Europe's Dark Ages Mathematics during this period was largely restricted to arithmetic and geometry until the mathematical knowledge from other cultures was reintroduced to Europe, paving the way for the Renaissance. Mathematics in the Islamic World The Islamic world was home to many great mathematicians during the Middle Ages, including Al-Khwarizmi and Ibn al-Haytham. Their work was hugely influential in the development of European mathematics. The Recovery of Ancient Knowledge Eager to recover the knowledge of the ancient Greeks, scholars in the Middle Ages sought out lost texts on mathematics and science. This laid the groundwork for the Renaissance's mathematical explosion. The Birth of Modern Geometry The work of mathematicians like Descartes and Euclid led to the creation of modern geometry, a branch of mathematics that deals with the properties of space and figures. The development of Calculus Perhaps the greatest achievement of the Renaissance was the development of calculus, a set of mathematical techniques used to study continuous change. The Introduction of Negative Numbers During the Renaissance, negative numbers were introduced into the mathematical lexicon, expanding the possibilities for mathematical reasoning. Integration with Art During the Renaissance, mathematics and art became intimately connected, with artists embedding mathematical principles into their work Impact on Architecture The development of new mathematical techniques allowed architects to create structures of incredible complexity and beauty, such as the dome of Florence’s Santa Maria del Fiore Application to reality In the Renaissance period, due to the needs of production and commercial development, practical mathematics represented by “commercial arithmetic” developed rapidly, resulting in “popular mathematics” that everyone could learn. The Golden Ratio The Golden Ratio, a mathematical concept that appears frequently in nature and art, was used throughout the Renaissance as an aesthetic principle. Perspective Drawing The development of perspective drawing, a mathematical technique used to create the illusion of depth and space, revolutionized painting during the Renaissace. The Fibonacci spiral The Fibonacci spiral reflects a geometric code found throughout the natural world. Named for the mathematician Leonardo Fibonacci, the spiral emerges from a sequence of numbers where each term is the sum of the two preceding ones (1, 1, 2, 3, 5, 8, 13, 21, etc.). As squares with areas of these Fibonacci numbers are connected, a spiral shape appears. Strikingly, the spiral represents the "golden ratio" of proportions that generates balance and beauty. In math and nature alike, the Fibonacci spiral signifies a vital archetype of harmony. Golden Ratio Triangle The Golden ratio triangle has proportions that match the golden ratio (1: 1.618). This ratio reflects a naturally pleasing balance between two unequal parts that many philosophers, artists and mathematicians believe represents an innate sense of beauty and symmetry in the cosmos itself. This divine proportion has been an object of fascination for millennia, coded in both the measurements of sacred geometries and the very frames that shape reality. Perspective Drawing - Vanishing Point: the spot at which parallel lines appear to converge in the distance. All lines that are parallel to the viewer's line of sight will seem to meet at the vanishing point. - The horizontal line: represents the viewer's eye level and is parallel to the horizon. Objects below the horizontal line will appear tilted upward, while those above it will seem tilted down. - Orthogonal are lines that travel from each corner of the object being drawn to the vanishing point. They establish the orientation and proportional diminution of the object within the perspective plane. Architectural Innovations Renaissance architects used new mathematical techniques to create structures of incredible complexity and beauty, such as the dome of Florence’s Santa Maria del Fiore How it fit to a bigger picture? - The development of mathematics during the Renaissance was deeply tied to the exploration and cross-cultural exchange of the era. As new trade routes opened and contact increased between Europe and the Islamic world and Far East, mathematics flourished through the spread and fusion of knowledge across regions. - The Islamic Golden Age had preserved and advanced ancient Greek and Indian mathematics, including the decimal number system, algebra, and geometry. This knowledge flowed into Europe through trade networks and the Moorish conquest of Spain. - The Arabic numeral system revolutionized commercial arithmetic with its ease of calculation relative to Roman numerals. Islamic algebra, trigonometry and geometry also infused Renaissance mathematics. Meanwhile, the Far East had a long mathematical tradition in areas from geometry to number theory. The abacus allowed fast computation with decimals and fractions. - Knowledge of Chinese mathematics came to Jesuit missionaries in China who translated works into Latin and brought some of the first samples of Chinese thinking to the West. The new sea routes also enabled wider trade and the exchange of ideas around the globe. From algorithm to algebra, geometry to the abacus, a vast array of Eastern and Islamic mathematics fused in the West during an age of global contact and exchange. The flows of knowledge across trade routes enabled by the Great Voyages of exploration fueled the renewal of mathematics in Europe and the advances in the arts of calculation, measurement, and theory that came to define the Scientific Revolution. Rene Descartes Rene Descartes was a French philosopher, mathematician and physicist. He made important contributions to the development of modern mathematics. In 1637, he invented the coordinate system, one of the fundamental tools of modern mathematics. He combined geometry and algebra to create analytic geometry. At the same time, he also derived the Cartesian theorem and other geometric formulas. It is worth mentioning that legend has it the famous equation of the heart-shaped curve was also proposed by Descartes. Isaac Newton The creation of calculus was Newton’s most remarkable mathematical achievement. To solve the problem of motion, Newton created a mathematical theory closely related to physics. Newton called it the ‘fluxion.’ Some specific problems it dealt with, such as the tangential problem, quadrature problem, instantaneous velocity problem and function maximum and minimum problem, had been studied before Newton. However, Newton surpassed his predecessors and stood at a higher level. He integrated previous scattered conclusions. He unified various techniques for solving infinitesimal problems since ancient Greece into two kinds of ordinary algorithms: differential and integral. He established the reciprocal relationship between the two kinds of operations. By doing so, he completed the most critical step in the invention of calculus. He provided the most effective tool for the development of modern science. He opened up a new era in mathematics. Newton was slow to publish his work on calculus. He may have developed it earlier than Leibniz, but Leibniz had a more rational form of expression, and published his work on calculus earlier. The discovery of calculus in the 17th century by Isaac Newton and Gottfried Leibniz led to a bitter dispute over who deserved credit for its development. Newton discovered calculus in the mid-1660s, though he did not publish his findings until 1687. Newton used calculus to solve the problem of finding tangent lines to curves and the maximum and minimum points of functions. He considered calculus a mathematical extension of his work in physics. Newton's approach was primarily geometric, conceiving of quantities as generated by flowing particles. Meanwhile, Leibniz developed his own system of calculus in the 1670s, publishing his first paper in 1684. Leibniz's calculus was more rigorously logical, using algebraic symbols to denote rates of change rather than geometrical arguments. Leibniz claimed his work was independent of Newton's, though he admitted reading and making notes on an early description of Newton's methods. The dispute erupted in 1711 when supporters of Newton accused Leibniz of plagiarizing Newton's methods. The debate deteriorated into a bitter nationalistic rivalry between British and Continental mathematicians. Newton himself insinuated Leibniz had stolen from his unpublished ideas. Leibniz defended himself from charges of plagiarism and accused Newton's allies of misconduct. Key scientists formed factions defending one thinker or the other. Ultimately, Newton's reputation and prominence as president of the Royal Society gave his version dominance in Britain, while Leibniz's prevailed on the Continent. Today, most mathematicians recognize Newton and Leibniz as independent inventors of calculus, with distinct notations and conceptualizations of the new discipline. Though the methods differed, both thinkers grasped the potential of the infinitesimal to capture rates of change and the relationship between areas and tangents that would come to underlie all of mathematical analysis. This dispute marked one of the first major priority struggles in the scientific revolution, revealing how rivalry and nationalism could fuel acrimony, even among thinkers dedicated to the pursuit of mathematical truth. The conflict cemented the legend and fame of calculus as a jewel in the crown of mathematics.
