NICOMACHUS 
107 
formula or summation. An a-gonal number with n in its 
side being 
n + ^n (n— 1) (a—2), 
it follows that the pyramid with that polygonal number for 
base is 
l-J-2-)-3-f-...-f*ri/-}-|r {a — 2) {1.2 + 2.3 + .,.+ {n — \}n} 
_n(n + 1) t a —2 (n — 1) n {n + 1) 
_ - + —. - 
A pyramid is KoXovpos, truncated, when the unit is cut off 
the top, SLKoXovpos, twice-truncated, when the unit and the 
next layer is cut off, rpLKoXovpos, thrice-truncated, when three 
layers are cut off, and so on (e. 14). 
Other solid numbers are then classified (cc. 15-17): cubes, 
which are the product of three equal numbers; scalene num 
bers, which are the product of three numbers all unequal, 
and which are alternatively called ivedges {a-^gvio-Koi), stakes 
{(T(f)T] klctkol) , or altars {^copiaKOL). The latter three names are 
in reality inappropriate to mere products of three unequal 
factors, since the figure which could properly be called by 
these names should tagger, i.e. should have the plane face at 
the top less than the base. We shall find when we come to 
the chapter on Heron’s mensuration that true (geometrical) 
PcopiaKOL and a(f)r]VL(TKOL have there to be measured in which 
the top rectangular face is in fact smaller than the rectangular 
base parallel to it. Iamblichus too indicates the true nature 
of ficopiaKOL and u^vlctkol when he says that they have not 
only their dimensions but also their faces and angles unequal, 
and that, while the 7tXlvO'ls or Sokls corresponds to the paral 
lelogram, the cr(j)rjvi(TKos corresponds to the trapezium. 1 The 
use, therefore, of the terms in question as alternatives to scalene 
appears to be due to a misapprehension. Other varieties of 
solid numbers are parallelepipeds, in which there are faces 
which are irepopijKt^ (obj-ong) or of the form n(n+ 1), so 
that two factors differ by unity; beams (SoklScs) or columns 
{a-rrjXiSes, Iamblichus) of the form m 2 {m + n)\ tiles {irXLvOiSes) 
of the form m 2 {m — n). Cubes, the last digit (the units) of 
which are the same as the last digit in the side, are spherical 
1 Iambi, in Nicom., p. 98.18, 94. 1-8.