178 7. ANCIENT NUMBER THEORYAlthough it is trivial to verify that this rule is correct using modern algebraic
notation, one would like to know how it was discovered.^12 Although the route
by which this discovery was made is not known, the motivation for studying the
equation can be plausibly ascribed to a desire to approximate irrational square
roots with rational numbers. Brahmagupta's rule with c = d — 1 gives a way of
generating larger and larger solutions of the same Diophantine equation ax^2 + 1 =
y^2. If you have two solutions (x, y) and (u, v) of this equation, which need not be
different, then you have two approximations y/x and õ/u for y/a whose squares are
respectively 1/x^2 and 1/u^2 larger than a. The new solution generated will have
a square that is only l/(xv + yu)^2 larger than a. This aspect of the problem of
Pell's equation turns out to have a close connection with its complete solution in
the eighteenth century.4.4. Bhaskara II. In his treatise Vija Ganita (Algebra), Bhaskara states the rule
for kuttaka more clearly than either Aryabhata or Brahmagupta had done and
illustrates it with specific cases. For example, in Chapter 2 (Colebrooke, 1817, p.
162) he asks, "What is that multiplier which, when it is multiplied by 221 and 65 is
added to the product, yields a multiple of 195?" In other words, solve the equation
195a; = 221y + 65. Dividing out 13, which is a common factor, reduces this equation
to 15a: = 17y+ 5. The kuttaka, whose steps are shown explicitly, yields as a solution
χ = 40, y = 35.
In his writing on algebra, Bhaskara considered many Diophantine equations.
For example, in Section 4 of Chapter 3 of the Lilavati (Colebrooke, 1817, p. 27),
he posed the problem of finding pairs of (rational) numbers such that the sum and
difference of their squares are each 1 larger than a square. It would be interesting
to know how he found the answer to this difficult problem. All he says is that
the smaller number should be obtained by starting with any number, squaring,
multiplying by 8, subtracting 1, then dividing by 2 and by the original number.
The larger number is then obtained by squaring the smaller one, dividing by 2, and
adding 1. In our terms, these recipes say that if u is any rational number, then(*'-"• K.)'± («-£)'-
is the square of a rational number. The reader can easily verify that it is (8it^2 —
l/(8u^2 ))^2 when the positive sign is chosen and (%u^2 — 2+ g^a)^2 when the negative
sign is taken.
Chapter 4 of the Vija Ganita contains many algebraic problems involved with
solving triangles, interspersed with some pure Diophantine equations. One of the
most remarkable (Colebrooke, 1817, p. 200) is the problem of finding four unequal
(rational) numbers whose sum equals the sum of their squares or the sum of the
cubes of which equals the sum of their squares. In the first case he gives |, |, |,
|. In the second case he gives -^, -^, -^, In both cases the numbers are in the
proportion 1:2:3:4. These three extra conditions (three ratios of numbers) were
deliberately added by Bhaskara so that the problem would become a determinate
one.
The characteristic that makes problems like the preceding one easy is that
the requirement imposed on the four numbers amounts to a single equation with
(^12) Wei{1984, pp. 17, 83, 204) refers to Eq. land the more general relation (x (^2) +Ny (^2) )(z (^2) +Nt (^2) ) =
(xz ± Nyt)^2 + N(xt + yz)^2 as "Brahmagupta's identity" (his quotation marks).