**The
Invention of Algebra**

Even though in the 16th century
mathematicians like Fiore and Tartaglia were gradually pushing back the
boundaries of equations, they had still not developed algebra as we understand
it today.

As we have seen, algebra is essentially
the ability to generalize, and the process of generalizing mathematics had
started way back with the Pharaohs, when Egyptian mathematicians started to talk
about an unknown number as a 'heap'. Plenty of other cultures took this step.
The Babylonians called an unknown number, 'ush' (which means 'length'), and
they had other words for things like the square of an unknown number, which
they called 'sagab' (which does in fact mean 'square'). The Indian
mathematician Brahmagupta (598-670 AD) used the abbreviated words for colours
to denote unknown numbers.

The next step was to apply
symbolization more widely. Again, this happened in various different places at
various different times.The Greek mathematician Diophantus, who lived from
around 200 AD to 284 AD, developed a shorthand for expressing mathematical
terms in approximately 250 AD. He had symbols to represent unknowns and
numbers, but the terminology was still quite complicated. He would have written
DggxibMq, which translates as x^{2 }3 x
12 units 9, or 3x^{2 }+ 12x + 9.

Because such terminology was so
complicated, it meant that it was difficult to see underlying patterns in the
way people solved mathematical problems, and it was impossible to express such
patterns in a simple way. In general, mathematicians would simply write down
many examples of how to solve a particular type of problem, and hope that their
readers would understand the theory behind it. In 1629, the famous
mathematician and philosopher Rene Descartes complained: 'Algebra, if only we
could extricate it from the vast array of numbers and inexplicable figures by
which it is overwhelmed, so that it might display the clearness and simplicity
which we imagine ought to exist in a genuine Mathematics.'*

It was a man called Francois Viete from
Poitou (1540-1603) who finally developed a sufficiently clear mathematical
language. Having spent his life falling in and out of favour with the French
kings of the second half of the 16th century, at a time when France was in
turmoil as religious groups fought against each other, Viete eventually gained
the support of Henry IV, who used his talents to try to decode the messages
that were being sent to his enemies by Philip II of Spain. Viete was
successful, so much so that Philip complained to the Pope that Black Magic was
being used against him.

During those periods of time when he
was not in favour, Viete devoted himself to his study of mathematics. In terms
of algebra, his major breakthrough was to generalize not just unknowns in an
equation, but also to develop symbols for operations like addition (he
introduced ．+・ for addition and ．-． for subtraction), and to generalize the
numbers in front of the unknowns. In other words, where previous scholars might
deal with a whole series of similar

equations separately (e.g. 2x + 3 =
10; 4x - 1 = 13; 6x + 23 = 132 and so on), Viete could
talk about all of these individual equations as being of a general type ax + b
= c, where a,b and c stand for numbers in the particular examples. He could
then explain, in general, how to solve all types of the same equation much more
concisely than his predecessors.

This might not seem overexciting, but
it was highly significant at the time as it enabled mathematicians to write
down generalized formulas to solve particular types of problems. Up to this
point, the Arabs and others had been able to solve any type of quadratic
equation that you might fancy giving them, but they had done this by
classifying quadratic equations into five different types, and coming up with
different methods for solving each. The Arabic techniques were very ingenious,
but due to their connection of algebra with geometry and apathy towards negative
numbers, they failed to see that all quadratic equations could be solved in the
same way.

Once Viete had developed his form of
algebra, however, it was possible to see that all quadratic equations could be
considered to be of one type: y =ax^{2}+ bx + c (where a, b and c are
the numbers in front of the x^{2 }term, the x term, and at the end of
the equation). From here, a general formula could be given for solving any
quadratic equation.

Again, I will forgive you for being
underwhelmed, but Viete's discovery had profound effects on the lives of his
contemporaries and everyone ever since. Previously, mathematics had been the
preserve of an educated elite made up of scholars or priests, and had been
viewed with suspicion by the average man, mostly because it was simply too
complicated for him to even begin to follow. However, with the growth of
commerce in Europe during the 17th century, the practical value of more
advanced mathematics became apparent. Algebra provided craftsmen and
businessmen with a way of accessing mathematics. They were able to use formulae
to solve practical problems, even if they did not fully understand where they
came from. Mathematics was beginning to be seen as a tool for the people,
rather than a mysterious black magic practised by the elite.

In addition to this process of
demystification, algebra allowed complex situations to be analysed and
manipulated. It would not be long before Galilee Galilei (1564-1642) started to
investigate laws that govern motion, or Johannes Kepler (1571-1630) studied the
movements of the planets, or Isaac Newton (1643-1727) watched apples falling
from trees. In our own centuries, we have managed to navigate to the moon and
to other planets, and Albert Einstein (1879-1955) has come up with his theory
of relativity. Without the simplifying power of algebra, and its ability to
highlight relationships and patterns, the work of these men would have been
impossible.

*Adapted from
．Mathematics Minus Fear・ by Lawrence Potter*

*Published in 2006 by
Marion Boyars Publishing Ltd*