A history of Greek mathematics - Wilbourhall.org

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THE TREATISE ON POLYGONAL NUMBERS 515
Pythagoreans, while Philippus of Opus and Speusippus carried
on the tradition. Hypsicles (about 170 B.C.) is twice mentioned
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 nth a-gon (1 counting as the
first)
is
i.
n{2 + (n-l)(a-2)}.
Theon of Smyrna, Nicomachus and Iamblichus 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
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
in {
2 + (n - 1) (a- 2) }
+ i{n— \)n
= in[2 + (n-l){(a+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.
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