HISTORY OF THE TERM ‘ GNOMON ’ 
79 
the figure similarly related to any parallelogram, instead of 
a square; it is defined as made up of ‘ any 
one whatever of the parallelograms about 
the diameter (diagonal) with the two com 
plements ’. Later still (5) Heron of Alex 
andria defines a gnomon in general as that 
which, when added to anything, number or figure, makes the 
whole similar to that to which it is added. 1 
(8) Gnomons of the polygonal numbers. 
Theon of Smyrna uses the term in this general sense with 
reference to numbers: ‘ All the successive numbers which [by 
being successively added] produce triangles or squares or 
polygons are called gnomons.’ 2 From the accompanying 
figures showing successive pentagonal and hexagonal numbers 
it will be seen that the outside rows or gnomons to be succes- 
sively added after 1 (which is the first pentagon, hexagon, &c.) 
are in the case of the pentagon 4, 7, 10 , . . or the terms of an 
arithmetical progression beginning from 1 with common differ 
ence 3, and in the case of the hexagon 5, 9, 13 .... or the 
terms of an arithmetical progression beginning from 1 with 
common difference 4, In general the successive gnomonic 
numbers for any polygonal number, say of n sides, have 
(n — 2) for their common difference. 3 
(e) Right-angled triangles with sides in rational numbers. 
To return to Pythagoras. Whether he learnt the fact from 
Egypt or not, Pythagoras was certainly aware that, while 
3 2 + 4 2 = 5 2 , any triangle with its sides in the ratio of the 
1 Heron, Def. 58 (Heron, vol. iv, Heib., p. 225). 
2 Theon of Smyrna, p. 37. 11-13. 3 lb., p. 34. 13-15.