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 x² 3 x 12 units 9, or 3x² + 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 =ax²+ bx + c (where a, b and c are the numbers in front of the x² 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