A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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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516 DIOPHANTUS OF ALEXANDRIA
4. The vertical columns are in arithmetical progression, the
common difference being the triangle in the preceding column.
Plutarch, a contemporary of Nicomachus, mentions another
method of transforming triangles into squares. Every triangular
number taken eight times and then increased by 1
gives a square.
•In fact, 8.£n(w+l) + l = (2?i+ l)
2
.
Only a fragment of Diophantus's treatise On Polygonal
Numbers survives. Its character is entirely different from
that of the Arithmetica. The method of proof is strictly
geometrical, and has the disadvantage, therefore, of being long
and involved. He begins with some preliminary propositions
of which two may be mentioned. Prop. 3 proves that, if a be
the first and I the last term in an arithmetical progression
of n terms, and if s is the sum of the terms, 2s = n(l + a).
Prop. 4 proves that, if 1, 1+6, 1 + 26, ... 1 + (n— l)b be an
A. P., and s the sum of the terms,
2s = n {2 + (n—l)b}.
The main result obtained in the fragment as we have it
is a generalization of the formula 8 . \n{n + 1) + 1 = (2 n + l) 2 .
Prop. 5 proves the fact stated in Hypsicles's definition and also
(the generalization referred to) that
8 P (a — 2) + (a — 4)
2
= a square,
where P is any polygonal number with a angles.
It is also proved that, if P be the nth. a-gonal number
(1 being the first),
1 . To
8P(a-2) + (a-4) = 2 {2 + (2n- 1) (a-2)}
i
Diophantus deduces rules as follows.
find the number from its side.
P =
{2 + (2 n-1) (a- 2) }
2 - (a- 4)
2
8(a-2)
2. To find the side from the number.
= 1 /y {8P(a-2) + (a-4) 2 }-2 v
" 2 V a — )
2
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