404 13. PROBLEMS LEADING TO ALGEBRA
This recipe is equivalent to the modern formula for the sum of the squares of the
first ç integers.
3. India
Problems leading to algebra can be found in the Sulva Sutras and the Bakshali
Manuscript, mentioned in Section 2 of Chapter 2. Since we have already discussed
the Diophantine-type equations that result from the altar-construction problems in
the Sulva Sutras, we confine ourselves here to a few problems that lead to linear
Diophantinc equations, determinate quadratic equations, and the summation of
progressions.
3.1. Jaina algebra. According to Srinivasiengar (1967, p. 25), by the year 300
BCE Jaina mathematicians understood certain cases of the laws of exponents. They
could make sense of an expression like am^^2 ", interpreting it as extracting the square
root ç times and then raising the result to the power m. The notation used was
of course not ours. The power |, for example, was described as "the cube of the
second square root." That the laws of exponents were understood for these special
values is attested by such statements as "the second square root multiplied by the
third square root, or the cube of the third square root," indicating an understanding
of the equality
„>/V/8 = a3/8.
3.2. The Bakshali Manuscript. The birchbark manuscript discovered in the
village of Bakshali, near Peshawar, in 1881 uses the symbol " to denote an unknown
quantity. One of the problems in the manuscript is written as follows, using modem
number symbols and a transliteration of the Sanskrit into the Latin alphabet:
C 5 _ 3 C 7+ ôçà ~
1 J yu rnu ÷ sa ÷ ÷ 1.This symbolism can be translated as, "a certain thing is increased by 5 and the
square root is taken, giving [another] thing; and the thing is decreased by 7 and the
square root is taken, giving [yet another] thing." In other words, we are looking
for a number ÷ such that ÷ + 5 and ÷ - 7 are both perfect squares. This problem
is remarkably like certain problems in Diophantus. For example, Problem 11 of
Book 2 of Diophantus is to add the same number to two given numbers so as to
make each of them a square. If the two given numbers are 5 and —7, this is exactly
the problem stated here; Diophantus, however, did not use negative numbers.
The Bakshali Manuscript also contains problems in linear equations, of the
sort that have had a long history in elementary mathematics texts. For example,
three persons possess seven thoroughbred horses, nine draft horses, and 10 camels
respectively. Each gives one animal to each of the others. The three are then
equally wealthy. Find the (relative) prices of the three animals. Before leaping
blindly into the set of two linear equations in three unknowns that this problem
prescribes, we should take time to note that the problem can be solved by imagining
the experiment actually performed. Suppose that these donations have been made
and the three people are now equally wealthy. They will remain equally wealthy
if each gives away one thoroughbred horse, one draft horse, and one camel. It
follows that four thoroughbred horses, six draft horses, and seven camels are all of
equal value. The problem has thereby been solved, and no actual algebra has been