[19847386_12_0] Galileo died in 1642. At very same year 1642 Issac Newton was born, born on Christmas Day, born to a mother whose husband had died just a few months earlier. Issac Newton would take the ideas that Galileo had started to pull together and he would bring them to fruition. Galileo had really recognized the five important threads that I've been talking about. Astronomy, algebra, geometry, mechanics, and the mathematics of motion. And throughout the seventeenth century various scientists would move these forward. And it would be at the very end of the seventeenth century that Sir Issac Newton as he eventually would become. Would take these threads and weave them into a masterpiece, that would solve the problem of celestial motion, that would solve the problem why it is we can be on an Earth that is travelling through the heavens at such incredible speeds without being aware of it.

And this great work that Issac Newton wrote is his Mathematical Principles of Natural Philosophy. The title itself is a reference to Descartes book, Principles of Philosophy. And Newton is a bit modest than Descartes was rather than laying out the foundations of all human knowledge. He only proposes to lay out the mathematical principles of what was then called natural philosophy, the natural world or the world of immediate experience the world of nature. And he took the tools that had been developed and really perfected them in this book that I will refer to from now on, more simply as Newton's Principia. The Principia almost didn't get written at the time that Newton began it in 1684, he really wasn't interested in questions of celestial mechanics, in questions of physics. At this point in his career he had moved on, he was trying to understand chemistry. He was also doing work in theology. But in 1684 Edmond Halley made a visit to Newton. And he raised the question why is it that the planets travel in elliptical orbits. At this point the astronomical observations were clear enough that it was obvious that Kepler's laws, that Kepler statements that the planets moving in elliptical orbits really had to be true. That agreed too closely to the observe data not to be true.

But the question was still out there why is it that the planets travelling in elliptical orbits. And Newton said to Halley, "well yes I can explain why that is true, and in fact I worked it out several years earlier." which as we now know he had done before. He tried to find the work that he done, he couldn't find it. He promised Halley that he would work it out over the next several months. He did work it oout, into a book that is now referred to as De moto. It was ever only a manuscript. And Newton realized though that as he put this together that there was a bigger story that needed to be told. That was a lot more investigation, a lot more pulling together of these threads of science and mathematics. And so he began work on the more ambitious work of the Principia. It would take him three years to write it. It was completed in 1687. And not only was this work that almost was not written, it's a work that almost was not published. Because the natural publisher for this book was the royal society in London. This was a [19847386_12_1] collection of the leading scientists in England at that time. It had been founded only a few years earlier and it was trying to publish important scientific books. But it had just finished publishing a history of fishes. And being a history of fishes it had a lot of plates, a lot of pictures within it. This is a very expensive publication. And the royal society had expanded all of its funds publishing a history of fishes and had not yet begun to collect any income from that. And so there was no money to publish the Principia. Fortunately at this point Halley stepped forward again and Halley personally put up the money that was needed in order to publish the Principia. And the Principia's an incredible work, because it begins by explaining inertia. And exactly what that means people at that time had a pretty good idea of what inertia was. But it was Newton who was the first person to clearly state inertia as we currently understand it, which is the tendency of a body that is at rest to stay at rest, but also the tendency of a body that is moving at a uniform velocity to maintain that uniform velocity, to keep moving in the same direction at the same speed. We also find in this book the first very clear statement of the relationship between force and acceleration, which is the rate at which velocity changes. And the very clear statement that force is directly proportional to the acceleration or change in velocity. So that if we double the force we're going to double the rate at which the velocity changes. And this would turn out to be foundational to Newton's understanding of celestial mechanics.

In this book, Newton's Principia, we also find calculus being used for the very first time. It shows Newton's great mastery of the ideas of calculus. To look at Newton's Principia doesn't look like he's using calculus because he's not using our modern terminology or modern notation. That would come later. That would be a product of Leibniz, who I'll about in the next lecture. But Newton was a master of the tools and ideas of calculus. And if Newton is described as one of the fathers of calculus it's because he understood it so thoroughly. The book itself is couched in the language of Euclidean geometry. Geometry is found throughout in a very sophisticated geometry. It uses Appolonius' conics in a very fundamental way. There also is all of the full understanding of algebra that's contained within the Principia. There is the mechanics of inertia, of force. And there is this mathematics of motion and of course there is astronomy. The entire third book is devoted to looking at actual astronomical phenomena and explaining them, showing how the model that Newton had managed to build, actually explains these phenomena.

Newton is quite known for stating "If I have seen a little further, it is by standing on the shoulders of giants." That's not a statement that was original to Newton. But it's certainly very true of him because he was building on the work of many earlier scientists, work of Napier, of Galileo, of Fermat and Descartes. But also many other scientists who followed Fermat and Descartes and I want to take some time to talk about their contributions. Because they are absolutely essential to understanding where Newton fits into this picture.

The first of the people I want to talk about is John Wallis. Wallis is born in 1616. He would die in 1703. He's someone who was very much caught up in the English civil war of the mid seventeenth century. In fact he worked for the Parliamentarian as a code breaker. And in 1647 he picked up a copy of Oughtred's Clavis Mathematicae, the Mathematical Key. It had been written in 1631, was a treatise on algebra that would be very influential. This is the book from which Newton himself would learn his algebra. And Wallis read this book on algebra and became fascinated with mathematics in the power of algebra. And two years after he started learning of algebra he became a professor of mathematics at Oxford University. I think that's probably one of the first cases in history of someone who two years after they learn algebra actually become a professor of mathematics. In 1649 he took over the Savilian Chair of geometry in Oxford. Six years later in 1655, he published Arithematica Infinitorum, Arithmetic on the Infinities. And this would be a very influential work for Newton. It has general methods for finding the slopes of curves. It has general methods for finding areas underneath curves. It also has incredible formulas, such as a formula for pi. Wallis realized that pi over two could be written as an infinite product: two over one, multiplied by two over three, multiplied by four over three, multiplied by four over five, multiplied by six over five. So for each fraction you either increase the numerator by two or increase the denominator by two. And you can show that as you take more and more terms in this product, it actually approaches the value pi over two. Wallis is also responsible for inventing our symbol for infinity. So the lazy eight, the eight on its side, this appears in his Arithmetic on the Infinities. It actually is a symbol that he had invented earlier than that. But it really is the Arithmetic on the Infinities that popularized this symbol for infinity.

Another one of the giants on whose shoulders Newton stood was Christiaan Huygens. Huygens was born in 1629. He would die in 1695. He was a Dutch scientist. His father was a scientist in Diplomat Constantin, also known as [19847386_12_2] a philosopher and he was close friend of Descartes and so Christiaan Huygens grew up knowing Descartes and being encouraged by Descartes. And Huygens' interests were all over the map. He became interested in astronomy and in 1655 he became the first person to discover a moon circling Saturn. The rings of Saturn were already known. But he became the first person to actually observe a moon that was circling Saturn. In 1656 he invented the pendulum clock. Clocks had existed before then, but not clocks that are regulated by a pendulum. And it was Huygens who realized that this would be a very effective way of key being a clock beating at a regular pace. Not only did he realize he could use a pendulum but he worked out a lot of the mathematics of a swinging pendulum and the mechanics that lies behind that. He is credited with having designed an internal combustion engine. He never actually built it. And it was an engine that was run on gunpowder. It's not clear if it actually would have worked or not. But he had this idea of an internal combustion engine. And he also was one of the first to pack in a pocket watch. That happened in 1675.

In 1661, Huygens met Wallis and this really marks the beginning of Huygens work in mathematics and also in physics. Huygens had a very clear idea of inertia although he never stated it as clearly as Newton eventually would. It's clear that Newton's understanding of inertia was also one that Huygens shared. Huygens is also often credited with being the person responsible for discovering the inverse square law of gravitational attraction. The fact that the force exerted by gravity decreases as you move farther away from the attracting body. And the force drops off as the square of the distance. So that if I double the distance I'm going to get a quarter as much force. If I triple the distance I'm going to get one ninth as much of the force. And this actually comes out of Huygens work. Considering the force that's necessary to keep a rock in its orbit as you swing it on a string. So you take a rock, tie it to a string, swing it around your head. You feel a force pulling on that string. What's actually happening as Huygens realized is that the rock is trying to travel in a straight line. This is the effect of inertia. The rock tries to travel in a straight line. What you're actually doing is constantly pulling on that string in order to keep it going around your head. And Huygens realized that this is the idea that lies behind the fact that the planets are travelling around the Sun. The planets are constantly trying to move off in straight lines and it's the Sun through gravitational attraction that's pulling them back in. This is an idea that Newton would come to independently and in fact the story of the falling apple probably is a true story. It's a story that was recorded by Newton's niece and she said that she actually heard it from her uncle, about one day he was sitting and he saw an apple fall and he began to think about gravity. Not gravity in the Aristotelean sense where gravity as an inherent property that causes things to try to move towards the center of the Earth. But rather gravity is a force in the same way that Huygens was beginning to think about gravity. And he thought about this force pulling on the apple and he wondered how far that force extends. Does it extend all the way to the Moon? If it does in fact extends all the way to the Moon, then perhaps what's really happening is that is the Moon travels around the Earth, it's constantly falling towards the Earth. But it's falling towards the Earth at the same time that it's moving. It's trying to move along tangent line. And so it moves along this tangent line and falls a little bit towards the Earth. It moves along a tangent line and falls a little bit towards the Earth. And compounding these motions, these different velocities as you pull them together you wind up with the actual motion of the Moon as it circles the Earth. This is something that Huygens probably realized but he never stated it as clearly as Newton eventually would.

[19847386_12_3] Huygens also will play a very important role in this story because he would become the mentor to Leibniz. Leibniz would travel to Paris, meet Huygens, and Huygens would teach Leibniz a lot of the mathematics that and physics also that Leibniz would need and would introduce Leibniz to much of what was known about the calculus at that time.

Another one of the giants on whose shoulders Newton stood was Isaac Barrow, born in 1630. He would become a professor of Greek at Cambridge University. And then a few years later while he was still holding his position as professor of Greek he would become a professor of mathematics at Gresham College in London. In 1663, he became the very first Lucasian Professor of mathematics. That's a very prestigious position. Today it is held by Steven Hawking. Barrow would become Newton's teacher of mathematics and in 1669 Barrow would go on to become the royal chaplin. And he would pass the position of Lucasian Professor of mathematics onto Issac Newton. In 1670 Newton prepared the Geometrical Lectures. These are lectures that Barrow had done but it was Newton who pulled them together. These are lectures that talk about the connection between areas and tangents. And so these are lectures that begin to tie together the two aspects of calculus, the differential calculus which deals with slopes of tangent lines, and the integral calculus that deals with areas and we see here that Barrow was beginning to understand the connections between these two aspects of calculus. What today we would refer to as the fundamental theorem of calculus. And Newton clearly understood this idea and was able to build on it very effectively as he wrote the Principia.

Another one of the giants is James Gregory, born in 1638, a Scotsman. He's someone who went to Italy. He studied with Angeli and Padua. And one of his important contributions is to figure out how to calculate the length of a curve. If you've got some curve traveling through space and you want to find how long it is. Gregory showed how to take that problem and translated it into a problem of finding the area underneath a different curve. And so that then could be done by the standard methods that we're known by that time for finding areas under curves, the integral calculus. Gregory also found general methods for finding power series. Now that's something that I'm going to be talking a lot more about later. Essentially what a power series is a polynomial of infinite degree. So I've got a sequence of polynomials, each one has higher degree than the previous one. And as I take polynomials of progressively higher degree I get closer to the actual function that I want to model. And in the limit we say that this polynomial of infinite degree which is called a power series is actually equal to the function in question. It has been said that Gregory might have been as great a mathematician as Newton himself. Unfortunately Gregory died at a relatively young age and so we do not know what his full potential might have been.

Newton had a very unfortunate childhood. As I said his father died before he was born. His mother remarried shortly after that. And Newton was packed off to live with her parents. And apparently he did not get along very well with them. He resented the fact that his mother had remarried greatly. And he went through life very much as an isolator. Someone who did not get along very well with others. And one of his students wrote of him, "Newton was of the most fearful, cautious, and suspicious temper that I ever knew." This was William Whiston, who would be a student of Newton's and eventually be his successor as the Lucasian Professor. Newton is someone who is constantly in conflict with the other scientists around him, arguing about who had come up with the key ideas that he had. Newton actually published very little about the calculus. The only real publication on calculus that Newton had was the Principia itself. And calculus is very hidden in that. You can't really sit down, read the Principia and learn calculus from it. So there were earlier manuscripts that he circulated with some of his ideas of the calculus. He had gone back home between being an undergraduate at Cambridge and then returning. During the year 1665 to 1667 there was a plague in Cambridge. He was forced to leave the city. He went home and that's when he really developed his ideas of the calculus. And he did circulate some of these ideas in manuscripts in 1666, in 1669 and in 1691. But it's not really true to say that Newton invented calculus in the sense of establishing the foundation on which others could build. That would be a job that would be left to Leibniz. Newton understood the calculus beautifully. He was able to use it very effectively. But he never really was really interested in [19847386_12_4] explaining it. And later on there would be a great controversy between Leibniz and Newton. Newton would claim that Leibniz had stolen his idea. Leibniz would claim that he in fact had been the first to come up with these ideas. And so there would be a great controversy not just between Newton and Leibniz but one that would spill over into a controversy between the English mathematicians and the continental mathematicians over who was responsible for laying the foundations of calculus.

Another example of Newton's inability to get along with others is his paper on optics. The first paper appeared in 1672. There were ten criticisms of that paper that were published. And one in particular 1675 Hook accused Newton of plagiarism, of stealing his ideas. Newton never forgave Hook for this and in fact at this point Newton threatened to stop publishing completely, that he was going to keep his ideas to himself. And this was part of the reason why in 1684 when Halley went to see Newton, Newton had all these ideas about celestial mechanics. But he had actually not gone forward and began to write them down, and in the form that actually could be published.

Newton had many other interests. He was also interested in chemistry or that time that really was alchemy, trying to understand the fundamental chemical nature of substances and he certainly spent far more time on chemistry than he ever did on physics or mathematics. But it was really too early to understand chemistry in a fundamental way. And nothing really ever came of Newton's work on chemistry. He was also fascinated with questions in theology and his works showed that he did a lot of exploring the bible, especially the Old Testament, looking for numerical patterns working the dates on which various events would have happened. Trying to see what the Old Testament could tell us about predicting the future. Newton eventually would leave his position at Cambridge. He would be elected to Parliament. He would become warden of the Royal Mint. And finally after Hook died in 1704, Newton would publish his other famous book on Opticks, an important work in which he explains reflection of light and he explains the idea of light as a particle, absolutely foundational work of understanding optics. Newton would be knighted in 1705, and then really spent the rest of his life in this battle with Leibniz over who had come up with the basic ideas of calculus first. As I said he was a difficult person to get along with. He would become president of the Royal Society. And he would maintain as president of the Royal Society until his death. And one of the things that he did was to encourage the Royal Society to make a statement about the fact that he, Newton, was the one who had invented the calculus, and that Leibniz had simply stolen Newton's ideas. And because of his great influence on the English scientific community of that time, Newton was able to ram this through the Royal Society and for this reason really alienated the continental mathematicians, the Bernoullis in particular I will talk about in the next lecture.

Newton is a complex figure. He was very much a loner. He never married. He never had close friends. He was totally focused on his science, his mathematics, his physics, his chemistry. And yet he was very reluctant to publish any of these. He was really a mystique. And one of the people who has been most fascinated with the story of Sir Issac Newton was the economist John Maynard Keynes. And in the early twentieth century Keynes began to pull together all of the manuscripts of Newton that he could find and he would pour over these and study them. And Keynes made a statement about Newton that I think is very much on the mark. We think of Newton as a modern scientist, the man who would set the stage for what modern science should be. But Keynes after looking at Newton's manuscripts, had a very different impression of Newton the man. And Keynes wrote, "Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10000 years ago. Isaac Newton, a posthumous child born with no father on Christmas day, 1642, was the wonderchild to who the Magi could do sincere and appropriate homage."

Next time we're going to turn to Leibniz and the Bernoullis. Newton truly understood calculus but he was not able to lay the foundation on which others could build from calculus. This task would fall to Leibniz and the Bernoullis and we will follow their stories in the next lecture. [End]

Wallis 使用了无穷符号。Wallis对于中学生最早接触可能是Wallis点和Wallis圆。这和我们之前上海很有名的一位老师还颇有渊源。

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av19847386 听写_Episode 13

Episode 13 - Newton Modeling the Universe

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