The Pell equation, by Edward Everett Whitford.

12

THE PELL EQUATION

Consequently, after subtraction,
AD2 = 2BD2.
Now erect on BD a square and draw its diagonal. Then
DF2 = 2BD2.
Therefore
DF = AD = 2A CD 2A BCD AB + BE.
Upon the prolongation of AD, lay off DG = BD, and
GH = DF; and on DH erect a square and draw its
diagonal HK. Then, by similar steps,
HK = BH = 2BD + GH = 2BD + DF.
It is clear that, in the continuation of this construction,
in each new square the diagonal equals the sum of the
diagonal and double the side of the previous square.
If we designate the sides of the successive squares by
s1, S2, S3,..., and the corresponding diagonals by di, d2,
d3, * * *, we have the double series,
Si, 2 = s + d1, s3 = S2 + 2,...
dl, d2 = 2sl + d, d3 = 2s2 + d2,...
in which the double square of each term in the upper line
is equal to the square of the corresponding term in the
lower line.
This geometric proof was, however, for the Pythagoreans the introduction to some fine considerations in
the theory of numbers. If they set s = 1, then d = /2
and this was, like 1/50 just referred to, an "unspeakable"
number; and in connection with this comes into view
the p'wnr 8tdaieIpoS, the nearest integer, the number 1.
Then they compared the square of the dpprT7o and the
p7 TB lCtd-epos, the first 2, the second 1, the difference 1.
If we now distinguish the successive p3rTa Stdl/eTpot
as 56, 62, 33,..., and substitute them in the double series