Physics Rediscovered #10: A tale of the plague
“Bring out your dead!
The shout stopped Samuel in his tracks. It was the searchers.
Wishing to avoid them, he would need to change his route. Samuel spotted a narrow alley and turned into it with little hesitation. Seeing that it was empty, he continued along with a vigorous pace.
The alley was devoid of life as were most of London streets. All the windows and doors were shut, with many carrying the marking of a cross and a few lines of prayer, all in vivid red. The occasional violet of lavender flowers hanging over the doorsteps, seemed to cast a shadow of vitality in an otherwise ghostly landscape.
Seeing the eerie streets, made Samuel sad. Only just recently were they so busy and animated, now to be this still. Suddenly, a familiar sound interrupted his nostalgia.
“Rats!“, he thought. Though he couldn’t see them, he could hear their unmistakable scurrying.
“Ever since all the cats and dogs were chased from the city, the rats seemed evermore present”, he noticed. Without sparing them another thought he marched on.
As he was about to turn the corner, a strange and haunting visage appeared before him. It was a figure with a face like a beak of a giant bird. At the base of the beak two round holes covered in thick and cloudy glass served as the eyes. Yet it wore a hat and a long, heavy robe, like a human.
“A plague doctor“, Samuel concluded.
“Stop!”, the doctor ordered, prodding Samuel with a stick. “Are you sick?“, he asked.
“No”, Samuel answered. “I am as best as can be, under the circumstances“, he continued.
“Good“, the doctor acknowledged. “Would have to report you, otherwise“.
“Wherever you are going, you will have to find another way“, the doctor informed. “This place has seen its fare share of sorrow and we are shutting it off.“
Glancing behind the doctor’s shoulder, Samuel noticed two men at work. One was tossing various herbs into a large brazier standing in the middle of the alley and setting fire to it.
The other one was preparing his horse to pull a heavy cart. The cart itself was filled with peculiar passengers, many more than it was designed to carry. Some of the passengers wore clothes, while many were completely naked.
However, they did not mind their indignity as they were dead and there wouldn’t be enough coffins in England to inter them all…
Medicine at its finest
I hope I got your attention with this joyful story. Things will get better from now on, I promise.
This intro takes you back to London in 1665. At the time, the city was growing rapidly, accommodating much more population than made sense. Multiple families were cramped in single housings and all sorts of filth and rot littered the streets. The sanitation was in the shitter, though no shitters were available.
In such pristine conditions, three hundred years after the Black Death, the bubonic plague came back with a vengeance. It wiped out at least 15% of London’s population, with thousands dying every week.
What was society’s response to such tragedy? Thoughts and prayers, of course. However, to make sure that all the bases were covered, we also burned some herbs to combat the bad odors and miasmatic air. Finally, if none of the above helped, there was always some good old-fashioned blood-letting to balance the humors.
Today, we know that the plague is caused by yersinia pestis. This bacteria is carried onto humans by fleas that feed on rats. Not cats and not dogs but rats. Yet we’ve decided that it’s the former two that are guilty and chased them out of London. Those we couldn’t chase away, we killed. Meanwhile, the rat population was booming. We couldn’t have gotten it more wrong.
So with London turning into a hellscape, where do you go?
Woolsthorpe, obviously.
Let there be Newton
While medicine was in absolute shambles, still clinging to ancient Greek concepts of humors and foul air, another branch of science thrived by detaching from them. That branch was physics. Shocking, I know.
Specifically, it was thriving in Woolsthorpe, north of Cambridge, where local asshole/genius, locked himself away, wanting to miss out on the plague.
Seeing social distancing as an absolute win, he locked the doors, closed the curtains, sighed in relief and got to work. The man in question was none other than sir Isaac Newton.
Ok, ok, he didn’t close the curtains all the way. He left a little bit of a slit to let some light in. Not that he needed the light personally, he was fine sitting in the dark. It was for an experiment, you see.
As you’ve probably noticed, everyone at the time was just repeating stuff after the Greeks. For example, that light is purely white and any dimming or coloration is a results of objects polluting the light with their properties. Newton thought this was very suspect, hence the experiment and it went like this:
Setup one prism and shine the unpolluted white light from outside.
Setup a narrow slit so that we can select a single color from the resulting rainbow.
Setup a second prism such that the selected color shines on that prism.
Remind yourself that other people are really dumb.
Observe the results.
And this is how it looked like:
Yo, why is the red light, going through the second prism, still red? Surely if the objects pollute the light with their filth then we should see a second rainbow or something after the second prism, right?
Wrong. Newton just showed that light is not originally white but composed out of multiple colors. He called it the spectrum and the name stuck.
To push the point even further he devised yet another experiment, where he shined the separated colors on a focusing lens, so that they all converged into white light again.
This is only the tip of the iceberg. Newton conducted multiple experiments in various configurations, wishing to torture the truth about light from reality. He published detailed descriptions of his work and its conclusions in Opticks, which you can read here, if you dare.
Already a branch of physics is being revolutionized and Newton is just warming up.
An apple a day keeps the Moon in orbit
The Greek bashing continues.
Newton could not perform all of his experiments in Woolsthorpe due to limited equipment and sometimes had to visit Cambridge to get access to the university’s lab. The problem was that universities is where other people are and that’s not so great. It’s not that Newton hated everyone. He had friends. He said it himself:
“Plato is my friend, Aristotle is my friend, but my greatest friend is truth.”
He had no problem demolishing Greek ideas because they were all buddies.
Also, this is the answer to the question of what motivated Newton in the first place. Why would he ever get involved in specific ideas of optics or celestial mechanics? Well, for the sake of it. Out of pure, unrestrained, violent curiosity.
Speaking of celestial mechanics, Newton got really into Kepler and his work on orbital motion (you’ve probably read about Kepler here). The German astronomer himself could not explain the fundamental driving mechanisms behind his findings, which got Newton’s attention.
For example, why planetary orbits are elliptical? This is bizarre. Descartes and Galileo already suggested that when objects are moving, they tend to stay in motion, unless something gets is the way. So shouldn’t a planet just shoot off into space rather than follow a strange orbit? There must be something acting on the planet.
But wait! Both of those guys were referring to earthly objects, like rocks and apples, and car keys, and cannonballs, while we are talking about giant worlds orbiting the Sun. These concepts are as far apart as can be. There was no obvious connection but Newton had a hunch.
He considered that maybe, just maybe, objects falling here, on Earth and planetary motions have something in common. What if there’s something like a force being exerted on the objects? What if that force can extend all the way up there, into space? What is the nature of that force? Why are people so obsessed with apples?
Ok, sensory overload. Let’s take this one step at a time.
As bodies travel in orbits, they should fly away but they don’t. This means that something, a mysterious force is constantly pushing them inwards. We will call it the centripetal force.
Let’s see if we can say something about it. We will investigate a circular motion for simplicity and take a look at two points separated by some time Δt. Follow along with the awesome video to see where this takes us:
Right, so we ended up with:
where Δv is change in velocity, r is the radius of the orbit and Δv/Δt is the change of velocity of some period of time, which is the acceleration (a). It’s something, but not entirely great. The radius is fine because we can invent an orbit of any size we want. The velocity is a problem, though. How much is it? Dunno. What now?
Wait! Didn’t Kepler say something about orbits? What was it? It was something here…
Got it. Loosely saying, is that the cube of the orbital radius was proportional to the square of the orbital period, or:
Maybe we can use that somehow…
First let’s do something about the velocity v. We know velocity is distance over time. In a circular motion that means that over the orbital period T, the distance traveled must be the circumference of the circle: 2πr. Therefore v = 2πr / T. Back to acceleration:
And now let’s use Kepler’s relation for T:
Whaaaat??? The acceleration is like the inverse square of the radius? Wow, this is so gravity, almost.
But this cannot be the full picture. We are still missing something. It cannot be just the distance that’s defining the motion. The Earth-Sun distance is the same both ways, so why is the Earth orbiting the Sun and not the other way around.
So maybe like this: bodies in orbit are pulled towards the center - we called it the centripetal force. If so, then something in that center must have something to do with it. Looking at the Solar system, all planets seem to orbit the Sun, which is kinda big. Maybe it is the mass of the thing that matters? Newton needed to take a leap of faith here and this is what he assumed. Let’s call that M.
With that, let’s get rid of the proportionality in the acceleration formula, in favor of an equality, by introducing some constant. Let’s call it G, since we are talking about gravity. We will also get rid of 4 and π by putting them into G. After all, a constant times a constant is a constant. Finally, we get:
Boooom!
Ok, now for the big finale. If this is correct then we should be able to use that to show that planets move in elliptical orbits, as Kepler observed.
For the sake of sanity and not alienating the 2 people that are going to read this, we will skip the proof. Its too long and complex, and easy to look up if you’re into it. Trust me that (spoiler alert!) they indeed are elliptical.
But wait, there’s more! The Sun not only pulls on the Earth, but the Earth also pulls on the Sun. This means that a body in orbit does not simply orbit another body, but both orbit a common center of gravity.
For example, the Earth and Sun both orbit common barycenter. The Earth’s orbit is spacious and almost circular. However, the barycenter is almost, but not exactly in the middle of the Sun, so the star has an orbit of its own. It’s very small but makes the Sun move, like so:
I could try, but the impact of this discovery is impossible to diminish. Not only does it resolve the age old mystery of celestial movement but shows that things happening here and things happening there are governed by the same set of mathematical laws.
Almost nothing means everything
There’s a very, very small thing I’ve omitted. It’s almost nothing. It’s a near negligible fact that Newton had no way of proving the ellipses thingy. Not at first, at least. The proof requires knowing how things change as other things change.
Now, this not the first time we would like to know such things. People have been studying the motion of various objects, celestial mechanics and other things that change all all the time. However, no one was able to tackle a general framework for that.
What the hell am I talking about - I hear you say. If we know, for example, that y = x*x, then obviously if x = 0, y = 0; x = 1, y = 1; x = 2, y = 4 and so on. That’s how it changes.
Yes, but no. If you change x, then y changes, but what is the degree of that change? You would need to calculate this for each 2 points and take the resulting difference. What if you wanted to do that for multiple points? Pretty quickly this would get unmanageable.
Many people tried to approach this by drawing tangent lines. The logic was: if you zoom in on something, like a fragment of a circle, that fragment becomes more and more like a straight line. If we could find what kind of a straight line, then we could say how much that fragment changes around a certain point. You would have to zoom in more than they did in those crazy CSI movies, but what you would find in the end is the tangent line.
Finding the tangent was cool, but no breeze. Many powerful, geometric minds of the time figured out smart ways to get them, but that wasn’t good enough. It was still too specific for the problem. However, someone did figure out a way to do this. Take a guess who it was and you cannot say Newton.
Ok, it was Newton.
This article is getting so predictable... Anyway, let’s try to follow his thinking. We will consider a specific case, just to illustrate. Let’s take y = x*x. We will use Newton’s notation for authenticity. It will look a bit weird but we will make it nicer later on.
Newton thought in terms of time. How do things change as time changes by just a little amount? That change in time he called o. If you think of y and x distances, then p and q are going to be the respective velocities in those dimensions. We start off by introducing the small time changes to y and x, recalling that distance is velocity times time:
Expanding the right hand side:
but:
so:
Now, from the very beginning we stressed that the change in time has to be very small. This means that o is very small (much less than 1) and o*o is even smaller still. With that, we will just ignore the last term, like it was never there. Finally, we divide both sides by o and we get:
That, my gentle readers, is a derivative and it is the first half of the theory of calculus. Working this out had a profound impact on the world that was to come. Calculus would become the basis of all of our calculations in all branches of science to this day.
So how come no one could figure this out before Newton? Well, we just solved a specific case, where y=x*x. However, solving a general case, where x is to some power n, requires general binomial expansion. Now, don’t worry, a binomial expansion is simply this:
Binomial because there are two terms: x is one “nomial“ and oq is the other one.
So what’s the big deal? The deal is that this one, to the power of 2, is trivial. But what about the power of 1/2 or -17. These are much more involved and, in fact, infinite.
Who figured out how to calculate those? You know the drill. It was Newton.
Like I warned you, the p/q solution looks a bit weird and I promised to make it nicer, but I need to deal with something first.
The force is strong with this one
As I’m writing this, Christiaan Huygens, Robert Hooke and Gottfried Leibniz are standing outside, throwing rocks at my window. That’s what I get for dealing in necromancy. Don’t believe me? Well, how do you think I got this insight into what all those guys were thinking? I’m too lazy to do research…
Why are these distinguished gentlemen so agitated? Because Newton finally published all his findings and everyone got mad. First there was the Philosophiæ Naturalis Principia Mathematica, published in 1687, where he laid out all the laws of motion and gravity. Where already touched on those here and there without naming them specifically - see if you can identify them (there are 3). Later, in 1708, there was Opticks - on optics, only spelled wrong.
Ok, Huygens, you’re first.
He has a problem with the first law of motion. The problem is that he might’ve worked it out before Newton. But you know what your problem is Christiaan? You never published it. Only in 1703, eight years after his death, in De Motu Corporum ex Percussione did his version saw the light of day:
Any body already in motion will continue to move perpetually with the same speed and in a straight line unless it is impeded.
A bit late to the party, there, Christiaan. That’s not all, though. Huygens also disagreed with Newton on the nature of light. The latter imagined light and its interaction with matter as small balls hitting and bouncing off each other. On the other hand, Huygens was pretty much into waves of all kinds and he took at approach to light as well. Long story short, Newton was wrong, Huygens was right, waves were weird, Newton was more popular, Huygens got sidelined.
Cheer up, Huygens! Maybe I’ll get you an episode of your own one day. Nah, but in all honesty he was more impressed than annoyed with Newton’s work so Christiaan was a good sport besides a friggin genius.
Hooke, you’re next.
At the time, both Hooke and Newton were members of the Royal Society. Hooke was more into biology and observations through the microscope, but not exclusively. In 1679 Hooke got appointed as manager of the Society’s correspondence and wanted to know what everyone is writing about and what they were thinking about each other, and what they were planning to research next, and what was their favorite color. Pretty intrusive, if you ask me, especially the color part.
At one point he got into a correspondence with Newton and one of the subjects was planetary motion. At some point Hooke suggested that the motion of planets might be split up into:
“…a direct motion by the tangent and an attractive motion towards the central body…”
This was a nice idea to have and certainly not an obvious one, at the time. In Principia Newton acknowledged that his correspondence with Hooke served as inspiration for his own ideas.
However, in that same book he published the inverse-square law of gravity and Hooke blew a gasket. He started telling everyone that it was his idea from the very beginning, but he had nothing to base this on. To make matters worse, while he was claiming the inverse-square law, he did not accept Newton’s elliptical curve calculations that result from the law. Like, make up your mind, dude.
Hooke, you overplayed your hand. You could’ve stuck to claiming rights to the idea of compound orbital motion and probably would’ve come on top. Instead you wanted it all.
Leibniz…
I mean, this chapter deserves a book of its own and it actually got several along with a legion of PhD dissertations. Let’s make it simple here.
Leibniz discovered calculus at about the same time as Newton. Lebniz wanted recognition of being the first. Leibniz was brilliant, but dare I say, naive. Leibniz wrote a letter to the Royal Society to settle the priority dispute. The Royal Society’s president at the time was Newton, because of course he was. Guess who won the dispute… I mean, what was Leibniz thinking?
Anyway, the whole affair involved almost the entire scientific society at the time slinging mud at each other. Academia back then was a state of mind. In the end, both gentlemen gave each other a respectful nod and soon after, Leibniz died.
Fear not, dear Leibniz! I will avenge you, with the power of equations!
To Leibniz, we owe the d notation of the derivate, for example: dx/dt, where dt is a small change in time and dx could be the associated small change in distance, resulting in the velocity of x.
Remember the weird p/q derivative from Newton, which I promised to make nicer?
Recall, that p was the velocity of y and q was the velocity of x. With Leibniz we could write it as:
The dt cancel out and we finally have:
Ahhh, so much better…
Also, Leibniz started from the other end. While Newton was busy figuring out derivatives, Leibniz came up with integrals, or summing up infinite, continuous elements of a curve, to get the area under the curve. This is the second part of calculus and an inverse operation to the derivative. We denote the integral by the symbol
∫
which you’ve probably seen here and there. This notation is also from Leibniz.
We already made an attempt at integrals here and we’re not going to go into it again. If you want to know how all that works, I will refer you to this series by Grant Sanderson from 3blue1brown. His videos are beautifully animated, intuitively narrated and in general fantastic. I hate him. Highly recommended.
Ok Leibniz, you are avenged! Now back to the afterlife with all you pompous, wig-wearing cutthroats.
Not all was well
We’ve discussed several revolutionary ideas in this article, which reshaped the world. However, they were pretty shaky, when you looked closer.
Remember when we were discussing derivatives? It’s where the whole argument stood on the basis that the changes we considered were very small. Well, how small? Small compared to what? Well, infinitesimally small.
But what the hell does that mean? No one really knew how to explain that properly. When you asked people, they said it was very small but not zero. Great…
Newton himself struggled with the concept and tried to work around it in Principia. He got smacked for it by critics, nonetheless. Only in the XIX century mathematicians were able to get a handle on that definition. So what was the answer? Let’s play a game…
A game that gets boring very quickly, yet we have to play it to infinity in order to win. Here are the rules: I propose a small number and you propose an even smaller number. Next, it’s my turn again and I propose a smaller number than yours, and so on… If for some reason we find that we can’t do that anymore, it means that we cannot do calculus. Otherwise, calculusate away. (again check 3blue1brown for details)
And what about gravity itself? Isn’t is super weird that planets are exerting forces on each other over unimaginable distances? That’s not how forces behave, right? Powder explodes and pushes the cannonball, the cannonball then touches someone’s head, the head then disappears. A force needs to be applied by something to something. Direct cause and effect. Not gravity though. Nothing is touching anything here.
Newton didn’t really want to attribute any causality to gravity and was satisfied with the mathematical description. Huygens, however was more annoyed and came up with the following. His idea was that space everywhere is permeated by this mysterious substance called the aether. The aether transfers gravitational interaction by creating a vortex that swirls and pulls planets along with it. Gotta love the imagination.
Note, that this is not the first or last time scientists introduce aether when they get desperate. It’s like saying that it’s magic, but without using the word.
Huygens idea didn’t really stick and it would take us centuries to get a grasp on what gravity could really be. This, however is a story for future episode of Physics Rediscovered.
See you there!