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1687, Newton published Mathematical Principles of Natural Philosophy. This masterpiece summarizes the research results of mechanics and marks the initial establishment of the classical mechanical system. This is the first large-scale synthesis in the history of physics, the product of the historical development of astronomy, mathematics and mechanics, and the crystallization of Newton's creative research. In this section, we mainly want to trace the origin of Newton's great achievements in human history and his creative process.

Newton lived in the background of the times mentioned above, and his life has been introduced in many monographs, so I don't need to go into details here.

The story of apple

The story of the apple landing has been widely circulated. According to Newton's letters, it can be proved that when he was young (1665- 1666), he did study mathematics and astronomy and thought about gravity. He wrote:

"At the beginning of 1665, I found the method of calculating approximation series and the law of simplifying any power binomial into such a series. In May of the same year, the calculation method of tangent was found, and the calculation method of differential was found in June 165438+ 10. In the second year, in June 5438+10, color theory was discovered, and the integral calculation method was studied in May. During this year, I began to think that gravity extended to the orbit of the moon. At the same time, after discovering how to estimate the pressure of celestial bodies moving in the celestial sphere on the surface of celestial bodies, I also deduced from Kepler's law that the period of planets is directly proportional to the 3/2 power of the center distance of their orbits, even though the force of planets in orbit must be inversely proportional to the square of the center distance between them and their operation. Then the force required to keep the moon in orbit is compared with the gravity on the earth's surface, and it is found that they are roughly equal. These findings were discovered in the two years of 1665 and 1666. "

This letter was written in 17 14. For more than 200 years, people have used this letter and other documents to explain Newton's creation. Although this letter does not mention the story of Apple, it shows that Newton began to think about gravity at least 22 years before the publication of Principles.

People have to ask: Since Newton had already calculated the inverse square law of gravity in 1665- 1666, why was it published more than 20 years later? Various explanations have been circulated in the past.

Some people say that Newton's calculation results at that time were too inaccurate because of radius of the earth's data, and he waited for 20 years out of caution.

Some people say that Newton's calculation only proved the motion of circular orbit, and the trajectory of the planet was elliptical, so he couldn't calculate it at that time. Only after he invented calculus himself can this problem be effectively solved.

It is also said that the story that Newton observed the apple falling to the ground may be true, because Newton told at least four people in his later years that he was really thinking about gravity. He must have thought of extending gravity to the moon.

Others say that Newton's letter 17 14 deliberately distorts history and is deliberately fabricated. Similarly, the story of Apple landing was fabricated by Newton himself and his relatives, probably out of the need of national defense priority.

For a long time (Newton's principles have been published for 300 years), there are few works about Newton. Newton's manuscript was set aside for neither study nor publication. It was not until recent decades that the research on Newton became active. Newton's letters and manuscripts were published one after another, books and periodicals about Newton came out, and several experts in the history of science and their schools who were famous for studying Newton appeared. They made textual research on some misinformation in the past, systematically studied the background of the book Principles, and analyzed Newton's life and creation. Now we can explain Newton's work more comprehensively, correctly and deeply. This paper only introduces the process of Newton's discovery of the law of universal gravitation. Readers may find that this process is more dramatic than the story of Apple's landing.

Newton's early research

Newton came into contact with Aristotle's theory of local motion during his college study, and later read the works of Galileo and Descartes. Under their influence, he began to study dynamics. Kepler and Brian (I. Bulliadus,1605-1694) stimulated his interest in astronomy, which led him to have the idea of proving the inverse square relationship of Brian's gravity. Brian once put forward a famous hypothesis in 1645: the force from the sun should be inversely proportional to the square of the distance from the sun. Kepler, on the other hand, guessed that the sun and planets depended on magnetism. 1664 in the first half of the year, Newton shook off Aristotle's influence and accepted Galileo's idea of attaching importance to experiment and mathematics. Descartes' thought of seeking "the first reason of nature" also greatly inspired Newton. The law of inertia, the law of collision, the conservation of momentum and the analysis of circular motion are all the results learned directly from Descartes' works.

Among Newton's manuscripts, it is particularly interesting that he wrote unpublished papers in the notebook of 1665- 1666. In these manuscripts, almost all the basic concepts and laws of mechanics are mentioned, the definition of speed is given, and the concept of force is clearly explained, which has actually formed a theoretical framework that was officially published later. He also derived the centrifugal force formula in a unique way.

The centrifugal force formula is the only way to deduce the gravitational inverse square law. Christian Huygens (1629— 1695) didn't publish the centrifugal force formula until 1673. Newton used this formula in 1665, which must be the result of his own independent work. However, the question is, from what angle did he understand centrifugal force at this time?

Let's trace back to his idea of deriving the centrifugal force formula from unpublished manuscripts.

1. Newton considered the motion of a small ball on a hollow spherical surface when analyzing circular motion to derive centrifugal force, as shown in figure 1-4. This object must be subjected to a force pointing to the center of the circle n. He first considers a half circle, and the force on the object can be found by two sides of an inscribed square. Newton put it this way:

Take a step forward, gain

Then extend it to any regular polygon, and get

So he wrote: "If an object is bounced by an infinite polygon (that is, the circle itself), the ratio of all bounced forces is equal to the ratio of all edges to radius."

In modern parlance, the ratio of centrifugal force to the integral of time to momentum is equal to 2π. The result is correct, but the meaning is vague, and the centrifugal force is not directly obtained. This is Newton's first attempt to deduce centrifugal force.

2. Then, Newton compared "centrifugal force" with gravity through circular motion and simple pendulum motion.

He used figure 1-5 to represent circular motion and simple pendulum motion. C moves along the circle Cgef, b swings along the arc with pendulum length ab=ad, and d is the center of the circle cgef. Newton wrote the following relationship:

"ad: DC = gravity: the force exerted on C by the center of gravity D."

3. In another manuscript of 1665, Newton wrote the following relationship:

"an object moving in a straight line under the action of centrifugal force is equal to a circular motion, and the radius of the circle is r, so when the circular motion passes the distance of r, the distance traveled by the object moving in a straight line is

This relationship is a special form of centrifugal force formula. Please see:

It is consistent with the results given by Newton, but Newton did not give the proof that led to the above relationship at that time.

4. In the manuscript of 1669, Newton's method of deducing centrifugal force formula was finally found. He adopted the number 1-6 and explained it as follows:

"When along the circumference AD, the force in the direction D from the center of the object A is as follows: in the time equivalent to AD, the object has a distance from the circumference, which is equivalent to the distance that the object can walk freely without force along the tangent.

"Assuming that this force acts in a straight line under the action of gravity, it will make the distance traveled by an object proportional to the square of time. In order to find the distance that ADEA has traveled in a circle, let's find a line segment. The ratio of this line segment to BD is exactly equal to the ratio of the square of the perimeter ADEA to the square of AD. "

Newton gave the answer in the manuscript, which is "equal to 19.7392 radius."

It is exactly equal to 19.7392R, so it can be seen that the relationship deduced by Newton is d=27π2R.

What does the above information show?

(1) confirmed that Newton mastered the centrifugal force formula at 1665, so it is entirely possible for him to deduce the inverse square relationship from circular motion;

(2) However, his idea of deducing centrifugal force is unique. Based on Descartes' collision theory and Galileo's time square relation, plus his brilliant mathematical ability, he got a mathematical relation with vague physical meaning. It can be seen that he did not clearly define the mechanical characteristics of circular motion at that time.

(3) Newton didn't realize the universality of gravity at that time.

Newton studied celestial bodies again.

1679, Newton put mechanical problems on hold for more than ten years. During this period, he founded calculus, a mathematical tool that made it possible for him to explore mechanical problems more deeply.

At the end of this year, Newton accidentally received a letter from Hooke, in which he asked about the whereabouts of objects on the earth's surface. Newton mistakenly regarded this path as a spiral line ending in the center of the earth in his reply. As Hooke pointed out, Newton admitted his mistake. But in reply to Hooke's second letter, he made another mistake. He deduced a kind of orbit, which was made when gravity was equal to a constant. Hook wrote back again, pointed out the mistake and said that he thought gravity was inversely proportional to the square of distance. These letters became the basis of Hooke's argument about the right of discovery. Newton thought that he had deduced the inverse square relation from Kepler's third law, but Hooke's views in the letter lacked a solid foundation, so he refused to acknowledge Hooke's achievements.

In fact, Hooke's advice was very important to Newton. Hook was the first person to correctly discuss circular motion and establish a complete concept. He regards circular motion as an unbalanced state, and thinks that some kind of force constantly acts on the object doing circular motion, destroying its linear motion and keeping it in a closed path. The communication between 1679- 1680 taught Newton a profound lesson. Later, he adopted the word "centripetal force" by Huygens, and proved in 1680 that an object in an elliptical orbit must be subjected to a force pointing to the focus, which is inversely proportional to the square of the distance from the focus. This work later became one of the cornerstones of the book Principles.

Inverse square law of elliptical orbit and the law of universal gravitation are different. By this time, Newton still didn't realize gravity. There is an example to prove that 1680 1 1 a big comet appeared in the eastern sky before dawn in June, and moved towards the sun until it disappeared; Two weeks later, another big comet appeared in the western sky after sunset, far away from the sun. J.Flamsteed, a British astronomer, insists that the two comets are actually the same, and the direction near the sun has changed by about 180. But he used a fantasy physics to deal with this problem, and regarded the interaction between the sun and the comet as the magnetic force between the magnetic poles, saying that the sun first attracted one pole of the comet and then repelled the other. Newton also observed those comets very carefully and made observation records himself. Interestingly, he claimed that these were two different comets. So there was a lot of communication between Newton and Franco. These letters show that Newton did not establish the concept of gravity, so he did not apply his theory to comets. At that time, like other physicists, he thought that only the solar system could be observed in inverse square law, and comets did not belong to the solar system, so they were not bound by this law.

Principle trilogy

As Huygens put forward the centrifugal force formula in 1673, more than one person successively deduced inverse square law from Kepler's third law, including edmund halley and Christopher Wren. At a party, Harley, Ryan and Hooke discussed the trajectory shape of an object in an inverse square force field. At that time, Hooke claimed that the laws of motion of all celestial bodies could be proved by inverse square relation. Ryan doubts Hooke's statement and offers a reward of 40 shillings if someone can prove it within two months. Hook insists that he can prove it, but he doesn't want to publish it first, in order to see who can solve it and then compete with it.

So Harley made a special trip to Cambridge to visit Newton in August 1684 and asked Newton about inverse square law's trajectory. Newton immediately replied that the trajectory should be an ellipse. Harley asked him, how do you know? Newton replied, I did the math. Harley wanted to see the calculation, but Newton pretended not to find it for fear of making a mistake like last time. But he recalculated according to Harley's request and sent the certificate to Harley. So, Harley soon received a nine-page paper from Newton. This paper has no title, and people usually call it "De motu". This is the predecessor of the book "Principles", which can also be said to be its first stage. Newton discussed the trajectory theory of objects under the action of central gravity, and derived Kepler's three laws from it. However, two key issues remain unresolved. One is the understanding of the law of inertia. In "On Motion", Newton still stays on the two basic concepts of internal force and external force. The "solid force" inside the object keeps the object in its original state of motion and moves in a straight line at a constant speed, while the external force changes the state of motion. He even combined these two forces with the parallelogram rule, and thought that the whole dynamics was based on the interaction of these two forces. This shows that Newton's theory also contains wrong concepts. The unit of measurement of "force" is mv, and the unit of measurement of force is ma. How do they combine into a force? This is against the law of inertia.

The second question is the nature of attraction. In "On Motion", Newton still called gravity gravity, but he didn't realize the universality of gravity, and he couldn't find the name of gravity.

However, Newton did not stop there. When he handed in the article "On Motion", deeper thinking led him to write a second paper, which was 65,438+00 times longer than the last one and consisted of two parts, named "On the Motion of Objects". It took him eight or nine months to write it and give it to Cambridge University Library as a handout, which is the second stage of Principles. Newton solved the problem of inertia in this paper. He admitted that circular motion is uniformly accelerated motion, which corresponds to uniformly accelerated linear motion. With the law of inertia, other problems will be solved. Another major development is the understanding of gravity. In On the Motion of Objects, he proved that a uniform sphere is attractive to every object outside the sphere, and the attraction is directly proportional to the mass of the sphere and inversely proportional to the square of the distance from the center of the sphere, and proposed that a uniform sphere can be regarded as the mass concentrated in the center of the sphere; Attraction is mutual; The operation of three-body proves the correctness of Kepler's law. He extended gravity to planetary motion and expounded the universality of gravity.

The second part of "On the Motion of Objects" was later included in the book "Principles" in the form of an appendix, entitled "On the World System", which highlighted the idea of gravity. He used a picture (as shown in figure 1-7) to explain why the planets kept their orbits unchanged under the action of centripetal force, and compared the projectile motion with planetary motion. He wrote:

"Because the centripetal planet will remain in a certain orbit, it is easy to understand if projectile motion is considered: when a stone is thrown, it is forced to leave a straight path due to the pressure of its own weight. If there is only one initial throw, it should move in a straight line. At this time, it draws a curve in the air and finally falls to the ground; The higher the throwing speed, the farther it flies before landing. So we can assume that when the speed increases to such a great extent, we will track an arc 1, 2, 5, 10, 100, 1000 miles long before landing, until it finally exceeds the limit of the earth and never touches the earth when entering space. "

This idea was put forward more clearly in Principles published by 1687. Newton finally understood the true meaning of gravity, unified the mechanics on the ground with the mechanics in the sky, and formed a mechanical system based on the three laws of motion.

Newton was interested in other forces in nature when he studied gravity. He considered the three forces known at that time-gravity, magnetism and electricity, and thought that they all acted within a perceptible distance, which he called long-distance force. He tried to find the laws of the other two forces, but failed. The result of magnetic experiment is not accurate enough. In the third principle, he wrote:

"The nature of gravity and magnetic force is different. ..... The magnetic force is not proportional to the amount of attracted substances. ..... As far as its relationship with distance is concerned, it decreases not with the square of distance, but with the cube of distance. This is the result of my rough test. "

As for electricity, he has also done experiments, but the movement of charged paper is too irregular to show the essence of electricity.

In addition to long-distance force, he thinks there is another force, which is called short-distance force. When he was doing optical experiments, he wanted to find the law of the force (short-range force) between light and matter, but it didn't come true. He even thinks that there are some other short-range forces, which are equivalent to polymerization and fermentation.

Newton stood on the shoulders of giants.

Newton wrote in his letter to Hook: "If I see farther, it is because I stand on the shoulders of giants." He refers to Hooke and Descartes here, and it goes without saying that it also includes Galileo, Kepler and Copernicus he mentioned many times. In fact, his comprehensive work is based on the numerous achievements of several generations of predecessors engaged in scientific research since the Middle Ages. We can make a table to illustrate the relationship between Newton and his predecessors:

Newton is good at inheriting the achievements of his predecessors, which is inseparable from his diligence and hard work. Someone asked Newton how he discovered the law of gravity, and he replied, "By thinking constantly." When he thinks, he forgets to eat and sleep. As I recall, he lived near the gate of Trinity College, Cambridge University. In the months after Harley visited him, he surprised many people to find that he was an eccentric. For example, he wanted to eat in the hall, but he took a wrong turn and walked into the street, forgetting why he came out, so he returned to his bedroom; In the hall, I sat there unkempt, absent-minded, with vegetable rice on the table, not knowing how to eat it. Colleagues in the university often see strange people on the gravel when they walk on campus. No one understands them, so they make a detour. Newton was preoccupied with celestial bodies.

Maybe some people think Newton is lucky. In his time, there were treasures everywhere and undeveloped virgin land everywhere, which was different from ours now. However, what we should learn is his spirit, and we must never regard him as a saint, thinking that he has achieved great achievements only by inspiration and genius. His pursuit of truth is not over yet, and it will never end. Please read his last words.

"I don't know what the world thinks of me, but in my opinion, I am like a child playing by the sea, happy to find a smoother stone or beautiful shell than usual from time to time; But the vast ocean of truth has not been discovered before me? "