Cao Zexian (researcher, Institute of Physics, Chinese Academy of Sciences, doctoral supervisor)

Sir isaac newton (1643- 1727) is an English mathematician, physicist and astronomer. Newton was a pioneer in mathematics and physics. His flow number and analysis based on infinite polynomials opened up the mathematical field of calculus, and the mathematical principle of his natural philosophy laid the foundation of classical mechanics. In addition, he was the first to observe and record the refraction of solar light with a prism and put forward the theory of light particles. Newton is regarded as the greatest scientist in human history. His epitaph was written by alexander pope, an English poet, imitating the first sentence of the Bible: "Nature and its laws are hidden in the night; God said, "Let Newton go! "Everything is light (the laws of nature and nature are hidden in the dark night; God said, "Let Newton come." The universe is bright. "

Newton is a familiar figure in today's world, so there is no need to waste more ink on his life and anecdotes. Let's look directly at his two great achievements, calculus and Newtonian mechanics, to see what inspired him and how Newton expanded the spark of inspiration into a learning system.

Binomial expansion and calculus

When we talk about calculus in English, we will use calculus, just like Euclid's Elements of Geometry. Use the definite article the to emphasize that the referred content is a unique existence that is respected. Calculus was once synonymous with profound knowledge for many people, and this situation may continue for a long time.

In English, the word integral is integral and differential is differential. Chinese translation of calculus into calculus. In fact, the original meaning of this word should be computing, a computing system. Finding the extreme value, the area of two-dimensional geometry and the volume of three-dimensional geometry are all old problems. Some achievements have been made in ancient China and Greece. /kloc-in the second half of the 0/7th century, there have been many viewpoints, methods and concrete discoveries about infinitesimal analysis, and it is time for someone to organize it into a brand-new knowledge. Leibniz, Germany published an article in 1684, in which the word calculi was used. When 1696 Frenchman Guillaume de lage? Spital wrote the first textbook in this field, Calculus, which became the name of this new knowledge.

Although Newton and Leibniz had an argument about the discovery sequence of calculus in history, one thing is certain: Newton studied and discovered calculus. So, what was the key step for Newton to discover calculus? It is a generalization of binomial expansion.

Binomial expansion, we began to learn in junior high school, the following formula is familiar to everyone:

As long as you like, you can get a binomial expansion of any power, where n is a natural number. Yang Hui Triangle (Pascal Triangle in Westerners) (Figure 1) summarizes the coefficients of all binomial expansions. Yang Hui Triangle is easy to remember: each line has one more item than the previous line, and always starts and ends with 1. The middle number is the sum of two adjacent numbers in the previous line. Don't think that you can easily know Yang Hui Triangle. This mathematical object contains rich and profound contents that ordinary people can't imagine.

Figure 1 Yang Hui triangle, the number in the nth row is the coefficient corresponding to each term in the expansion.

For most of us, this binomial expansion of knowledge is a strict dogma. However, in Newton's eyes, knowledge can be expanded, developed and used for transcendence. Great Newton extended the binomial expansion above to the case that the exponent is a fraction or even a negative number, that is, he will not only expand such binomial, but also expand such polynomial. Newton gave a general expression of the expansion, in which p and q are arbitrary real numbers and m/n is a fraction, that is

Here, a, b, c, D… stands for the one before the letter appeared. Of course, this extension includes an infinite number of projects. Newton used right expansion to verify whether his expansion was correct. The expansion he found was. Squaring the right side of this formula, you can find that the result is infinite series, which you can verify yourself. This is a well-known proportional series, and its sum is 0, which proves that the above expansion is correct. The world is wonderful, and this wonder needs people like Newton to lift the veil on it.

In this way, the form of infinite series can represent the general function f(x). Newton further developed the method of finding inverse series, that is, starting from infinite series, the series was obtained. The extension of binomial expansion and the method of finding inverse series are important tools for Newton to develop calculus.

With this binomial expansion, Newton has to prove that the curve is from 0 to any x (x >; 0) Floor space. Newton's argumentation process on this issue gives people a sense of disorder and contains a lot of logical loopholes, so it is known as "a concise and incomprehensible form". In any case, Newton's argument can be expressed in modern mathematical language as follows: For any real number A, the first differential of the function is. With this relationship, the development of calculus has embarked on a smooth road.

In Newton, differential is called flux and integral is called the inverse method of flux. Fluxion, like other English words expressing flow, such as flow, continuous and flux, is a cognate word, which is related to flow or speed. Take the change of position with time as a function of time, and the flow number or differential of this function is speed. Everything is flowing, and the equation of physics is essentially the equation of flow.

Apples and gravity

There is a magical legend about Newton, saying that one day Newton was sitting under an apple tree, and an apple [1] happened to fall on his head, which made him realize the mystery of gravity. Some people say that this legend is fictitious, but it also appears in the later works of his acquaintances. However, due to Newton's great influence, people would rather believe that this legend is true. Newton's alma mater, Trinity College of Cambridge University, planted such an apple tree (as shown in Figure 2), which is said to be the descendant of the apple tree that inspired Newton. This is a memory symbol about the moment of great discovery or the inspiration of great discovery. It is easy to introduce the descendants of Newton's apple tree, but the apple that can be inspired has not waited for Newton's brain that can be inspired.

Fig. 2 Apple trees planted later by Trinity College, Cambridge University.

Not to mention whether there is such an apple tree, an apple on that tree fell on Newton's head, which inspired Newton, so that he could understand the mystery of gravity. To be sure, when Newton studied the laws of planetary motion, he noticed the falling motion on the earth, and the falling of ripe fruits was a natural free falling motion. In fact, long before Newton was born, Galileo had drawn the law of falling body, and Kepler had realized the three laws of planetary motion.

Long ago, people thought that force led to movement. A great progress in the history of human cognition is the understanding of the law of inertia. All objects have inertia, and objects that are not subjected to external forces remain stationary or move at a uniform speed (this was later expressed as Newton's first law, but it was actually recognized long before Newton)-force is the reason for the change of motion. At that time, the forces people talked about were contact forces such as pressure, friction and thrust.

The motion of planets in the sky makes countless people curious. Many ancient civilizations in history have observed the movement of planets. Kepler took the sun as the reference point of planetary motion from 1609- 16 19, thus summing up the famous three laws of planetary motion. The first law says that the planet moves in an elliptical orbit with the sun as one of its focal points, and the second law says that the planet sweeps the same area relative to the sun in unit time. Why is this? Or what kind of force makes the planet take this form of motion? People want to answer such challenging questions.

Planets fly forward and constantly change the direction and speed of their movement. An intuitive idea is that the forward pull of the stock index will push the earth forward. But where does this power come from? If there is, then the source of this force must not be contact force, but should be action from a distance. Recognizing the existence of the function of distance is a great progress in the history of human cognition. So what should this distance force look like?

Perhaps the falling apples inspired Newton. Once the apple broke away from the tree, it immediately fell straight to the ground, indicating that the distance effect of the earth on it has always been there. Perhaps the earth has such a distance effect on the moon, the sun and those stars in the sky, and of course the sun should also affect the movement of planets with such a distance effect. On the other hand, the apple fell on its head and hurt it because it blocked its way. If you don't touch a person's head, it will always fall to the ground. If there is no ground or a well is dug in the ground, the apple will keep falling. God, that apple will always fall to the center of the earth. That distance force, exactly the attraction of the earth to apples, always points to the center of the earth! At this time, Newton should have realized the true meaning of gravity or gravity: gravity exists between all objects, is a function of distance and a central force.

So, if the gravity between the sun and the planet is centripetal, can this explain the observational nature of the planet's orbit, that is, Kepler's three laws? Newton assumed that the gravity between objects is a centripetal force along a straight line between them, and its magnitude is inversely proportional to the square of the distance. He used plane geometry to prove that such planetary orbits are indeed ellipses with the sun as the focus. With this conclusion, Kepler's second and third laws can be easily proved. Newton's proof of Kepler's first law is contained in his book Mathematical Principles of Natural Philosophy. Figure 3 shows a simplified diagram of Newton's proof on a pound note. When the author looks at this picture to prove it, like a destitute person facing a diamond weighing two kilograms, he is surprised and at a loss. If someone thinks he is good at plane geometry, try to understand Newton's proof process. Later, Chandraseka rewrote this proof. Of course, it is much longer and not necessarily easier to understand.

Figure 3 Newton and his typical deeds on the back of a pound note. The pattern in the upper left part is the same as that in Newton's book, which is a geometric proof that the planetary orbit is oval under the action of heart force in the mathematical principle of natural philosophy.

With calculus and gravity, the science of classical mechanics was finally established. It is important that Newton's work is a model of rational thinking. When Newton was born, science in the western world had not yet reached the dominance of medieval ignorance. By the time of his death, the west had entered a rational era, and Newton made great contributions to it.

Newton's deepest revelation to the author is that a great scientist should not only have profound and bold ideas, but also have the ability to prove himself right. Weaving new ideas and evidence (or demonstrations) together is a knowledge system.

Interestingly, there was a saying of "Newton" in ancient China. There is a saying in Cao Cao's Autumn Hu Xing: "Newton can't afford it, and the car crashed into the valley."

This article was originally the sixth article of Cao Zexian's "The Difference of One Thought-How Great Scientists were tempered" (Foreign Language Teaching and Research Press, 20 16.5), and it was slightly supplemented when it was published.

To annotate ...

1. This is the second most famous apple in the history of human civilization. The first is the apple that the snake lured Eve. The third apple was stained with potassium cyanide, and the genius alan turing took a bite and ended his creative career.

This is another great event in the history of human civilization. Humans have moved the reference point of celestial motion from their own feet to other places.

refer to

1. Gottfried Wilhelm Leibniz, a new method of maximization and minimization, unreasonable fractional quantization morality. Strange e proilli calculigenus, Journal, 467–473, 1684. The new method of finding the maximum, minimum and tangent is also applicable to the calculation of fractions or irrational numbers and a special case (Latin).

2. Paris, Paris, 1696, hospital, infinite small article analyst. Understand the infinitesimal analysis of curves (French).

3. William Dunham, Calculus Gallery: Masterpieces from Newton to Leberg, Princeton University Press (2005). The Chinese translation is Calculus Course-From Newton to Leberg.

4. Richard Vestfold, Endless: A Biography of isaac newton, Cambridge University Press (1983).

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The preface of The Wholehearted Author is extraordinary.

The view that human history was created by a few heroes has been severely criticized. But if "the history of science is created by a few heroes", it is estimated that there will not be too many people who oppose it. Because, when it comes to some great scientific achievements, don't say that you dare to fight for credit. Ordinary people like me are proud just because they can only understand scales and claws. Of course, when I say science here, I mean mathematics, physics and other disciplines that have formed a rigorous and heavy system.

The history of science is the glorious history of a few giants. Some people, such as Galileo, Hamilton, Euler and Sir Kelvin, can penetrate the unknown fog of existence and bring intellectual enlightenment to mankind. Some people, such as Poincare and Gamow, came to this world to show the world that genius exists. Every time I read the thoughts of these masters and appreciate their genius achievements, I always feel inexplicable excitement and sigh-why hasn't such a scientific giant appeared in this oldest land that has nurtured the largest part of mankind?

Maybe it's not that we were born mediocre, but that we didn't see the power of example when we were hungry, and no wise man gave us an enlightenment baptism? ! Those giants of science got full and high-quality education and enlightenment in their youth. Teenagers don't know what really great ideological achievements are, what kind of talents are real masters of science, and naturally they don't have the desire and ambition to become masters. So, where is the possibility of producing scientific giants from among them?

I don't know since when, I have always had a wish to share with my friends, especially those energetic young friends, the ideas and achievements of those scientific giants I know. However, it is far beyond the author's ability to correctly understand and accurately convey the thoughts and creative achievements of these scientific masters. With the idea of settling for second best, the author wrote this little book, introducing to readers the extraordinary idea of those scientific giants to achieve scientific status-perhaps just an accidental idea, but it later became a landmark event in the history of science, which brought unexpected impetus to science and further human civilization. I sincerely hope this little book can help young friends, from middle school students to young scholars who have just entered the scientific research career.

This book focuses on people and things, limited to philosophy, mathematics and physics. There is no other reason, just because some masters in the fields of philosophy, mathematics and physics have different charms, and their achievements have aroused special reverence in my heart. According to Kant, this is "the more I think about them regularly and persistently, the more they fill my heart with new and growing surprises and awe." Readers will notice that some of the people mentioned in this book are the trinity of philosopher, mathematician and physicist, and some are even linguists or other experts at the same time. Although it may not be correct, I still think that as far as the components of mental work are concerned, the requirements of physics in intelligence and personality are not as good as those of philosophy and mathematics. Physicists who are not mathematicians are always separated from physics. It is an indisputable fact that there are very few philosophers and mathematicians who are not worthy of the name, let alone those who cheat the world and steal their names.

This book contains 30 short articles, of which the first 25 generally introduce an extraordinary idea of a great scientific giant when he made great achievements; Article 26 is about the arrogant loneliness as a scholar's character-only those who meditate in loneliness can catch a glimpse of the faintest light in the dark; The next three articles tell how a middle school teacher, a factory apprentice and a farmer influenced science with an extraordinary idea. The last one is about how ordinary people enjoy learning and participate in scientific creation. While keeping the content easy to understand, I still insist on adding some profound things, including mathematical formulas. There is a saying that for every extra formula, the readers of the book will be reduced by half. According to this statement, the number of readers of this book will be less than 1. But I don't quite believe this statement. Every healthy person likes a challenge. The reason why mathematical formulas can scare away readers may be because the formula appears in a blunt and bluffing way. Just like some people translate German philosophy into obscure texts, the problem lies in the immorality of the author or translator rather than the incompatibility of mathematics or philosophy. If readers are not interested in the mathematical formulas in this book, they might as well skip them without affecting the whimsy of those scientific giants. However, I hope readers can try their best to understand these seemingly difficult things and don't regret returning to Baoshan empty-handed. After some chapters, I will list some very professional references. I don't think it is necessary to list the contents of these majors in such a book. Although these documents may be written in a language you don't know yet, or their contents are not easy to understand, if these contents arouse your passion to become a master of science, then these documents may become your stepping stone. One day, you can read it-maybe you still think it's simple.

This little book is the author's study notes, his experience and a sigh from the bottom of my heart. Because the author's skills are contemptuous, although we can show our friends the remains of scaffolding in the process of great academic creation, it is not enough to give people a glimpse of the road of scientific creation. Friends who are interested in the scientific cause, please try to read the works of the scientific giants as soon as possible, and reach an understanding of the great achievements of the masters from a unique perspective as soon as possible.

One advantage of understanding the people and achievements of science masters is that it will make you humble-sincerely and sincerely.

The reading object of this book is anyone who is interested in it. The author has no intention or ability to control the difficulty at a certain level; In fact, the author simply does not accept that knowledge can be divided into different levels that middle school students, college students, graduate students and professors can understand. If a book can make different readers gain more or less, even if it is just a knowing smile when reading, it is success. Besides, if the books we read are not difficult at all, where does our progress come from?

You must have noticed that there are no flashes of many great ideas mentioned here. It's okay. This is an open series, and more content will be added in the future.

The nation to which I belong, though full of disasters, is endless, and will make substantial contributions to science in any case. To this end, we should learn how to learn and how to create. Do it, Anglefen, Softer!

Cao Zexian

2065438+2003 Spring Writing

20 15 finalized in Beijing in autumn.