The Renaissance is one of these catchphrases we learn at school, which conveniently breaks the historical mosaic into course-length segments. However, like 'Bronze Age' and 'The Enlightenment', any degree of investigation reveals that it is very location and definition dependent. There was no opening ceremony with a ribbon with a dignitary announcing 'I hereby declare the Renaissance open'.

It is generally interpreted as being a loosely connected series of movements, in art, philosophy, and science, which heralded the phasing out of the medieval world. What that means is also difficult to define.

The Renaissance is popularly associated with the art of Italy, embodied by people like Leonardo da Vinci, whose lifespan (1454 - 1519) saw the Renaissance gain dominance in Italy, and also in many other parts of Europe, to varying degrees. However, ideas which are associated with the Renaissance certainly predated this era. One of the reasons the Renaissance had such a momentum in the latter half of the 15th century is the invention of the printing press.

The understanding of science and mathematics in Europe had been held back by a number of factors, the prime one being religion. In other parts of the world, the mathematics, for example, of the Greeks, which had been effectively lost to Europe with the fall of Greece to the Romans, had been revived, and at times reinvented in the Arab world, India, and China. Algebra, for example, is derived from an Arabic phrase.

The Greeks gave us geometry, but the Arabs gave us the algebraic language of mathematics which allowed the Renaissance to flourish. The zero had been invented by the Babylonians, but the Greeks were having 'none' of it ('nothing will come of zero' they said, 'it will never add up to anything'). So, it was up the Indians to rediscover the zero, and with it, by the act of division, discovered infinity. Without these concepts, the infinite division of integrals which leads to calculus, the most powerful tool in mathematics, would never have been found.

Luca Pacioli, 1445 - 1514, was an Italian mathematician and teacher, who published books in the Italian vernacular to popularise mathematics.

Pacioli was the teacher and then life-long friend of Leonardo da Vinci, who went on to provide sketches for his books. His mathematical works brought the work of previously little known mathematicians, such as Fibonacci, to a greater pubic awareness.

With Leonardo he explored the Golden Ratio, which led to its popularization in architecture and art. Leonardo was inspired by Pacioli to draw the famous 'Vitruvian Man' on the basis of the Golden Ratio as described by Vitruvius, the ancient Roman architect.

Pacioli developed and taught accounting, and popularised the techniques used at the time, which are still standard in companies around the world. For his presentation of the double-entry accounting technique he is often referred to as 'The Father of Accounting', although he probably did not actually invent it, as such.

He was also interested in mathematical applications to chess, magic tricks (including card tricks), probability, and art. At least two of his books had been lost and rediscovered only recently.

The story of Pacioli and Leonardo's extraordinary collaboration is recounted in Andrew Bone's 2010 novel 'Vitruvian Boy'.

One of the consequences of the rise and thousand-year dictatorship of the medieval Church in the period between the fall of the Greek empire and the Renaissance was the ban on the adoption of any other number system than the Roman numerals. This held back the development of algebra in Europe, while the rest of the world was able to drive forward with number theory, algebra, and concepts like zero, infinity, and negative numbers.

When Fibonacci brought arabic numbers (1, 2, 3, ...) to Europe from North Africa, the church declared them to be work of the devil, as they wouldwith anything they did not understand. However, a few independent researchers defied the ban, and mathematics began to take hold, even if science remained locked to Aristotle's fallacious philosophical think-products. Empirical science, employing the necessity of verification through experiment, was developed on the cusp between the Renaissance and the Enlightenment. I select to interpret the invention of Empirical Science (synonymous with Galileo, so around the 1580s) as being a natural consequence of the Renaissance, and which, as a scientific revolution, challenged and defeated religious dogma, and defines the core of the movement known as the Enlightment.

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