In 16th and 17th century Europe, the long Middle Ages had come to an end, the Renaissance had brought about an awakening, and the dogmatic authority of cumbersome philosophy and theology, which had constrained the free development of people's minds, was gradually destroyed. Feudal society began to disintegrate, and in its place came capitalist society, and the productive forces were greatly liberated. The prosperity of the capitalist factory crafts and the transition to machine production led to the rapid development of technical science and mathematics.

In the history of science, this period saw many significant events that raised new questions to mathematics. First of all, Copernicus put forward the geocentric theory, which fundamentally shook the geocentric theory, which was an important theoretical pillar of theology. His disciple Retticus saw that astronomical observations were becoming more sophisticated and that it was urgent to calculate detailed tables of trigonometric functions, so he began to make tables of sine, tangent, and secant every 10". It was all by hand, and Retticus and his assistants worked diligently for 12 years, only to be completed after his death by his disciple Otto.

During the Renaissance, projective geometry gradually developed due to the perspective method created by artists; After Fibonaccci's Abacus, a number of mathematical works appeared in Europe, which promoted the development of the theory and operation of decimal fractions; At the beginning of the 16th century, the most outstanding mathematical achievement was the discovery of algebraic solutions to cubic and quadratic equations by Italian mathematicians, some using imaginary numbers, and improving the mathematical notation of the time; In terms of the development of trigonometry, the Europeans also separated trigonometry from astronomy, making it an independent discipline, and redefined the concept of various trigonometric functions, and also compiled very precise trigonometric tables. In the Middle Ages, European mathematics was gradually developed after absorbing and digesting the mathematical knowledge of Greece and Arabia.

The discovery of cubic equations in Europe was made in Italy in the 16th century, when mathematicians often kept their discoveries secret and challenged their peers to solve the same problems. Presumably, this is an intellectually demanding and fascinating competition, and this is how the solution of cubic equations was discovered. Originally, a man named Fior, who had learned some cubic equations from someone else's secret tradition, challenged another man known as Tartaglia. He was intelligent and industrious, having mastered Latin, Greek, and mathematics by self-study. This time he succeeded in solving all the cubic equations proposed by Fior, but Fior could not answer the question he proposed. Caldán, who was famous at the time, begged him to teach him how to solve cubic equations and swore to keep it a secret, so Tartaglia wrote his method into an obscure poem and gave it to Caldan. Later, Caldan reneged on his word and published this method in a book published in 1545. In the book he writes: "Ferro of Bologna discovered this method almost thirty years ago and passed it on to Fior. Fiol gave the latter a chance to discover it in his match against Tartalia. Tartaglia told me the method at my pleading, but kept the proof. I found out all forms of proof of it with help. It's hard to do. Tartaglia was outraged by Caldan's perfidy, and he immediately wrote a book vying for the priority of this approach. He had an open conflict with Ferrari, a student of Caldan. Later, Ferrari solved the formula solution of the quadratic equation.

In 1545, the Italian scholar Caldan published the formula for finding the root of the cubic equation X^3+pX+q=0, Caldan was the first mathematician to write negative numbers in the quadratic root number, and thus introduced the concept of imaginary numbers, and later developed into the theory of complex numbers through the efforts of many mathematicians.

In numerical calculations, Steven systematically expounded and used decimals, and then Napier created pairs

number, which greatly speeds up the calculation. Later, Pascal invented the addition machine and Leibniz invented the multiplication machine, which, although not practical, opened up a new way of mechanical calculation.

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