Ch 5 - The discovery of gravity This chapter is about how gravity was discovered. Nowadays gravity is something we take for granted, but it actually took thousands of years for humans to figure it out. The people who finally did figure it out, Isaac Newton and Albert Einstein, are revered as two of the greatest minds of all time. Gravity is easy enough for a sixth grader to understand, but *figuring out* gravity in the first place is really, really hard. I want to walk you through the discovery of gravity in some detail because seeing how the scientific sausage is made is useful if you are going to try to make some of it yourself. In particular, it's helpful to remember that while you are in the midst of trying to figure something out, progress can at times be painfully slow, even for the sharpest minds. Unfortunately, in order to talk about gravity, we are going to have to do a little bit of math. I promised earlier to keep the math to a minimum, but in this case it's unavoidable. We are going to have to crunch a few numbers. I promise to make it as painless as possible. --- Suppose I asked you what is the next number in the following series: 1 2 3 4 5 6 You would probably say "7". Now suppose I asked you what is the next number in this series: 1 2 3 4 5 6 13 50 Or this one: 1 2 3 4 5 6 7 8 9 10 11 12 2134 14 15 16 Those are a little harder. You might even suspect that the third one is some kind of a trick question, that it doesn't actually have an answer. And you would be half-right. It is a trick question, but not because it doesn't have an answer. It does. Before I tell you the answers I want you to stop and think deeply about this, but not as a series of math puzzles. Remember, this is a book about the scientific method, which is the process of coming up with the best explanation that accounts for all the data. Imagine that these numbers were the results of a series of experiments, the result of some instrument measuring some physical process, and you were trying to come up with an explanation for why you got the results that you did. One reasonable possibility for the first series is that the system you are studying is some kind of a counter. Every time you do an experiment, the counter is incremented, and the result you get is the value of the counter. This explanation would lead you to expect that the next result should be a 7. So imagine that you actually did that next experiment and instead of 7 the result was 13. Your simple explanation is falsified. It is not possible that the system you are studying is just a counter. There must be something else going on. What is it? Well, one possibility is that the 13 was just an anomaly, a glitch, an experimental error. Maybe you were hallucinating. Maybe someone tampered with your measuring device. Maybe the battery is low. These are all possibilities that you have to consider if what you are studying is a real physical system. But let me stipulate here that you were not hallucinating. The measuring device you were using is working properly. You do the experiment 1000 times and every time you get the same sequence of results: 1 2 3 4 5 6 13. So now what? Maybe this hypothetical situation is not possible? That's obviously not the case. It is obviously possible to write a computer program that could produce that sequence, indeed *any* sequence, simply by putting a copy of the sequence into the program. But that doesn't help figure out what the next term would be. If this sequence was produced simply by copying some random pre-defined sequence, then the next term could be *anything*. One possible way to make progress on this puzzle is to cheat and look up the answer on the internet. There is a dedicated search engine specifically for this purpose at https://oeis.org/ and if you look up 1 2 3 4 5 6 13 you will find that there is more than one answer. Ten of them, in fact. And depending on which answer you choose, the next term could be 17 or 12 or 34 or any one of seven other possibilities. And in fact you could have guessed this simply by knowing that the second puzzle had an answer! Because if the second puzzle has even one answer, then the first puzzle must have at least two answers: 7 and 13. And if the first puzzle has more than one answer then the second one probably does too. What about the third puzzle? If you look that up on OEIS it will tell you that there isn't an answer, at least not one that it knows about. But I have already told you that it does have an answer. I have also told you that it is a trick question! Let's table that puzzle for a moment and talk instead about something more science-y. We will return to the puzzle at the end. === I picked gravity as an example because it is less controversial than the thorny existential questions of the previous chapter. There are people who deny that the earth is round and there are people who deny the lunar landings, but as far as I know there is no one who denies the existence of gravity. It's just *obvious* that gravity exists. You can *feel* it. If you find yourself beginning to doubt the existence of gravity all you have to do is go bungee jumping or skydiving and your faith will be restored. But it was not always so. The existence of gravity is actually far from self-evident. In fact, the common idea taught in high school physics classes that gravity is a force that pulls massive objects towards each other actually turns out to be wrong. Gravity is not a force but an effect of the curvature of space-time, but we are getting ahead of ourselves. If you were to go back to ancient times and proposed that gravity was real you would have been laughed out of the room. It is just *obvious* that this "gravity" theory cannot possibly be true because of something that is self-evident to anyone: if gravity were real the stars and the moon and the planets would fall to earth, and obviously they don't. If you tried to offer up the modern explanation to counter this objection you would have been laughed out of the room even more definitively because, again, it is just *obvious* that the laws that govern the behavior of the heavens must be radically different than those that apply here on earth. The stars move with constant speed and never speed up or slow down. Nothing on earth exhibits that behavior. Tracing the evolution of ancient thought about the movements of the heavenly bodies makes a fascinating historical study. Entire books can be (indeed, have been) written about it, and so I'm not even going to attempt even a cursory review. But there are a couple of highlights that are important to know. Keep in mind that the telescope was not invented until 1608, so the only data about the heavens that anyone had to go on before that was what could be seen with the naked eye, namely: the sun, the moon, the stars, five planets (Mercury, Venus, Mars, Jupiter and Saturn) and the occasional comet and meteor. That's it. Of these heavenly objects, the stars are obviously the most numerous and their behavior is very regular: they move in (what appears to the naked eye to be) perfect circles centered around Polaris, the north star. At least that is what you see in the northern hemisphere. In the southern hemisphere things are a little different, but most of the world's ancient civilizations just happened to be in the north. The movement of the other objects in the sky are all considerably more complicated. The sun, moon, and planets all move relative to the stars in semi-regular but still fairly complex patterns. Two of the planets, Venus and Mercury, are never seen at more than a certain angle away from the sun (about 20 degrees for Mercury, 45 for Venus), and so can only be observed near dawn and dusk. The other three planets, Mars, Jupiter and Saturn, are not so constrained. They can appear at any angle from the sun. Jupiter and Saturn always move in the same direction relative to the stars, but not at a constant speed. Sometimes they move faster, sometimes slower. Mars *mostly* moves in the same direction as Jupiter and Saturn, but with even more extreme variations in speed, and about every two years Mars will stop and move *backwards* for a few months before resuming its more usual direction. Explaining this "retrograde motion" of Mars was a major project of ancient astronomers. The person first credited with figuring it out was a Greek polymath named Claudius Ptolemy, who lived in or around Alexandria in Egypt (then a province of the Roman empire), between about 100 and 170 CE. Ptolemy advanced a theory called *epicycles*, which goes something like this: everything in the heavens moves in circles. (This is obviously true of the stars.) The sun, moon and planets also move in circles but their motions appear more complicated because of two things: first, the circles are tilted so that we see them edge-on or at an oblique angle rather than top-down, and second, the circles can be *nested*, so that the actual motion is not necessarily a simple circle, but a circular orbit whose center moves in a larger circular orbit. Sometimes these "epicycles" can be nested more than two levels deep. The Ptolemaic epicycle hypothesis is *almost* a geo-centric model but not quite. It was the conventional wisdom in those days that the earth was located at the center of the universe, and that the reason the stars move in circles is because they were physically attached to a sphere made of crystal or some other transparent material. The planets moved differently because they were attached to their own crystal spheres. The problem is that if you try to make the math work out, it doesn't. Ptolemy had to offset the earth from the center of rotation, and also add epicycles in order to produce accurate predictions. But having done that, the ptolemaic model did produce very accurate predictions for over 1000 years. In the mid-13th century, though, the predictions of the Ptolemaic model began to diverge from observations by as much as 1-2 degrees. Astronomers tried to correct these errors by adding more and more smaller and smaller epicycles. This actually worked, but made the task of carrying out the computations even more difficult than before. (Remember, no electronic computers back then. All of the calculations had to be done by hand.) Still, the Ptolemaic model went unchallenged for another 300 years, until 1543 when a book by Nicolas Copernicus called "De revolutionibus orbium coelestium" was published -- posthumously. The idea presented in that book was so radical that Copernicus refused to allow it to be published while he was alive. It was that the sun, not the earth, was at the center of the universe. The argument that Copernicus advanced in support of this position was very complicated in its details (he wrote an entire book about it) but conceptually trivial: putting the sun at the center made some of the math easier. This was *not* a persuasive argument in its day. Copernicus was not the first to advance a heliocentric model. That idea went back as far as the ancient Greeks, and it was *obviously* wrong to the point of being laughable. For example: you can do the math on how fast the earth is moving, and the result is an unfathomably large number -- tens of thousands of miles an hour. The idea that *anything* could move that fast, let alone the entire world, was patently ridiculous in 1543. The winds that would result would instantly sweep everything from the face of the earth! Today we know that Copernicus was mostly correct despite getting many of the details wrong. The earth is not the center of the universe, and it is moving. But at the time, mere simplification of the math seemed like a weak argument. It was just *obvious* that the sun moved around the earth and not the other way around. You could *see* the sun moving. You could *see* and *feel* that the earth was stationary. So how can we be so confident today that the earth is moving? For one thing, we have a lot more data than was available in Copernicus's time. But the main argument is the one that Copernicus first advanced: it makes the math easier. You *can* put the earth at the center of the universe and you can use all of the tools of modern physics to make predictions about the motions of the planets, and those predictions will be every bit as accurate as the ones made in the more conventional way. But the effort you will have to make to do the calculations will be vastly greater than what you would have to do if you put the sun at the center. So: "the sun is the center of the solar system" is not a *fact*, it's a *convention*. In fact, the sun being at the center is not quite right either. The "real" center of the solar system is the center of mass of all the objects in the solar system, a place which astronomers call the "solar barycenter". Depending on the positions of the other planets, the solar barycenter can actually be outside the sun at certain times. (Homework: is "the earth is round" a fact, or is it merely a convention that makes the math easy?) The Copernican model remained controversial for another 60-70 years until Johannes Kepler made yet another major breakthrough: you could simplify the math even further if you assumed that the planets moved in ellipses rather than circles. Kepler's model fit the data much better than Ptolemy's did, almost perfectly in fact, and with vastly fewer calculations to crank through to produce predictions. Furthermore, by that time the measurements of planetary motion had become much more accurate because of the invention of the telescope, which allowed measurements of the positions of the planets to be much more accurate. Ironically, the data from early telescopes also led astronomers astray. Early telescopes had optical flaws that made stars appear to be disks with a measurable diameter rather than points. This indicated that the stars were relatively nearby, otherwise they would have to be unfathomably huge. But if the stars were nearby and the earth was moving, then they should be seen to move relative to each other due to a phenomenon called *parallax*. But no parallax movement was observed, and so this was evidence that the earth was not moving, at least not in relation to the stars. Kepler's discovery that the planets moved in ellipses rather than circles led to a vexing question: *why* did the planets move in ellipses? Circular motion made a lot more sense. There was an obvious physical mechanism that could produce circular motion, namely, actual physical spheres rotating out in the heavens. Elliptical motion was a lot harder to explain. There are physical mechanisms that will produce ellipses, but they are complex and contrived. Nothing in the heavens could be seen, or even imagined, that was remotely like what it would take to cause a planet to move that way. The puzzle was solved, of course, by Isaac Newton in 1687. His book "Philosophiae Naturalis Principia Mathematica" describes his famous laws of motion and gravitation, which were phenomenally successful in predicting planetary motion. No deviation from Newton's predictions would be detected until 1859, and even then it was a single tiny anomaly in the orbit of Mercury. But Newton's theory did something much more important than just provide more accurate predictions. For the first time in human history, the behavior of objects in the heavens could be explained by the *same physical laws* that apply here on earth. This unification of the physics of heaven and earth is the reason that Newton's Principia is considered the dawn of modern science. Before Newton, astronomy was completely separate from any earthly concern. After Newton these became unified in a single field which came to be called "physics". And it cleared up a myriad of mysteries. Why did the moon not fall down? Because it is moving fast enough to remain in orbit. What are meteors? They are rocks, made of the same stuff that rocks here on earth are. Why do the moons of Jupiter (discovered by Galileo in 1610) appear to move as if they were attached to springs? It's because they are actually in orbit around Jupiter, governed by the same laws that control all of the other heavenly bodies, and we are seeing them edge-on. Newton's laws were, of course, phenomenally successful. But they, too, had a problem, one which Newton himself recognized: they did not actually explain *how* gravity worked. Just as with Kepler's elliptical orbits, gravity had no apparent physical mechanism. It was a mysterious "spooky action at a distance", black magic. But it accurately predicted the motions of everything, both in the heavens and on earth, to the limits of mankind's ability to measure them at the time. It went even further than that. In 1846 Urbain Le Verrier (the same person who would go on three years later to detect the previously mentioned anomaly the orbit of Mercury) used small anomalies in the observed orbit of Uranus (discovered by William Herschel in 1781) to predict the existence of a previously unknown planet: Neptune. Even today, NASA uses Newton's laws for spacecraft navigation. Note that predicting the future position of a planet or spacecraft has a very similar structure to the abstract numerical puzzles that I posed at the beginning. You observe the planet (or spacecraft) for a while, write down your observations, i.e. a sequence of numbers, and try to guess what the next one will be before you actually observe it. The only difference is the source and quantity of the numbers. In the case of predicting planetary motion the numbers come from physical observations whereas in the case of the puzzles I just made up the sequences in order to make a point. Also, in the case of physical observations, there are a lot more numbers: tens or hundreds of thousands in Newton's day, countless billions today. But the fundamental structure of the challenge is the same: find a rule that explains all of the given numbers, and predicts the next ones. === So let us now consider those puzzles again. The correct answer to the first one is obviously "7". But *why* is this obvious? It's because there is a simple rule that explains the sequence: every number is one more than the one before, and applying this rule to the data that we have results in 6+1=7. What about the second and third sequences? The rule that worked for the first one fails for those. There are two "anomalous readings" in the second sequence and one in the third. So we have to modify our last-number-plus-one theory somehow. One way to do it is to write off the anomalies as "experimental error", a glitch. This actually seems plausible in the third sequence because the number that "should" be there instead of 2134 is 13, and it's not hard to imagine that I might accidentally hit two keys at once if my hand were in slightly the wrong position as I was trying to type "13". The fact that the anomalous digits are physically adjacent to the expected ones lends credence to this theory. But this only works if you already know that this sequence was generated by someone typing at a keyboard. You happen to know that in this case because I've told you so, but I've also told you that the *intent* of this puzzle is not to figure out what *actually* happened, but to imagine what you would think if these numbers were produced by observations of some physical process. To which you might counter: but recording keystrokes *is* a series of "observations of some physical process", namely, the process of typing. And so this answer isn't wrong, but it is missing the point. You might think that solving this puzzle requires a flash of brilliant insight. Newton's work certainly qualifies. And brilliant insights rarely hurt, but in this case none is required. It turns out that there is a *procedure* for generating a mathematical formula that will perfectly match *any* finite sequence of numbers. Once you have that formula you can use it to extend that sequence arbitrarily far into the future. In fact, there is more than one such procedure. If you think about it you might even be able to come up with one yourself. I'm going to show you one of these procedures. It is a very simple one, but with a few changes it can be used for actual scientific research. To motivate the particular procedure that I'm going to show you, consider this sequence: 2 5 10 17 26 37 ... You can probably figure this one out without too much difficulty, but suppose you couldn't. Look at the sequence of *differences* between successive terms. 2 5 10 17 26 37 ... 3 5 7 9 11 ... I have written the differences between successive terms between and below the terms. Just in case it's not yet obvious what is going on we can repeat the process: 2 5 10 17 26 37 ... 3 5 7 9 11 ... 2 2 2 2 ... And now we are obviously done. To generate the next term in the original sequence we just extend the bottom row of 2's and add going back up. So the next term in the middle sequence is 11+2=13 and the next term in the original sequence is 37+13=50. That is a mechanical procedure that we can apply to *any* sequence of numbers in order to produce a plausible next number in the sequence. In the worst case, you have to go all the way down to the bottom until you get a single number, but then you can extend that number to the right and go back up the chain to produce new terms. So this is cool and all, but you might have noticed that there is a more elegant solution to this particular sequence, namely, that the Nth term in the sequence is N squared plus one. Is there a way to produce that formula directly? Yes, in fact there is, but I'm going to leave that as an exercise for you. Here's a hint: try picking some random polynomial formulas, generating a few terms in the sequences that they generate, go through the successive-differences process above, and see if you notice any patterns emerging. What does any of this have to do with gravity? Well, it turns out that the *reason* that Ptolemy's epicycle theory worked as well as it did is that epicycles turn out to be one of these procedures for producing mathematical formulas that can fit any sequence of numbers. Ptolemy didn't know this, of course. It would be nearly 2000 years (in 1822 to be exact) before a French mathematician named Jean-Baptiste Joseph Fourier would figure that out. (Fourier, incidentally, also discovered the greenhouse effect which causes anthropogenic climate change.) Today Ptolemy's epicycle theory is considered quaint. It has been falsified. Ptolemy was wrong and Newton was right. And it is tempting to judge Ptolemy from the intellectual high ground of our post-Newton era and think that if only he had been a little bit more clever he might have seen what is obvious to us. But this is to miss a very important point: despite the fact that epicycles are in some sense "wrong", they were the received wisdom for nearly 2000 years. And this is *not* because humans were idiots before Newton came along, it's because *there was not enough data*. Epicycle theory produced correct predictions to within the margin of error of what could be observed with the available technology (the naked eye) for over 1000 years. There was simply no reason to believe that it wasn't correct. There was no *problem* that needed to be solved. All scientific inquiry must start with such a problem, a divergence of observation and the current-best explanation. The falsification of epicycles as a theory of planetary motion actually leads to such a problem: if epicycle theory is wrong, why did it work as well as it did for as long as it did? One would not a priori expect it to take over 1000 years for an incorrect theory to diverge from reality, particularly not one that is, in retrospect, as egregiously wrong as Ptolemy's. Answering *that* question leads to a deep and profound insight: it's because there are procedures for constructing mathematical formulas that will fit any data set, and epicycles happen to be one of those procedures. Moreover, some of those procedures actually provide quite a bit of predictive power, at least for a while, and again, epicycles happen to be one of those procedures too. The fact that epicycles turn out to be one of those procedures is actually a source of additional deep and profound insights, though it took literally centuries for mankind to figure this out. It turns out that a lot of the universe actually *can* be explained with math that is very much akin to epicycles. Today we call it Fourier analysis, but it amounts to the same thing. There are a *lot* of ways to construct theories that are a perfect fit to a finite amount of data, which is all we humans with our finite life spans can ever have. There are in fact an infinite number of such techniques, and there are an infinite number of resulting theories, all of which are a perfect fit to the data. You don't even have to get mathematical about it. Consider the way in which we debunked the flat-earth hypothesis at the end of the last chapter. Is that really a definitive debunking? If you presented this to a flat-earther, would it persuade them? Almost certainly not. Instead, a committed flat-earther would almost certainly respond by "adding a term" to their hypothesis, some additional proviso that explained how the sun could be observed to be at the horizon at different times despite the "fact" that the earth is flat. It would probably have something to do with atmospheric diffraction or some such thing. If you didn't dig in to the details, it might even sound persuasive. Apparently, unless flat-eartherism is all just a big practical joke, *some* people are persuaded by these arguments because there are people who profess to believe them. === Just because a theory is at odds with the scientific consensus does not mean that it is wrong. All scientific progress starts with an idea that is at odds with the current scientific consensus in one way or another. So how can you tell the difference between a crackpot theory and a potential breakthrough? You can't do it simply by looking at how the theory compares with the data. We humans are finite creatures with finite lifespans and so we can only ever have access to a finite amount of data, and for any finite amount of data there are an infinite number of theories that are a perfect fit to that data. In fact, seeking a *perfect* fit to the data might even lead you astray! Consider this sequence: 10 40 90 160 249 360 490 640 810 Here is a formula that produces a perfect fit for this data: -(1/576)x^8+(5/72)x^7-(335/288)x^6+(1525/144)x^5-(32773/576)x^4+(26365/144)x^3-(5295/16)x^2+(1325/4)x-126 This formula makes the following predictions for the next five terms in the sequence: 874 160 -3510 -15635 -48090 On the other hand, the formula 10x^2 is not a perfect fit for the data -- it is off by -1 in the fifth data point -- but it seems a lot tidier. And its predictions for the next five terms: 1000 1210 1440 1690 1960 seem somehow more plausible. Having the sequence suddenly switch from increasing all the time to decreasing seems like a high price to pay to account for such a small anomaly. On the other hand, if we actually did the additional measurements and the predictions of the complicated formula were confirmed, that would be a very strong indication that the "anomaly" was not actually an anomaly, but a clue! How do we sort out all this complexity in the real world if it's already so complicated in the puzzle world? With a lot of hard work, and a lot of data. The foundations of modern science were laid down mainly in the 19th and 20th centuries using data sets curated by hand and comprising thousands upon thousands of observations. The math was mainly done by hand, occasionally with the aid of some kind of special-purpose mechanical device like a slide rule. Modern science uses electronic computers to crunch trillions upon trillions of numbers and try to figure out the simple rules that generate them. In this endeavor, science has been spectacularly successful. Modern science rests on a foundation of just two sets of rules, each with a few dozen parameters. With this conceptually simple scaffolding we are able to make accurate predictions to within the limits of our technological capabilities of making physical measurements of all the phenomena that happen here on earth. The only data which our present theories cannot account for come from outer space. This is not to say that our present theories are *correct*. In fact, we know they are not (more about how we know that later). But what it does mean is that if you want to try to *improve* on our present theories, you have a tough road ahead of you. Many have tried to improve our scientific understanding over the years. Only a very few have ever succeeded. But that is not what this book is about anyway. This is not a book about how to become a famous scientist whose work rocks the foundations of physics, it's about how to apply the scientific method to everyday life, and one of the ways to do that is to understand that although the current-best theory may not be *right*, it is nonetheless advantageous to act as if it were right until something better comes along. One of the reasons this works is that, at the foundations of science, better things come along only very rarely. Other people have done the scientific heavy lifting for you. All you need is the ability to identify who those people are so you don't get taken by charlatans. There are a few basics that are useful to know, including a general idea of how the scientific sausage is made. That is what I've tried to describe to you here. The next few chapters are going to take you on a very cursory tour of the well-established foundations of science that have been built up over the last 400 years. There are a few basic ideas that you can get a lot of leverage out of in everyday life. For example: many people spend a lot of money on homeopathic remedies and essential oils. These are both multi-billion-dollar industries. All of the people buying these products are wasting their money. Psychics, dowsers, and purveyors of perpetual motion machines are likewise all charlatans. And you can know this by understanding just a few basic scientific principles. What is less obvious is that you can use the same principles to identify other things that you probably ought not to be spending your money on, like "money managers". But it will be a long time before we get to that. Let's start with atoms.