Different subject. Now I'm gonna just take you through Horologium Oscillatorium for a couple of minutes. I'm gonna initially be relatively brief comments, but I wanna give you a picture of the thing you of course you can read this. I've given you the whole book now but this was part of the reading.
So it starts with three hypotheses. The first one says, if there were no gravity and if the air did not impede the motion of bodies, then any body will continue its given motion with uniform velocity in a straight line. Our principle of inertia. Second, by the action of gravity, whatever its sources, it happens that bodies are moved by a motion composed both of a uniform motion in one direction or another, and of a motion downwards due to gravity.
So we're gonna combine, we're gonna have two motions together, one of which is in any direction uniformly, and the other is gravity. Third, these two motions can be considered separately with neither being impeded by the other. That's a big deal remember that's the one that Simplicio challenged and it doesn't hold in the case of air resistance.
And notice how he does it. These are, of course, parabolas, if you look at them carefully. But he's doing it at oblique projections. He's read Torricelli so he doesn't have to do just the Galileo case. He can do all the different oblique projections, and say the same thing.
Proposition one. In equal times, equal amounts of velocity are added to a falling body. And in equal times, the distances crossed by a body falling from rest. Are necessarily increased by an equal amount. That's the one three five type progression. Now, he thinks he can prove uniform gravity.
I once received a paper as a referee that said Huygens is better than Newton. Huygens proved gravity is uniform, and my comment back to the author, it's wonderful to prove something we know is false. So the only question here is where did he go wrong, and I've underlined it.
Which clearly is the same in the second unit of time as in the first. Okay, if you assume the action of gravity is exactly the same amount in each unit of time, yeah, you can prove uniform acceleration. Okay, you're totally begging the question. The whole point is the action of gravity depends on where you are, and therefore is not the same.
In each unit of time, okay? This is an example to show you Huygens' mathematical proofs are not always proofs. Okay. You have to read them carefully to see the physics principles coming in. Now the next sequence of slides give you proposition six through ten. It begins with the following.
From this, it will not be difficult to prove the following proposition, which Galileo asked that we accept, as in a sense, being self evident. For the demonstration which he tried to give later, and which appears in the later edition of his works, does not seem to me to be too strong.
The proposition is the following. The velocities acquired by bodies falling through variably inclined planes are equal if the elevations of the planes are equal. This is path wise independence. Just before this proposition five and proposition three develops the mean speed theorem. And gives you the first rigorous proof of the mean speed theorem.
I chose to skip that. So, he does it for this case first. You can go read what the proof is, but the proof ends up invoking Torricelli's Principle. That is, you're gonna contradict Torricelli's Principle unless this is true. Then, two propositions later, he lets a sequence of planes.
And that then lets him go up, as well, and finally, he does it completely curvilinear. In other words, he ends up giving us pathwise independence not in an incline planes, on any surface whatsoever. That's what it ends up saying. If a body falls perpendicular through any surface, and if it later moves upward by the acquired impetus through any other surface, then it was always had the same velocity or at points of equal height in its descent and ascent.
So that's gonna be, that couple to the prior principles gives us what we really want. It doesn't matter what the path is. You're gonna acquire a certain velocity and that velocity will be exactly what you need to put you back up. And at every height, the velocities will be the same.
Zero at the top going down, coming up, et cetera, regardless of path. So that's where the pathwise independence principle finally becomes very much established. It presupposes his version of Torricelli's Principle, so he can't literally prove it. But he can prove it based on a weakened or g principal.
This is proposition 25, it's the result for a cycloidal pendulum, on a cycloid whose axis is so much trouble with these glasses. I guess it's erected on the perpendicular, and who's vortex is located at the bottom. That's this curve, it's the cycloid. The times of descent in which a body arrives at the lowest point, at the vertex, after being departed from any point on the cycloid are equal to each other.
And those times are related to a perpendicular fall through the whole axis of the cycloid with the same ratio by which the circumference of a circle is related to its diameter. Remember earlier when I gave you the law of the cycloid of the pendulum, there was a pi in there.
That's where the pi is coming from. So he stated that law in words. I want you to appreciate, I've often had even physics major look at it and, how can that be? So if on a cycloid I start this thing a hundredth of an inch from the bottom, And I start another one at the very top, let's say 20 feet away.
They're gonna arrive at the same time about. That's what's being said. It's truly isochronous. Doesn't matter how large the arc. Now that shouldn't shock you that much cuz every musical instrument is isochronous in that way. The amplitude doesn't alter the frequency. That's what it's all about. But at any rate, that's the result.
And it's quite beautiful. It's a breakthrough. Watch the isochronous curve, this is the isochronous curve, or Merseinne posed the problem. Huygens solved it. He had solved it in 1659. This is the publication. Now this next I have to spend a little more time with. Try not to be too long.
You've already seen this. Namely he asked what shapes do the cheeks have to be in order to produce. The cycloidal pendulum. And this is the proof. I'll read the proposition out. This is in part three of Horologium Oscillatorium now. If a straight line is tangent to a cycloid at its apex and if on that line as a base another cycloid similar and equal to the first is constructed starting from the point.
Of the apex just mentioned, in any straight line which is tangent to the lower cycloid will meet the arc of the higher cycloid at right angles. Now, let me say what that means. You've pictured earlier, the cycloidal pendulum with the cheeks going vertical. Here the cheeks are down here and this is the line.
What is this line to cycloid? Answer, it's its radius of curvature. This line is exactly perpendicular to the cycloid, and if you drew a circle around this point It's first and second derivatives would exactly match those of the cycloid. This is what's called the curvature is one over the radius of curvature.
So what he has done is to find a new way to represent curves. Give the curvature at every point and the trajectory of the center of the radii of curvature and that has a name invented by him, it's the evolute of a curve. The way to think of it, what shape does a wall have to have for a string unrolling from it, with a weight at the end, to describe any curve?
He's got the solution for the cycloid. The evolute of a cycloid is itself. Okay? Now one other thing about this. If this is the circle at this point and it's moving, what's the tension in the string from the motion at that point? Well, instantaneously it has to be v square over r.
It's exactly the same as if the circle were there because that's the tension in the string at that point. It has to be. I'll say it differently. Here's a way to think of this motion. It's a series of circles, one after another, changing in size. That's what the evolute is giving you.
Okay. Now I'll just pass these two around. This is a classic. This is my undergraduate text in differential geometry. The first section is on curves. It's all about evolutes and Huygens. This is a very recent book. A quite advanced book a retired professor at Harvard, Schlomo Stamberg, on the importance of curvature in mathematics and physics.
Huygens has initiated something new. Newton did this independently of Huygens. Newton did it in the calculus. But they're both landing at the same thing. Namely, here's a way of representing curves. It's a series of motion in a curve. It's a series of motions in circles. At any given moment, the perpendicular component has to be v square over r, and then some other force is making a change from one circle to the next.
Okay. That's fundamental to Newton's Principia. He knew all about it before he got to the Principia. He knew about it before he read Horalogium, but there sitting in Horalogium part four is not just the evolute for cycloids. It's the evolute for ellipses, the evolute for hyperbolas, the evolute for parabolas, a whole series of curves.
How to represent curves by radii of curvature and the trajectory of the center of curvature. And it's striking, Huygens invents the idea of an evolute and pretty much completes the theory all in one single part of the Horologium. And that's why Gioiella called it unrolling time. Because it's this picture of the string unrolling from a surface that's at the heart of all of this thinking here.
So this is a whole new element. Just the foundation of modern differential geometry. Though in Newton's hands, it's done with derivatives, so it's really transparent. But Huygens is managing to do it geometrically with evolutes.