IA 
= 64, making 
re' 
re 
acting the two 
22528 m. 
071—22528 as 
sible value for 
ieter p = 64. 
n— 2048, and, 
= |o m+ 1024) 
i, we have to 
= 4 2 4 5 8 ( or TTe)' 
te, though the 
) 
or >^xy.) 
3ers. 
di Diophantus 
io- back to the 
o 
THE TREATISE ON POLYGONAL NUMBERS 515 
Pythagoreans, while Philippus of Opus and. Speusippus carried 
on the tradition. Hypsicles (about 170 B.c.) is twice men 
tioned by Diophantus as the author of a ‘ definition ’ of 
a polygonal number which, although it does not in terms 
mention any polygonal number beyond the pentagonal, 
amounts to saying that the 7ith a-gon (1 counting as the 
first) is 
%n{2 + {n-l) {a-2)}. 
Theon of Smyrna, Nicomachus and larnblichus all devote 
some space to polygonal numbers. Nicomachus in particular 
gives various rules for transforming triangles into squares, 
squares into pentagons, &c. 
1. If we put two consecutive triangles together, we get a square. 
In fact 
%{n— l)n + ^n{n + 1) = n 2 . 
2. A pentagon is obtained from a square by adding to it 
a triangle the side of which is 1 less than that of the square ; 
similarly a hexagon from a pentagon by adding a triangle 
the side of which is 1 less than that of the pentagon, and so on. 
In fact 
{2 + (n — 1) {a — 2)} + i(n— l)n 
= i^[ 2 + (n~l) {(u+l)-2}]. 
3. Nicomachus sets out the first triangles, squares, pentagons, 
hexagons and heptagons in a diagram thus: 
Triangles 
1 
3 
6 
10 
15 
21 
28 
36 
45 
55, 
Squares 
1 
4 
9 
16 
25 
36 
49 
64 
81 
100, 
Pentagons 
1 
5 
12 
22 
35 
51 
70 
92 
117 
145, 
Hexagons 
1 
6 
15 
28 
45 
66 
91 
120 
153 
190, 
Heptagons 
1 
7 
18 
34 
55 
81 
112 
148 
189 
235, 
and observes that: 
Each polygon is equal to the polygon immediately above it 
in the diagram plus the triangle with 1 less in its side, i. e. the 
triangle in the preceding column. 
l 1 2