Rasmussen: Thank you, Lyn. Do you have time for more questions?

LaRouche: Have you got a good one?

Rasmussen: Just a small thing: So, the cycloid is not the catenary?

LaRouche: What?

Rasmussen: A catenary is not a cycloid; is that what you just said?

LaRouche: Yeah!

Rasmussen: Hmm.

LaRouche: It never was. That's the famous proof of the principle of least action by Leibniz.

Rasmussen: Do you know how many classes, where I was wrong, that I've taught?

LaRouche: The cycloid was an approximation in the process of--see, the cycloid came--. The genius of Huyghens (not Galileo, who's a faker; he's a fraud and a fake, all the way through), but when Huyghens is confronted with the evidence of Fermat's work on quickest time, simply argued, when he was trying this navigational clock, is: How can you build a clock, which  will eliminate the errors of the pendulum, hmm? And therefore, he came to an isochronic curvature, which corresponded with the derivative of the cycloid, and has a very obvious relationship to circular action, in terms of the complements of circular action, because the cycloid is a complement of a sine function, hmm?

So then, he discovered (it was actually he, that discovered it), the [null?] and the double [inaud] spiral problems, one of the things that led to this discovery: that it could not be that. Then, you had the 1690s' discussion among Leibniz, Jean Bernouilli, and others, of the issue of quickest time. And, even there, Bernouilli would try to generalize this principle, from the standpoint of the cycloid, as the isochronic pattern.

Then Leibniz and Bernouilli continued the discussion, on the issue of the catenary. And it was Leibniz's definition of the principle of universal least action, was based on this discussion of the catenary, which we've done fairly well on. Bruce [Director] has done a good job in pulling together the system of geometries, which define that.

And then, you look at Gauss's proof against Euler and Lagrange, on the question of the fundamental theorem of algebra, the complex domain. Then you see exactly, that from the standpoint of curvature--forget the so-called formal mathematics; look at the curvature. What are the curvatures, which combine to define the catenary function?

Now, locate this function within the complex domain. Now, go ahead to Leibniz: Applying Gauss's evolution of the idea, into a general theory of curved surfaces, and use the principle of least action, as the notion of a geodesic principle, which applies to an n to n+1 order Riemannian manifold.

And then it's clear: That the principle of least action, as defined by Leibniz, in this discussion of the catenary, the same place where he discovers and proves the significance of natural logarithms--by that proof, which Euler did not have--and, as a matter of fact, Euler rejected it. Euler wrote his attack on Leibniz, based on rejecting Leibniz's discovery of the principle of least action, and, in particular, Leibniz's concept of natural logarithms.

But Leibniz was right. And this shows most clearly, when you come to the question of "What do you mean by a ‘geodesic,' in the transformation of curvature of a system, from a Riemannian order n, to a Riemannian order n+1?" Then it becomes indispensable.

And, this is absolutely indispensable in economics. You can have no understanding, whatsoever of physical economy, except from this standpoint. [background laughter]

Rasmussen: I have some work to do.

LaRouche: Well, good! Have fun: It's great fun. And Bruce has done a lot of this, and Jonathan [Tennenbaum] has done a lot. But, it's great fun. And it's exciting. I mean, this is the beauty of scientific work--even abstract scientific work, in these kinds of problems, which belong in the same class, as Platonic dialogues. They're exciting, and they... [tape break]

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