How And Why Did Mathematics Develop In Europe During The Renaissance, 1300- 1700?

"Renaissance in art, reformation in religion, Columbus discovered the new world, Copernicus demonstrated that the earth need not be considered the centre of the universe. Intellectual world was full of new possibilities, and in this context a mathematical revolution was not unexpected." 1 After the so-called Dark Age -"`a time of ignorance and superstition` " 2 which held back the development of science in Europe, mathematics was reborn. By the 13th century, many ancient Greek texts had been rediscovered and translated into Latin and as a result, Medieval universities had started cropping up all over Europe, with a great emphasis on scholastic studies - particularly medicine, law, theology and the arts. 3 4 Scholastics believed in supporting the Roman Catholic Church through study, such as using the recently translated works of Aristotle and, many other Greek and Islamic mathematicians, including Euclid, Ptolemy and al-Khwarizmi 5 6 These universities, however, were not specifically interested in science and therefore mathematics (other than to help students with religious development), which meant there were very few new mathematical discoveries for many centuries in Europe. 7 Further ancient scientific texts were discovered after the Fall of Constantinople in 1453.8 This followed the invention of the printing press in 1450, 9 which in turn facilitated the quick and easy spread of new scientific theories throughout Europe. The disciplines of algebra, geometry, applied mathematics and calculus were either born or revitalised during the Renaissance, thus leading to the modern mathematics we know today. The development of algebra is one of the most important factors when considering the change of mathematics at this time. By the end of the 13th century in Italy, international trading had taken precedent and merchants required new means of problem solving, as opposed to the Quadrivium mathematics being taught at universities. Therefore, due to demand, Abacus Schools began to educate the sons of rich merchants, i.e. boys of the `middling sort`. These schools paid particular attention to the teaching of algebra and finding an unknown numerical quantity, 10 based upon the work of Fibonacci, and especially his Book of the Abacus (1202) which used the Hindi-Arabic number system.11 The release of Book of the Abacus was essential to the development of modern mathematics as it introduced the concept of zero to Europe. Despite the syllabus being practical, the schools introduced numbers to the merchant class and therefore to the masses, and hence it was a milestone in further mathematical advances. 12In addition, around this time, international banking started to develop due to war in Europe, (for example the Hundred Years War between England and France) and the desire to fight - using credit and paying later. 13­ Hence, a mathematical understanding was more vital than ever before. The introduction of algebraic notation meant that mathematics was faster to compute and easier to handle. Before, mathematics was written in words - making it laborious and time consuming. It is also easier to see solutions when written in a formulaic fashion and it therefore assists logical thinking. For example, Johan Widermann used both the `+` and `-` signs in Mercantile Arithmetic (1489). Francois Viete in In artem analyticam isagoge (1591) was possibly one of the first mathematicians to use letters to represent numbers and Thomas Harriot - the first mathematician to write purely symbolically - introduced many symbols, such as a dot to represent multiplication, as well as `<` and `>` in Artis Analyticae Praxis (1631). 14 The advancement of algebraic notation had a significant impact on mathematical development during the Renaissance, due to its role in making mathematics more user-friendly, well known and accessible. However, it is important to remember that it was not widely accepted until the beginning of 19th century, and mathematicians such as Fermat and Hobbs were sceptical of it - Hobbs described it as a ­"scab of symbols". 15 As Michael Atiyah said, "Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.`" 16 This makes it clear that many leading mathematicians did not trust algebra, unlike geometry which had a clear axiomatic foundation. 17 In addition, during the Renaissance, practical geometry was important for example, for use in warfare, navigation and architecture and therefore there was a necessary increase in the production of machines. This, alongside the creation of clocks introduced people to technical concepts, which eventually led to the birth of many geometric instruments, as well as the study of motion as a science. 18 By the mid 16th century, Greek geometry texts, from writers such as Euclid, Ptolemy, Appolonius which had been recently rediscovered were translated and available in Europe. 19There would have been a feeling of excitement in Europe after so many years of faith and superstition, as they had now finally discovered a discipline that was based on axioms and proof, as opposed to authority or religion. 20 At the time, people practised mathematics in their free time as an art. 21 The rediscovery and translation of these texts would have further increased the enjoyment of mathematics, as if evident through, Euclid`s work, first translated into English in 1572. As one of the first mathematical texts in English, this would have built confidence, due to removal of the language barrier, and made mathematics more accessible. 22 In the 15th and 16th centuries, the masses believed in witchcraft and magic, particularly in Germany- each court even had its own astrologer. As a result, there was a phenomenal demand for ephimerides (planetary tables) and hence Ptolemy`s Almagest was studied, and his tables improved and rewritten. The most notable of these texts was Epitome of the Almagest, written by Regiomontanus (1436-1476), which quickly made astrology/astronomy (classed as the same subject) available to the masses, due to new printing methods and the comfort people seeked in astrology and its use in prognostication. Alchemy was the hobby of the educated man and soon, astronomy was wide-spread, thus establishing it as a separate subject from astrology. 23 Further study of the universe eventually led to some of the most important foundations of modern astronomy by a few revolutionaries, during the scientific revolution. Astronomical discoveries about the universe by Copernicus, Kepler and Galileo during this so-called Age of Enlightenment, led to the realisation that mathematics could be used to describe the universe. 24 As a result, a more powerful discipline was needed to assist the study of motion and dynamics, and hence in 1637, Descartes`s Geometrie linked the art of geometry with algebra, by introducing the Cartesian coordinate plane and expressing straight lines as equations `y=mx + b`. 25 This made it easier to think mathematically and made geometry easier to handle. 26 In turn, this is likely to have encouraged more people to study mathematics, and encouraged people to work with curves. Perhaps the most significant development of mathematics during the Renaissance was the invention of infinitesimal calculus by Newton and Leibniz, at the end of the 17th century. This refers to the study of change based around limits, differentials and integration. 27 At the time, the study of motion and change was advancing quickly, and therefore the invention of calculus came at a perfect time. The Fundamental Theorem of Calculus was discovered, i.e. the theorem that the integral of a function is its anti-derivative 28and as a result, it was now possible to calculate the velocity of a body, for example the speed at which a rocket travels. Newton wrote about this in his book Principia (1687) - after Leibniz had published his work on calculus. This led to a huge controversy about who had invented calculus first, however nowadays it is widely accepted that they both probably came up with the idea independently. 29 Newton is believed to have "distinctly advanced every branch of mathematics" and applied calculus to his other work, however Leibniz contributed more to its notation. Although the discipline was invented to solve the tangent line problem, it had many practical applications, for example in the development of physics. 30 Problems that were unsolvable before, such as, Kepler's failure to calculate velocity=distance/time (because distance and time have different dimensions), were now solvable 50 years later thanks to calculus. 31 This development was revolutionary and truly marked the beginning of the modern era of mathematics because it enabled the further development of so many branches of mathematics and aided key mathematicians, such as Bernoulli, Lagrange and Laplace and even mathematicians to this day. 32 It was now possible to calculate the speed of motion, as well as prove Kepler's Laws of Motion. 33 Although, more work was needed on calculus after the renaissance by figures (such as Cauchy), the groundwork had already been laid, and so modern mathematics was born.34 The introduction of calculus was evidently significant, because Principia was one of the last books written geometrically, thus implying the take-over of mathematics by calculus. 35 In conclusion, the development of mathematics during the renaissance led to declarations about the universe which questioned everything previously taught by the Roman Catholic Church. 36 Within a few centuries, it developed from a hobby into a tool that could be used to prove things about the universe. The most important reason for the quick spread of mathematics across Europe is the printing revolution, which allowed ideas and interest to spread quickly throughout the continent. This also meant increased literacy levels, which was likely to spark a general passion for education. The popularity of mathematics at the time is particularly evident in Italy where over 200 mathematics books were published over 30 years in the late 15th century to meet readers' demands. 37 This interest would have been unlikely otherwise and therefore key academics would not have travelled all over Europe to attend universities, such as the University of Bologna to read mathematical subjects. 38 During the Renaissance, humanism was emerging and people were questioning the Roman Catholic Church, with cases of corruption by important figures, such as Pope Alexander in 1511.39 As a result, people were questioning everything they had known, and likely to have desired proof after so many years of faith. As mathematics was able to provide clear, succinct proofs, it slowly developed into a science, from an art.