Question: Hello, I'm from Los Angeles. And, I have a weak understanding of geometry, but if I understand it correctly, a logarithmic spiral, as, in particular, defined by the Golden Ratio is one that becomes more complex as it progress? And I don't understand--first of all, how is this--is this only a property of a spiral defined by the Golden Ratio? Or is it a property of all logarithmic spirals?

And, how is it, that a spiral, which is bound by this ratio, grows in complexity, while it has to stay faithful to this--rule? To this ratio? How is it that it grows complex?

LaRouche: Well, the problem here is a pervasive mis-education, on all these related questions, under the influence of Euler and Lagrange, and their followers, such as LaPlace, Cauchy, and so forth, in the 19th Century. So therefore, the idea about the role of spirals is wrong.

The solution was already given, out of the principle of quickest action--that you had two basic discoveries, which are generative, for all modern, competent forms of modern mathematical physics: One, is the development of modern mathematical physics, at least in its elementary form, by Kepler. The second was the development provoked by Fermat, of this principle of quickest action.

Now, through the continued work, beyond Fermat, of Christian Huyghens, and Leibniz, and Jean Bernouilli: Leibniz developed this concept, which is called the "principle of universal, physical least action."  The principle of least action is expressed characteristically by the catenoid/catenary principle. As Leibniz showed, contrary to Euler--and this is contrary to Leonhard Euler; Euler's thing came later, as a fraudulent effort to try to bury Leibniz's discovery. But, Leibniz's understanding: The logarithmic function, the natural logarithmic function, is an expression of what is reflected by the catenary principle. It's the principle of universal, physical least action: the Leibniz principle.

Now, this principle is defined in modern mathematics, after Leibniz, only by the successive work of Gauss, in defining the complex domain and the general principles of curvature, in that connection, as by his Copenhagen lectures; and also, by Riemann; but, you have also work of Abel, for example, the young Abel, who died much too prematurely, and of Dirichlet, who was one of the collaborators, out of the Ecole Polytechnique, of Alexander von Humboldt, and was an associate of Gauss, and one of the patrons and teachers of Riemann. So, the Riemannian conception of space, is the only one, which we have now, which we know, which corresponds to reality, as opposed to something other.

The problem here, in part, as was emphasized by the ancient pre-Euclidean Pythagoreans, and is emphasized by Gauss, is, that the focal point of mathematics is not linear action. And any conception of spirals, which is based on the notion of linear action, is a very limited usefulness. That, to understand the process, you have to look at the question of the paradoxes of a spherical approach to primitive action.

Remember, the concept of the universe, with ancient man, as the Pythagoreans, for example, with their concept of spherics: Ancient astronomy, apart from this crazy variety of so-called Euclidean and so forth, was based on the notion of spherics. That is, looking up at the nighttime sky, what you're seeing is the inside of a sphere, of unknown but large radius. And, you think the stars, at first glance, the stars you see in the sky, are as if located on the interior surface of this sphere, whatever it is. So, this was the approach.

So, the idea of least action, which involves this, as Gauss presented it, incorporates this. So, what you have to do, you start from the spherical view of the universe. And, then you say: The spherical universe, as such, this universe does not define the way it actually works. There are other actions, which are impinging upon the trajectory defined by the apparent spherical universe, and these other actions act upon the so-called spherics universe. And the two things combined, become the complex domain.

So therefore, Leibniz's definition of the catenary, or the catenoid, which comes from that, in connection with the principle of physical least action, is the point of definition for which all the logarithmic functions are appropriately derived. And, implicitly, we can say, therefore, that Leibniz's treatment of the catenary, in this respect, is actually the most characteristic feature of the complex domain, as defined by Gauss and by Riemann.

That's the better way to look at the whole problem.

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