The solution of numerical higher equations for
approximate values of roots has been known for a long time in China. It has
been called the most characteristic Chinese mathematical contribution. That it
was well developed in the work of the Sung algebraists has long been known, but
it is possible to show that if the text of Han Jiuzhang [Suanshu], (Nine
Chapter of the Mathematical Art) is very carefully followed, the essentials of
the method are already there at a time which may be dated as of the -1st
century.
``The
Chinese also developed a similar method to find the cube root of a number and
in so doing were able to solve a cubic equation of the form x3
+ ax2 + bx = c, where a, b and c
are positive. Through the method of finding a side of a square or cube with the
rod numeral system, the Chinese invented a notation to represent quadratic and
cubic equations and were able to solve numerically a
particular type of such equations. One could say that at this stage there were
indications that arithmetic was forging into the yet to be known field of
algebra.'' (AHES 47 (1) 1994, 1-51).
``The ancient Chinese performed the four fundamental operations on fractions
and the extractions of square and cube roots as easily as we perform these
operations using our numeral system. More important, in spite of the two
thousand years' gap and the difference in media, the procedures that they used
bear parallel resemblance to the ones we use.
[...] (AHES, 37 (4) 1987, 365 - 392).
It means
that the principles and methods of extraction of square roots and cube roots as
explained in Jiuzhang Suanshu have gradually evolved into a method of solving
polynomial equations, a method known as "zeng cheng kai fang"
(extraction of roots by addition and multiplication) in the Song period (9th --
11th century), finally capable of dealing with equations of arbitrarily high
degrees and coefficients positive, negative, and zero. Such numerical methods
are indeed spigot algorithms which give a positive root of a polynomial
equation digit by digit. In practice, each of the digits is determined by
deliberation.
In
Jiuzhang Suanshu and other Chinese texts, the digits obtained in this way are
called `shang', the same character for the word `quotient'. This character
`shang' (deliberation) captures the essence of the division process, (unless
one insists on performing division by repeated subtraction).
``Among
the problems in the Jiuzhang, one, the finding of a cube root, may be written
x3 = 1,860,867, and another, involving the
square and the simple term,
x2 + 34x + 71,000 = 0. Naturally, these
equations were not written as such, but set up with counting rods on a board. The
uppermost line in the table is to contain the radical solution sought, the
second line (shi) is the constant term before the operation, the third (fa, or
later in the operation, ting fa) is for numbers obtained in its successive
stages, the fourth (zhong) receives another number found at an intermediate
state, and the fifth (jie suan) is the coefficient of x3
before the operation.
Numerical
equations of degrees higher than the third occur first in the work of QIN
Jiushao around 1245 AD. He deals very clearly with equations such as:
-x4
+ 763,200x2 - 40,642,560,000 = 0.
|
The extraction of cube roots
on the counting board.
--- 1
860 867. Make an
estimate of some A such that A3
does not exceed 1, in this case take A=1. Subtract
1*106 from the number to obtain 860 867. The next
estimate of some B is made such that (3*A2*102+3*A*B*10+B2)*B
does not exceed 860, in this case take B=2
( because 364*2 = 728 while 369*3 = 1107 ).
Continue in a similar fashion. In this
particular case, the third estimate gives C=3 and the
procedure ends, yielding the answer 123. In general, the
result can be calculated with an accuracy to within any decimal place. The
mathematical basis is the identity |
|
A
comparison of the method on the counting board (given in Jiuzhang Suanshu) is
described below. It is remarkable how similar they are. The
extraction of cube root is carried out on a counting board, in five rows. The
first row is for the root. In the second row one puts down the digits of the
dividend (the number whose cube root is to be extracted). The third and
fourth rows are left blank in the beginning. The
method begins by moving the borrowed rod to the left, jumping two places at a
time. This has the effect of dividing the digits of the dividend into blocks
of threes. Suppose
the number is not the cube of one of the digits 1, 2, ... 9. (The problem
would then be trivial). Figure out the first quotient (digit of the root). ``Multiply
it twice to the number represented by the borrowed rod; use this number as
divisor (fa) to perform division in the first block. Treble this and leave it
as ``fixed divisor'' (ding fa, in the third row). Once more, divide this
divisor by the quotient and leave the result in the All
these operations have the effect of (i)
removing the cube of the quotient from the first block, this first quotient
is so chosen that its cube is just no greater than the three - digit number
in the leftmost block), (ii)
putting 3 times the first quotient in the fourth row as ``the middle
number'', (iii)
putting 3 times the square of the first quotient in the third row as divisor, (iv)
leaving the borrowed rod in the bottom row. ``Now,
shift the fixed divisor one place to the right, the middle number two places,
and the borrowed row in the bottom row three places. This is
illustrated in Problem IV.22 of Jiuzhangthis where the cube root of
1,937,541(17/27),
``Figure
out the quotient again; multiply it to the middle number; multiply it twice
to the bottom number; add these (two) to the fixed divisor''. 12
quotient
12
quotient ``Shift
the fixed divisor one place to the right, the middle number two places, and
the bottom number three places''. 12
quotient ************* At this
point, the text indicates that ``if the division does not terminate, then the
cube root cannot be extracted precisely. It is intended, of course, that the
above process (of choosing a digit and adjusting the bottom, middle numbers
and the fixed divisors) be carried out up to the block containing the units
digit. Then it deals the case when the dividend has a fractional part. LIU
Hui has remarked (in his commentary on the square root process) that this can
be continued as long as required. ************* 124
quotient _____________________ 124
quotient _____________________ 124
quotient _____________________ To work
beyond the decimal point, one would shift the numbers in the three bottom
rows as follows, and choose 6 for the next quotient: 124
quotient Wang
Xiaotong (fl. 625) Mathematician and astronomer. Wrote Xugu suanjing
(Continuation of Ancient Mathematics) of 22 problems. Solved cubic
equations by generalization of algorithm for cube root. According
to J. Needham, Science and Civilization in China, vol 3, p. 65ff, the square
and cube root algorithm already existed in the Chinese text: Chiu Chang Suan
Shu (mathematical treatise in nine sections) by about 100 BC to 100 AD. The
implementation on the abacus apparently occurred around the 13-th century
(Yuan = Mongol Dynasty) when the abacus became the *calculator of choice*
replacing the *counting rods*. The algorithm anticipated Horner's method. The
above method does of course depend on the formula: N = (a + b + c + d + ...)3 = a3 + (3a2 + 3ab + b2)b+ {[(3a2 + 3ab + b2) + (3ab + 2b2)] + [3(a + b)c + c2]}c + {[3(a + b)2 + 3(a + b)c + c2] + [3(a + b)c + 2c2] + [3(a + b + c)d + d2]}d + ...
When
extracting (cube) root, the above formula is reserved. At each step, each
term (first a3, then (3a2 + (3a + b)b)b, etc.) is
removed from the number N. The
above method requires more than one counting board to find the double or
triple of a term (while keeping the table). Jia Xian improved the method so
as to use only one counting board. |