A new episode of my podcast is out: my review of Isaac Newton’s Opticks. And I have a new microphone!
My new podcast episode is here, and it is my review of Isaac Newton’s Principia. To listen, click below:
It is shown in the Scholium of Prop. 22, Book II, that at the height of 200 miles above the earth the air is more rare than it is at the surface of the earth in the ratio of 30 to 0.0000000000003998, or as 75,000,000,000,000 to 1, nearly.
Marking this book as “read” is as much an act of surrender as an accomplishment. Newton’s reputation for difficulty is well-deserved; this is not a reader-friendly book. Even those with a strong background in science and mathematics will, I suspect, need some aid. The historian of mathematics Colin Pask relied on several secondary sources to work his way through the Principia in order to write his excellent popular guide. (Texts by S. Chandrasekhar, J. Bruce Brackenridge, and Dana Densmore are among the more notable vade mecums for Newton’s proofs.) Gary Rubenstein, a math teacher, takes over an hour to explain a single one of Newton’s proofs in a series of videos (and he had to rely on Brackenridge to do so).
It is not that Newton’s ideas are inherently obscure—though mastering them is not easy—but that Newton’s presentation of his work is terse, dense, incomplete (from omitting steps), and at times cryptic. Part of this was a consequence of his personality: he was a reclusive man and was anxious to avoid public controversies. He says so much himself: In the introduction to Book III, Newton mentions that he had composed a popular version, but discarded it in order to “prevent the disputes” that would arise from a wide readership. Unsurprisingly, when you take material that is intrinsically complex and then render it opaque to the public, the result is not a book that anyone can casually pick up and understand.
The good news is that you do not have to. Newton himself did not advise readers, even mathematically skilled readers, to work their way through every problem. This would be enormously time-consuming. Indeed, Newton recommended his readers to peruse only the first few sections of Book I before moving on directly to Book III, leaving most of the book completely untouched. And this is not bad advice. As Ted said in his review, the average reader could gain much from this book by simply skipping the proofs and calculations, and stopping to read anything that looked interesting. And guides to the Principia are certainly not wanting. Besides the three mentioned above, there is the guide written by Newton scholar I. Bernard Cohen, published as a part of his translation. I initially tried to rely on this guide; but I found that, despite its interest, it is mainly geared towards historians of science; so I switched to Colin Pask’s Magnificent Principia, which does an excellent job in revealing the importance of Newton’s work to modern science.
So much for the book’s difficulty; on to the book itself.
Isaac Newton’s Philosophiæ Naturalis Principia Matematica is one of the most influential scientific works in history, rivaled only by Darwin’s On the Origin of Species. Quite simply, it set the groundwork for physics as we know it. The publication of the Principia, in 1687, completed the revolution in science that began with Copernicus’s publication of De revolutionibus orbium coelestium over one hundred years earlier. Copernicus deliberately modeled his work on Ptolemy’s Almagest, mirroring the structure and style of the Alexandrian Greek’s text. Yet it is Newton’s book that can most properly be compared to Ptolemy’s. For both the Englishman and the Greek used mathematical ingenuity to draw together the work of generations of illustrious predecessors into a single, grand, unified theory of the heavens.
The progression from Copernicus to Newton is a case study in the history of science. Copernicus realized that setting the earth in motion around the sun, rather than the reverse, would solve several puzzling features of the heavens—most conspicuously, why the orbits of the planets seem related to the sun’s movement. Yet Copernicus lacked the physics to explain how a movable earth was possible; in the Aristotelian physics that held sway, there was nothing to explain why people would not fly off of a rotating earth. Furthermore, Copernicus was held back by the mathematical prejudices of the day—namely, the belief in perfect circles.
Johannes Kepler made a great stride forward by replacing circles with ellipses; this led to the discovery of his three laws, whose strength finally made the Copernican system more efficient than its predecessor (which Copernicus’s own version was not). Yet Kepler was able to provide no account of the force that would lead to his elliptical orbits. He hypothesized a sort of magnetic force that would sweep the planets along from a rotating sun, but he could not show why such a force would cause such orbits. Galileo, meanwhile, set to work on the new physics. He showed that objects accelerate downward with a velocity proportional to the square of the distance; and he argued that different objects fall at different speeds due to air resistance, and that acceleration due to gravity would be the same for all objects in a vacuum. But Galileo had no thought of extending his new physics to the heavenly bodies.
By Newton’s day, the evidence against the old Ptolemaic system was overwhelming. Much of this was observational. Galileo observed craters and mountains on the moon; dark spots on the sun; the moons of Jupiter; and the phases of Venus. All of these data, in one way or another, contradicted the old Aristotelian cosmology and Ptolemaic astronomy. Tycho Brahe observed a new star in the sky (caused by a supernova) in 1572, which confuted the idea that the heavens were unchanging; and observations of Haley’s comet in 1682 confirmed that the comet was not somewhere in earth’s atmosphere, but in the supposedly unchanging heavens.
In short, the old system was becoming unsustainable; and yet, nobody could explain the mechanism of the new Copernican picture. The notion that the planets’ orbits were caused by an inverse-square law was suspected by many, including Edmond Haley, Christopher Wren, and Robert Hooke. But it took a mathematician of Newton’s caliber to prove it.
But before Newton published his Principia, another towering intellect put forward a new system of the world: René Descartes. Some thirty years before Newton’s masterpiece saw the light of day, Descartes published his Principia Philosophiæ. Here, Descartes summarized and systemized his skeptical philosophy. He also put forward a new mechanistic system of physics, in which the planets are borne along by cosmic vortices that swirl around each other. Importantly, however, Descartes’s system was entirely qualitative; he provided no equations of motion.
Though Descartes’s hypothesis has no validity, it had a profound effect on Newton, as it provided him with a rival. The very title of Newton’s book seems to allude to Descartes’s: while the French philosopher provides principles, Newton provides mathematical principles—a crucial difference. Almost all of Newton’s Book II (on air resistance) can be seen as a detailed refutation of Descartes’s work; and Newton begins his famous General Scholium with the sentence: “The hypothesis of vortices is pressed with many difficulties.”
In order to secure his everlasting reputation, Newton had to do several things: First, to show that elliptical orbits, obeying Kepler’s law of equal areas in equal times, result from an inverse-square force. Next, to show that this force is proportional to the mass. Finally, to show that it is this very same force that causes terrestrial objects to fall to earth, obeying Galileo’s theorems. The result is Universal Gravity, a force that pervades the universe, causing the planets to rotate and apples to drop with the same mathematical certainty. This universal causation effectively completes the puzzle left by Copernicus: how the earth could rotate around the sun without everything flying off into space.
The Principia is in a league of its own because Newton does not simply do that, but so much more. The book is stuffed with brilliance; and it is exhausting even to list Newton’s accomplishments. Most obviously, there are Newton’s laws of motion, which are still taught to students all over the world. Newton provides the conceptual basis for the calculus; and though he does not explicitly use calculus in the book, a mathematically sophisticated reader could have surmised that Newton was using a new technique. Crucially, Newton derives Kepler’s three laws from his inverse-square law; and he proves that Kepler’s equation has no algebraic solution, and provides computational tools.
Considering the mass of the sun in comparison with the planets, Newton could have left his system as a series of two-body problems, with the sun determining the orbital motions of all the planets, and the planets determining the motions of their moons. This would have been reasonably accurate. But Newton realized that, if gravity is truly universal, all the planets must exert a force on one another; and this leads him to the invention of perturbation theory, which allows him, for example, to calculate the disturbance in Saturn’s orbit caused by proximity to Jupiter. While he is at it, Newton calculates the relative sizes and densities of the planets, as well as calculates where the center of gravity between the gas giants and the sun must lie. Newton also realized that gravitational effects of the sun and moon are what cause terrestrial tides, and calculated their relative effects (though, as Pask notes, Newton fudges some numbers).
Leaving little to posterity, Newton realized that the spinning of a planet would cause a distortion in its sphericity, making it marginally wider than it is tall. Newton then realized that this slight distortion would cause tidal locking in the case of the moon, which is why the same side of the moon always faces the earth. The slight deformity of the earth is also what causes the procession of the equinoxes (the very slow shift in the location of the equinoctial sunrises in relation to the zodiac). This shift was known at least since Ptolemy, who gave an estimate (too slow) of the rate of change, but was unable to provide any explanation for this phenomenon.
The evidence mustered against Descartes’s theory is formidable. Newton describes experiments in which he dropped pendulums in troughs of water, to test the effects of drag. He also performed experiments by dropping objects from the top of St. Paul’s Cathedral. What is more, Newton used mathematical arguments to show that objects rotating in a vortex obey a periodicity law that is proportional to the square of the distance, and not, as in Kepler’s Third Law, to the 3/2 power. Most convincing of all, Newton analyzes the motion of comets, showing that they would have to travel straight through several different vortices, in the direction contrary to the spinning fluid, in order to describe the orbits that we observe—a manifest absurdity. While he is on the subject of comets, Newton hypothesizes (correctly) that the tail of comets is caused by gas released in proximity to the sun; and he also hypothesizes (intriguingly) that this gas is what brings water to earth.
This is only the roughest of lists. Omitted, for example, are some of the mathematical advances Newton makes in the course of his argument. Even so, I think that the reader can appreciate the scope and depth of Newton’s accomplishment. As Pask notes, between the covers of a single book Newton presents work that, nowadays, would be spread out over hundreds of papers by thousands of authors. The result is a triumph of science. Newton not only solves the longstanding puzzle of the orbits of the planets, but shows how his theory unexpectedly accounts for a range of hitherto separate and inexplicable phenomena: the tides, the procession of the equinoxes, the orbit of the moon, the behavior of pendulums, the appearance of comets. In this Newton demonstrated what was to become the hallmark of modern science: to unify as many different phenomena as possible under a single explanatory scheme.
Besides setting the groundwork for dynamics, which would be developed and refined by Euler, d’Alembert, Lagrange, Laplace, and Hamilton in the coming generations, Newton also provides a model of science that remains inspiring to practitioners in any field. Newton himself attempts to enunciate his principles, in his famous Rules of Reasoning. Yet his emphasis on inductivism—generalizing from the data—does not do justice to the extraordinary amount of imagination required to frame suitable hypotheses. In any case, it is clear that Newton’s success was owed to the application of sophisticated mathematical models, carefully tested against collections of physical measurements, in order to unify the greatest possible number of phenomena. And this was to become a model for other intellectual disciples to aspire to, for good and for ill.
A striking consequence of this model is that its ultimate causal mechanism is a mathematical rule rather than a philosophical principle. The planets orbit the sun because of gravity, whose equations accurately predict their motions; but what gravity is, why it exists, and how it can affect distant objects, is left completely mysterious. This is the origin of Newton’s famous “I frame no hypothesis” comment, in which he explicitly restricts himself to the prediction of observable events rather than speculation on hidden causes (though he was not averse to speculation when the mood struck him). Depending on your point of view, this shift in emphasis either made science more rational or more superficial; but there is little doubt that it made science more effective.
Though this book is too often impenetrable, I still recommend that you give it a try. Few books are so exalting and so humbling. Here is on display the furthest reaches of the power of the human intellect to probe the universe we live in, and to find hidden regularities in the apparent chaos of experience.
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My rating: 4 of 5 stars
My Design in this Book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiment
Newton’s masterwork is, unquestionably, his Principia. But it is neither an easy nor a pleasant book to read. Luckily, the great scientist wrote a far more accessible volume that is scarcely less important: the Opticks.
The majority of this text is given over to descriptions of experiments. To the modern reader—and I suspect to the historical reader as well—these sections are remarkably dry. In simple yet exact language, Newton painstakingly describes the setup and results of experiment after experiment, most of them conducted in his darkened chamber, with the window covered up except for a small opening to let in the sunlight. Yet even if this doesn’t make for a thrilling read, it is impossible not to be astounded at the depth of care, the keenness of observation, and the subtle brilliance Newton displays. Using the most basic equipment (his most advanced tool is the prism), Newton tweezes light apart, making an enormous contribution both to experimental science and to the field of optics.
At the time, the discovery that white light could be decomposed into a rainbow of colors, and that this rainbow could be recombined back into white light, must have seemed as momentous as the discovery of the Higgs Boson. And indeed, even the modern reader might catch a glimpse of this excitement as she watches Newton carefully set up his prism in front of his beam of light, tweaking every variable, adjusting every parameter, measuring everything could be measured, and describing in elegant prose everything that could not.
Whence it follows, that the colorifick Dispositions of Rays are also connate with them, and immutable; and by consequence, that all the Productions and Appearances of Colours in the World are derived, not from any physical Change caused in Light by Refraction or Reflexion, but only from the various Mixtures or Separations of Rays, by virtue of their different Refrangibility or Reflexibility. And in this respect the Science of Colours becomes a Speculation as truly mathematical as any other part of Opticks.
Because I had recently read Feynman’s QED, one thing in particular caught my attention. Here is the problem: When you have one surface of glass, even if most of the light passes through it, some of the light is reflected; and you can roughly gauge what portion of light does one or the other. Let us say on a typical surface of glass, 4% of light is reflected. Now we add another surface of glass behind the first. According to common sense, 8% of the light should be reflected, right? Wrong. Now the amount of light which is reflected varies between 0% and 16%, depending on the distance between the two surfaces. This is truly bizarre; for it seems that the mere presence of second surface of glass alters the reflectiveness of the first. But how does the light “know” there is a second surface of glass? It seems the light somehow is affected before it comes into contact with either surface.
Newton was aware of this awkward problem, and he came up with his theory of “fits of easy reflection or transmission” to explain this phenomenon. But this “theory” was merely to say that the glass, for some unknown reason, sometimes lets light through, and sometimes reflects it. In other words, it was hardly a theory at all.
Every Ray of Light in its passage through any refracting Surface is put into a certain transient Constitution or State, which in the progress of the Ray returns at equal Intervals, and disposes the Ray at every return to be easily transmitted through the next refracting Surface, and between the returns to be easily reflected by it.
Also fascinating to the modern reader is the strange dual conception of light as waves and as particles in this work, which cannot help but remind us of the quantum view. The wave theory makes it easy to account for the different refrangibility of the different colors of light (i.e. the different colors reflect at different angles in a prism).
Do not several sorts of Rays make Vibrations of several bignesses, which according to their bignesses excite Sensations of several Colours, much after the manner that the Vibrations of the Air, according to their several bignesses excite Sensations of several sounds. And particularly do not the most refrangible Rays excite the shortest Vibrations for making a Sensation of deep violet, the least refrangible the largest for making a Sensation of deep red, and the several intermediate bignesses to make Sensations of the several intermediate Colours?
To this notion of vibrations, Newton adds the “corpuscular” theory of light, which held (in opposition to his contemporary, Christiaan Huygens) that light was composed of small particles. This theory must have been attractive to Newton because it fit into his previous work in physics. It explained why beams of light, like other solid bodies, travel in straight lines (cf. Newton’s first law), and reflect off surfaces at angles equal to their angles of incidence (cf. Newton’s third law).
Are not the Rays of Light very small Bodies emitted from shining Substances? For such Bodies will pass through uniform Mediums in right Lines without bending into the shadow, which is the Nature of the Rays of Light. They will also be capable of several Properties, and be able to conserve their Properties unchanged in passing through several Mediums, which is another conditions of the Rays of Light.
As a side note, despite some problems with the corpuscular theory of light, it came to be accepted for a long while, until the phenomenon of interference gave seemingly decisive weight to the wave theory. (Light, like water waves, will interfere with itself, creating characteristic patterns; cf. the famous double-slit experiment.) The wave theory was reinforced with Maxwell’s equations, which treated light as just another electro-magnetic wave. It was, in fact, Einstein who brought back the viability of the corpuscular theory, when he suggested the idea that light might come in packets to explain the photoelectric effect. (Blue light, when shined on certain metals, will cause an electric current, while red light will not. Why not?)
All this tinkering with light is good fun. But the real treat, at least for the layreader, comes at the final section, where Newton speculates on many of the unsolved scientific problems of his day. His mind is roving and vast; and even if most of his speculations have turned out incorrect, it is stunning simply to witness him at work. Newton realizes, for example, that radiation can travel without a medium (like air), and can heat objects even in a vacuum. (And thank goodness for that, for how else would the earth be warmed by the sun?) But from this fact he incorrectly deduces that there must be some more subtle medium that remains (like the famous ether).
If in two large tall cylindrical Vessels of Glass inverted, to little Thermometers be suspended so as not to touch the Vessels, and the Air be drawn out of one of these Vessels thus prepared be carried out of a cold place into a warm one; the Thermometer in vacuo will grow warm as much, and almost as soon as the Thermometer that is not in vacuo. And when the Vessels are carried back into the cold place, the Thermometer in vacuo will grow cold almost as soon as the other Thermometer. Is not the Heat of the warm Room convey’d through the Vacuum by the Vibrations of a much subtiler Medium than Air, which after the Air was drawn out remained in the Vacuum?
Yet for all Newton’s perspicacity, the most touching section was a list of question Newton asks, as if to himself, that he cannot hope to answer. It seems that even the most brilliant among us are stunned into silence by the vast mystery of the cosmos:
What is there in places almost empty of Matter, and whence is it that the Sun and Planets gravitate towards one another, without dense Matter between them? Whence is it that Nature doth nothing in vain; and whence arises all that Order and Beauty which we see in the World? To what end are Comets, and whence is it that Planets move all one and the same way in Orbs concentrick, while Comets move all manner of ways in Orbs very excentrick; and what hinders the fix’d Stars from falling upon one another? How came the Bodies of animals to be contrived with so much Art, and for what ends were their several Parts? Was the Eye contrived without Skill in Opticks, and the Ear without Knowledge of Sounds? How do the Motions of the Body follow from the Will, and whence is the Instinct in Animals?
I set little store by my own opinions, but just as little by other people’s.
—Michel de Montaigne
Although nobody is free from self-doubt, I have long felt that I have this quality to an inordinate degree. The problem is that I can’t decide whether this is a good or a bad thing.
On the one hand, doubting yourself is one of the keys of moderation and wisdom. If you think you already know everything you cannot learn. If you are sure that your perspective is right you cannot empathize. Dogmatism, selfishness, and ignorance result from the inability to doubt the truth of your own opinions.
There are no such things as self-doubting fanatics. The ability to question your own opinions and conclusions is what prevents most people from committing atrocities. I couldn’t kill somebody in the name of an idea, since there is no idea I believe in strongly enough.
And yet, this tendency to doubt my own beliefs and conclusions so often makes me hesitating, indecisive, and occasionally spineless. Never mind killing anyone: I don’t even believe in my political ideals enough to stand up to somebody I find offensive. I doubt the worth of my dreams, the reason of my arguments, the virtue of my actions; and I am not terribly sure about my professional competency or my literary skill.
No matter what I do, I have this nagging feeling that, somewhere out there, there are people who could make me appear ridiculous by comparison. So often I feel out of the loop. I hesitate to submit my writing anywhere because I think a professional editor would cut it to pieces. I hesitate to put forth arguments because I think a real expert could see right through them. I hesitate to commit to a profession because I doubt my own ability to follow through, to perform difficult tasks, and to do my duties responsibly.
My nagging self-doubt is more of a feeling than a thought; but insofar as a feeling can be expressed in words, it goes like this: “Well, maybe there’s something big out there that I don’t know, something important that would render all my knowledge and standards inadequate.”
The odd thing is that I have no evidence that this fear is justified. In fact, I have evidence to the contrary. The more I read and travel, the more people I meet, the more places I work, the less surprised I am by what I find. The contours of daily reality have grown ever-more familiar, and yet this fear—the fear that, somehow, I have missed something big—this fear remains.
The perilous side of self-doubt is that it can easily ally itself with baser qualities. I can argue myself out of taking risks because I am unsure whether I really want the goal. I can argue myself out of standing up for what I believe is right by doubting whether it really is right, and whether I could prove it. Self-doubt and fear—fear of failure, fear of rejection, fear of being publically embarrassed—so often go hand-in-hand.
I can’t say why exactly, but the thought of writing a flawed argument, with a logical fallacy, an unwarranted assumption, or a sloppy generalization, fills me with dread. How mortifying to have my mental errors exposed to the world! Maybe this is from spending so much time in a school environment wherein the number of correct answers was used as a measure of my worth. Or maybe it is simply my personality; being “right” has always been important to me.
This fear of being wrong is particularly irrational, since some of the greatest minds and most influential thinkers in history have been wrong—Galileo, Newton, Darwin, Einstein, all of them have erred. Indeed, the fear of being wrong is not only irrational, but counterproductive to learning, since it sometimes prevents me from exposing my thoughts, increasing the likelihood that I will persist in an error.
Despite the negatives, I admit that I am often proud of my ability to doubt my own conclusions and change my opinions. I see it as a source of my independence of mind, my ability to think differently from others and to come to my own conclusions. After all, doubting yourself is the prerequisite of doubting anything at all. As Plato illustrated in his Socratic dialogues, from the moment we our born our minds are filled with all sorts of assumptions and prejudices which we absorb from our culture. The first step of doubting conventional opinion is thus doubting our own opinions.
But just as often as my self-doubt is a source of pride, it is a source of shame. I am sometimes filled with envy for those rare souls who seem perfectly self-confident. In this connection I think of Benvenuto Cellini, the Renaissance artist who left us his remarkable autobiography. Cocksure, boastful, selfish, prideful, Cellini was in many ways a despicable man. And yet he tells his story with such perfect certainty of himself that you can’t help but be won over.
Logically, self-confidence should come after success, since otherwise it isn’t justified. But so often self-confidence comes beforehand, and is actually the cause of success. In my experience, when you believe in yourself, others are inclined to believe in you. When you are confident you take risks, and these risks often enough pay off. When you are confident you state your opinions boldly and clearly, and thus have a better chance of convincing others.
Confidence is often discussed in dating. Self-confident people are seen as more attractive, and tend to have more romantic success since they take more risks. The ability to look somebody in the eye and say what you think and what you want—these are almost universally seen as attractive qualities; and not only in romance, but in politics, academics, business, and nearly everything else.
The charisma of confidence notwithstanding, this leads to an obvious danger. Many people are confident without substance. They boast more than they can accomplish; they speak with authority and yet have neither evidence nor logic to back their opinions. The world is ruled by such people—usually men—and I think most of us have personal experience with this type. I call it incompetent confidence, and it is rampant.
As Aristotle would say, there must be some ideal middle-ground between being confidently clueless, and being timidly thoughtful. And yet, in my experience, this middle-ground, if it even exists, is difficult to find.
I suppose that, ideally, we would be exactly as confident as the reach of our knowledge permitted: bold where we were sure, hesitating where we were ignorant. In practice, however, this is an impossible ideal. How can we ever be sure of how much we know, or how dependable our theories are? Indeed, this seems to be precisely what we can never know for sure—how much we know.
For the world to function, it seems that it needs doers and doubters. We need confident leaders and skeptical followers. And within our own brains, we need the same division: the ability to act boldly when needed, and to question ourselves when possible. Personally, I tend to err on the side of self-doubt, since it easily allies itself with laziness, inaction, and fear; but now I am starting to doubt my own doubting.