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From Precalculus to Calculus

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Calculus is the greatest achievement of mathematics in human history as fellows. And it’s a significant turning point in the development of human civilization. Now, all science and technology can be said to rely on calculus. Of course, others, such as non-Euclidean geometry, especially Riemannian geometry and group theory, are also very important.

Now, it’s accepted as a historical fact that: calculus was established independently by [English]Isaac Newton(1643-1727) and [German] Gottfried Wilhelm Leibniz (1646-1716). Newton was the first person to invent calculus(c.1665-1668) on earth. Leibniz invented calculus ten years after Newton(c.1675-1686), but he first published the differential paper in 1684, then published the integral paper in 1686. Newton did not reveal any his achievements about calculus until he published Philosophiae Naturalis Principia Mathematica in 1687.

Newton regarded calculus more as a mathematical tool for solving physical problems. And his superb application in physics showed that he had a profound understanding of calculus. Newton was a genius, and his book “Principles” had a great influence in the world.Leibniz, on the other hand, started study calculus from the direction of geometry, and he introduced calculus through analytics. Leibniz invented the calculus symbols that have been widely used now. His strictness, standardization and systematicness are especially praised.

But Newton-Leibniz’s calculus is very simple, John Bernoulli, Nopital, Lagrange, Euler, d ‘Alembert, Cauchy, Dirichlet, Riemann, Weierstras and other generations of mathematicians have made immortal contributions to the development and improvement of calculus after them.

1. The Giants Before Calculus

1.1. Ancient Dynasty

The development of mathematics has a long history, and the thought of calculus has been throughout it.

People of Ancient Egypt and Ancient Babylon probably formed the earliest knowledge of mathematics. they knew simple counting methods, simple solutions to equations and simple geometry. Later, Ancient India and Ancient China alse developed their own basic mathematical systems. However, it was only Ancient Greece that had the greatest impact on the entire human civilization and produced the highest mathematical achievement. Ancient Greece’s geographical ports, democratic political system, and open atmosphere of speech gave birth to great and free thoughts. Philosophy rose, and science such as mathematics, physics and astronomy also developed vigorously.

Ancient Greek map
Ancient Greek map

1.1.1. Thales and Milesian School

The first philosopher, Thales (c. B.C.624-B.C.546) of ancient Greece, founded the glory of the Melisian School.

His philosophy was that “All things are made of water” and he explained earthquakes by the movement of the sea. This plain naturalism weakened the influence of the religious worldview represented by Poseidon. He not only correctly predicted the solar eclipse, but also took its explanation out of theology. Thales also used similar triangles to figure out the height of a pyramid.

Thales measured the pyramids
Thales measured the pyramids

Thales’ contributions to geometry and astronomy were of great importance.

He paid attention to theory, gave consideration to application, and believed in thinking, which set an excellent example for the research methods of later Greek scholars.

He discovered basic geometric concepts such as the base angles of an isosceles triangle are equal, vertical angles are equal, and a circle is bisected by its diameter. He introduced the idea of proposition proof and proposed Thales’ Theorem, making him a pioneer of ancient Greek science.

Thales' theorem
Thales’ theorem:The inscribed Angle of the diameter is 90°

1.1.2. Pythagoras and His School

After Thales, a young man of great talent —— Pythagoras (c. B.C.580-B.C.500) left his native Samos and began to travel through the countries. He eventually formed the philosophy of “All is number” and created the Pythagorean school with his disciples.

Pythagoras was called “the Wise Man”, but he refused the title, saying: “I am not a wise man, but a philosopher.” From that time on, the term “philosophy” became popular.

Pythagoras
Pythagoras

Pythagoras was fascinated by “numbers”, and the more he studied numbers, the more he realized that everything is possessed by the power of numbers. His philosophical exploration led to his extraordinary attainments in mathematics.

On algebra, he found odd, even, prime, composite, perfect, amicable, square, and so on. His contribution to mathematics lies in his pioneering and development of the research method of “combination of numbers and shapes”.

The number of triangles beginning in Pythagoras
Triangles number beginning in Pythagoras

In the field of geometry, he found that the Sum of the inner angles of the Triangle is $ 180^\circ$(Triangle Sum Theorem) and used the reduction to absurdity method to prove that there are only five Platonic solids. That is, regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron and regular icosahedron.

The sum theorem of the inner angles of a triangle
The sum theorem of the inner angles of a triangle

The Pythagorean School already had a certain understanding of the golden section.The most famous achievement of Pythagoras is the proof of the Pythagoras theorem with deductive reasoning. In astronomy, he rejected the idea that orbicular sky and rectangular earth, and could even be said to be the first person to question the geocentric theory.

Pythagoras was the first high point in the development of human natural science. Of course, he was very mean to the series of actions of the disciple Hippasus (B.C.6th) who found $ \sqrt{2} $.

pythagoras theorem
pythagoras theorem: in $ \mathrm{Rt}_\triangle ABC$,if $ a $、$ b $ are two right-angle sides,$ c $ is hypotenuse ,then, $ a^2 + b^2 = c^2 $.

1.1.3. Other Scholars of The Same Period

At the same time, Herakleitos (c. B.C.535-B.C.475) and his disciple Kratylos (c. B.C.5th) insisted that “all things change”, and founded the dialectics of unity of opposites.

The Leucippus (about B.C.4th) and Demokritos (c. B.C.460-B.C.370), who initiated the “atomism”, had enlightening contributions in atoms, segmentation and limits and other scientific thoughts .

Demokritos'atoms thoery
Demokritos’atoms thoery

Whereas Protagoras (c. B.C.480-410), Gorgias (c. B.C.483-B.C.375), Hippias of Elis (c. B.C.5th), Bryson of Heraclea(c. B.C.5th), Antiphon (B.C.426-373), formed the Sophists, which mainly studied three major geometric problems in mathematics: Trisection of an Angle, cubic equation, and squaring the circle.

The quadratrix was invented by Hippias to solve the problem of turning a circle into a square, and the Method of Exhaustion proposed by Antiphon can be regarded as the originator of limit theory.

Circular cutting curve
Circular cutting curve

1.1.4. Three Masters of Greece

Then,Known as the “Greek Three”—— Socrates (B.C.470-B.C.399), Plato (B.C.427-347) and Aristotle (B.C.384-B.C.322)——have had a huge influence.

Plato made remarkable achievements in the field of philosophy, while Aristotle was great in the fields of philosophy, cosmology, dynamics, photology and so on. His authority had an influence throughout the Middle Ages and even the Renaissance. He focused on Logic and devoted himself to linking mathematics and physics.

Raphael Sanzio<the School of Athens>
Raphael Sanzio the School of Athens

1.1.5. Eudoxus

During this period, Eudoxus of Cnidus(B.C.410-B.C.355) invented a new theory of proportions to solve the problem of non-commensurability.Many people think that Eudoxus is the greatest Greek mathematician after Archimedes, because most of the content of the fifth book of Euclid is attributed to him.

method of exhaustion
method of exhaustion

Meanwhile, Hippocrates of Chios (B.C.470-B.C.410) also contributed to the study of turning circles into squares. Hippocrates may have been the first ancient Greek to write a systematic textbook on geometry, literally translated as Elements. A century after Hippocrates, at least four other mathematicians wrote their own books on the Elements, including Euclid’s version, which shows how influential this book was.The period of the Three Great Masters in Greece marked the second peak of the development of human natural science.

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