Introduction
The
history of finding roots of polynomials goes a long way back in the history
of man, and it can be divided into two sections. In the beginning some
mathematicians sought numerical approximations for roots, while others
concentrated on finding algebraic formulae for the solutions. The two
approaches may even seem like two separate subjects, but they are very
closely related. A
lot is written about the early mathematics but we do not find any advanced
steps until we reach the mathematics of the Babylonians and the Egyptians of
about 3000 BC. A fundamental problem of the Babylonian algebra deals with
constructions leading to a quadratic equation. They had in fact the formula
for solving the quadratic equation but did the Babylonians (about 2000 - 400 BC) solve cubic
equations? The
answer is from algebraical point of view - no. What they used was an
algorithmic approach to solve problems, which in modern terminology, would
give rise to quadratic or cubic equations. The method is essentially one of
completing a square and using tables of values of roots. The answer however
was always positive as the answer had to be a length. But
maybe the most interesting question is: What kind of mathematical problem
could possibly at that time give rise to a need of a method involving
quadratic or cubic roots? Or was it possibly so that they were interested in
the "algebra" for itīs own sake? An
important source of early Indian
mathematics derives from a class of ritual literature dealing with the
measurement and construction of various sacrificial altars. The Sulbasutras
(conservative dating: 800BC - 500BC) provided such instructions for two types
of rituals, one for worship at home and the other for communal worship.
Square and circular altars were sufficient for household rituals while more
elaborate altars involving combinations of rectangles, triangles and
trapeziums were required for public worship. One of the most elaborate of the
public altars was shaped like a falcon, or rather like the shadow of a
falcon, just about to take flight . It was believed that offering a sacrifice
on such an altar would enable the soul of the supplicant to be conveyed
by a falcon straight to heaven. Fairly complicated mathematical complications
might occur in connections with altering the size of altars. In
Greece
..."During the time of Plato, a plague befell the citizens of Delos, the
island where Apollo was born. The oracle of Apollo was consulted, in the hope
that a solution for the pestilence would be found. The response was that the
situation would resolve itself if the Delians would simply construct a cubic
altar exactly twice the volume of the existing one. However, the Delian
craftsmen were confounded by the problem, which is far more complex than it
initially seems because simply doubling each side produces a cube with eight
times the original volume. Eratosthenes reports that the Delians
"therefore went to ask Plato about it, and he replied that the oracle
meant, not that the god wanted an altar of double the size, but that he
wished, in setting them the task, to shame the Greeks for their neglect of
mathematics and their contempt for geometry." According
to Plutarch, a priest of Apollo at Delphi, there was another important reason
why Apollo exhorted the Greeks to "double the cube," and
hence study geometry. Namely, "he was ordering the entire Greek nation
to give up war and its miseries and cultivate the Muses, and by calming their
passions through the practice of discussion and study of mathematics, so to
live with one another that their intercourse should be not injurious, but
profitable." The problem of solving the cubic equation
has also led to the myth of the "Curse of the Cubic" :
One
period of considerable interest is that between the decline of Greek
mathematics, coinciding with the collapse of the western Roman Empire in the
fifth century, and the rise of European mathematics in the fifteenth century.
Mathematics professor Morris Kline of New York University's Courant Institute
of Mathematical Sciences expressed a common view of that period in his 1972
book Mathematical Thought from Ancient to Modern Times. "The
Arabs made no significant advance in mathematics," he wrote. "What
they did was absorb Greek and Hindu mathematics, preserve it, and ultimately,
... transmit it to Europe." In
other words, Islamic scholars should have done no more than put Greek
mathematics into cold storage until Europe was ready to accept it. Historian
George G. Joseph challenged that view in his provocative book The Crest of
the Peacock: Non-European Roots of Mathematics. He asserted that
mathematical knowledge originated in many parts of the world, and much of
this knowledge was transmitted over the centuries to Europe, where it
inspired further developments. However, Joseph had scant concrete evidence to
back his claim. In
Liber abaci, Fibonacci (1180-1250) treated the cubic equation x3+2x2+10x=20.
He showed that the equation could not be solved using rational number or
roots and assumed that it was not possible to solve by algebraic methods. He
did however find an approximate value for the positive root with 10 decimals
accuracy. Historians
of mathematics now generally agree that scholars in China, India, and the Islamic world
produced remarkably sophisticated mathematics during this period. And most
would probably still argue that Europeans in later centuries were unaware of
this work and made advances with minimal help from the earlier efforts. Root extraction in the West D.
E. Smith in his book History of Mathematics, points that the
Western mathematicians did not have a general method for extracting roots
which the Chinese did at about the first century. The Western mathematicians
did treat each problem in finding square root or cube root of a number
separately. In fact, they did not know how to extract the cube root
until the sixteenth century. ( Smith, p148.) It
is also interesting to point out that the Chinese mathematicians already knew
how to extract roots of arbitrary degree at the first century. The method is
stated in the Jiuzhang
suanshu . The method can easily be extended to solve
first, quadratic and higher numerical equations. This method is the same
method that Horner discovered at 1819 for solving higher numerical equations.
No one noticed that the Chinese had this knowledge for a long time until Wang
Ling and Joseph Needham's paper on "Horner's Method in Chinese
Mathematics" appeared in T'oung Pao, 1955, 43:p345-401. Wang
and Needham discuss the similarities between the ancient method of extracting
roots in China and Horner's method for solving numerical equations. The
method was used by mathematicans in the Sung dynasty (960 - 1279AD) for
solving numerical equations. Maestro Dardi of Pisa in a work from 1344 extended a list to 198 types of equations of
degree up to four, some involving radicals. He gave an example of how to
solve a particular cubic equation, but the methods would not generalize. During
the 16th century the work with the cubic equations came to take place in
Italy, the most known mathematicians were Scipione dal Ferro (1465-1526),
Pacioli who declared that cubic equations were not possible to solve, dal Ferro who succeded to
solve cubic equations of the form x3+ mx = n around 1515
but kept his work a complete secret until just before his death, in 1526,
when he revealed his method to his student Antonio
Fior. About the same time Niccolo of Brescia, known as Tartaglia meaning 'the
stammerer', managed to solve equations of the form x3 + mx2
= n. In 1545 Cardan
published Ars Magna the first Latin treatise on algebra, in which
he presented Tartaglias and dal Ferros work with cubics. What
anyhow is clear is that there has, in China, India, Babylonia and Greece been
a lot of work done in order to solve equations of higher degree, and a
remarkably great deal of the work has been done hundreds of years BC. This
work is an attempt to follow the development of the treatment of the
"cubic", from early numerical calculations to the development of
algebraic solutions of cubic equations.
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