Ele Munjeli's List: Newton

• Near the origin, the hyperbola passes from one asymptote to the other in a smooth curve. There are two branches of the hyperbola, starting from opposite ends of the asymptotes. For most practical purposes, the hyperbola can be considered as the asymptote itself except in the neighborhood of the origin.
• The hyperbolic functions mentioned above are combinations of exponentials and their connection with the hyperbola is not obvious. From their names, they are analogous to the trigonometric functions. In fact, hyperbolic functions are related to the unit rectangular hyperbola x² - y² = 1 just as the trigonometric functions are related to the unit circle x² + y² = 1.

• Knowing a and b, we know the parabola, and can find the angle between the asymptotes, and thus the deflection D. D = π - 2θ, so tan θ = cot (D/2) = bv_o²/K. This is the relation between the impact parameter b and the deflection D. The distance of closest approach is q = a (e + 1), where e can be found from a and b. This is how Rutherford determined an upper limit on the size of the nucleus, from the maximum observed deflection of the alpha particles. The solid angle between the axis and the deflection D is Ω = 2π(1 - cos D), while the area within the impact parameter b is A = πb². Therefore, the area dA for scattering into a solid angle dΩ is given by dA/dΩ = C / sin⁴(D/2), where C = (zZe²/8πεmv_o²)². This is the famous Rutherford scattering differential cross section, proportional to the inverse fourth power of half the angle of deflection. C is the cross section for scattering directly backwards (D = 180°).

    

  Rutherford's discovery of the nucleus led soon after to Bohr's atom, and from there to quantum mechanics, revealing our modern view of matter. Rutherford received a peerage and a Nobel Prize, which were richly deserved. Discoveries in mathematics and physics may lead to understanding; discoveries in most other sciences lead only to knowledge.
• n the 17th century, there was great interest in improving telescopes, which were severely limited by the spherical aberration of the spherical refracting surfaces available, which meant that they did not bring parallel rays to a point focus. Descartes worked out the forms of surfaces that would bring rays coming from a point source (or infinity) to a point focus; that is, which would provide stigmatic imaging. These were the famous Cartesian Ovals, and included among them were hyperboloidal surfaces. Their form depended on the index of refraction and on the object and image distances, so they were inflexible in application, but worst, they could not be manufactured with sufficient accuracy. For optical work, the surfaces must be correct to less than a wavelength, and this was simply impossible for other than spherical (or cylindrical) surfaces. It is still largely impossible, though good aspheric optical surfaces can be made in certain cases, such as corrector plates for Schmidt and Maksutov telescopes. Approximate shapes are good enough for non-imaging lenses, such as condensers and illumination, and aspheric surfaces are quite popular for these applications, though the surfaces may or may not be hyperboloids. As it happens, there are other problems besides spherical aberration, such as chromatic aberration, and field of view, that cannot be solved with aspheric surfaces, though several coaxial spherical surfaces (which can be very accurately produced) and glasses of different index and dispersion, can solve the problem very well.

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  Certain mathematical models suggest that until the early 1970s the world population underwent hyperbolic growth (see, e.g., Introduction to Social Macrodynamics by Andrey Korotayev et al.). It was also shown that till the 1970s the hyperbolic growth of the world population was accompanied by quadratic-hyperbolic growth of the world GDP, and developed a number of mathematical models describing both this phenomenon, and the World System withdrawal from the blow-up regime observed in the recent decades. The hyperbolic growth of the world population and quadratic-hyperbolic growth of the world GDP observed till the 1970s have been correlated by Andrey Korotayev and his colleagues to a non-linear second order positive feedback between the demographic growth and technological development, described by a chain of causation: technological growth leads to more carrying capacity of land for people, which leads to more people, which leads to more inventors, which in turn leads to yet more technological growth, and on and on.^[3] Other models suggest exponential growth, logistic growth, or other functions.

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  This pedagogical exercise is part of an ongoing series on ``Riemann for Anti-Dummies.'' For more articles like this, visit the Schiller Institute Pedagogy List, which is updated frequently. To contact the authors, or Mr. LaRouche, who commissioned and directs these these pedagogical exercises, send an email to schiller@schillerinstitute.org .

   

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• hen the Delians, circa 370 B.C., suffering the ravages of a plague, were directed by an oracle to increase the size of their temple's altar, Plato admonished them to disregard all magical interpretations of the oracle's demand and concentrate on solving the problem of doubling the cube. T
• l, exercises for economics.
• As the accompanying diagrams illustrate, the parabola possesses the characteristic of one mean between two extremes, while the hyperbola embraces two
• f significance for this discussion is the discontinuity revealed by this projection between the parabola and the hyperbola. The hyperbola stands on the other side of the infinite, so to speak, from the ellipse and the circle, while the parabola has one side toward the infinite and the other toward the finite.
• uyghens recognized that the curved and the straight expressed themselves in the hyperbola differently than in the other conic sections. His insight was based on the same principle recognized by Menaechmus, that the hyperbola, when projected onto a plane, was formed by a series of rectangles whose area was always equal. As one of the sides of each rectangle got longer, the other side got inversely smaller. Huyghens focused his attention on the area bounded by the hyperbola and the asymptote, which is the area formed by an ever-changing rectangle whose area is always the same (Figure 6). Areas between the hyperbola and the asymptote, formed by rectangles whose sides are in proportion, are equal. Consequently, as the diagram illustrates, those sections of the hyperbola, formed as the distance along the asymptote from the center increases geometrically, are equal. Thus, as the areas increase arithmetically, the lengths along the asymptote increase geometrically.
• he catenary. Leibniz demonstrated that the catenary was formed by a curve, which he called ``logarithmic,'' today known as the ``exponential.'' This curve is formed such that the horizontal change is arithmetic, while the vertical change is geometric. The catenary, Leibniz demonstrated, is the arithmetic mean between two such ``logarithmic'' curves (Figure 7).
• rom here we are led directly into the discovery of Gauss and Riemann through Leibniz' and Bernoulli's other catenary-related discovery: The relationship of the catenary to the hyperbola [1]. This relationship is formed from Huyghens' discovery. The equal hyperbolic areas define certain points along the hyperbola, that are ``projected'' onto the axis of the hyperbola, by perpendicular lines drawn from axis to those points. These projections produce lengths along the axis, that are the same lengths that, as Leibniz showed, produced the catenary!

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• Ken Schmarge

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  Neugebauer suggests that the origin of the concept is in the theory  of sundials, since the sheaf of light rays involved in the design of  sundials is a cone which is cut by the plane of the horizon in a  hyperbola, and a portion of that hyperbola is then marked out on the  sundial.
• These works share  a common thread-they require the extensive use of the properties of  parabolas, Archimedes' specialty amongst the conic sections
• Let AB be the principal axis of a conic.

     

  Let P be any point on the conic.

     

  Let Q be the foot of the perpendicular to AB.

     

  Mark off a distance AR, perpendicular to AB by a distance  now known as the latus rectum or parameter of the curve.

     

  Apply to the segment AR, a rectangle having for one side AQ and an  area equal to (PQ)².

     

  If the rectangle exceeds the segment AR, then the conic is a  hyperbola.
• The final topic of Apollonius' Conic Sections to be considered is  his treatment of tangents. He develops this topic in both Book I and  Book V. Book V introduces the idea of "maximum" and "minimum" lines  to refer to tangents and normals, respectively. This book, considered  by Eves to be "the most remarkable and original" of the seven we have  today, quickly becomes very difficult to read and follow. The  propositions and relationships it proves, which today are more easily  shown using differential calculus, are rigorously explored in the  classic Greek geometric fashion (Heath, 1961, pp. lxxv-lxxvi).   Preliminary theorems, however, are not terribly difficult to follow.   First we will look at two propositions from the first book concerning  tangents (one will be stated and discussed, the other formally  proved), and then we will look at one Book V theorem.
• The culmination of  the subject came at the hands of Apollonius, who, in eight volumes,  rigorously developed all that was known about conic sections before  him, and added a multitude of propositions that were original (we  believe) to him, so much in fact that Eves notes, "The treatise is  considerably more complete than the usual present-day college course  in the subject".
• Though the work  of scientists and mathematicians, like Kepler who was both, conics  evolved from a novel intellectual exercise in Ancient Greece, to a  powerful modeling tool for explaining the physical laws of the  universe.

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