NICOMACHUS 
111 
a fact which, according to Nicomachus, was not generally 
known. Boëtius 1 mentions this proposition which, if we 
take a + d, a, a — d as the three terms in arithmetical pro 
gression, may be written a 2 = (a + d) {a—d) + d 2 . This is 
presumably the origin of the régula Nicomachi quoted by 
one Ocreatus (1.0’Great), the author of a tract, Prologus in 
Helceph, written in the twelfth or thirteenth century 2 
( : Helceph ’ or c Helcep ’ is evidently equivalent to Algo- 
rismus ; may it perhaps be meant for the Al-Kdfi of 
Alkarkhi ?). The object of the régula is to find the square 
of a number containing a single digit. If d = 10 — a, or 
a + d — 10, the rule is represented by the formula 
a 2 = 10 {a—d) + d 2 , 
so that the calculation of a 2 is made to depend on that of d 2 
which is easier to evaluate if d<a. 
Again (c. 24. 3, 4), if a, 1), c be three terms in descending 
geometrical progression, r being the common ratio {a/h or h/c), 
then 
a—h_a_h 
h — c b ~ c 
and {a — b) — {r—l)b, {b — c)={r—\)c, 
{a — b) — (b — c) = (r — 1 ) {b—c), 
It follows that 
b = a—b {r— 1) — c + c (r— 1). 
This is the property of three terms in geometrical pro 
gression which corresponds to the property of three terms 
a, b, c of a harmonical progression 
7 a c 
b — a = c + - s 
n n 
from which we derive 
n = (a + c) / (a—c). 
If a, b, c are in descending order, Nicomachus observes 
(c. 25) that < = > - according as a, b, c are in arith 
metical, geometrical, or harmonical progression. 
1 Boëtius, Inst. Ar. ii. c. 43. 
2 See Ahh. zur Gesch. d. Math. 3, 1880, p. 134.