The modular composition of roman water-wheels

Abstract

The m etrical analysis of the Rom an w ater-w heel from the m ine in Rio Tinto, now exhibited in B ritish Museum, and of the draw ing of a sim ilar wheel from San Domingo offers the explanation of the Roman approxim ation of irra ­ tional values. The ratio circum ference : diam eter, which is the irrational n, can be approxim ated w ith the ratio 22 : 7, the half of which (11 :7) appears in the sequence of term s in the second Fibonacci series (1—3—4—7—11— ...) and in the ratio betw een the heptagonal num ber (7) and its gnomon (11). The diam eter of the first w heel is 7 m odules of 7 trientes, and th e corres[1]ponding circum ference is 22 m odules of th e sam e size. In fact th is w heel has 22 spokes. It would be very complicated to devide 360° in 22 parts geom etrically; the arithm etical calculation of a circum ference, implying irrational it , and its di[1]vision into equal p arts is also too dem anding for a simple craftsm an. B ut m odular composition sim plifies th e problem. The diam eter of the second w heel is again 7 modules of 7 trientes, but its circum ference is not devided in 22 intervals of 7 trientes (7 trientes = 28 un[1]ciae), but in 28 m odules of 22 unciae. E quation 22 M38”—28 M ^” is obvious. The sam e com positional principle has been adopted for the P antheons dome. The Domes diam eter equals 7 modules of 14 cubiti, but the corresponding circum ­ ference (22 M 14 cui,iti) is devided in 28 intervals of 11 cubiti. T he equation 22 Mj4 cubit; — 28 Mu cubit; is analogous to the previous one and differs only in extent. The practical value of the described principle for the composition of a cir[1]cum ference is its sim plicity, facilitating th e w heelm akers or builders task. This and other compositional properties probably m ade num ber 7 exceptionally famous and 28 a perfect num ber.