popular in Islamic and medieval times, and a tradition claimed that it was originally due to- Pappus does not note that an accurate water-clock, compared with a sundial, would show the variation in the length of solar
days; but this is not surprising, since elsewhere he criticizes water-clocks for their inaccuracy. The variation (‘equation of time’) is
derived by Ptolemy purely on theoretical grounds—see below.