I want to accomplish just as now summarizing on the night. In a 21-year period from 1652 to 1673, Huygens answered 12 important questions that had not been answered before. I'm just gonna run through them. What's the distance of vertical fall in the first second in the absence of a resisting medium?
Second, what rules have impact for perfectly hard spheres in head-on collision? What rules actually work in contrast to those proposed by Descartes? What agree with experience? Third is Descartes' quantity of motion conserved in head-on impact of perfectly hard spheres? And if not, what quantity or quantities are conserved?
We've got what we now call momentum and kinetic energy. What is the strength of the tendency, the to recede from the center in uniform circular motion? v square over r. What is the tension in the string retaining a body moving in uniform circular motion? v square over r, times the weight of the body.
What is the law fully characterized in the relationship between the dimensions of conical pendulums and their periods? I'm not gonna state that. Where must a 90 degree circular arc pendulum be intercepted for its bob to reach the vertical with its string remaining taut in ascent? From what principles can Galileo's claim a path lies independence of speed acquired in the absence of a resisting medium be derived?
And does it hold regardless of the trajectory of descent? Given that the circular arc is not the answer what is the isochronous path in descent? How can an isochronous pendulum be constructed? With gravity, as in number nine, what is the law fully characterizing the relationship between the dimensions of simple isochronous pendulums and their periods?
And where is the center of oscillation of a circular arc pendulum with multiple small bobs? Those were all outstanding problems. Riccioli had answered one of them. All of the others were still hanging. All got answered in a 21-year period by Huygens, with our current answers. That is, there's nothing up here that isn't current textbook physics.
It's a reasonable accomplishment, right? By this time, by 1673, when Horologium was published he was 44 years old. Which is about the age Newton was when Newton began The Principia, he was 43. All right, we still have a few minutes. Now, what we've done here is, we've gone beyond Galileo in a very striking fashion.
When I finished Galileo a couple weeks ago, I tried to emphasize, I don't know how well I communicated it. I tried to emphasize that Galileo had only a fragment of a theory. He had vertical fall, he had the parabola. He couldn't do pendulums, he couldn't do arbitrary curvatures.
He could do very, very little. What Huygens has done is to use Galilean principles to solve circular motion, to solve even impact cuz he's using Torricelli's principle as the basis. And energy principle derived from Galileo, whatever velocity is acquired is exactly what you need to get to the same height.
He's using those to solve a whole range of further problems, all tied together primarily by Galileo's sublimity principle, and path-wise independence. Now, I'm not saying this is a complete theory. What I'm saying is we now have a very rich theory of motion under uniform parallel acceleration. Enough so that for the rest of this course, both semesters, I'll start calling it the Galilean, or Huygensian theory of motion under uniform parallel acceleration, okay?
In the process, we've added two new forms of evidence that nobody had come upon before. One of them he didn't emphasize, but Newton did. Namely, if you have multiple theoriate mediated means for measuring fundamental quantities like constants of proportionality, then you can play them off against one another.
Both to expose sources of systematic error, or when they're in tight convergence with one another you conclude that something fundamentally is right among the assumptions building to them. My students from last year learned about this with a measurement of the mass to charge ratio of the electron. Those who ever looked at the history of microphysics, the key moment was multiple measurements of Avogadro's number from very different theories.
That was the turning point in people finally accepting molecular theory. And it's all over the place. This is the first place I know of where anybody did it. Ptolemy comes close. Because he'll do a theory mediated measurement of an element and then suggest a second way to get the same element.
So it's not totally new here, but for Ptolemy it's a crosscheck procedure, in effect, safe guarding against systematic error. Here I think it's much richer. And then this thing I said at the end, and extending the theory beyond its initial idealizations, Huygens has opened the way to another new form of evidence.
Namely by showing that the initial theory is a first approximation that can be extended to cover deviations from the initial theory without requiring new hypotheses. The same theory can be extended beyond the simple cases to more complicated cases. Now the remark I make at the bottom here, and it leads in to the next slide.
What you've just seen in the last two and a half hours is the birth of what we call theoretical physics. What I mean by that, the practice of theoretical physics is to take theoretical principles and see how many disparate problems you can solve with the same principles. And you keep building up this collection of problems.
People sitting at a desk using the same principles trying to get a solution to a new problem. That's what theoretical physics is, and that's what Huygens has done better than anybody before him. By extending a core theory to lots and lots of new problems. Okay, and it becomes a model for future physics.
And that's my last slide. This is the attempt to describe how we got what's called now, Newtonian mechanics. And there are four traditions feeding into it. At the top is the magnetical philosophy, purely in England. Following Gilbert, you have a group of people, Wilkins, you'll learn about him next week.
You've already heard about John Wallace, Christopher Wren, and Robert Hooke. They're all starting to talk about gravity and things like that as like magnetism. You've got an orbital astronomy tradition that reaches it's height with Kepler and Tycho and Horrocks. Following it you have Street, the person from whom Newton learned his astronomy.
He's also the one who brought Horrocks to the public. That feeds into Newton. And that feeds into Newton's Principia, the double arrows are the really strong ones. Quite separate from that tradition is so-called rational mechanics. The term is Newton's, but I'm gonna describe it as the way Clifford Truesdell describes it.
Take everyday common sense, join it to mathematics, and you can produce most of the history of mechanics. Continuum mechanics, Truesdell wants to do everything that way. The point is, it's a combination of mathematical tractability in good sense and careful observation. I think Truesdell's wrong about only one aspect of that.
Add in the capacity to make predictions that are really striking that can be tested. Huygens and Galileo were the model for it. The thing is, Galileo didn't carry it far enough to be a really clear model. Huygens, it's all over the place. This agrees with reason and it also coincides with experiment, place after place.
That tradition starts from Galileo with all those other people tied to him. Comes to Huygens, Huygens has a huge influence on Leibniz and the two Bernoullis. They in turn have their influence on the next generation. All of these people are looking for the suitable fundamental principles to solve all problems, and they know Newton's laws can't do it.
That's what Lagrange's Analytical Mechanics does, it proposes, this is the right way to solve all these problems. Meanwhile, Euler working off of this tradition but using Newtonian force, develops the Euler equations of motion. That's what we actually call Newtonian mechanics, Euler's six equations of motion. Newton's influence is weak to that.
The main influence is through these people, into Euler's equations of motion. Where Newton's influence is gigantic is in celestial mechanics. I'll say it a different way, when you study Physics you learn almost nothing about how to solve problems involving inverse square gravity. They're messy. What you learn is the part of mechanics that doesn't have inverse square gravity.
That part of mechanics comes overwhelmingly out of Galileo, Huygens, through these people. Okay, so this is a map of influences. I'll make one last point about it. I mean, obviously any drawing like this, no drawing like this is perfectly right. But I'll make two points about it. It's not clear we wouldn't have most of what we call mechanics if Newton had never lived.
Because Huygens had been in a position and started a tradition where most of it would have occurred. What we're missing inverse square of gravity. That, somebody else had to do. Huygens was not about to do it, and you'll see the extent to which that's true. That's one comment.
The other comment, and this is probably the most useful thing to end with. When we talked about Kepler and Horrocks I made the point, until the next generation picks it up and assimilates it, meaning they start doing something with what came before, nothing much happens. Horrocks died five years into doing that with Kepler.
Huygens lived 40 years doing that with Galileo. And you can see the difference, okay? There's a monstrous development in the hands of one person and people around him, working off of a tradition he started from Galileo. The obvious question, if Horrocks had lived 40 years, what would astronomy have looked like?
I don't know, okay? But I hope you'll appreciate this. This is the plot I want you to remember into the next semester. Because as important as Newton's Principia was, it's remarkably unimportant to what we call Newtonian mechanics. It's indispensable to celestial mechanics.