THE ’EITAN0HMA (‘BLOOM’) OF THYMARIDAS 95 
From these equations we obtain 
x + y + z + u = (a + 1) (z + u) = (b + 1) (u + y) — (c + 1) (y + z). 
If now x, y, z, u are all to be integers, x + y +z + u must 
contain a + 1, b + 1, c + 1 as factors. If L be the least common 
multiple of a+ 1, b -f 1, c+ 1, we can put x-+y + z + u — L, and 
we obtain from the above equations in pairs 
x + y 
a 
a+ 1 
L, 
x + z 
h 
b+1 
L, 
C T 
x -j- u — A, 
c + 1 
while x + y + z + u = L. 
These equations are of the type to which Thyrnaridas’s rule 
applies, and, since the number of unknown quantities (and 
equations) is 4, n — 2 is in this case 2, and 
t / a h c\ 
'a+l^ft+l^c + l/ 
x — 
2 
The numerator is integral, but it may be an odd number, in 
which case, in order that x may be integral, we must take 2 L 
instead of L as the value of x + y + z + u. 
Iamblichus has the particular case where a = 2, b = 3, c = 4. 
A is thus 3.4.5 = 60, and the numerator of the expression for 
x becomes 133 — 60, or 73, an odd number; he has therefore 
to put 2A or 120 in place of A, and so obtains x=7o,y~7, 
z— 17, u — 23. 
Iamblichus goes on to apply the method to the equations