A history of Greek mathematics - Wilbourhall.org

2
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),
8P(a-2) + (a-4) 2
Diophantus deduces rules as follows.
= {2 + (2n- 1) (a-2)}
2
•
i
1 . To
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 —
)