december 2008 Vol 3 No. 3
Transcription
december 2008 Vol 3 No. 3
ΔΥΝΑΜΙΣ The Journal of the LaRouche–Riemann Method of Physical Economics december 2008 Vol 3 No. 3 December 2008 Vol. 3 No. 3 www.seattlelym.com/dynamis ΔΥΝΑΜΙΣ The Journal of the LaRouche-Riemann Method of Physical Economics EDITORS Peter Martinson Riana St. Classis Jason Ross LAROUCHE YOUTH MOVEMENT OFFICES: Boston, MA: 617-350-0040 Detroit, MI: 313-592-3945 Houston, TX: 713-541-2907 Los Angeles, CA: 323-259-1860 Oakland, CA: 510-251-2518 Seattle, WA: 206-417-2363 Washington, D.C.: 202-232-6004 For submissions, questions, or comments, please email [email protected] - or [email protected] - or [email protected] 2 From the Editors 3 The Calling of Elliptical Functions By Michael Kirsch 26 On the Subject of ‘Insight’ By Lyndon H. LaRouche, Jr. 38 Third Demonstration of the Theorem Concerning the Decomposition of Integral Algebraic Functions into Real Factors By Carl Friedrich Gauss 41 Letter from Carl Gauss to Wilhelm Bessel December 18, 1811 44 The First Integral Calculus Johann Bernoulli 48 Exclusive Interview: René Descartes What’s the Matter with Descartes? By Timothy Vance On the Cover Raphael’s Alba Madonna. An instant, which encompasses all eternity. “...God, like one of our own architects, approached the task of constructing the universe with order and pattern, and laid out the indivitual parts accordingly, as if it were not art which imitated Nature, but God himself had looked to the mode of building Man who was to be.” Johannes Kepler Mysterium Cosmographicum 2 Editorial At the moment this issue of Δυναμις is being put to print, there is a vast, immediate battle being waged for the future of the human race. On the one side, Lyndon LaRouche is mobilizing a team of national governments to erect a new international credit system, based on the best aspects of the American System of Economics. On the other side, an old oligarchical empire, centered in London, England, is unleashing the terrorist hordes it has been fostering through the international drug trade for years, to preserve its power and create chaos, potentially leading to the assassination of America’s president elect. To think clearly in a time of crisis such as this, one needs to be creative. August of this year past marked an historic turning point in the development of the scientific capabilities of humanity. Concluding a month long seminar series on the conceptions associated with Carl Gauss’s discovery of the orbit of Ceres, the Gauss “Basement” team passed the baton to the next team, who was tasked with blazing the trail into Riemann’s conception of higher hypergeometries. Simultaneously, a selection of former Basement dwellers produced an hour long Δυναμις video, taking Harvard University, and the rest of so-called academia, to task, for graduating uneducated boobs (such as George W. Bush) into positions of influence in our society. This video, The Harvard Yard (http://www.larouchepac.com), presents a real history of Kepler’s discovery of our harmonic solar system, as opposed to the Newtonian witch-quackery of “action at a distance” heaped out in its place today. The new Riemann team has already made history, by making several new discoveries about Riemann’s early development. In looking back at Riemann’s own written draft manuscripts on Geistesmassen, or thought-objects, they found that several key sections had been kept out of the published version in his collected works, edited (selectively) by Heinrich Weber. In re-translating the entirety of Riemann’s writing on this subject, they put together an initial picture of how Riemann’s work on the mind informed his conceptions of related physical phenomena such as gravitation, light, heat, and magnetism, through to the end of his life. A crucial insight was made by the team leader, Sky Shields, who had begun investigating Wolfgang Köhler and his school of Gestalt Psychology. Köhler found that the mind does not assemble its ideas from small parts and sense impressions, but that all thoughts are derived from whole gestalts. In attempting to find a mapping between this activity of the mind and the human nervous system, he ran into a conceptual block which came from physics. Assuming that thoughts are just epiphenomena of mechanical or chemical (i.e. nonliving) processes in the brain, implicitly eliminates the possibility of a higher mental principle that can order lower thought phenomena. Köhler had thought that the most appropriate physical model, where a higher principle or- Δυναμις Vol. 3 No. 3 ders the lower, observed phenomena, was what Carl Gauss, and later Riemann, called potential. The 20th Century introduction and promotion of the science of little hard particles, called atomic science, had virtually replaced the study of potential. Köhler saw that the observed relationship between the mind and the senses can only exist in a universe composed of gestalts – which can, in cross section, be expressed as what we might call individual thoughts, much like a potential field expresses itself, in cross section, as the motion of the bodies in that field. This notion was in exactly the same spirit as Riemann, who declared that all of what we explain as “forces” are expressions of the motion of a space-filling matter, not a Newtonian war of nut-tugging. Nobody who thinks, really believes in the particles-acting-at-a-distance hoax, anyhow. It is merely a form of social control. To help break this control, the Riemann team has produced a video, called The Matter of Mind (http://www.larouchepac. com), laying out the argument that, only the science from Kepler through Gauss, Riemann, and Köhler is real – the other stuff is what makes slaves. During the transition to the Riemann team, one of the outgoing members of the Gauss team, Michael Kirsch, prepared a report about the conceptual history of elliptical functions, from Nicholas of Cusa up through Gauss and Riemann. His report forms the centerpiece of this issue of Δυναμις, and is supplemented by translations of Carl Gauss’s third proof of the Fundamental Theorem of Algebra and a letter he wrote to his student Bessel on the true nature of physical functions, and also a translation of Johann Bernoulli’s first lecture on the integral calculus. Also included is a report written by Lyndon LaRouche, to aid in clarifying a crucial issue in how humans generate discoveries. Last, Δυναμις has obtained an exclusive interview with none other than René Descartes. The reader should be reminded, that the reports contained inside Δυναμις represent the rebirth of true science, which has virtually died out over the past century. The true mission of a young person today, must be to become a true scientist (which also means learning how to perform classical choral counterpoint). The next real scientific breakthroughs must be made in this generation, and not wait for some distant point in the future. The survival of the human race depends on breakthroughs in our understanding of the universe, so don’t waste your time. Anybody can read a popular science magazine, but it was only a few driven individuals who went further than anybody else ever had. Peter Martinson Riana Nordquist Jason Ross December 2008 3 The Calling of Elliptical Functions Kirsch The Calling of Elliptical Functions How a Lemniscate is Not Other than a Riemann Surface Michael Kirsch H ear now from those who sought to tame the principles of transcendental physical pathways, bringing into mankind’s understanding the principles which were uncertain. Gauss’s discovery of the characteristics of functions, related to those processes which Johann Bernoulli defined as elliptical, and Riemann’s later explicit reworking of his concept of an elliptical function, led to a new degree of conceptual freedom for mankind—an ability to understand that the reality of a process can be understood by the internal unseen characteristics of the process itself, rather than by any predicates. The importance of this realization, is that it defines the historical arc of investigating transcendental processes of nature, as a continous development of demonstrating the human mind’s ability to conceive of invisible principles in their own domain, without depending on false shadows and images. This is not to say that the invisible principles are now seen, sensed, or even understood directly, but rather, the principles are come to be known not in terms of what effects they produce, but how those effects are produced. The specific principle here goes far beyond the subject matter involved; it is as Kepler’s captive, the physical pathway of the Mars orbit, which, when brought into the understanding of the human mind, is applied to increase mankind’s power and effectiveness. Just as certain principles, when applied to society as a whole, have the effect of increasing the power of labor through technological advancement, this principle of understanding how processes can be known and conceptualized, is as important as it is extensive. This important realization defines a continuity through the foundation of modern science to its maturity. It leads one to understand something fundamental about the human mind, why it must be defended, and why civilization’s main object is the pursuit of its development. The Foundation of Modern Science Cardinal Nicolas of Cusa defined the method of investigating processes in his dialogue On Not-Other. The following reasoning from his dialogue serves to introduce his concept: What causes us to know, what most gives us knowledge of the world? Definition. How is definition known? The definition is expressed from the defining of a thing---definitions define all things. And if definition defines everything, does it not define itself? Then the definition defining everything is not other than the defined. Δυναμις Vol. 3 No. 3 What Cusa presents to the reader of On Not-other, is that principles of nature, such as light, or heat, define themselves. From this characteristic, principles do not have pieces from which they are made; they are indivisibles, units. The distinguishing character of men then becomes whether they, in investigating the effects of light, for example, keep their mind on the principle of light which shares in this Not-other characteristic, that of being self-defined. Only those who ask, “What is the cause of this process,” can come to discover this characteristic, a realization only obtained by having a clear concept of the cause. The history of modern science is founded upon this method. The lack of this approach is due to the fact that it has been replaced by a priori assumptions about physics, geometry, and mathematics; their presence acts in such a way, that although there have been many discoveries which have overthrown these assumptions, the shackles are continually re-fastenend, and new discoveries are re-stated in terms of the old ones, re-explained by what was known before. Like the old flame that keeps translating you back into the person which their fantasy first thought up when you just met, for the reductionists, nothing is really new. The Error of Archimedes This method of approach was founded in Cusa’s correction of the error of Archimedes in his attempt to square the circle. Cusa shows in his paper, On the Quadrature of the Circle, that the curvatures of the angles inside and outside the circle have an invisible difference which nonetheless exists, an essential characteristic which is the result of the kind of action which generates a circle, as the action which Kepler discovered generates the ellipse corresponded to a physical relationship present at every infinitesimal moment of the orbit. This defines the curved and straight lines partaking in those angles, as incomparable with one another. After noting this ‘species’ difference, he shows that from straight lines, it is impossible to determine the radius of a circle which would have an equal perimeter to a triangle, but, that one can come seemingly very close. This error of precision is pointed out by him as fundamental to the way in which the human mind, seeking to measure truth, must approach principles.1 1 “The measure with which man strives for the inquiry of truth has no rational proportion to Truth itself, and consequently, the person who December 2008 4 The Calling of Elliptical Functions Kirsch Cusa wrote later in his Theological Complement, that the defining difference of his approach with Archimedes’ is based on the fundamental distinction between the rational and intellectual parts of the mind. He states in that location, that the gravest mistake of Archimedes was his reliance on his rationality to measure a principle only graspable in his intellect; the problem was not that in his mind he sought to measure the circle with a straight line, but that he endeavored to manifest this rationally.2 His point is best expressed by two different responses to the following question: how do you find the perimeter of a circle, whose measure is a straight line? Archimedes’ reply was to use an exhaustive method of approximation through bisection of arcs in comparing the straight to the curved. Cusa, however, answered that the real circle whose area is measured by a straight line exists only in the infinite. “The ancients sought after the squaring of a circle.3 If they had sought after the circularizing of a square, they might have succeeded…a circle is not measured but measures… Given a finite straight-line, a finite circular-line will be its measure. “Thus, given an infinite circular line, an infinite straight-line will be the measure of the infinite circular-line… Because the infinite circular line is straight, the infinite straight-line is the true measure that measures the infinite circular-line… “Therefore, the coincidence of opposites is as is contented on this side of precision does not perceive the error. And therein do men differentiate themselves: these boast to have advanced to the complete precision, whose unattainability the wise recognize, so that those are the wiser, who know of their ignorance.” Nicholas of Cusa, from Toward a New Council of Florence, translated by William F. Wertz, Jr. (Schiller Institute: Wasthington, DC 1993) 2 Cusa writes: “But the coincidence of those features in every polygon in terms of a circle, ought to have been sought intellectually; and [then those inquirers] would have arrived at their goal.” The rationality determines the properties of a subject, such as a radius of a sphere, or the geometrical properties of a curve. The intellect, can conceive of the concept of an infinite sphere, where the center and circumference coincide, or, as the infinite circle, whose measure is a straight line. Cusa, ibid. 3 Such as Archimedes Δυναμις Vol. 3 No. 3 the circumference of an infinite circle; and the difference between opposites is as the circumference of a finite polygon.” [emphasis added] Cusa’s solution is outside the comprehensibility of the rational, but exists in the intellect; in conceiving of the essence of a thing, the intellect brings the relations between different species to clarity; bringing the boundaries of a species into the understanding, it thereby illuminates the concept of a generating principle.4 In other words, a process such as a circle is the projection of an unseen principle, and thus the essence determining its uniqueness will be unseen. This process is known to the mind, not by describing the process (the arc of the circle), with its effects (the straight lines of the polygon), but only through the paradox of the infinite. Cusa’s concept of the infinite is that the mind must ascend to the generating principle, the principle of the not-other, to see that the principle of the circle defines itself. 1. Descartes’ Fraud Holding this approach of Cusa in the mind, travel forward to beyond Kepler’s rigorous demonstration of Cusa’s method for the elliptical orbit, his own paradox in measuring such a curve, and his discovery of universal gravitation bounding the system of our sun.5 In his New Astronomy, Cusa’s follower Johannes Kepler had uncovered the physical paradox of constant change in the universe. This paradox he captured by determining the constant physical relationship governing the relation between the physical cause of the sun, and the sense perceptible observations of Tycho Brahe. As this relationship was found to take the form of an ellipse, it was accompanied by a problem he called upon future geometers to solve. In the aftermath of Kepler’s mortal death the networks of Descartes had attempted to remove the method of investigating physical paradoxes. By means of his limited powers of mind, Descartes announced a ban on such physical paradoxes as Kepler left for geometers, or, what he termed, ‘mechanical’ problems. 4 Cusa had made this point in De Docta Ignorantia when he brought the infinite to mathematics. Cusa used the example of the infinite line to demonstrate that the maximum is in all things and all things are in the maximum. Each finite line could be divided endlessly and yet, a line would always remain. Thus the essence of the infinite line was in a finite line. Likewise each line, when extended infinitely, became equal, whether it was 4 feet or 2 feet. Thus the essence of each finite line was in the infinite line, although participated in by each finite line in different degrees. Here, the circle is in every polygon, in such a way that each polygon is in the circle. “The one is in the other, and there is one infinite perimeter of all.” http://cla. umn.edu/sites/jhopkins/DI-I-12-2000.pdf 5 See http://www.wlym.com, and the LaRouche PAC video The Harvard Yard December 2008 5 The Calling of Elliptical Functions Kirsch “Probably the real explanation of the refusal of ancient geometers to accept curves more complex than the conic sections lies in the fact that the first curves to which their attention was attracted happened to be the spiral, the quadratrix, and similar curves, which really do belong only to mechanics, and are not among the curves that I think should be included here [in geometry], since they must be conceived of as described by two separate movements whose relation does not admit of exact determination.”6 [emphasis added] Box 1 Bernoulli shows an example of Leibniz’s description of this transcendental relation in his lectures on the integral calculus. Since the expression for the circumference of a circle is 2 times Pi times the radius of the circle, then two circles, having radii which are in proportion to each other as two lines which are incommensurable, such as 1 and square root of 2, will have perimeters which are likewise incommensurable. Imagine a cycloidal curve, produced by two circles in such a relationship described, one rolling about the other. In such a case there will be no number of times the rolling circle will come back around to the same position, such that it would trace out a finite number of cycloidal curves; rather an infinite number of curves will be traced. The consequence of this is that an equation produced by the intersection of a line with these infinite number of curves will have an infinite number of roots, be of infinite degree, i.e. transcendental, and non-algebraic. This means that geometry was not to be used in determining physical paradoxes. But, to say you can’t use geometry is to say you can’t know it, since geometry is the means by which we come to measure what something is and is not. Gottfried Leibniz, who like Kepler before him, had been in training to be a priest before devoting himself to the study of mathematical physics, exposed this fraud. The ‘mechanical’ curves, as Descartes defined them, were indeed constructible, but only by rising above the assumptions of Descartes. Leibniz’s reply was the following: “If you wished to trace geometrically (that is by a constant and regulated motion) the Archimedean spiral, or the Quadratrix of the Ancients, you could do it without any difficulty by adjusting a straight line to a curve, in such a way that the rectilinear motion would be regulated from the circular motion. And that is why, contrary to what Descartes has done, I will not exclude such curves from geometry, because the lines which are so described are exact, and they involve properties which are very useful, and are adapted to transcendental magnitudes.”[emphasis added] In an earlier letter on the subject to Antoine Arnauld in 1686, Leibniz further elaborates and clarifies this sentiment, saying that his new method provides the means of reducing transcendental “curves to calculation, and I hold that they must be received into geometry, whatever M. Descartes may say. “My reason is that there are analytical prob6 From The Geometry of Rene Descartes Translated by Smith and Latham Δυναμις Vol. 3 No. 3 lems which are of no degree…e.g. to cut an angle in the incommensurable ratio of one straight line to another straight line….”7 “… I show that the lines which Descartes would exclude from geometry depend upon equations which transcend algebraic degrees but are yet not beyond analysis, nor geometry. I therefore call the lines, which Descartes accepts, algebraic, because they are of a certain degree in an algebraic equation. The others I call transcendental.” [emphasis added] Circular and Exponential Transcendentals What made Leibniz’s method of constructing transcendentals with geometry possible, was, that unlike Cartesian geometry, which treats a geometrical image as a self-evident fixed object, Leibniz discovered the characteristic of change of a process to be its most essential nature. With this conception, combined with experimentation by means of moments of change in the geometry, the mind can discover patterns and lawfulness that lead to a realization about the constant relationship which is present at every moment of change, one that exists in each infinitesimally small interval, guiding the process of the sense perceptible effect. Inversely, the curve could then be constructed as an expression of that relationship acting as though infinite to each moment of generation of the curve, as from above. From this, the curve is known not as an a priori sense perceptible object, but as an expression of the unfolded relationship maintained at every moment.8 7 See Box 1 8 See Box 2. A basic demonstration of the application of infinitesmal calculus can be easily be understood by means of the simple December 2008 6 The Calling of Elliptical Functions Kirsch Box 2a Box 2b Take the hyperbola (top diagram). What is the equation for finding the simple commensurable points of the curve? Y = X −1 or Y = X1 So if Y = X1 1 The equation for the parabola is ob = cb2 . ob = oa oa' = ob' If cb = w and oa = z, then w:2 z::dw:dz Since z = w , then dz = 2 w dw, which is the expression for d(z) = d(w2 ) . The relacb tionship 2oa is maintained for any tangent to the parabola. Inversely, the constant expression of that relationship is the parabola. 2 1 then X is to 1 as 1 is to X, or X1 = X1 . For example, if X = 4, then ¼ is to 1 as 1 is to 4 Think, what is the geometric relation of this curve? It is such that it always has 1 or a constant as the geometric mean between X and Y. Therefore, how do you create a geometric mean? Have you ever doubled a square? If yes, then you know what a geometric mean consists of and therefore will know where the constant would be between X and Y on the hyperbola. Did you figure it out yet? Ahh, well, I’ll give you the answer. It’s at half the angle between X and Y (in the top diagram, it is 45°). If you don’t know why, double a square, until you create a spiral. Then look at the properties of the spiral. This then remarkably gives us not only any X and Y relation, but also a tangent to the hyperbola of ½ X and Y, meaning that the point of the curve which is the tangent is at the coordinate ½ X and ½ Y (see the top diagram, in this case it is 2 for X and ⅛ for Y) Lets translate this now into mathematical mumbo jumbo and see why this works out according to Leibniz’s calculus of differential. Supposedly, the curve we found is the tangent, which means that it is dy −2 in X terms of mathematics. the characteristic triangle of the curve or the dx = Now, by way of the mathematics of the calculus, we know that the differdy = X −2 . We also see that the triangle we find ential of Y = X1 is dy = X −2 dx or dx geometrically as the tangent has a length of X and a height of X −1 ; which means 1 dy −2 that the proportion is X1 = X1 . But our calculation says that dx = X −2 That’s ok, because its actually X over 1, which we find is a similar tri- This geometrical differential of change blasts the empiricism of object-fixated analysts9, and defines the calculus as a language of change, capable of communicating ideas about indivisible principles which lay invisible to processes, and yet exist as their cause. As a corollary of this general method, Leibniz showed how such ‘transcendental’ and other curves could be constructed by different means, for example, not di- angle (bottom diagram) to rectly, but by the quadrature of geometric curves. 1 X 1 = 1 X. geometric parabolic and hyperbolic curves. 9 Anal-ists Δυναμις Vol. 3 No. 3 1 December 2008 1 The Calling of Elliptical Functions 7 Kirsch Leibniz’s noBox 3 tation for the integration of curves showed Bernoulli shows that the exponential curve, that the expression for a transcendental of no degree, can be constructed by both the quadrature of means of the quadrature of the hyperbola. the hyperbola and the Starting with the equation for the curve of conarc length of a circle, stant subtangent(the characteristic of the exponential as Cusa had indicated curve), dw:dz :: w:a, one sets up the equality a dz = a2 earlier, was indeed dw: w. transcendental, and Bernoulli then shows that when the equality of not able to be inte- this relationship is maintained through all possible valgrated algebraically10; ues of z and w, the exponential curve will be constructhowever, although ed. 2 transcendental, the a dw : w = az . The integral To conintegrals of the circle struct the left side of the equation, the z axis of the 2 and hyperbola could hyperbola has the lengths, a . This multiplied by w be related to the phys- the infinitesimally small change in w, gives us the area under the hyperbola. The area of the ical function to which hyperbola grows arithmetically in proportion to the geometric growth of the w axis (as investhey correspond, and tigated by St. Vincent de Gregoire,1 to used to construct othwhom Leibniz gives credit as one of his er transcendentals rethree major inspirations.) The right side lated to them. Thereof the equation, a dz, is a simple rectanfore, although they gular growth, whose sum would be az, a themselves were not constant times the z azis. algebraically solvThis area az maintaining its able with the Leibniz equality to the area of the quadrature method of integration, of the hyperbola sum of a2 dw : w, will the functions were therefore, in arithmetic rectangles along known, and thus the the z axis, project geometric growth in concept of the conthe w axis, which is the exponential struction was clear. curve. In this way So, the sum of a2 dw : w, the curves, such as conquadrature of the hyperbola, is the logachoids, could be rithm of w, that is, the logarithm of the constructed from an geometric growth, w. equivalent area made up of a hyperbola and 1 See Bill Ferguson’s article, in the Upcoming Issue of Dynamis for more on St. Vincent. a circle, or, many othing quadrature of one geometric curve on one axis, and the er transcendental curves, such as the exponential curve, could be equality of a rectangular area on the other axis, produces two constructed with two simultaneous actions. simultaneous actions which construct the relation on which the Maintaining a relationship between a constantly growtranscendental curve depends. An example of such a construction 10 The concept of integration depended on the relationship between the function and its change at every moment. Without such a relationship, the integral could not be found, e.g., in Box 4, the differential of the quantity under the square root, –2 x dx, is not the differential part of the expression, that is dx, and thus there is no means to integrate the expression, since there is not a direct relation between a quantity and its differential. (See Bernoulli’s Lectures on the Integral Calculus, Translated By Bill Ferguson, in this issue of Δυναμις.) For certain algebraic functions, many methods were used to solve integrals; however, for the transcendental curves, the only recourse was the ability to relate the differential to the actual function to which it corresponded. Δυναμις Vol. 3 No. 3 of a transcendental curve is the method of constructing the exponential curve by means of the quadrature of the hyperbola (Box 3).Within this method, for calculation, the arclengths closely related to the quadrature of the circle could be converted to Arcsine and Arccosine, while the quadrature of the hyperbola could be converted to logarithms. This method, related to a whole class of transcendentals, was able to solve many problems not otherwise solvable using the algebraic rules of the integral calculus.11 11 See Box 4 and 5 December 2008 8 The Calling of Elliptical Functions Kirsch Box 4 CONVERTING ARC LENGTH OF CIRCLE TO RADIANS 1 as = Quadrature of the Circle 2 1 as = Quadrature of the Circle 2 s = Arc length of Circle vπ a dx length of Circle √s = Arc = s= Arc Cosine x= 360◦ a2 − x2 Physical Transcendentals The expression for the arc length of the circle cannot be algebraically integrated using the rules of the integral calculus; this underscores its transcendental nature, as there is no way to capture the true meaning of the arc length by means of the sine and cosine expressed as the algebraic magnitude, x. What is shown here, is that the expression for the arc length can be simply turned into a certain amount of radians which are equal to the arc length. This is not ‘solving’ the integral from within the functions, but rather, simply stating what the arc length equals in terms of the circle itself. It is, in a sense a tautology: the arc length equals the arc length. This is easily seen if one substitutes Cosine v for x in the integral √ expression for s; the ratio of the value υ)a : a2 − cos υ 2 becomes simply dv, thus, for d(cos υ = dυ . As an example, if the angle is 60°, then the arc π length in radians is 3 . Box 5a1 Now, for an Equilateral Hyperbola √ AC = CH. √ With this underx+ √x2 −1 AH = (HC) 2 AF = 2 standing of the infinitesimal √ √1 F H = (IH) AG = calculus, the understanding 2 2 of how to conceptualize AB = a, a = 1 1 dx Since AF*IF = AG*BG, then ABG = ADE and √ ABD = processes, such as Cusa had AC = x √ 2−1 2 x 2 thus, ABD = IFBG.2 indicated much earlier, was CD = y = 1 − x 1 1 dw dx brought to a higher degree = = ABD IF BG = 2 w 2 x2 − 1 of maturity; however, the real breakthrough for sci1 1 dw By Box 3: = log w ence came with Leibniz’s 2 w 2 investigation of physiBut, for this axis, since AB is taken as the concal transcendentals. The stant log √12 must be subtracted. boundary of this domain of Thus:3 transcendentals was defined √ 1 1 1 1 w dw by the investigation of the = log w − log √ = log (w 2) 2 1 w 2 2 2 physical principle existing ontologically outside the Therefore: domain of the ‘geometric’ dx transcendentals. √ log (x + x2 − 1) = x2 − 1 This superior campaign waged by Leib1 For more on the relation between definite integrals niz in the method of conand logarithms, see Bill Ferguson’s article, which will appear in a future issue of Δυναμις, for a more ceptualizing processes, and generalized study of definite intergrals involving the capturing them for manhyperbola and logarithms. kind, necessilarily leads us to introduce what may 2 Here, log w means the natural logarithm. seem to some, an unrelated 3 See Box 5b subject. Leibniz had taken bodies which is not contained in extension, for which he is falseit upon himself personally, since leaving Paris in 1676, to wipe ly praised. With Descartes’ science of ‘extension’, substance (in out the interrelated Cartesian fraud which pertains to the domain the sense of Plato, Cusa, and Kepler) was banned from thought. of physics. It is Descartes’ religious belief, that there is nothing in Δυναμις Vol. 3 No. 3 December 2008 1 1 9 The Calling of Elliptical Functions Kirsch Box 5b are necessary in physics, but rather, everything is deduced from a priori axioms about geometry. This is the battle field upon which Leibniz must be understood. Taking this into account, we will look at two physical transcendentals, one which ontologically defined the whole domain of transcendentals just discussed, and a second, posed by Leibniz, which leads us into the discoveries of Gauss and Riemann. 2. Experimental Metaphysics Here, it is quite easy to see, that log w + log ww = log www. What is more, it is generally true that log b + log d = log (bd) ea = b, a = log b ec = d, c = log d log b + log d = log ea + log ec = log e(a+c) = log [(ea )(ec )] = log (bd) He held that all phenomena are only modifications of extension and should be explained by their properties, such as form, position, and motion---explaining all phenomena in terms of sense perceptible quantities. This “salvation from mysticism”12 must be understood, as having been done with a full consciousness of Kepler’s already established method for defining causes beyond extension. Descartes doctrine of extension thereby reduced the study of bodies in motion to purely geometric analysis, as Ptolemy had done before him. Causes were, not outrightly ‘denied’, but, in end effect, located, without reason, in the bodies themselves; ‘causes’ explained by mathematical laws such as Descartes rules for motion or Ptolemy’s epicycles.13 In this way, no experiments 12 It is common jargon, that Descartes modernized physics to have replaced what was seen as the ancient Greek, mystical explanations of the ‘substance’ of bodies. 13 This is an entirely different meaning of ‘self-defined’. From Descartes’ reasoning one might be led to believe that after mankind was freed from his body, they would be free from his ideas. If only this were so in the case of Descartes! (See Δυναμις exclusive interview with Descartes, this issue) For a concise exposition of the absurdity of Descartes’ rules of motion based on his maxim, the Δυναμις Vol. 3 No. 3 It is important to make the point here at the outset, that the following detailed elaboration, seeming to some as an extended tangent to our current arc of thought, is in fact a crucial element defining the entire pathway of our methodological discussion. Leibniz’s method, described above, of finding the relationship maintained throughout the curve – the differential relationship – took on a different challenge in application to the catenary, the curve formed by a hanging chain.14 As with curves such as the parabola, a constant relationship is sought which is present throughout the curve; however, in the case of the hanging chain, the constant relation is a physical one, not geometric. The catenary is formed by the tension between two tensions: the weight of the chain tending in the vertical direction15, and the horizontal tension amongst the links themselves. The distance between the end links that hang the catenary defines the amount of horizontal tension that exists throughout the chain, and thus the differentiation between catenaries for any particular length of chain. Through physical experiment, the proportion of vertical and horizontal tension at a given point in the chain is found to be directly related to the direction which the chain is tending at that point. What is added to this derived fact, is the most essential property of the catenary: the horizontal tension is constant for any one particular catenary hung between two points, regardless of the amount of chain that is removed. These experimentally derived relations lead the physicist to traverse from the seen to the unseen, in order to investigate quantity of motion, see the October 2006 of Dynamis: Experimental Metaphysics, Kirsch, Yule, ft: 17. Also, in the same Issue, Inertia of Descartes Mind, Ross. See Fermat Book on Light, Ross. 14 Jacob Bernoulli, posed this challenge at the end of his response to Leibniz’s Isochrone challenge of 1689, which was put forward to expose the Cartesian fraud in physics and geometry. 15 There are assumptions brought to investigating physical curves from geometry with which we must dispense. First of all, directions are not arbitrary, lying on an infinite space of many directions, rather, they are physically defined. What is down? Down corresponds to the direction which a weight falls, and horizontal is that which is perpendicular to a falling weight. Any other definition is arbitrary. December 2008 The Calling of Elliptical Functions 10 Kirsch Box 6a Box 6b the changing relation at every moment of change. If the direction at two moments of the chain are extended by means of tangents, and the weight of the chain is placed where the tangents meet, the problem of measuring the relation, of horizontal and vertical tension in the chain, can be turned into another relation proportional to trigonometric functions created by the tangents.16 Employing this, a unique singular moment where the direction of one of the tangents is completely horizontal, turns the just mentioned relation into one relating the vertical tension of 16 See Box 6 Δυναμις Vol. 3 No. 3 the whole chain, to the horizontal tension.17 The physical relationship of tensions guiding the change in direction at every moment of the curve is discovered: for any given length of chain, the proportion which represents the vertical and horizontal slope of the curve at that point, is equal to the proportion of the weight of the chain to the constant horizontal tension.18 Once the physical differential was discovered, both Leibniz and Johann Bernoulli saw its relation to the quadrature of the Hyperbola. From there however, their methods were entirely different. The most essential characteristic of the curve, the constant horizontal tension, makes the catenary the best expression of how nature performs the least action pathway in traversing space, thus defining, in its expression, the curvature of gravity and tension in bodies moving around the sun. This characteristic plays the main role in unfolding the unseen, physical paradox of this curve of least action. Here we come to the moment of inflection which leads to a completely new, ontologically defined geometry. H o w do you define a process? With what axioms do you approach an unknown process in order to measure it? The discovery of the catenary allows none. What is the algebraic relationship which defines the coordinates? Here, there are none to be found. As Carl Gauss would later demonstrate the method 17 See Box 6b 18 See Box 7. The constant horizontal tension represented by the letter a, can be derived experimentally with a pulley attached to the bottom link of the chain, or, as will be shown below, geometrically, once the physical principle is clear. December 2008 11 The Calling of Elliptical Functions Kirsch Box 6c = Length of Chain a = Constant Horizontal Tension S dx S = Tan w = a dy of ‘intrinsic curvature’ in his method of determining the curvature of a surface, the principle of the catenary can only be discovered through utilizing the physical relations of the curve, dumping all axiomatic systems in the garbage. Going to the immediate point: how would one measure the height of a catenary? In order to measure the coordinates of the curve, where does one place the abscissa, the horizontal line beneath the curve? Should it be tangent to the bottom point? Somewhere below? If so, where? There is only one non-arbitrary distance below the curve to construct and measure the changing heights of the catenary: the length, one. But what length is one? The relation of vertical and horizontal tensions in the chain, to the vertical and horizontal direction, leads to a singularity at a tangent of 45°, where the proportion of vertical and horizontal direction is equal to one.19 This length of chain is thus equal to the constant horizontal tension, the unit length, defined by the physical principle of the catenary itself. Only this intrinsic measurement, constructed under the catenary as the height of the abscissa, led Leibniz to discover the relation of the catenary to the quadrature of the hyperbola. As soon as he investigated, experimentally, the growing lengths of chain laid down straight as lengths at the bottom of the catenary the physical differential of the tensions was revealed in a new light. Investigating the diagonals of the triangle whose two sides are 1) the horizontal constant, which is set as the height at the bottom of the curve, and 2), the length of chain laid down straight, he then discovered these diagonals to be equal in length to the heights corresponding to that given length of chain. This points to the most essential corollary of these relations. In utilizing the square Pythagorean theorem, for these three sides—the diagonals (heights), constant tension (constant side), and length of chain (long side)—the physical differential relation20 was transformed into one expressing the differential in terms of the quadrature of the hyperbola.21 S dx 19 Here, our realationship a = dy = 1, thus by our above relation =1, or in otherwords S (length of chain) = a (horizontal tension). = 20 S a dx dy 21 See Box 7a S a Box 7a S = Length of Chain a = Length of Chain equal to Constant Horizontal Tension Height = S 2 + a2 = x dx S = a dy a dx a dx = √ dy = S x2 − a2 Since a is equal to 1, dx dy = √ x2 − 1 However, Box 7b unlike Bernoulli’s construction, Leibniz, dx employing his knowl√ y= From Box 7a x2 − 1 edge of the relation of the exponential curve to the quadrature of y = log (x + x2 − 1) From Box 5 the hyperbola,22 inverted and translated this particular relation ey = x + x2 − 1 From Box 3 for y, in the above diagram, into one for ey − x = x2 − 1 x, showing the height ey + e−y x= of the catenary to be 2 Thus the arithmetic mean between two exponential curves.23 In other words, Bernoulli showed the catenary could be drawn with the quadrature of the hyperbola, but Leibniz showed that the inverse function for the quadrature of the hyperbola is the catenary.24 By this means, the substance of the catenary defined its own predicates, by its physical principle alone. Every effect of the physical differential is given definition, as a function of two transcendental functions.25 22 See above Box on Relation of Quadrature of Hyperbola and Exponential Curve. 23 The idea for the last step is credited to Bill Ferguson. See box 7b 24 Johann realized, which his brother Jacob failed to do, that it was indeed a transcendental curve, and not algebraic, a discovery which recruited him to Leibniz’s method of seeking for the ironies in nature, as for example, his Brachristicone; however, his construction was inferior to Leibniz’s and did not leave the domain of the earlier, = dx dy “geometric” transcendentals discussed, thus failing to capture a substantial irony involved in the paradox of physical least action. 25 Δυναμις Vol. 3 No. 3 1 1 It is highly of note that the exponential curves which are de- December 2008 1 1 1 The Calling of Elliptical Functions 12 Kirsch In the fantasy of his own mind, Descartes imagined a physics based only on extension, one which required only a priori geometrical deduction without experimentation, i.e. an infinitely boring universe in which nothing new ever happens. On the contrary, in physics, a concept of the individual substance, or monad, is possible, a concept which is “so complete that it is sufficient to make us understand and deduce from it all the predicates of the subject to which the concept is attributed”26—provided one exists in the real universe of experimental metaphysics. The Power of Leibniz’s Construction Shattering the domain of Cartesian geometry and space, Leibniz’s construction captures, in the most accurate paradoxical metaphor, the unique irony between geometry and physics. All curves which could be described by one ‘coordinate system,’ an action defining a geometric space, are now seen to be inferior to those like the catenary which require two independent geometries. His construction points one in the direction of asking: doesn’t the catenary substantiate and define the existence of the exponential curve? That is, since the catenary is generated so easily by nature, and from it is derived two exponential curves, doesn’t only the physical least action of the catenary define the domain in which this relation of exponential curves exists? And further, doesn’t the creation of two exponential curves in opposite directions mean that one exponential curve is only a special case of the two, making the two together a single higher function, and in fact primary, defined ontologically by their generation by least action in the field of gravity and tension? In this sense, one is not putting geometric curves together in Cartesian space, but rather, a single unified physical principle is generating a geometric space—a different geometry of physical space altogether. It is not that the catenary can be drawn by the hyperbola or exponential, but rather, the catenary, is what draws the other curves. The physical principle thus defines geometry in a paradoxical manner, a geometry which goes beyond rived from the catenary are those whose logarithms are natural, i.e. the subtangents of the exponential curves are equal to one, or better said, whose subtangents are equal to the constant tension in the catenary itself. 26 Gottfried Leibniz, Discourse on Metaphysics, Section VIII Δυναμις Vol. 3 No. 3 Descartes. To elaborate this point further, think back to Johann Bernoulli’s demonstration of constructing the exponential curve by means of the quadrature of the equilateral hyperbola. The right cone, to which this equilateral hyperbola would belong as a conic section, would, in order to create the double exponential construction of Leibniz, have to be joined with another right cone. Therefore, whereas in sense perceptible, descriptive geometry there exists one cone, in physics, there exist two cones set at 90 degrees to one another. Looking back upon geometry and ‘geometric’ transcendentals from the standpoint of Leibniz’s catenary, one asks: doesn’t this mean that all prior analysis that didn’t involve physics was only a special case of sense perceptible geometry, not the kind which subsumes the true nature and characteristics of the functions? Even more specifically: Leibniz’s discovery redefines ontologically, the whole domain of the quadrature of the circle and hyperbola, and all the lower geometric curves which could be constructed by quadrature.27 The physical action is primary and generates, as a lower domain, the predicates of geometrically related quadrature. In this way, the circular and logarithmic are in a sense geometric transcendentals; they are not related to physical transcendentals directly, but only indirectly, existing as their effects and projected shadows. The catenary’s relation with the quadrature of the hyperbola shows that the transcendental quadratures were sufficient for describing certain processes, but Leibniz shows their limitation, by bringing the concept of the transcendental to a higher domain, a domain which corresponds to least action, expressing principles organizing the space of gravity and tension. Above all, Leibniz’s method of discovery demonstrated the Not-other characteristic of the catenary---a discovery possible only with the intrinsic physical geometry defined by the process 27 This is also demonstrated in the case of Leibniz’s quadrature machine, where the tractrix, the evolute of the catenary, is used as a generalized quadrature principle. See Extension of geometric measurements using an absolutely universal method of realizing all quadratures by way of motion: accompanied by different procedures of construction of a curve from a given property of its tangents. (A. E. September 1693, M. S. V p. 294-301). Latin Title: SUPPLEMENTUM GEOMETRIAE DIMENSORIAE SEU GENERALISSIMA OMNIUM TETRAGONISMORUM EFFECTIO PER MOTUM: SIMILITERQUE MULTIPLEX CONSTRUCTIO LINEAE EX DATA TANGENTIUM CONDITIONE. Translated by Pierre Beaudry, ftp.ljcentral.net/unpublished/Pierre_Beaudry/. December 2008 13 The Calling of Elliptical Functions Kirsch itself. He demonstrated the way in which the mind can be elevated to an understanding of this characteristic, by discovering the geometrical principle which expresses ironically the physical principle at every moment. The truest expression of reality is the irony with which the relation between the predicates, points to the truth that the process defines the predicates. And thereby, it is this type of domain in which intellectual ideas truly exist. It is the transformation between domains, which gives a power to man, and pushes his ability to new degrees. In this sense, there is not an object which one can see as the transcendental, only an effect is seen. The transcendental is only present in the mind, seen as an anomaly between what is yet undiscovered and what is known and yet transformed ontologically by the fact of the existence of the transcendental. In this sense, for those who do not ask “why?” when confronted with a paradox to their preconceived notions, the transcendental does not exist. 3. Descartes’ Fraud, Again We saw above that the differential expressions for the quadrature of the circle and hyperbola could not be integrated but could be related to their known physical functions. Yes, these transcendental expressions were a rigorously true definition for the area of the curve, but not indicative of how they were generated, i.e. of the domain to which they actually corresponded. However, there were other physical transcendentals which couldn’t be related in any way to known functions, not even in the way that the catenary is ‘related’ to quadrature of the hyperbola. Accordingly, Leibniz had his own challenge for scientists which, in his lifetime, did not become solved to a sufficient degree. He posed a challenge of constructing a curve which couldn’t be related in any way to the quadrature of the circle and hyperbola, and thus posed a completely new test for physicists. Like Kepler’s elliptical orbit, what Johann Bernoulli called the integrals of “elliptical” curves, left a challenge. A true solution to them would not be solved in his life time by him nor the Bernoullis and would lead to the paradox of what Johann Bernoulli called the ‘elliptical’ integral. As Carl Gauss himself would later say, it is this domain beyond the shadows of the quadrature of the circle and hyperbola that is of essential interest to those who seek to open up new domains of thought.28 28 “In the computation of integrals I have always had little interest in matters that simply follow from substitutions, transformations, etc.--in short, making use of a certain mechanism in an appropriate way to transform integrals into algebraic, logarithmic, or circular functions; instead, my real interest has been a more careful and deep consideration of transcendental functions that cannot be transformed into those named above. We can now deal with logarithmic and circular functions as we can with 1 times 1, but that lovely goldmine that contains the higher functions is still almost completely terra incognita. …One stands in awe before the overflowing treasure of Δυναμις Vol. 3 No. 3 However, the gateway to this new domain of study involves an often missed, but crucial, historical irony. Among most scholars of science and mathematics today, there is little understanding that higher analysis in geometry originated from the overturning of Descartes’ ban on transcendentals. But what is even less understood today, is that the development of the analysis of what came to be known as ‘elliptical’ integrals, was born directly out of the instance of Leibniz’s incorporating his method of transcendental physical curves into his fight against the Cartesian ‘physicists’. The weight of this historical truth is necessary for comprehending the tensions within the body of knowledge discovered by Carl Gauss and Bernhard Riemann. Leibniz’s Dynamics In 1686, Leibniz exposed the fraud of Descartes’ quantity of motion, causing an irrational freak out by a well known Cartesian, Abbott Catalan29; despite further correspondence on the subject, his cult-like belief in Descartes would not be challenged.30 Understanding the steps one takes to determine a new paradoxical proposition, Leibniz consciously recruited others to his method by posing a challenge whose solution required the discovery of a new principle or the application of a new method. In 1689, Leibniz challenged those infected by Cartesian methods to solve both the simple isochrone, (the curve a body would take traveling equal vertical distances in equal times)31, and the paracentric isochrone (the curve a body would take receding equal distances from a given point in equal times), knowing that only those who surpassed the dogma of Descartes could approach the problem. The political guts of Leibniz, and the realization of the uniqueness of his method, inspired the Bernoullis to take up the challenge of these isochrones, and led in end effect to a movement of scientists who could demonstrate, simultaneously, the fallacy of both Descartes’ physics and his ban on applying geometry to mechanical curves.32 new and highly interesting truths and relations which these functions offer.” –Carl Gauss, Letter to Schumacher, as translated in Carl Friedrich Gauss, A Biography, Tord Hall, p. 135 29 Gottfried Leibniz, Brief Demonstration of the Error of Descartes, Leroy Loemker (Kluwer Academic Publishers 1989) 30 See Leibniz’s correspondence with Arnauld, translated by Montgomery (Open Court Publiching 1901) 31 This curve led to the cubic parabola, which Jacob Bernoulli divided in such a way as to play a unique role in inspiring Fagnano’s Discoveries. 32 Leibniz recounts in the Acta Eruditorum of 1697, that it was his isochrone challenge which led Jacob Bernoulli at the conclusion of his solution published in the Acta Eruditorum to pose the Catenary challenge; Leibniz comments that this was the first true application of the power of his method, allowing them “to later accomplish marvels with this calculus, so much so that, from now on, this method is December 2008 14 The Calling of Elliptical Functions Kirsch Box 8a The Paracentric Isochrone The paracentric isochrone is the curve formed by a body which moves away from a given point in equal distances corresponding to equal times. To construct such a curve requires a method of inversion. Rather than having a curve, and determining its physical properties, as with the catenary or elastic curve, this curve is found as the solution to the aforesaid physical properties. The body is dropped from a height i, the radius of the circle in the diagram. Since the property is traveling equal distances in equal times away from a point, once the body reaches the start of the curve at A, it will fall the same distance away from A at the first moment of the curve, as it will in a moment along any other part of the curve. This important fact is combined with two other known laws of motion, to lead to transcendental relation which is to be constructed. First, the relationship between the speed at the first moment to the speed at the second moment is proportional to the proportion of the two distances traveled at those moments. Secondly, the squares of the speeds are proportional to the vertical distances fallen. This means that the squares of the proportion of the distances traveled at two moments, are directly proportional to vertical distances from which they have fallen. Obtaining a representation of this proportion is our first step. In our diagram, call Aw the infinitely small distance fallen at the first moment of the curve, δα distance traveled in a moment at some other point δ of the curve. The distance βα, is the change in the distance of the body from the given point A, in the time it travels form from point δ to an infinitely small distance away point α. From what was said, βα this distance is equal to Aw. Therefore, as Aw δα equals the speed of the body at A to the speed at δ, so δα equals the same βαβα Ai Ai relation. And, from what was also stated above, βαβα . Since βαβα + δβδβ = δαδα , this can be coverted to δαδα = Ai+αγ δαδα = Ai+αγ βαβα Ai . To express this we take into consideration βαβα a simple geometric application of the differential calculus. Ai = δαδα = αγ t, βα =αγdt, yt = az Let Aγ = x, γα = y, ζ = z, A = a, Aδ =δαδα t dz a dz By Box 4, o = √aa−zz , thus, βδ = √aa−zz . Therefore, our relation above βαβα δβδβ = At αγ , becomes, dt dt : tt dz dz aa−zz =a: tz a . √ dt √ a dz √ a dz 2 t = = Rearranged and reduced, this becomes √ . Integrating, we have . How to conaaz−zzz aaz−zzz t struct this relation? In Bernoulli’s 1694 paper on the elastic curve, he compares the relation of the tension in an elastic band, to its length, and the width of the band to its stretching. He also compares the length of the band to a small element, and the radius of curvature to its stretching. By comparison of these two relations, he determines a transcendental relation, which he proceeds to construct by an elaborate method of quadrature, similar to that in Box 3, but much more compliaaaa du cated. Through this, he is able to construct the elastic curve, the arc of which AQ, is found to be equal to √aaaa−uuuu . In comparison of his two constructions, one for the paracentric, and one for the elastic curve, he finds that the mean z is the third proportional of a and u, the abscissa of the elastic curve, words, z = uu a . By substituting √ orin other aaaa du this in the equation above for the parcentric isochrone, we are left with at = √aaaa−uuuu . Δυναμις Vol. 3 No. 3 December 2008 The Calling of Elliptical Functions 15 Kirsch However, due to the non-constant ‘law of tensions’ which the elastic curve obeys, anJacob Bernoulli’s construction other curve would have to be used.35 Jacob found a saving grace. He writes “as if from a prayer, a of the Paracentric Isochrone1 curve of four dimensions presents itself” shaped OA==xx OA like “the bow of a French ribbon.” This curve AB==yy AB was the lemniscate, which, by employing it inOA OA OA = OA' stead of the elastic curve, since the expression OM 1a OM==√ for their arclengths are identical, he could turn a OD = a OD = 2 quadrature into a rectification.36 In relating these Arclength of Lemniscate OA′ 2 curves to one another, Jacob came upon an intera dx √ esting relation between the ellipse, lemniscate, 4 4 a −x and elastic curve.37 But, as the elastic curve and lemniscate Applicata of Elastic Curve AB were themselves only described in terms of the x2 dx functions they produced, relating such curves √ a4 − x4 to the paracentric isochrone did not reveal the identity of such an integral. The problem was, all the relatives shared in the family secret; they Arc Length of Ellipse CD 2 2 only posed new paradoxes, leaving as a mystery (a + x )dx √ the principle involved in these higher transcena4 − x4 dentals. One is left asking: what was the prin1 Bernoulli’s original diagram included a hyperbola on either side of the lemniscate. ciple which created the process of the paracenBy the acceptance of this challenge, physical transcentric isochrone, and its relatives? What was the dental curves were now opened to be studied. What this signified process to which these predicates correspond? was a decisive defeat for the Cartesian method, and also an openSuch was the boundary of this higher transcendental. But, just as the quadrature of the hyperbola needed to ing to a new domain. be ontologically defined by a higher transcendental of a different Jacob’s solution to “Leibniz’s Curve” species, the catenary, so the ‘elliptical’ integrals of the lemniscate and elastic curve, related to the paracentric isochrone, demanded The differential relationship of the paracentric isochrone to be defined by a higher domain. What seemingly was limited led to a form similar to the differential expression for the quadrato physics was in fact an impassable barrier into the realm of the ture of the circle and hyperbola, i.e., a transcendental one.33 Howprinciples of space. ever, unlike those, and related others, there was no way to relate Looking back from the origin which led to the study of the paracentric isochrone to any known, constructible functions. ‘elliptical’ integrals, as Bernoulli referred to them, it is seen that In the construction of such a curve, Jacob tried to uncovDescartes’ absurd physics had an ironic effect of rendering the er the identity of this curve by enticing relatives of the curve, and investigation of transcendental physical processes, and the funcseeing if they would, as though spilling the family dirt, help in tions related to them, impossible. Leibniz’s method of dynamics identifying its true character. In this way, he found the integral of Box 8b the paracentric was directly related to the elastic curve, the curve of a bent flexible rod, whose integral rises to the fourth degree.34 gungen der ebenen elastischen Kurven Von Jakob Bernoulli (1691, 1694, 1695) und Leonh. Euler (1744) (Leipzig: Verlag von Wilhelm Engelmann) as much theirs, as it is mine.” (ftp.ljcentral.net/unpublished/Pierre_ Beaudry/) 35 Jacob Bernoulli, Complete Works 1740, Volume I no. LXIV, p. 627, G.G.L Construction Propia Problematis De Curva Isochrona Paracentrica 33 See Box 8a. Jacob Bernoulli, Complete Works 1740, Volume I, No. LIX, p. 601 Jacobi Bernoulli Solutio Problematis Leibnitiani: De Curva Accessus and Recessus aequibalis a puncto dato, mediante rectificatione Curva Elasticae 36 The elastic, like other transcendentals noted in Bernoulli’s lectures, required for its construction, the double action of maintaining an equality of area between a quadrature of a geometric curve, and 34 Jacob had begun studying this curve upon working on the a growing rectangle equal to it. Rectification is simply a geometric Catenary chain. His paper was published in 1694, the same year as 1 curve being drawn, as in the case of the lemniscate. his ‘solution’ to Leibniz curve. See H. Linsenbarth’s 1910 German translation Abhandlungen ueber das gleichgewicht und die Schwin37 See Box 8b Δυναμις Vol. 3 No. 3 December 2008 1 16 The Calling of Elliptical Functions Kirsch Box 9 Fagnano’s Discoveries Complement Principle Multiple Arc Principle Arc OZ = 2 Arc OW when √ 2 OW 1 − OW 4 OZ = 1 + OW 4 Arc OA = – Arc BM When √ 1 − OA2 OB = √ 1 + OA2 The Complement Principle discovery may very well be connected with the following: Let OA = z, AE = y, OE = x 2 Since (xx + yy) = xx − yy and zz = xx + yy √ √ z 1 + zz z 1 − zz x= y= 2 2 CM y 1 − zz = = x 1 + zz OM If OM = 1, then CM = OB ! which means that DM is also = OA Box 10a Comparison between Euler’s and Gauss’s view of the Lemniscate Euler A= Gauss Complement Arc Principle 1 − aa 1 + aa coslemn p = 1 − sinlemn pp 1 + sinlemn pp Double Arc Principle Corda Arcus dupli = Corda complementi dupli = Sehne des doppelten Bogens Cosehne des doppelten Bogens 2aA 1−aaAA AA−aa 1+aaAA 2sc 1−sscc cc−ss 1+ccss Multiple Arc Principle corda arcus (n + 1) cupli = corda complementi = aB + bA 1 − abAB AB−ab 1−abAB sinlemn sinlem (p + ± q) = (sinlemn p)(coslemn q) ± (sinlemn q)(coslemn p) 1 ∓ (sinlemn p)(sinlemn q)(coslemn p)(coslemn q) coslemn coslem (p (p + ± q) q) = (coslemn p)(coslemn q) ∓ (sinlemn q)(sinlemn p) 1 ± (sinlemn p)(sinlemn q)(coslemn p)(coslemn q) Δυναμις Vol. 3 No. 3 1 December 2008 1 1 1 The Calling of Elliptical Functions 17 Kirsch So inspired by ‘Leibniz’s curve’, Fagp = Arc OM = Arc om nano discovered funq = Arc ON = Arc on Euler Gauss damental principle’s. First, what is the algebraic relation between OM = a the functions related to ON = b complementary (oppoOm = A site) arcs of the curve? On = B This involves a geometrical relation with OM = sinlemn p the tangent taken at a OM = a ON = sinlemn q unique singularity of ON = b Om = coslemn p the curve, allowing one Om = A On = coslemn q to compare the angles of On = B the relevant functions. Box 10c Second, as multiple arcs of the lemniscate are traversed, OM = sinlemn p what is the algebraic relation between the functions of the Gauss comparedON arcs=on the lemniscate, with arcs on the circle. As sinlemn q curve? 38 Gauss noted in his notebook, the tangent of an angle from the cenOm = coslemn p The origins of Fagnano’s discovery are still to ter of the lemniscate, theq lemniscatic cosine at that angle. On =equals coslemn be uncovered by intrepid discoverers; only passing clues are left in his two part paper, Metodo per Misuare La Lemniscata. In his first work, he presents a relation between the equilateral hyperbola, ellipse, and lemniscate, in a diagram very similar to Jacob’s construction for the paracentric isochrone. Fagnano states that his complementary arcs principle was discovered by him in relation to rectifiying the arcs of a certain parabola. In connection with his second principle of multiple arcs, he states that the measure of the lemniscate depends on the extension of the equilateral hyperbola and a type of ellipse, while the measure of the cubic parabola depends on the extension of the lemiscate. This echoes Jacob’s construction noted above.39 With the combination of these two principles, thus opened the door to the confrontation with the essential, and Fagnano showed how to divided the Lemniscate into equal parts, aggravating problem of dealing with physically related transcensuch as 2,3,5 parts; and generally divisions which fall under 2 dental pathways of four dimensions. times 2 to the m, 3 times 2 to the m, 5 times 2 to the m, where the exponent m represents any positive whole number. A Great Frustration These discoveries by Fagnano were crucial, as they made The one crucial, actual discovery in the study of ellipticlear certain relations between the curve itself, and the functions cal integrals occurring before the time of Gauss, was made by the produced by the curve. This at least allowed the shadows, which Italian, Giulio Carlo Fagnano, who had been studying philosophy the higher transcendental generated, to form a pattern with which and theology before devoting himself to the study of geometry, to work. However, although Leonard Euler generalized the reparticularly, the investigations of the Lemniscate. lation of multiple arcs to their functions, which became known He begins his most famous work Metodo per Misuare as the ‘addition theorem’, the principle of organization was still La Lemniscata, thus: Box 10b “The two greatest geometers, the brothers 1 Giacomo (Jacob) and Giovanni (John) Bernoulli have made the lemniscate famous, using its arcs to construct the paracentric isochrone”. Δυναμις Vol. 3 No. 3 38 See Box 11. 39 Fagnano then references its relation to a paper in the Acta Eruditorum of 1695, concering the division of the Cubic Parabola. The cubic parabola was the solution to the first isochrone challenge given 1 1689, and Jacob Bernoulli had investigated dividing it by Leibniz in into equal arcs. December 2008 1 The Calling of Elliptical Functions 18 Kirsch ering this function, and simply utilized the methods of series for a different task. For Gauss, the paradox of elliptical integrals served as a tool to open up a new domain of truths, a new instrument for experimentation, not a paradox to be reduced back into something relatable to so-called known domains, as Euler had exhaustively attempted to do. Series approximations of the Lemniscate integral In this spirit, Gauss by Stirling and Euler, as noted in Carl Gauss’s notebooks deviated from the prior method of Fagnano and treated the lemcompletely unknown40; the cause which generated the process, niscate problem, not as an algebraic relation of functions whose created a great frustration looming over the cognizant minds who complementary arcs are arithmetically related, but as a periodic attempted the search. function (what he called the sine and cosine of the lemniscate).42 4. Removing the Training Wheels With this understanding of the periodic Sine and Cosine of the lemniscate, Gauss proceeded along a new path, untread. Carl Gauss’s daily log and notes from January 1797 Gauss, unlike all his earlier predecessors, investigated not how through fall of 1798, maps out the discovery of a new field of the ‘elliptical integrals’ could be expressed in terms of their funcscience involving the nature of the Lemniscate. 41 tions, but rather, what is the process such that it defines these Gauss began with the methods of his predecessors, but particular predicates. looked at the same function with completely different eyes. His To clarify the point: just as the sine and cosine of a circle notebooks show that he read everything by Euler on the Lemare the trigonometric functions of the arc of the circle, so the sine niscate related to his elaborations of Fagnano’s discoveries, such and cosine of the lemniscate, an ‘elliptical’ integral, are the ‘elas his Observationes de Comparatione Arcuum Curuarum Irrecliptical’ functions of the arc of the lemniscate. tificabilium. He also tackled the many works on determining Therefore, the question restated in another way is: rathvalues of transcendentals with infinite series. For example, John er than investigating how the elliptical integral could be defined Stirling’s De summatione et interpolatione serierum and Leonas a function of its functions, Gauss asked, how can the elliptiard Euler’s, De Miris Proprietatibus Curvae Elasticae, the latter cal function be defined as a function of elliptical integrals? The written specifically on finding infinite series for the lemniscate process is thus investigated not by what it generates, but how it integral. generates it, i.e., by looking at the way the function of elliptical However, he wasn’t interested in these methods because integrals can express the elliptical function, he asked the questhey were only good for finding numerical values. Unlike Euler, tion: how does the process, define itself? who treated the expression for the lemniscate as a complicated algebraic expression, which merely begged a numerical approximation by series, Gauss had something else in mind when consid40 When Fagnano was nominated to the Berlin Academy in 1750, he sent the Academy a copy of his Produzione Matematiche which reached Euler’s hands on 23rd December 1751, a day described by Jacobi as “the birthday of elliptical functions”. However, in the papers he would present to the academy in the following years on the subject of the lemniscate addition theorem, Euler’s work was nothing more than an elaboration of the implications of Fagnano’s discoveries; Euler himself made no original discoveries of his own. Euler, who did nothing more than take the insights of Fagnano and fill them out, was led to great fame, and is today considered one of the great geniuses in mathematics. Fagnano, an actual genius, is held in obscurity, barely known by the recipients of his original discoveries. 41 Werke, Volume III p. 404-480, Volume X pages 145-206, 509543 Δυναμις Vol. 3 No. 3 Carl Gauss’s experimentation with the Lemniscate function, expressed as quotients of infinite products and series, from his notebooks 42 See Box 10 on Lemniscatic Sine December 2008 The Calling of Elliptical Functions 2 19 x −4=0 x2 −Kirsch 4=0 Box 11 Zeroes and Infinities x x x ) sin x =x x(1 − x )(1 − x )(1 − 3π There) )(1the − 2π )(1 lobe − π of sinitxtraverses = x(1 − one every time lemniscate. 3π 2π π fore, when x is 0, Π, 2Π, x 3Π, . x. . , −Π, x−2Π, −3Π, . . . , ) · · · etc. )(1 + )(1 + · · · (1 + x x x the lemniscate function is2π zero. 3πOur function ) · · · etc. + π )(1 + · · · (1 + )(1 π look2πmuch 3π should, therefore, like the sine function: x x x ) sinlem x =x x(1 − x)(1 − x)(1 − 3Π sinlem x = x(1 − )(1 − Π )(1 −2Π ) Π 2Π 3Π x x x ·x· · (1 + x)(1 + x)(1 + ) · · · etc. 3Π · · · (1 + )(1 + Π )(1 +2Π ) · · · etc. 3Π 2Π Π Let’s see what Gauss really said: 2 x x2 x2 sin x =x2x(1 − x2 2)(1 − x2 2)(1 − 2 ) sin x = x(1 − 2 )(1 − π 2 )(1 −4π 2 ) 9π π 4π 9π 2 2 x x x2 ·x·2· (1 + x2 2)(1 + x2 2)(1 + 2 ) · · · etc. · · · (1 + 2 )(1 + π 2 )(1 +4π 2 ) · · · 9π etc. π 4π 9π x4 x4 (1 − π 4 ) π4 ) Gauss was more interested in finding how the function itself acts, rather than finding infinite series approximations. In other words, Gauss was searching for singular characteristics of the function, to distinguish it from others. Take an average, everyday transcendental function, such as the sine of an angle. What are some singular properties of the sine function? The function continuously changes, as its radius moves around the arc of a circle, but it has a maximum and a minimum – positive 1 and negative 1, respectively. It also traverses zero twice in its period around the circle. Its change is non-constant, so it is impossible to determine its exact length, in terms of the diameter of (1 − its circle, except at those singular points and a few others. We can begin to develop a mathematical expression of x2 − 4 = 0 those singular points, in hopes of defining our function with them. x x x ) )(1 − sin x = x(1 − )(1 − In 1799, Gauss proved beyond a doubt that, since 3π 2π π algebra is merely a description of a real physics, an alx x x gebraic equation was always decomposable into a num) · · · etc. )(1 + · · · (1 + )(1 + 3π 2π π ber of factors equal to the highest degree of the equation. Each of these factors represent where the equation is x x x zero. So, the equation x2 − 4 = 0 has two zeroes: at posi) )(1 − sinlem x = x(1 − )(1 − 3Π 2Π Π tive and negative 2. Therefore, it can also be writtenxas OK, x now this looks a bit different than what we x x ) analyze it. First, )(1 − Let’s sin x = x(1 − )(1 − found. (x − 2)(x + 2) = 0 . x−Π x x (1 ) , Gauss instead of 3π 2π π ) · · · etc. )(1 + )(1 + · · · (1 + 2 4 4=0 x 3Π 2Π Π Every algebraic equation has a finite number2 of x − x − x (1 − Π4 ) . Second, if you look closer, Gauss actux 4 = 0has x zeroes, and thus factors, but our sine function is transcenx x with xdenominator · · · etc. )(1 +ally )has )(1 + · · · (1 + written down a ratio, terms. ) x2 )(1 − )(1 sin x = x(1 3π 2π π xx2 of x2 − x −whole dental, thus having an infinite number of zeroes. Therefore, x Still, if thesin arcxis= equal to some number lobes, Gauss’s 3π 2π π )(1 − x(1 −= x(1 sin x − 2π2 )(1 )(1 − − 3π 2) )(1 − 2 ) π we can attempt to approximate our sine, using an alge4π function equals zero. But, in the dex x x x factors 9π xany of the x what ifπ braic equation with an infinite number sinlem of factors. ) · · x·zero? )(1 − equal x =Since x(1 −the)(1 −nominator ) · · · etc. )(1 + )(1 + (1 + Then, this function expresses where x x 2Π Π 2π x2) · · ·3π etc.x2 · ·3Π · (1 + )(1 +π x2 )(1 + the lemniscate function is infinity! Where on the lemniscate, 2 zeroes are all at integral numbers of half-circumferences · · · (1 + )(1 + )(1 + ) · · · etc. 3π 2 2π2 π x −4=0 2 x π 4π is Gauss 9π 2 doing here? canxwrite: x − 4 = 0( 0, π, 2π, 3π, . . . , −π, −2π, −3π, . . . ),· ·we Nowhere! What x x x ·4 · · etc.infinite? )(1 +is its )radius · (1 + )(1 + x Π x x )(1 sinlem = ,x(1 2Π x4 x is−equal) to the x − arc)(1 xor −when (13Π − πx4sinlem ) when =x−1 x − x − x − )(1 ) )(1 sin x = x(1 4 x(1 3Π 2Π Π the Π ) )(1 − )(1 − x = − sin x = x(1 − )(1 −π )(1 − 2π ) 3π Π fourth root of negative 1 times Π.2Π What is 3Π the fourth root of π 2π 3π 2 2 x x x x1?2 (In· other x x xsin x = x(1 − x )(1 − xnegative ) · ·root · etc.of nega)(1 + )(1 +the biquadratic ·x · (1 +words, x x )(1 − ) x + x + · ·x · (1 + )(1 )(1 ) · · · etc. 2Π 2 · 3Π · etc. )(1 + for) ·the )(1 +Π ·Gauss ·2(1 + would π2 4π tive one.)·9π also write, good old circular etc. · · · (1 + )(1 +π )(1 + 2π ) · · ·3π 3Π 2Π Π 3π 2π π sine,2 2 2 Now, whenever the arc x to+a xdenominax2 x2 2 x2 x )(1 + x )(1 + x ) · · · etc. sin x = x(1 x· · · (1 x is equal x have x and x −zero, 2 2 x2 − 2 )(1 )(1 − π 2a) zero 4π )(1 − we sinlem xbecome = x(1 x2 − 2 )(1 x − 2) 9π tor, that factor will of π ) 3Π − sinlem x = x(1 − )(1 −Π )(1 2Π sin x = x(1 − 2 )(1 − )(14π − 2 ) 9π 3Π 2Π Π some π 4π 2 9π the function. Try it out(1for non-singular point, such x4 )x ◦ 2 2 x x − π4= x x x2 2π radians 360 ). Our function gives as x =1 radian ( x + x + ) · · · etc. )(1 · · ·x(1 + )(1 2 + 2 + 2 + · · · (1 )(1 )(1 ) · · · etc. x x x 2 2 2 ) · · · etc. )(1 + )(1 + 1 · · · (1 + 1 1 1 1 1 3Π Π ) · · · (12Π π 1(1 − π )(1 − 2π )(1 + 3Π · · · (1 + 2 )(1 + )(14π + 2 ) · ·9π · etc. Π − 3π2Π π )(1 + 2π )(1 + 3π ) · · · 2 π 4π 9π 1 etc., which will be, approximately, 0.866054044. The ac- (1 − x4 ) In this function, 1 which actually gives a much better 4 4 2 2 2 π x 2like 0.8414709848, x x 2 tual sine of one which (1 − πx4 ) 2 x2 is−more x x sinradian x = x(1 )(1 − )(1 − ) approximation to the sine function, the first factor (1 + πx2 ) sinclose. x = x(1 − 2 )(1 − −2 2 ) 9π 2 π 2 2 )(14π is pretty π 4π 9π equals zero when x equals positive or negative π times Gauss applies2 this reasoning to2 the lemniscate 2 1 the the square root of negative one. Where is that zero on x x x 2lemniscate 2 is the radius of the function. If the x2 +function x · ·x · (1 + 2 )(1 )(1 + ) · · · etc. 2 circle? Notice, also, that the circle has no infinity points, · · · (1 + )(1 + )(14π +2 2arclength ) · ·9π · etc. π lemniscate, and of the lemπ 2its variable 4π 2 is the 9π 4 or poles, but the lemniscate does. Where are those poles? x 4 ) x, then we can find its zeroes. The radius is zero (1 −niscate 1 − πx4 ) π4 Δυναμις Vol. 3 No. 3 December 2008 20 The Calling of Elliptical Functions Kirsch Box 12 Lagrange’s Inverse Function Continuing to utilize the mathematical apparatus of his predecessors, Gauss capitalized on the Lagrange inverse theorem to get an expression for the inverse function for the lemniscate, i.e. if by some variable y, we denote the arc length of the lemniscate which is a function of some variable x, then how can we express x, in terms of a function of y? At left is an algebraic infinite series approximation of the inverse function, revealing nothing too profound about the nature of x. Where Lagrange used his inverse function to simply get a numerical expression for the function he was looking at, Gauss was using the inverse function as a means, a stepping stone for a more elaborate investigation into the nature of the function, using calculations to determine what the inverse function does. Since Gauss looked upon the sine and cosine of the lemniscate as real functions, not simply numerical quantities, he experimented with them, and looked at how they change taking their derivatives, and reciprocals, to determine the properties of the function. This stands in contrast to Euler, who looked at the function as a magical box, where one puts something in, and gets something out. The uniqueness of Gauss’s investigation here can also be seen as he finds, in an analogy to the circle, the tangent and the derivative of the tangent. But wait – on the visible lemniscate, as we saw above in Box 9 on Fagnano’s use of the tangent, there is no ‘tangent’ of the lemniscate, because it equals the sine and cosine of the lemniscate! Δυναμις Vol. 3 No. 3 December 2008 The Calling of Elliptical Functions 21 Kirsch One asks, “But how “On the lemniscate we have found out the most elegant in beauty and roundness and can these functions be defined things exceeding all expectations and that by methods in a moment all truths, which by the process, if the process otherwise would be universally which open up to us a whole new field ahead.” is the principle you do not yet valid, are necessarily yoked to – Gauss 1798 know and for which you are the most cumbersome limitasearching?” That is exactly the tions.” [emphasis added]44 difficulty. In this respect, one might exclaim, looking back at the earlier calculus, “Ah, training wheels!” 44 This lack of universality arising as a consequence of dismissing ‘imaginary’ values, is not limited to higher analysis. Gauss demTo accomplish the fullfillment of this method, Gauss onstrates two years later in his 1799 doctoral dissertation, that equageneralized analysis for the first time. He took into consideration, tions have imaginary roots, and likewise, that no general theorem the arc of the lemniscate which was not visible on the geometrical can be stated about algebraic magnitudes, without their physical lemniscate: a ‘complex’ one.43 Gauss, knowing the crime which significance understood. This statement is also reflected in one who Euler perpetrated in this regard, wrote that, by the neglect of looks back upon the study of quadratic residues from the standpoint imaginary magnitudes, the field of analysis “forfeits enormously of biquadratic residues;what may seem as a paradox, such as -1 be43 Gauss never breathed a word about how he incorporated complex numbers into his analysis of higher functions, such as the lemniscate, except to his collaborator Bessel in 1811.[see letter in this issue] Δυναμις Vol. 3 No. 3 ing a quadratic residue of all prime numbers of the form 4n+1 and not 4n+3, is clear as day to one who incorporates ‘imaginary’ numbers into the field of arithmetic, and contemplates the geometry of the complex modulus. http://www.wlym.com/~animations/ceres/ index.html December 2008 The Calling of Elliptical Functions 22 Kirsch By this, Gauss discovered a whole new set of relations for the numerator and denominator of the sine of the lemniscate, relating the arcs of the lemniscate, the cirThe value of the angle which produces cle, and the exponential functions. Using a given sine on a circle, is ambiguous Derivative of the Arc Length this, he continued to investigate the nature from the standpoint of the sine. 1 ds of the lemniscatic function as expressable =√ dz 1 − xx through the infinity and zero points, but in ways which revealed deeper truths. x ∞ 0 1 −1 √2 −2 √ etc . . . ds Applying this new found relation, 0 1 ∞ ∞ 3i 3i dz he investigated which values of the sine of the circle make the sine of the lemniscate Two values of x for every one value ds equal to zero, and which make it equal to of dz infinity, representing the lemniscate now by a quotient of infinite products whose variables were sines of the circle; later on, he converted this to a trigonometric series of multiple angles of circular sines. Another way to think of it: About the implications of this, he Two values for x, for each value of y writes of this in late July 1798 in his day y = xx √ book: “On the lemniscate we have found y=x out the most elegant things exceeding all expectations and that by methods which open So this was true indeed for the higher functions of the up to us a whole new field ahead.” 4th degree. Without this, their nature, and a new field, was not What was involved with this, was his study of the interepossible. lated properties of the numerator and denominator themselves.48 Gauss examined the quotient of periodic functions from this standpoint and determined which values of the periodic funcA Priest’s Calling tions make the numerator equal to zero, giving a value of zero for Gauss investigated the domain in which the nature of the the elliptical function , and which values of the functions make process generating elliptical functions exists. He investigated the the value of zero for the denominator, giving a value of infinity characteristics which uniquely define elliptical functions. Howfor the elliptical function. ever, Gauss never made clear in his lifetime how he conceptualWithout expressing the sine of a real arc plus an ‘imagiized the principle involved in these higher functions, and thereby, nary’ arc, the infinty points were not representable, indeed they did not bring their potential fully into access for use by the human could not be found; nor was the characteristic that the elliptical mind in conceptualizing the universe, and its processes. 45 function is found to be periodic in two different ways. Bernhard Riemann, who like Kepler and Leibniz before Once Gauss obtained the knowledge of the zero and him, had been in training to be a priest, inverts the problem altoinfinity values of the of the elliptical function, he could then gether, defining ontologically what Gauss never stated. (employing some of his earlier inductive researches related to In Riemann’s lectures on Elliptical Functions, he begins 46 Lagranges algebraic method of inversion ) represent them, first, with the characteristic that elliptical functions are doubly periby means of a quotient of infinite products, and, subsequently, by a quotient of power series, leading to a basis for investigating 48 This study of the numerator and denominator of the quotient further into the nature of the lemniscatic function.47 For example, expression of the elliptical function led to various paths. In 1799, Gauss finds that the logarithm of the numerical value he approxiGauss’s first entries about his use of the arithmetic-geometric mean π mates for the denominator is equal to 2 , which he says “is most arise out of this research of the relations between the numerator and remarkable, and a proof of which promises the most serious indenominator of the elliptical function. Also, the development of his crease in analysis” (See Figure 1) study of the numerator and denominator of the elliptical function Box 13 A Simple example of Multi-Valued functions 45 See Appendix 46 See Box 12 47 For a related expression of infinite products see Box 11. Δυναμις Vol. 3 No. 3 1 1 becomes the basis for Gauss’s 4th proof of quadratic reciprocity. Jacobi, in his own studies of elliptical functions, references this paper by Gauss, when employing the same series, which he called the Theta Function. Later in Riemann’s lectures on elliptical functions, he discusses the Theta Function, seen here in a unique way in Gauss’s notebooks, as one of many expressions for elliptical functions. December 2008 The Calling of Elliptical Functions 23 Kirsch Thus, as Cusa defined the characteristic of Notother, which all true principles in the universe share, their nature is that they, like the definition defining everything as being not other than the defined, define themselves. So, man is freed from the domain of the shadows, depending on predicates to express curves. Rather, the process which generates the sense perceptible curve is conceived in the mind; a concept of the substance so clear that, by the interconnected leaves of the Riemann surface, every value, every predicate is defined, as a result of the action of the substance itself. From the standpoint of this conception of elliptical functions---realizing that these are not sense perceptible curves, but have their nature and are derived from The doubly-periodic function, from Bernhard Riemann’s the characteristic of double periodicity---it is then clear that the former method of dealing with geometric curves, lectures on Elliptical Functions which for the most part looked at the characteristics of the odic. From this fact, he then derives what the consequent chareffects of the process, was entirely inferior. acteristics would have to be, representing the zero and infinity Looking back from Riemann’s conceptual discovery, points on a complex plane tiled with parallograms, whose sides it becomes clear that with the correct understanding of higher represent the two periods which were originally set forth. From physical functions, all of sense perceptible geometry exists as a these characteristics alone, the total of all the possible values special case, a projection of some higher function which can be of the function to which a single parallelogram on the complex conceptualized but not sense perceptibly represented. plane corresponds, is found to be equivalent to an elliptical integral. Thus, the elliptical integral is shown to be derived simply from the double periodicity. He then shows that since for doubly periodic functions of the second order, there are two values of the differential of the elliptical integral for every value of the inverse function, therefore, in order to conceptualize this inverse function, the two values have to be made distinguishable and unique. To achieve this, Riemann imagines two separate sheets, one for each value, and once more employs the characteristic zero and infinity points, but, this time, as discontinuities of the inverse function, to reunite the sheets in such a way as to obtain a connected surface, where one can traverse the whole range of possible values of the function continuously. However, the concept of the way in which these sheets are connected, representing the different values Branch Cuts, from Riemann’s lectures on Elliptical Functions of the function, is not a concept which allows itself to be visually represented in sense-perceptible, three-dimensional space; Ah! But the question remains: this defines the nature of this class of higher transcendentals, to If the catenary was the necessary physical process which which the lemniscate and all of its related functions belong as defined the lower geometric transcendentals ontologically, and special cases, as existing in such a domain, outside of sense perthe multiply-connected Riemann surface is the concept which deceptible physical space, conceivable in the intellect alone.49 fines the geometric lemniscate, to what physical process does this concept correspond? 49 It is unclear if Gauss had come to this conception but simply failed to present it. Two years after Gauss’s introduction of complex magnitudes into the study of higher transcendental functions, he demonstrates complex functions as a multiply connected geometry of two dimensions, in his 1799 Fundamental Theorem of Algebra. In letter to Bessel in 1811, Gauss writes of a proof for path independence for integration in the complex plane, that this path independence holds as long as integration is not through zero points, and Δυναμις Vol. 3 No. 3 that integrating through infinity points leads to multiple values of the function. Five years later, in 1816, in his third proof of the Fundamental Theorem of Algebra, he again discusses integration of areas on a plane, and makes an almost identical point that he earlier made to Bessel concerning the ambiguity of infinity points, with respect to values of an integral. December 2008 24 The Calling of Elliptical Functions Kirsch Appendix: Imaginary Arcs In 1839, C. G. J. Jacobi read a paper at the Berlin Academy of Sciences devoted to the use of complex magnitudes in arithmetic. As introduction, he said this: “Gauss, in his investigations of biquadratic residues, √ introduced the complex numbers of the form a + b −1 as moduli or divisors... But, however simple such an introduction of complex numbers as moduli may now seem, it belongs nonetheless to the most profound notions of science; indeed, I don’t believe that arithmetic alone led the way to such concealed notions, but rather that it was derived from the study of elliptical transcendentals, and indeed the particular type that are given by the rectification of the arcs of the lemniscate... [J]ust as the arcs of the circle can be divided into n parts through the solution of an equation of the nth degree, the arcs √ of the lemniscate can likewise be divided into a + b −1 by the solution of an equation of degree aa + bb .” Jacobi was on to something, but he did not have the insight of Bernhard Riemann, or his teacher Carl Gauss. How did Gauss find the poles of the lemniscate function? Nowhere on its arc, does the lemniscate attain an infinite radius, although those infinite radii are represented in the denominator of the quotient Gauss has in his notebooks. Soon after this entry, Gauss demonstrates more of how he constructed a notion of functions: Here, we see Gauss determining the lemniscatic sine of √ an arc t plus an imaginary arc u −1 . How long is an imaginary arc? Use your imagination! Up until Gauss’s time, the predominant word on complex magnitudes was that, though they are impossible, and therefore imaginary, we need to use them to make the math come out alright. But, don’t go around thinking that they are actually real – we mathematicians can just make up anything we want, so our systems work out. Gauss thought that idea was not just perverted and lazy, but downright damaging to the progress of human science. In 1799, he blasted the academics on this point, and showed that imaginary magnitudes were, in fact, more real than so-called real magnitudes. Gauss represented those complex magnitudes on a surface, where one direction was an increase in the real part of the number, and the perpendicular direction was an increase in the imaginary part of the number. So, an arc of a+bi must have two perpendicular components – but that doesn’t make any sense. Δυναμις Vol. 3 No. 3 1 First, let us see how this imaginary arc will represent itself through our sinlem equation. Using Gauss’s series approximation for the sinlem, let us see what happens when we make the arc imaginary. 1 1 9 11 13 φ −i φ + ··· sinlem iφ = iφ − i φ5 + i 10 120 15600 1 1 9 11 13 φ − φ + · · ·) = i(φ − φ5 + 10 120 15600 = i sinlem φ How about for the coslem? 1 3 7 coslem iφ = 1 − i2 φ2 + i4 φ4 − i6 φ6 + i8 φ8 − · · · 2 10 40 1 3 7 = 1 + φ2 + φ4 + φ6 + φ8 + · · · = ? 2 10 40 Well, we know how long the coslem is in terms of the sinlem, so, 1 − (sinlem iφ)2 coslem iφ = 1 + (sinlem iφ)2 1 + (sinlem φ)2 = 1 − (sinlem φ)2 = 1 coslem φ This should seem to not quite correspond to observed physics. While the physical lemniscate has only real arcs, our math equations are perfectly happy with imaginary arcs. But, where are they? One gets the sneaking suspicion, that when we answer this question, we will no longer be looking at a lemniscate curve. Now, let us force the addition formula to cough up some answers about this lemniscate function. First, we have the regular addition formula: (sl a)(cl b) + (sl b)(cl a) sinlem (a + b) = 1 − (sl a)(sl b)(cl a)(cl b) Let us smoothly introduce our “imaginary arc” which, using the relations we just found, (sl t)(cl ui) + (sl ui)(cl t) 1 − (sl t)(sl ui)(cl t)(cl ui) sl t cl u + i(sl u)(cl t) = 1 − (sl t)i(sl u)(cl t) cl1u sinlem (t + ui) = = (sl t) + i(sl u)(cl t)(cl u) (cl u) − i(sl t)(sl u)(cl t) Now, our lemniscatic sine of a complex arc is expressed as the lemniscatic sines and cosines of purely real arcs. Let’s put 1 in some values, to see what happens. We know the minimum and maximum values of the lem1 December 2008 1 1 1 1 The Calling of Elliptical Functions 25 Kirsch niscatic sine and cosine of real arcs. If Π represents one half of an arc (or one lobe), then we can calculate several values. Thus, at the origin, the sinlem is zero and the coslem is 1. At an arc equal to one half Π, the sinlem is one, but the coslem is 0. At an arc of Π, the sinlem again equals zero, but the coslem has continued to the other side, to equal –1. At three-halves Π, the sinlem has passed over to the other side, and equals –1, but the coslem has returned to the center, and equals zero. Finally, at two Π, the sinlem is again zero, while the coslem is one. Now, we can see what kinds of values our “imaginary arcs” formula will give us. Let’s set u equal to one half Π, and let t cycle through our extreme points: for t = 0 0+i·1·0·1 =0 1−i·0·1·1 for t = one half Π 1+i·0·0·1 =∞ 0−i·1·1·0 for t = Π 0 0 + i · −1 · 0 · 1 = =? 0 − i · 0 · 1 · −1 0 for t = three halves Π −1 + i · 0 · 0 · 1 = −∞ 0 − i · −1 · −1 · 0 for t = 2Π 0+i·1·0·1 =0 1−i·0·1·1 senting the growth of u. Let a surface of varying height be placed upon this sphere, whose heights correspond with those values of the lemniscatic sine in our table. What would this look like? Where is the lemniscate, now? Perhaps it is only an effect of a higher, unseen function! This is getting interesting. We have found some infinite lemniscatic sines! The lemniscatic sine for t = Π is a bit ambiguous, but that is OK, since we are just experimenting. Here is a table, with the values of sinlem for all values of u and t between zero and 2Π: u 1 3 0 Π Π 2Π 2 2Π 0 0 0 0 0 0 1 Π 1 ∞ −1 ∞ 1 2 t 0 0 Π 0 ? 0 ? 0 0 0 3 −1 −∞ 1 −∞ −1 2Π 2Π 0 0 0 0 0 The industrious reader will recognize that, if this table is continued to higher angles than 2 Π, it will simply repeat, in both the t and the u directions. Gauss recognized that, since the characteristics of repeating is different for the two directions, what we have here is a function that is doubly–periodic. It has two periods: one real, the other “imaginary.” One final step we can take, imagining what Gauss must have thought, a well-behaved function must be smooth everywhere, with the only exceptions at its poles. Imagine a sphere with infinite radius. Imagine that on that sphere is a set of perpendicular lines, one representing the growth of t, the other repre- Δυναμις Vol. 3 No. 3 December 2008 1 26 On the Subject of ‘Insight’ LaRouche Science in its Essence: On the Subject of ‘Insight’ Lyndon H. LaRouche, Jr. This article first appeared in the May 9, 2008 issue of Executive Intelligence Review In my Sir Cedric Cesspool’s Empire,1 I emphasized the importance of the concept of “insight” as key for, among other things, understanding the mechanisms of evil which characterized the most notable writings of the leading Fabian Society figure H.G. Wells. Here, I return to that notion of insight for conceptualizing the root-causes of the present plunge of world civilization, into the prospect of an immediate new dark age of mankind, a prospect caused by the role of the same standpoint of Wells in his threatening the planet as a whole, with what has now become its currently accelerating plunge toward an abyss. In real life, one never really knows what has been done, until one knows not only why and how it was done, but is capable of replicating the formation of the concept. A s I have indicated within written and oral reports published earlier: looking back from today, the most crucial event in my life, has been my surefooted rejection of the concept of Euclidean geometry on the first day of my encounter with it in my secondary classroom. The most crucial implication of that for my later life, has been, that, in rejecting Euclidean geometry as intrinsically incompetent, as I did that day, I had actually made a decision which was to shape the essential features of my life over the seventy years which have followed that event. To repeat what I have said repeatedly on the subject of that event, over the intervening years, the following should be noted as an entry-point into the discussion to follow here. My fascination with the Boston, Massachusetts Charlestown Navy Yard, had been centered in the ongoing construction-work there. This had forced my attention to the fact of the challenge of understanding the geometric principle of construction through which the ratio of mass and weight of supporting structures to the support of the total structure, is ordered. This repeated experience, on both my several relevant visits there, and my haunting possession of the fact of that experience, had already See LaRouche, H. G. Wells’ ‘Mein Kampf’: Sir Cedric Cesspool’s Empire, 2008 http://www.larouchepac.com/ established the meaning of “geometry,” as physical geometry, for me, that already prior to my first encounter with secondary school geometry.2 The continuing importance of my flat rejection of socalled Euclidean geometry at first classroom encounter with it, is typified by considering the way in which this reverberating experience led, a decade and more later, to my flat rejection of the sophistry of Professor Norbert Wiener’s presentation of so-called “information theory,” of the still wilder insanity of John von Neumann’s notions of “economics,” and von Neumann’s matching, pervert’s view of the principle of the human mind. These latter goads, and related experiences, prompted me, in 1953, to discover and adopt the appropriate consequence of Leibniz’s work, as the standpoint of Bernhard Riemann’s 1854 habilitation dissertation. In that light, this adolescent experience, with its outcome, is the best illustration from my experience of the proper technical meaning of the term “insight.”3 In fact, it was an integral feature of the process which had led me, during adolescence, to adoption of the work of Gottfried Leibniz as the chief reference-point of my intellectual life, then, and, implicitly, to the present day. From that point in my youth, onwards, the chief philosophical reference-points in my intellectual development, were wrestling against the sophistry of Immanuel Kant’s series of “Critiques,” and the systemic sophistry of both Aristotle and his follower Euclid. It was against that background—those rejections, which had been fully established already for me during the course of my adolescence, that I came to recognize, and to rely upon the concept of insight per se: Insight as being the Platonic domain of hypothesizing the higher hypothesis, a concept of the nature of the human species and its individual member, which is central to all of the discoveries of principle by Plato. The LYM Science Project Presently, three relevant, major projects by the LaRouche 2 This development was associated, during that same period of my life, with my father’s principal intention in selecting those visits, the ritual tour of the U.S.S. Constitution; my own attention was focused on the mysteries of the construction in other parts of that yard. 1 node/10610 Δυναμις Vol. 3 No. 3 3 Wolfgang Köhler: please forgive me; it was necessary! December 2008 On the Subject of ‘Insight’ 27 LaRouche Investigating the shape of space with Youth Movement (LYM) have preceded that association’s presently approaching treatment of the implications of Riemann’s 1854 dissertation. The first of those three had been based on a West Coast team, which had worked through some crucial features of the ancient origins of modern European science, as located in the related work of the Pythagoreans, Plato, and the modern reflection of this treatment of dynamics in the work of Leibniz. A second team had worked through the main features of the founding of modern European science by Cardinal Nicholas of Cusa’s and by Leonardo da Vinci’s follower, Johannes Kepler. The LYM’s thorough-going, published report on the Kepler project, is a uniquely competent treatment, as similarly expressed in the work of Albert Einstein, as by relevant others, but is not competently taught in known university programs otherwise available today. In the second study, that of the uniquely original discovery of gravitation, by Kepler, the difficulty, highly relevant to the matter of insight, is that secondary sources on Kepler’s work have been (see http://www.wlym.com/~animations), chiefly, viciously fraudulent evasions of the actual development of Kepler’s original and crucial discovery of a principle of Solar gravitation, Δυναμις Vol. 3 No. 3 a discovery which is maliciously denied to exist, as such, in conventional academic and related programs today. This is the aspect of Kepler’s work which was strongly upheld by Albert Einstein, against those TwentiethCentury Max Planck-hating thugs of the modern positivist tribes associated with the pathetic Ernst Mach, and with the worse Bertrand Russell of Principia Mathematica notoriety. In the third case-study, the work of Carl F. Gauss, I had proposed to the incoming team, from the outset, that Gauss rarely presents the history of his actual processes of discovery, but, rather, presents the results, and also provides a plausible approach to study of the way in which he might have effected the relevant discovery. The mission assigned to the incoming team was, therefore, to discover how Gauss’s mind actually worked in his making his key discoveries. Obviously, that assignment for the incoming team had been crafted by me as a challenge within the realm of epistemology, the domain of insight properly defined. Kepler This frankly original approach to the study of Gauss’s work, has produced some uniquely useful findings, findings which provide a uniquely original approach to taking up the unique revolution effected by Bernhard Riemann, from the point of his 1854 habilitation dissertation, the change which launched the Riemann revolution in science, through those challenges which Riemann posed to such among his successors as the Italy school of Betti and Beltrami. To explain the significance of those listed, four initial stages of work for understanding human scientific creativity in general, I proceed now with reference to the relevant implications of what I define, once more, ontologically, as the principle of insight. This will clear the pathway for the study of the uncompleted projects of Riemann, as the case is only illustrated by the work of Betti and Beltrami, as by the challenges posed by V.I. Vernadsky and Albert Einstein, later. Here, comprehension demands the more precise treatment of the notion of insight which is included in the following pages. The importance of treating that subject in this fashion here, is to be located, in significant part, in the fact that the third in a continuing series of science projects conducted by teams from the LYM is nearing the point at which the team’s study of December 2008 28 On the Subject of ‘Insight’ LaRouche the mystery of Carl F. Gauss’s career is now entering its completion, a point at which a comprehensive treatment of the work of Bernhard Riemann will be undertaken by a new team, the essential contributions to advancing the frontiers of modern science to be found in the work of Bernhard Riemann and his immediate associates and other collaborators. T 1. Man as Man, or Beast? he quality of insight, as I define it, again, here, is a specific potentiality which is fairly defined as being unique to all those individual human beings who are not victims of relevant physical or psychological damage. The present definition of human, as distinct from beasts, is the specific power of the human species to alter its behavior, as a species, to the effect that the potential relative population-density of the members of a culture is increased willfully, as this is illustrated not only by a human culture’s ability to increase its potential relative population-density willfully, but by the manifest transmission of such specific qualitative changes from one, to other members of the human species, as, for example, through stimulation of discovery of a physical principle by individuals presented with the appropriate intellectual stimulus. This quality is demonstrated, crucially, by the willful increase of the relative population-density of the human species, as expressed in the quality of antientropic increase of the mass of the Earth’s Noösphere, that relative, functionally, to the specific masses of the Biosphere and the mass of matter originally generated as part of the abiotic domain. Thus, there is no species of ape, or other beast, which is capable of meeting the standard of this test. On this account, there is only one human race, and no essential human differences in species, or variety, within the ranks of humanity so defined.4 This functional distinction in the potentials of human behavior, whether expressed by individuals, or by societies as a whole, is properly approached for examination from the vantage-point established by Plato, both respecting Plato’s refined definition of the concept of hypothesis, and the systemically related subject of the quality of the individual human soul, as that subject was treated by Plato and Plato’s follower Moses Mendelssohn.5 4 Any deviation from that rule is “racism, per se,” which is, in itself, the expression of an impulse tantamount, under natural law, to crimes against humanity. 5 I.e., both Plato’s Phaedo and the treatment of Phaedo by Δυναμις Vol. 3 No. 3 Bernhard Riemann (1826-1866) In general, the Classical term hypothesis, when employed in any approximation of a meaningful, Platonic way, is already a reflection of specifically human potential for creativity. The simplest expression of that distinction is the difference between reason and Sophistry. For the purposes of our discussion here, Sophistry is typified by the reductionist method, opposed to reason, which was shared among Aristotle, Euclid, and the hoaxster Claudius Ptolemy, as typical of the Aristotelean form of the method of lying called “Sophistry,” or, in current argot, “spin.” The typical expression of corruption of the human mind in contemporary, globally extended European culture, is AngloDutch Liberalism, otherwise known as the legacy of the New Venice faction of Paolo Sarpi. The extremely degenerate expressions of Liberalism (e.g., empiricism) today, are extreme expressions of Liberalism’s intellectual degeneracy such as positivism Mendelssohn. This is also the method of Nicholas of Cusa, as in De Docta Ignorantia, his follower Leonardo da Vinci, Johannes Kepler, Pierre de Fermat, Gottfried Leibniz, and Bernhard Riemann. December 2008 29 On the Subject of ‘Insight’ LaRouche and existentialism.6 Therefore, we shall proceed with our exposition here by taking up the case of Aristotle’s follower Euclid, as in the case of the work titled Euclid’s Elements. Minds Blinded by Sight The Aristotelean form of Sophistry represented by the Euclid of Euclid’s Elements, is premised upon so-called a-priori presumptions, assumptions which are associated with reliance upon the believed absurdity that “seeing is believing.” For example, it would be impossible to discover the universal principle of gravitation, as characteristic of the organization of the Solar System, except by relying, as Johannes Kepler did, upon the clear evidence of a systemic contradiction between the Solar System viewed from the standpoint of an assumed paradigm of sight, rather than the fruitfully paradoxical solution provided by contrasting the characteristic of hearing, as Johannes Kepler did, with the characteristic, linear presumption usually associated with a naive notion of the characteristic of sight.7 The entirety of the purely arbitrary presumptions underlying Euclid’s Elements, was located in a naive presumption respecting the assumed ontological elementarity of the characteristic of vision. Thus, true insight sees vision as such as representing the primitive level, sees that one’s opinions on this level, are products of a foolish belief in the reality of simple sense-experience. The lowest level of actual human intelligence, the level of actual insight, is the recognition of the fact that one’s opinions respecting sight alone, are being formed in the grip of a kind of form of mass-insanity such as “sense-certainty,” which is to be recognized as a mind blinded, thus, by blind faith in sight. For matters of science, and also history, naive seeing as such must be superseded by insight.8 6 Typically, mathematical formulations, such as mere statistics, are substituted for actual physical principles, and even for simple truth. 7 Kepler’s reflection on the apparent role of the series of Platonic solids in locating the organization of the planetary orbits, led him, by aid of reflections on the preceding work of Nicholas of Cusa, Luca Pacioli, and Leonardo da Vinci, to recognize the composition of those Solar bodies then known to him as being an harmonic ordering. It was this recognition that led Kepler to his principled discovery, through recognition of the paradoxical juxtaposition of the assumptions of sight and the assumptions of harmonically ordered hearing. 8 As in the distinction of Max Planck’s actual discovery from that positivists’ perversion (e.g., Ernst Mach, et al.) known as “quantum mechanics.” Δυναμις Vol. 3 No. 3 Kepler’s discovery of the principle of general gravitation, provides a typical kind of crucial proof of the fallacy of sense-certainty. In his Harmony of the World, the discovery of general gravitation within the Solar System required the juxtaposition of two notions of senses, those of sight and hearing (i.e., harmony), for the derivation of a general principle of gravitation among the planets. This leads to the recognition that our powers of sense-perception are to be regarded as the natural experimental instruments which “come in the box of accessories”: when the infant is delivered from “the manufacturer.” A similar insight into the fallacy of “sense-certainty” was expressed by the ancient Pythagoreans and Plato, as this was typified then in a crucial way by the construction of the doubling of the cube by Plato’s friend from Italy, the Pythagorean Archytas. Similarly, the significance of Eratosthenes’ praising that construction, was shown afresh through Europe’s EighteenthCentury conflict between the work of Gottfried Leibniz and the Anglo-Dutch Liberals (a.k.a. empiricists) Voltaire, Abraham de Moivre, D’Alembert, Leonhard Euler, and Euler’s dupe, Joseph Lagrange.9 The modern history of that conflict begins with the Eighteenth-Century algebra of Ferro, Cardan, Ferrari, and Tartaglia, on the subject of quadratic, cubic, and biquadratic geometries, and continues through, and beyond, the work of Carl F. Gauss in such matters as the evolution of his treatment of his Fundamental Theorem of Algebra and related matters. Gauss’s Personal Situation Carl Gauss suffered the misfortune of having come to maturity in the aftermath of the French Revolution, a time which Friedrich Schiller identified as expressing a lost, great moment of opportunity in history (the American Revolution and the great work of Abraham Kästner, Gotthold Lessing, Moses Mendelssohn, Gaspard Monge, Lazare Carnot, et al. as a moment which had fallen prey to “a little people.” Thus, although Gauss’s achievements themselves were to be essentially a continuation of the legacy of Cusa, Leonardo, Kepler, Fermat, and Leibniz, Gauss’s professional career depended upon his avoiding the appearance of support for all things which might suggest indifference to the alleged genius of the hoaxster Galileo, Sir Isaac Newton, and of such Eighteenth-Century enemies of Leibniz and Leibniz’s follower Abraham Kästner as Voltaire, de Moivre, D’Alembert, Euler, Lagrange, and their Nineteenth-Century successors such as Laplace, Cauchy, Clausius, Grassmann, and Kelvin. Thus, once more, the early Nineteenth Century had brought on a period in which the minds of most were blinded by 9 Lagrange, in the last years of his life, edified the tyrant Napoleon Bonaparte, an effort used by Napoleon to disperse the leaders of the Ecole Polytechnique into technical duties in the tyrant’s military service. It was Laplace and Cauchy who destroyed the educational program of the Ecole, on orders from London. December 2008 On the Subject of ‘Insight’ 30 LaRouche Carl Friedrich Gauss (1777-1855) sight. Thus, when I first introduced the LYM’s current “basement team” to the challenge of their present work (presently nearing completion) on the work of Gauss, I forewarned them, that, whereas Gauss’s work is brilliant, and his post facto account of the discoveries plausible; such was the nature of his time, that his actual method of discovery was tucked, as in the case of his personal preference for non-Euclidean geometry, behind a protective screen of intellectual camouflage. The implied duty laid upon him, or his successors, on account of that carefully crafted, protective screen, included the complementary obligation to uncover what lay, awaiting today’s attention, behind the camouflage imposed by those hoaxsters who represented the reputed embodiment of the alien, Newtonian tyrant. However, today, the present result of adopting that implied mission, is, that, to the degree Gauss’s discoveries are now being presented as finished reports from the standpoint of Bernhard Ri- Δυναμις Vol. 3 No. 3 emann’s frankness in this matter, the results, thus far, are, increasingly, most agreeable. Thus, the true genius of Carl Gauss could be recognized by students today, only when the fact is considered, that much of what Bernhard Riemann said and wrote, was indebted to what Gauss, in his adult years, rarely dared to say publicly. Therefore, to really understand Gauss, it is necessary to know Riemann, and then to see how much of Riemann’s wonderful work, his habilitation dissertation and beyond, had been made possible by what Riemann recognized as having been lurking within the shadows of what Gauss had permitted himself to say. Gauss’s repeated treatments of the subject of his doctoral dissertation, on the subject of The Fundamental Theorem of Algebra (as complemented by the related paper on the law of quadratic reciprocity), are to be recognized as a recurring theme in much of the span of Riemann’s work.10 10 Gauss’s Fundamental Theorem was first presented in 1799, December 2008 31 On the Subject of ‘Insight’ LaRouche T 2. The Infinitesimal hat much said thus far: shift the choice of subsuming topic, back from the account of Gauss’s role as such, to the ontological implications of insight per se—the point of reference, the ontological standpoint, at which Gauss’s published accounts of his discoveries, are, for reasons noted above, often met at their relatively weakest expression. Gauss’s recurring, fresh treatment of the subject of his first three statements of what he would come to call his “Fundamental Theorem of Algebra,” and the intimately related, higher subject of “the law of quadratic reciprocity,” is typical. Nonetheless, Gauss’s intention, however bounded by the ugly peer-review pressures of his time and place as a young adult, onward, is nevertheless to be seen as persistent in his effort to provide his more sensible readers crucial evidence leading them, hopefully, toward the relevant conclusions which Gauss dares not state explicitly.11 Once Riemann’s 1854 habilitation dissertation and his treatment of Abelian functions are taken into account, and the preceding writings of Gauss viewed from this standpoint, the debated matter of Gauss’s ontological intention, contrary to D’Alembert, Leonhard Euler, and the crooked British imperial assets Laplace and Cauchy, et al., should be clear to any qualified student of such matters. 12 Gauss’s treatments of the subject of the Fundamental uttered as a direct rebuttal of Euler’s 1760 publication on that subject and the closely related matter of the law of quadratic reciprocity. In all of his published work on this subject, the underlying theme which Gauss references, but does not state explicitly, is the Leibniz notion of the ontologically infinitesimal, a connection made implicitly clear in Gauss’s work. See Bernhard Riemann, Über die Hypothesen. Welche der Geometrie zu Grunde liegen (New York: Dover reprint edition, 1953): Sections numbered I. (Begriff einer nfach ausgedehnten 11 Grösse), p. 273, and II. Massverhältnisse, deren eine Mannigfaltigkeit von n Dimensionen fähig ist ...), p.276. 12 With the defeat of the Emperor Napoleon Bonaparte, the French intention of electing Lazare Carnot President of a French Republic was defeated by action of the relevant British occupation authority, the Duke of Wellington, sticking a wretched Bourbon on a London-controlled French throne. Under this British reign over occupied France, the scoundrels Laplace and Cauchy were installed to uproot the educational program of the Ecole Polytechnique’s Gaspard Monge. Monge was dumped, and his associate Lazare Carnot went to die as an exiled hero, in Magdeburg. The mental disease called positivism, thus grabbed control, but for a relatively few stubborn heroes, of the official French scientific intellect. Cauchy’s role as a hoaxster, and plagiarist of the work of Abel, was finally exposed by examining Cauchy’s post-mortem files. Carnot was a fellow member, with Alexander von Humboldt, of the Ecole. Δυναμις Vol. 3 No. 3 Theorem of Algebra and its crucial, correlated reflection of that “Theorem,” as reflected in what he defines as a “law of quadratic reciprocity,” point the alert student toward the ontological issue which he wishes to argue, but, considering the auspices, he dares not do that too explicitly. The often referenced parallel, related case of what is actually anti-Euclidean geometry, is to be considered in this light, as being a correlative of that view of the Fundamental Theorem. The relevant argument to that effect, is as follows. Once we acknowledge, as the Pythagoreans and Plato already knew, that the objects of sense-certainty are never better than shadows cast by an unsensed, but nonetheless efficient reality, and, when the same matter is then reviewed from the standpoint of Riemann’s work, the issues are much clearer. The crucial point, as I have repeatedly emphasized in earlier locations, is the fact that the enemy of Leibniz, of Gauss, of Riemann, et al., in science, has been the pack of hoaxsters typified by the Eighteenth-Century Liberals such as Antonio Conti, Voltaire, de Moivre, D’Alembert, Leonhard Euler, and Euler’s dupe Joseph Lagrange. With that British victory over France which Britain secured through, successively, the siege of the Bastille, the French Terror, Napoleon Bonaparte’s reign, and the British monarchy’s triumph at the Congress of Vienna, young Gauss had now entered the Nineteenth Century, entering a world in which official science was oppressed by the top-down enforcement of that moral, intellectual corruption known as the Liberalism of Euler and Euler’s followers. If we, then, take into account the specific issues of scientific method posed, still today, by that same Liberal political corruption, of the reigning official opinion in science of that time, and ours, too, we are enabled to distinguish what Gauss clearly intended, from what the same fear of reactions by powerful adversaries prevented him from stating clearly, as was the case in his suppression of reports of his own discoveries in anti-Euclidean geometry. To present this case, it is necessary to restate here the related point made in locations published by me earlier. The Roots of Science When we trace the history of European science from its roots, in Sphaerics, from the ancient maritime culture which settled Egyptian civilization (including that, notably, of Cyrenaica), we must recognize what can be competently termed “science” as being rooted essentially in the development of the navigational systems of the ancient, seafaring maritime cultures of the great periods of glaciation, rather than such silly, but popular academic myths as attempting to trace civilization from “riparian” cultures as such. It was the observation of both seemingly regular and anti-entropic cycles in the planetary-stellar system, which is the only supportable basis for the notion of “universal,” as that term could be properly employed for grounding the notion of science per se today. December 2008 On the Subject of ‘Insight’ 32 LaRouche The case of the settlement account, banned the knowledge of Sumer and its culture, from the of science (e.g., “fire”) from the sea, by a non-Semitic people’s minds of those mortal men and sea-going, Indian Ocean culture’s women such as Lycurgan Sparta’s colonizing of southerly Mesopohelots, the lower, subjugated social tamia, is indicative.13 In any case, classes. the very idea of science would have It is to be emphasized now, no secured basis in knowledge unas we contemplate the global wave less very long spans of ocean-going of mass-starvation which has been maritime cultures were taken into caused by the spread of the massaccount for crucially relevant feamurderous, neo-Malthusian model tures of ancient calendars. of that British lackey otherwise In short, the notion of known as former Vice-President universal, which does not exist as Al Gore, that virtually all of the a functional conception in Liberalgreat crises of known civilizations ism, is the essence of any compehave been the result of those same tent effort at developing actual scipolicies of practice which are fairly identified as pro-Satanic attempts entific knowledge. Only long-rangto ban scientific knowledge and its ing ancient maritime cultures could practice from the great majority of have been impelled to produce the the world’s human populations.14 elementary considerations underlying the Sphaerics from which all of Such has been the accelerating competent strains in European, or decline of the physical economy other science has been derived. The of the U.S.A., per capita and per idea of a universal physical princisquare kilometer, since the terrible ple, on which all competent science developments and aftermath of is premised, could not come into 1968.15 existence for mankind in any other The upshot of that line of way, unless we were to presume inquiry, is that we exist within a the source of this opinion to be, stellar universe which is governed arbitrarily, colonists arriving from by what Albert Einstein, for examJohannes Kepler (1571-1630) “outer space.” I emphasize, that the ple, emphasized as being univerdiscoverer of Universal Gravitation true concept of universal, does not sal physical principles of change. actually exist as a scientific concep14 Former U.S. Vice-President Al Gore, a British agent against tion within the bounds of empiricism or its spin-offs. the U.S.A.’s American System of political-economy, who walks in What we know with certainty, respecting contrary views the footsteps of the de facto traitor to the U.S.A., and sometime U.S. on the possibility of the existence of a practice of science, is that Vice-President Aaron Burr, is a typical advocate of the “oligarchical the contrary views are all either implicitly “malthusian,” or are model.” President Andrew Jackson of “Trail of Tears” notoriety, had products of a type of culture, such as the typical “oligarchical been an accomplice of Burr’s anti-U.S. conspiracy, and had served model,” congruent with malthusianism. I emphasize, that all as U.S. President as a lackey and accomplice of Land-Bank swindler such latter types known to us generally now, belong to a category and later U.S. President Martin van Buren. known to ancient through modern European cultures as “the oligarchical model,” a model to be recognized as being congruent 15 It is not merely the actions of the trans-Atlantic “sixty-eightwith Aeschylus’ representation of the Satanic-like figure of that ers” and the U.S. Richard Nixon Administration which have caused Delphic Olympian Zeus. This was the Zeus, who, in Aeschylus’ the pattern of accelerating physical decline of the economies of the 13 Suspected to have been an offshoot of a maritime culture of the Dravidian, or closely related language-group. Herodotus indicates a kindred maritime-cultural origin for Ethiopia. So, Bal Gangadhar Tilak back-traced the origins of Sanskrit to a colonization, across land, from the north coast of Siberia, through mid-Asia, into Iran and northern India (Orion, and Arctic Home in the Vedas). Δυναμις Vol. 3 No. 3 Americas and Europe since 1968. Trends do not perpetuate themselves, except as the relevant trend takes life, as a form of “tradition,” within the culture of those who are shaping the policy-making proclivities of the society. To free the U.S.A., in particular, from the grip of forty years of self-destruction, we must free control over our society’s policy-shaping from the hands and minds of those who embody the “68ers” tradition. December 2008 33 On the Subject of ‘Insight’ LaRouche These principles are presented to us in this capacity, as they were to long-ranging ancient maritime cultures, presented so in their astronomical expression, as a combination of both ostensibly regular and antientropic universal physical principles of change. Some cycles, such as the equinoctial cycle, are long-ranging, and may appear to be fixed. However, contrary to the neo-Aristotelean fraudster Claudius Ptolemy, and to Clausius, Grassmann, and Kelvin, the universe is not, ontologically, a domain of cycles of repeatedly fixed no-change: the universe is essentially anti-entropic. In the latter case, that universe of change, the universe is finite, but anti-entropic, in the respect that nothing exists outside it. Thus, rather than the foolishness of a ignorant believer’s assumption of an Euclidean or Cartesian, limitless space, the universe is not Euclidean, nor Cartesian, but a dynamic system in the sense of dynamic employed by the ancient Pythagoreans and Plato, or such as Leibniz, Riemann, Max Planck, and Einstein, in modern science. This notion of a physically efficient universality which I have just presented here so, is, as Albert Einstein emphasized, indispensable for modern universal science; without this notion, no competent notion of the work of Kepler, Fermat, Leibniz, Gauss, or Riemann can be reached. This notion which I have just so emphasized, is crucial for understanding the great Nineteenth-Century crisis in science which Gauss and Riemann addressed. The interwoven conceptions of a “Fundamental Theorem of Algebra” and “law of quadratic reciprocity” in the work of Gauss, are typical of this. Riemann’s remedy for what is lacking in the work of Gauss, addresses precisely this conceptual problem, a problem which continues to underlie not only the ongoing essential work of all modern science, but the systemically dynamic form of social crisis menacing the very existence of world society today. Cupola of Santa Maria del Fiore, in Florence, Italy Our Universe That aspect of the efficiently existing universe which is accessible to our sense-perceptual powers, is the passing footprints of those powers which generate such shadows themselves. As Albert Einstein made this point in his own fashion, it is through the relevant power of insight, like that of Kepler’s uniquely original discovery of universal gravitation, which is, manifestly, uniquely specific to the human species, that we are enabled to adduce the eternal motion of that great unseen entity which has left those footprints upon our heavens. Such is the implication of Riemannian dynamics, as also that of Leibniz before him. As emphasized here earlier, the fact that the organization of the Solar System is fairly regarded as in conformity with Δυναμις Vol. 3 No. 3 Kepler’s harmonic approximation, as Albert Einstein emphasized the principle involved, defines a universe which is ontologically finite. That is to say, that principles, such as the principle of gravitation as discovered by Kepler, principles which envelop our universe, are discoverable, and provable, only through the kind of method of dynamics which Gottfried Leibniz revived from the earlier discoveries of the Pythagoreans and Plato. We owe comprehension of the implications of that fact, as Albert Einstein emphasized, chiefly to the work of Johannes Kepler and Bernhard Riemann. However, that discovery had already been made implicitly by Cardinal Nicholas of Cusa, in such among his works as the seminal De Docta Ignorantia, but it had also been known, earlier, by the Pythagoreans and Plato. To restate this same point: the principled form of ac- December 2008 34 On the Subject of ‘Insight’ LaRouche tion which is expressed to our senses as a predicate of universal principles, is the universal principle on which all manifest forms of apparently principled actions depend for their expression. The universe of experience is defined, thus, as Einstein defined it, as self-bounded. Thus, it is a finite universe in that sense, but without any external boundary but the principle of anti-entropic, creative powers associated with the notion of a Universal Creator. The human faculty upon which such higher-ranking knowledge of that higher, efficiently necessary existence depends, is the object of insight in the fullest sense of Plato’s presentation of that notion. Thus, all competent modern science depends upon the view of this matter by Nicholas of Cusa. To summarize that point: the notion of an ontologically existing universe, as opposed to some Euclidean or kindred sort of Sophist’s fantasy, depends upon the notion of universal lawfulness, as Einstein’s view of Kepler’s work illustrates the crucial point of all this present discussion. To illustrate that point, take the case of the history of the modern European discussion which led into Gauss’s first statement of what was to become known as his view of the challenge of the Fundamental Theorem of Algebra. Go back to the previously referenced, Sixteenth-Century treatment of the subject of the relations among quadratic, cubic, and biquadratic residues, as by Cardan et al. The ontological implications of this Sixteenth-Century treatment of those matters must be considered against the background of Archytas’ duplication of the cube. Against that historical background of Sphaerics, the principled nature of the systemic fallacy of the method employed by Cardan et al. should have been obvious. What should have been the obvious remedy for that had been supplied, during the Fifteenth Century by the work of Filippo Brunelleschi,16 Nicholas of Cusa, and Luca Pacioli, as also by the surviving known fragments of the work of Leonardo da Vinci. In brief, the necessary approach would have been the same concept of physical geometry on which I had insisted during my adolescence, or, much more appropriately, Riemannian physical geometry, rather than the ivory-tower formalities of an implicitly pro-Euclidean algebra. In other words, when the empiricist followers of Descartes and Antonio Conti employed the fallacy of the hoaxsters de Moivre and D’Alembert, in crafting the hoax of so-called “imaginary numbers” for the fraudulent attack on Leibniz by themselves, Leonhard Euler, et al., they were not merely constructing a fraud against physical science. They were behaving as a-priorist incompetents in refusing to grasp the readily accessible, physicalgeometry implications of the uniqueness of Archytas’ method for constructing a process of duplication of the cube, rather than the intrinsically incompetent, Sophist method of Aristotle, Euclid, 16 As in Brunelleschi’s employment of the catenary as a principle of physical geometry which had been the required principle of design for the construction of the cupola of Santa Maria del Fiore. Δυναμις Vol. 3 No. 3 and Claudius Ptolemy. Admittedly, this erroneous presumption reflected a crucial oversight which had been made by the Sixteenth-Century set of Cardan et al., prior to the experimentally crucial discovery of least action by Pierre de Fermat. However, the discoveries by Kepler and Fermat were an integral feature of both the uniquely original discovery of the calculus (ca. 1676) by Leibniz, but, more emphatically, Leibniz’s taking into account the crucial principle of Fermat in Leibniz’s own crafting, in collaboration with Jean Bernouilli, of the concept of a universal physical principle of least action. This “imaginary number” fraud by de Moivre, D’Alembert, Euler, et al., was not merely a reflection of their apparent ignorance of elementary principles of physical geometry known since no later than Archytas and Eratosthenes. It was to be seen as an echo of the “malthusian” oligarchical-model hoax expressed by the Olympian Zeus of Aeschylus’ Prometheus Trilogy. When that aspect of the matter is taken into account, the difficulty which threatened Carl Gauss in the matter of the Fundamental Theorem of Algebra, ought to become transparent. Gauss’s third statement of that case ought to have made it clear, retrospectively, to all modern mathematical physicists re-considering Gauss’s proof, once the publication of Riemann’s habilitation dissertation had made clear the essential issue lurking in the shadows of Gauss’s own argument. From the appearance of Riemann’s habilitation dissertation and his Theory of Abelian Functions, onward, the deeper implications of the history of modern science since Nicholas of Cusa’s De Docta Ignorantia should have been clear, as Albert Einstein located the root of competent modern physical science in those methods which Kepler had attributed to Cusa’s work, the work which, chiefly, founded competent forms of modern European science. Such is the nature of true insight. A 3. Insight Reviewed t the close of July 2007, the world as a whole entered a phase-shift into chronic hyperinflation, into what has been, ever since that date, a general breakdown-crisis of the present world system as a whole. Since that time, the entire world’s presently existing, post-August 1971 monetary-financial system, has been doomed to its extinction, in one way, or another. There are alternatives, but these mean abandoning what has become the 1971-2008 world monetary-financial system. It means putting the present system under a juridical system of reorganization-in-bankruptcy, and replacing it with an echo of the principles and intentions of President Franklin Roosevelt’s policy for a Bretton Woods world monetary system free of those vestiges of British imperialism which, unfortunately, reign, and ruin us all, still today. December 2008 35 On the Subject of ‘Insight’ LaRouche It is important to recognize that we are obliged to use that term, “British Imperialism,” because that is the name by which it goes. The content of what that term connotes, is an international financial tyranny whose appropriate technical term of description is Anglo-Dutch Liberalism, which means the present form of organization of a network of financier and closely associated interests which was built up in northern maritime Europe by Venice’s Paolo Sarpi and his followers. “British” in “British Imperialism” marks that empire-in-fact, the leading single The author, presenting an imperial power in the world today (since the 19711972 betrayal of the U.S.A. by the Administration of President Richard Nixon), which had first been established as the imperial power of a private company, the British East India Company through the implications of the Paris Peace of February 1763. Such is the great challenge to the creative powers of the members of mankind today. Thus, on July 25th, I spoke: “... this occurs at a time when the world monetary system is now currently in the process of disintegrating. There’s nothing mysterious about this; I’ve talked about it for some time; it’s been in progress, it’s not abating. What’s listed as stock values and market values in the financial markets internationally is bunk! These are purely fictitious beliefs. There is no truth to it; the fakery is enormous. There is no possibility of a non-collapse of the present financial system— none! It’s finished, now! The present financial system cannot continue to exist under any circumstances, under any Presidency, under any leadership, or under any leadership of nations. Only a fundamental and sudden change in the world monetary-financial system will prevent a general, immediate, chain-reaction type of Δυναμις Vol. 3 No. 3 international webcast on July 25, 2007 collapse. At what speed we do not know, but it will go on, and it will be unstoppable! And the longer it goes on before coming to an end, the worse things will get. And there is no one in the present institutions of government who is competent to deal with this. The Congress—the Senate and the House of Representatives—is not currently competent to deal with this. And if the Congress goes on recess, and leaves Cheney free, then you might be kissing the United States and much more good-bye by September. “This is the month of August; it’s the anniversary of August 1914. It’s the anniversary of August 1939. The condition now is worse,objectively, than on either of those two occasions. Either we can make a fundamental change in the policies of the United States now, or you may be kissing civilization good-bye for some time to come....”17 17 From the original transcript of my remarks on that occasion. (For the complete transcript of LaRouche’s July 25, 2007 webcast, see EIR, Aug. 3, 2007.) December 2008 36 On the Subject of ‘Insight’ LaRouche The Individual in History As I have said repeatedly, of late, the history of mankind is not event-driven; it is man-driven. The most essential decisions which drive the actually crucial changes in the course of history have often been what was deemed impossible by conventional opinion-makers earlier. It is not what happened in yesterday’s usually fraudulent leading press reports which drives history; it is men or women of a special kind of influence, such as our Benjamin Franklin, or the great historian and dramatist Friedrich Schiller, who choose to lead nations in one direction or another. It is rarely a matter of choosing from among multiple choices on the table; the most momentous turns in history have been the changes, changes made by the initiative of a seemingly tiny minority, changes like the founding of our Constitutional republic which had seemed, in July 1776, to the world at large, not merely impossible, but an ill-fated conceit of a few. The greatest decisions in history are made by men or women, as individuals, decisions which have seemed virtually impossible to conventional institutions and public opinion even a relatively short time before. All great turns in history of that quality come as the unique innovation in thought and will by relatively rare individuals. So, President Abraham Lincoln saved our republic, virtually despite itself; so, the greatest poets and scientists did what no one else had dreamed before. The greatest of all such deeds occur in such times as those of which the great English Classical poet, Percy Bysshe Shelley wrote in his In Defence of Poetry. There are times when much of a people is overcome by a marvelous increase in the power of imparting and receiving profound and impassioned conceptions of man and nature, as by the inspiration of the then already deceased Friedrich Schiller in calling forth the great initiative of the German people led by Scharnhorst in organizing, according to the principle of strategy defined by Schiller’s studies of the religious wars in the Netherlands and the Thirty Years War, to accomplish the otherwise seemingly impossible defeat of the tyrant Napoleon Bonaparte in Russia and in that tyrant’s desperate effort to return to France to raise a new army and a new general war. So, a Genoese sea-captain working in the service of Portugal, the greatly talented and inspired Christopher Columbus, was led by his continuing study of the testament of the founder of modern science, Cardinal Nicholas of Cusa, one of the greatest geniuses of all modern history, to devise a plan for realizing Cusa’s program, for great strategic voyages across the great oceans, to rescue a corrupted European culture by extending its reach to distant lands. This was Cusa’s intention, as actually adopted, with full consciousness of that intention, by Columbus from about 1480 onward, which created the Americas, and brought about that subsequent colonization of New England which gave birth to what became our United States. This was the object of the actual founding of our republic, the U.S.A., whose morality was defined, first, by the crucial Δυναμις Vol. 3 No. 3 Christopher Columbus (1451-1506) passage of a work denouncing the evil slaver John Locke, the passage, “the pursuit of happiness,” from Gottfried Leibniz’s New Essays on Human Understanding, which is the core principle of our Declaration of Independence and the root of the principle of moral law of our republic which is elaborated, as in the spirit of the Peace of Westphalia, as also reflected in the great Platonic and Christian principle of agape, in the Preamble of our Federal Constitution. Thus, the true history of mankind is only that which is defined by the actuality of the perfectly sovereign creative powers which can be expressed only by the individual creative personality. These are the same creative powers, unique to sovereign individual minds, which are expressed by uniquely great discoveries of scientific principle, as by the Pythagoreans, Plato, Cusa, Kepler, and Leibniz, or Classical qualities of artistic principle, such as those of Friedrich Schiller, or the combination of initiatives rooted in a concurrence of scientific craft and moral inspiration in the achievement of Christopher Columbus. The contrary implication to be considered, against that December 2008 On the Subject of ‘Insight’ 37 LaRouche background, is that the chief source of the ugliest failures of humanity is a certain kind of popularized stupidity of the type demanded by the Olympian Zeus of Aeschylus’ Prometheus Bound, as demanded by the creature of the British Foreign Office’s Jeremy Bentham, Thomas Malthus, or as the lame-brained perversions uttered by that pathetic puppet known as the incumbent President of our U.S.A. Popular opinion, such as that induced by our presently, inherently corrupt and lying major news media, is the deadliest of the Trojan Horses inserted into the domains of mankind today. In that sense, the issue of the development of the creative powers of the individual young member of society is, in the final analysis, the most crucial political, and also moral issue of the existing cultures of this planet, most notably our presently dumbed-down, Boomer-ridden U.S.A. Our present educational systems have assisted greatly in making our people stupid enough to be influenced by the opinions uttered by the proverbial “paid prostitutes” of our presently popular “yellow” press. The Relevant Paradox The power of creativity, as I have presented the case summarily in the preceding chapters here, is, as I have already emphasized, not only a built-in natural potential of the human individual, a potential absent in all animal species; it is unique to all persons who are not victims of relevant damage to their potential range of human powers. In broad terms, therefore, every individual should be developed as a truly creative personality. As the case may be, as cows do not make for intelligent citizens, it is wrong to attempt to train people to become cows, as the latter has been done, in effect, to most of the human population in most known cultures to present date. The subject, therefore, is, once more, the case of the suppression of knowledge of “fire” by order of the archetypical Malthusian (or, present-day Malthusian and lying former Vice-President Al Gore). Only under artificial conditions such as those prescribed by Britain’s leading anti-humanist, the World Wildlife Fund’s Prince Philip, is the natural, human intellectual potential of the person suppressed in ways—pro-Malthusian ways—which turn children into the virtually half-witted cattle of today’s neo-Malthusian movements. Consider what caused the legendary Olympian Zeus to cook up this anti-human role of “environmentalism.” There are two, complementary motives. First, actually creative and brave people will not willingly submit to either a legendary Olympian Zeus, or a Prince Philip or Al Gore. Second, since mankind’s creativity is typically expressed through its realization as scientific and related progress in developing prevalent human conditions, the continuation of the progress which man’s true nature demands, “uses up natural resources” in ways which only the natural advances in the science-driven and related creative productive powers of mankind could remedy. Δυναμις Vol. 3 No. 3 On the latter account, of the Earth’s total mass, the portion corresponding of pre-biotic masses is shrinking as a percentile relative to the product of living processes, while the rate of increase of the portion of the mass generated by human activity is increasing, relative to both abiotic residues and residues of other kinds of living processes. Thus, to keep large populations sufficiently stupefied to be reigned over by the tyrannical likes of the Olympian Zeus, it is necessary (for the sake of that tyranny) to keep subject populations as stupid as possible, and, therefore, to prevent actual increases in the productive powers of human labor, or, even, as has been done in the U.S.A., and in western and central Europe since 1989, to reverse previous economic progress absolutely.18 For that reason, nominal American citizens such as former Vice-President (and traitor) Aaron Burr and former VicePresident-turned-British-lackey Al Gore do not like honest patriots of our U.S.A. very much. However, on the opposite side of that matter, the potential for developing true scientific creativity, and also artistic creativity in the individual member of society, is there. It exists, and can be promoted, if we come to understand this subject-matter, and are willing to make its achievement the essential goal for the development of our future individual citizen. My own dedication to that mission is multifarious; but, my most essential, relevant skill is in the field of those expressions of physical-scientific creativity which are coincident with my special competence in the domain of physical economy. To this end, I have promoted an approach to the students’ replication of the development of the principal valid currents of physical science, ranging, explicitly, and most typically, from the Pythagoreans and Plato through Cusa, Leonardo, Kepler, Fermat, Leibniz, Gauss, the Monge-Carnot phase of the Ecole Polytechnique, Dirichlet, and Riemann. Those who work in relevant forms of teams, to relive the acts of discovery which are most relevant for re-experiencing first-hand knowledge of the most-relevant discoveries, can generally succeed in one significant degree or another. With great science and great Classical art, combined, we can generate among us new generations sharing the quality of temperament we should require for those generations of our new citizens. The benefit would be, not only skills, but the fostering of the truly creative powers of the human mind, upon which progress depends. Best of all, once one knows that expressed quality of potential in oneself, which distinguishes one from an ape, or brutalized slave, insight comes naturally, because it is natural, for as long as people are developed for what the human individual is, and is intended to become. 18 As in the pattern set by the predatory, dictatorial, ThatcherMitterrand “conditionalities” imposed upon Germany. December 2008 m−1 X m−3 = xm +M, Bxm−2 + m−2 Cxm−3 + m−3 ... + + Lx...++M, m−1 X = xm + Axm−1 + Bxm−2 + Cx ++... + Lx X =Ax xmm−1 ++ Ax + Bx + Cx Lx + M, m m−2 m−3 m m−1 m−2 m−3 m m−1 m−2 m−3 X = x + Ax + Bx + Cx + ... + Lx + M, x + Ax + Bx + ... + Lx + M, X = x + Ax + Bx X+=Cx + ... + Lx + M, + Cx ϕ 38 ϕ ϕ Third Demonstration ϕ Gauss ϕ ϕ ϕ rm cos mϕ + Arm−1 cos (m − 1)ϕ + Brm−2 cos (m − 2)ϕ Third Demonstration of the Theorem Concerning the m−3 m−2+ Ar cos (m cos −cos 1)ϕ cos −cos 2)ϕ r +cos m−1 + Cr (m+ 3)ϕ++Br ... m−2 +(m Lrcos ϕ+ Brmϕ cos (m+ −Ar 2)ϕ rm cos mϕ + Arm−1 cos (m − 1)ϕ mϕ (m −−Br 1)ϕ (m −M 2)ϕ= t rm cos m m−1 m−2 m r cos mϕ m−1 m−2 m−3 cos m−1 m m−1 m−2 + Ar (m − 1)ϕ + Br cos (m − 2)ϕ m m−2 m−3 cos+ mϕ − 1)ϕ + Br+ cos −ϕLr 2)ϕ r 1)ϕ Cr cos − 3)ϕ + Lr cos +sin M(m m−3 Ar cos (m − Br cos (m −(m 2)ϕ r cos mϕ + Cr sin mϕ + Ar sin (m −... 1)ϕ +(m Br 3)ϕ ... + +rAr Lr+ cos ϕcos + M =(m t cos + Cr (m − 3)ϕ + ... + cos ϕ=+t−M2)ϕ =t m−3 m−3 m m−1 cos m−3 m−2 m−3 + Cr (m − 3)ϕ + ... + Lr cos ϕ + M == t u m m−1 m−2 Cr cos (m − 3)ϕ + ... + Lr cos ϕ + M = t r+ mϕ + Ar sin (m − 1)ϕ + Br sin (m − 2)ϕ m m−1 m−2 Cr cos(m (m − − 1)ϕ 3)ϕ +sin ... + + Lr cos ϕ + M = t + Cr sin (m − 3)ϕ + ... + Lr sin ϕ r sin mϕ + Ar sin Br sin (m − 2)ϕ r sin mϕ + Ar sin (m − 1)ϕ + Br sin (m − 2)ϕ m m−1 m−2 m−1 m−2 m−3 sin (m − 1)ϕ m−2 m m−1 m−2 r sin mϕ + Ar + Br sin (m − m−3 rm1)ϕ sin mϕ + Ar sin (m − 1)ϕ + Br sin (m − 2)ϕ + Cr sin (m − 3)ϕ + ... + Lr sin ϕ = u2)ϕ m−3 rm sin mϕ + Cr Arm−1 sin (m − + Br sin (m − 2)ϕ mr cos mϕ + (m − 1)Ar cos (m − 1)ϕ + (m cos (m 3)ϕ ... Lr sin ϕ = + uCr sin (m − 3)ϕ + ... + Lr sin ϕ−=2)Br u m−3 m−3 m m−1 m−2 m−3 m−3 + Cr sin (m − 3)ϕ + ... + Lr sin ϕ = u + Cr (m − +m−1 ... Lr sin+ ϕ = u mr(m+cos mϕ +sin (m −= 1)Ar cos (m+ −cos 1)ϕ −++ 2)Br cosm−2 (m −t2)ϕ m Cr − 1)Ar sin m−1 (m −cos 3)ϕ Lr ϕ u(m + (mm−2 −3)ϕ 3)Cr (m −(m 3)ϕ ... + Lr ϕ =cos mrm cos mϕ + (m −...mr 1)ϕ + (m −sin 2)Br cos (m −cos 2)ϕ cos mϕ + − 1)Ar (m − 1)ϕ (m − 2)Br (m m m−1 m−2 mmr cos mϕ m−1m−2 m−2 m−3 cos m−2 m+ m−1 (m − 1)Ar (m − 1)ϕ + (m − 2)Br cos (m − 2)ϕ m−3 m−3 mrcos cos + (m − 1)Ar cos (m − 1)ϕ + (m − 2)Br cos (m − 2)ϕ + (m − 3)Cr cos (m − 3)ϕ + ... + Lr cos ϕ = t mrm cos mϕ + (m − 3)Cr 1)Arm−1 (mmϕ − 1)ϕ + (m − 2)Br cos (m − 2)ϕ mr sin +cos (mϕ −= 1)Ar sin (m (m − − 3)ϕ 1)ϕ + + ... (m+−Lr 2)Br 3)ϕ ... mϕ + Lr t + (m − 3)Cr cos cos ϕ = sin t (m m−3 Carl F. Gauss m−3 = t m−1 cos m−2 m−3 mrm sin + − + (m − 3)Cr (m − 3)ϕ + ... + Lr cos ϕ m−3 m m−1 m−2 (m − 3)Cr cos (m 3)ϕ + ... + Lr cos ϕ = t mϕ + (m − 1)Ar sin (m − 1)ϕ + (m − 2)Br sin (m − m m−1 m−2 + (m − 3)Cr cos (m − 3)ϕ + ... + Lr cos ϕ = t +(m (m− −1)Ar 3)Cr sin (m−−1)ϕ 3)ϕ++(m ... + Lr u2)ϕ mr sin mϕ 1)Ar sin (m − mr 1)ϕ +sin (mmϕ − 2)Br sin (m −sin 2)ϕ + (m − 2)Br ϕ =sin (m m m−1 m−2 m m−1 m−2 m−3 m four proofs of the Fundamental m−1 m−2 mr sin mϕ + (m − 1)Ar sin (m − 1)ϕ + (m − 2)Br sin (m − 2)ϕ 2 m 2 − m−1 2− m−2 This, the third of Carl Gauss’s m−3 mr sin mϕ + (m − 1)Ar sin (m 1)ϕ + (m − 2)Br sin (m 2)ϕ + (m − 3)Cr sin (m − 3)ϕ + ... + Lr sin ϕ = u m−3 mr sin mϕ + (m − 3)Cr 1)Ar sin (m −m1)ϕ (m 2)Br sin (m −sin 2)ϕ r +cos +sin (mϕ− −=3)Cr 1) cos 1)ϕ − 2) 3)ϕ ... mϕ +−Lr u Ar + (m (m(m −− 3)ϕ ++ ...(m + Lr sin Br ϕ = u cos Theorem of Algebra, appeared in 1816. Here, Gauss makes a2 rare interm−3 2 m−3 m−1 2 ϕ = m−2 m−3 2cos m−3 + (m − 3)Cr sin (m − 3)ϕ + ... + Lr sin u 2 m m−1m2 rm cos2+ m−2 (m − 3)Cr sin (m − 3)ϕ + ... + Lr sin ϕ = u mϕ + (m − 1) Ar (m − 1)ϕ + (m − 2) Br cos − 3)Cr sin ...mϕ + Lr ϕ− + (m −= 3)u2 Ar Crm−1 cos cos (m−−1)ϕ 3)ϕ++(m ... + Lr ϕ(m = cos t 2)( m r cos mϕ + (m − 1) ArMervyn cos(m (m− −3)ϕ 1)ϕ 2)sin Br 2)ϕ m rm+ cos + (m 1) (m −m−2 2)2cos Brm−2 ventiaon for a physical basis of mathematics. It was translated by 2 m 22 m−1 2 2 m 2 m−1 2 m−2 m−3 2 m m−1 2 (m 2cos m−1 2(m m−2 r (m cos mϕ 1) Ar (m − 1)ϕ 2)(m Br−ϕcos (m − 2)+ 2 m−3 m−3 rm cos mϕ −(m 1)+− cos − 1)ϕ + (m+ −(m 2)+−Br + (m −Ar 3) Cr (m − 3)ϕ + ... Lr =cos tcos r cos mϕ + (m 1)further Arm cos − 1)ϕ 2)(m Br 2)ϕ Fansler, who offers this disclaimer:m “Since my German is much along m+ rm + sin + (m − 1) Ar sin (m − 1)ϕ + 2) Brϕ− ( − 3) Cr 3)ϕ ...mϕ Lr cos ϕm−2 =2(m tcos + − 3) Cr cos (m − 3)ϕ + ...cos + Lr =2)ϕ tsin 22 m−3 2 m−3 2 m m−1 2 m−2 m−3 − than my Latin, this translation derives of the 2 m−3 + (m − 3) Cr cos (m − 3)ϕ + ... + Lr cos ϕ = t 2 mfrom a German translation 2 m−1 2 m−2 + (m − 3) Cr cos (m 3)ϕ + ... + Lr cos ϕ = t m mϕ + (m (m −+ 1)(m Ar 1)ϕ −++ 2)(m sinϕ (m m−2 3) Ar Cr cosr(m (msin −21)ϕ 3)ϕ ... + Lr cos ϕ1) =2 Ar tsin + (m − 3) Crm−1 sin (m+ 3)ϕ ...Br + Lr =− u ,( m r sin mϕ + (m − 1) sin − + − 2) Br (m − 2)ϕ m rm sin mϕ − sin (m −−(m 1)ϕ −m−2 2)2sin Br sin2) original published by E. Nette in Ostwald’s Klassiker der2 exakten 22 m−1 2 2 m m 2 rm sin mϕ + (m − 2 1) m−1 2 2) m−2 m−3 (m 2 m m−1 2 m−2 2 2 2 2 Ar sin (m − 1)ϕ + (m − Br sin − 2)ϕ m−3 sin mϕ (m+−(m − 1)ϕ (m 2)+ sin (m 3)(m Cr 3)ϕ + Lr ϕLr =+ uuuϕ ,−)=2)ϕ m−3 + uϕsin )(tt uu )+ +(m (tu − utBr ) ...sin −+(tt (t2) m r Latin sin mϕ 1) Cr Armin rthesin (m −+ 1)ϕ Br sin,+ (m − 2)ϕ + (m − appeared 3) 3)ϕ ...1) +−Ar Lr sin =2(m uCr Wissenschaften, B. 14. The original version, which + − 3) sin −−... 3)ϕ + sin u , 2 m−3 = y. 2 ++ 2 )2+ Lr 223)m−3 2(m − r(t u 2... sinTϕT= 2 + 2 m−3 + Cr sin (m − 3)ϕ u , 2 2 2 2 U U + u )(tt + uu ) + (tu − ut ) − (tt + uu ) (t + (m+ Cr 3)ϕ Lrutsin Commentationes recentiores... and is the version in+Gauss’s 2+sin + ... + 2ϕ = u , + (m 3) Cr 3)ϕ + Lr = (m u + ,−uu +published u− )(tt uu sin ) +(m (tu−− ut ) −... −3) (tt(t ))(tt (t +uu uϕ2sin ) + (tu − ) − (tt +=uuy. )2 = 2 2 y.2 2 22 2 2 2 = y. 2 + u 2 )22 (t Gesammelte Werke, was consulted as a reference. (t2 + u2 )(tt + uur(t u )(tt + ) + (tu − ut ) − (tt +)uu ) u 2 2+ + uu )uu 2+ (tu r(t + u ) 2 2 2 + )(tt − ut ) − (tt + uu (t ) + (tu − ut ) − (tt + uu ) r(t + u ) = y. r = R = y. = y. 2 2 2 r(t2 r(t + u2+ )2u ) r(t2 + u2 )2 fter the previous treatise was already printed, continued meditations upon the same subject led me to a new proof The factor r can evidently be canceled out of the deof the theorem, which, similar to the preceding, is purely nominator of the last formula, since t’, u’, t’’, u’’ are divisible analytical, yet is based upon entirely different principles, and, by it. Finally, let r be a determined positive magnitude, which, with respect to simplicity, appears by far to be superior to the forthough indeed arbitrary, should still exceed the highest of the mer. To this third proof are the following pages now devoted. magnitudes √ 2 √ 3 √ 4 √ 1. mA 2, mB 2, mC 2, mD 2, ...; m m−1 m−2 Decomposition of Integral Algebraic Functions into Real Factors A for these magnitudes the signs of A, B, C, D, ... should be neglected; that is, any negative signs occurring should be changed given: m−1m−1 m−2m−2 m−3m−3 m m m−1 m−2 m−3 X =Xxm + Ax +m−1 Bx +m−2 Cx +m−3 ...++...Lx + M, + uu m = x + Ax + Bx + Cx + Lx + M, into positive. Following this that tt preparation, I claim X = x + Ax + Bx + Cx + ... + Lx + M, √ √ 4 √ √ √ √ √ 2 3 2 3 4 m m m−1 m−2 m−3 assuredlyobtains a positive setmC = R, 2, which also 2, gives m−1 X = x=m−3 + Ax + Bx + Cx + ... + + M, 1r2, mB 2, mC mA −1 mBif we 2, mD ...; 2, mD mA 2, value + Ax +m−1 Bxm−2 +m−2 Cxm−3 +m−3 ...√Lx ++Lx + M, m ◦ m−1 ◦ m−2 ◦ + Bxm−2 +X Cx xX + ++ LxAx + M, =... xm + Bx + Cx ... + Lx + M, √ √ √ ϕ .(45√√ cos 45 + AR cos + ϕ) + BR cos (45 +4√ 2ϕ) 3√ √ 4 2 √ R 3 √ 4 √ to a (real) value √ 3√ 2 3 4 2√ m−2 −1 mB 2, mC mD ...; 2,2,...;2 mB1 mA 2, xm + + ... Ax+m−1 + Cxm−3 mA + ... + Lx +2,M, 1 2, 2, mB mC 2, 2, mD mA + Bxm−2 +X Cx=m−3 Lx + Bx M, mB 2, mC 2, 2, mC 2, mD 2, ...; 2, mD mA 1 m−3 ◦ ◦ Demonstration. Setting ... + CR4 √ cos (45√+ 3ϕ) + +√ LR cos (45√ + (m − 1)ϕ) ϕ whose √ √ √ ϕ ϕ coefficients A, B, C,... are determined real √magnitudes. 2 3 4 2√ 3√ 1 √ + M cos √ 3 4 follows: mB 1mC 2,◦ 2, mCitm−1 2, mD 2, ...;2,12 mB mA 2,set-m2, mB mD 2, ...; mA 2, Considering r, ϕ to be other indeterminate magnitudes and ◦ m−2 ◦ 2, mC 2, mD 2, ...; mA +√ϕ) + BR √ sin (45√+ 2ϕ) ϕ ϕ √sin (45 √R 3sin 45√+ AR √ √ 2 2 3 4 4 tingϕ mB 2, +mC 2, sin mD mA 2, ...; m−3 ◦ 2, mC + 2, mD 2, mA 2, mB (45 + 3ϕ) ... +m−1 LR (45◦2,+...; (m − 1)ϕ)m−2 + M sin ϕ √ m3 CR √◦ sin √ 2 m√ ◦ ◦ m−1 m−2◦ 4 R cos 45 + AR cos (45 cos +m−2 ϕ) BR R cos 45 + AR cos (45 + ϕ) + BR (45+◦ + 2ϕ)◦ cos (4 mB 2, mC 2, mD 2, ...; mA 2, m m m−1m−1 m−2m−2 m ◦ m−1 ◦ m m−1 m−2 + Ar+ Ar cos (m − + Br cos (m − r cos mR 45 m−2 + (m m−2 − 1)AR cos (45 ◦+ ϕ) m−3 + (m − 2)BR cos (45 + 2ϕ) cos coscos (m1)ϕ − 1)ϕ + BrBr coscos (m2)ϕ −− 2)ϕ r rmϕ m mϕ m−2 m− ◦ 2)ϕ cos ◦ m−3 m ◦+ ◦ m−1 m−1 cos mϕ + Arm−1 (m 1)ϕ (m m◦ + ◦ + m−1+cos ◦◦ + m−2 m ◦(45 m−1 ◦ + m−2 ◦+◦ ... R cosR 45 + 45 AR cos (45cos +(45 ϕ)◦++BR cos +R 2ϕ) CR (45 3ϕ) + cos +cos CR cos (45 3ϕ) ... (45 + LR cos (45 (mLR − 1)ϕ) cos + AR ϕ) +cos BR (45 2ϕ) cos 45 + AR cos + ϕ) + cos (45( R 45 + AR cos (45 + ϕ) + BR cos (45 ++ 2ϕ) m m m−1 m−2 m−3 m−3 ◦ ◦BR m−3 m−1 m−2 m−3 mϕmϕ + Ar coscos (m − 1)ϕ + Br cos (m − 2)ϕ r rcoscos Cr 3)ϕ ... + Lr cos ϕ + M = t m−2 + (m − 3)CR cos (45 + 3ϕ) + ... + LR cos (45 + (m − 1)ϕ) + Cr cos (m − 3)ϕ + ... + Lr cos ϕ + M = t m−3 m m−1 m−2 + Ar (m − 1)ϕ + Br cos (m − 2)ϕ m−3 ◦◦2)ϕ ◦ m−1 ◦ ◦◦ ◦ m − 1)ϕ + Br − 2)ϕ cos (mm− 1)ϕ m ◦ m−1 m−2 m ◦ ◦ m−2 ◦ m−3 ◦ Cr 3)ϕ ... + Lr cos ϕ + M = t mϕ(m + Ar + Br cos (m − r cos cos m◦ ◦ m−1 ◦ m−2 ◦ m−1 m−2 ◦ m−3 m−3 ◦+ ◦ ◦2ϕ) + CR cos (45 3ϕ) +BR ...R+ LR cos (45 + (m − 1)ϕ) (45 + mϕ) R sin 45 + AR sin (45 +cos ϕ) + BR (45 45 + AR sin (45 + ϕ) + sin (45 + m... ◦ m−1 ◦M m−2 + CR 3ϕ) +sin + 45 LR cos (45 +2ϕ) (m − 1)ϕ) ++BR M cos (45 += mϕ) T, R45 cos 45 AR ϕ) + BR cos (45 + R+ ... cos +m−2 AR cos (45cos+ +(45 ϕ) ++ cos (45 +m−1 2ϕ) + CR cos (45 + 3ϕ) +T, ...+= +(m LR cos (4 + CR cos 3ϕ)cos ...BR + LR (45 −sin 1)ϕ) m−3 m m m−1 m−2 m−2 m ◦(45 m−2 ◦2ϕ) R + AR cos cos (45 + m−2 mmϕ m−1 m−2 + Cr cosm−1 (m − 3)ϕ + Lr cos ϕm−2 + = t= sin Ar sin (m − + Br sin (m − 2)ϕ cos (m − 1)ϕ Br cos (m −M 2)ϕ rrm mR sin 45◦m−2 + (mcos − 1)AR sin (45 + ϕ)+m−3 +ϕ) (m+− 2)BR sin (45 + 2ϕ) r ... mϕ +m−3 Ar sin (m1)ϕ −m− 1)ϕ + Br sin (m − 2)ϕ m m−1 m− − 3)ϕ 1)ϕ + Br cos (m − 2)ϕ + mϕ Cr cos 3)ϕ + 1)ϕ ... +◦+ Lr cos ϕM + t◦M m−1 ◦ ◦ m−3 m + + sin Lr cos ϕ + Mm−3 = tcos ◦ ◦ ◦ m ◦ m−1 ◦ m−2 ◦ rsin Ar sin (m Br sin (m − 2)ϕ + Cr 3)ϕ ... + Lr cos ϕ + = t m−3 ◦ ◦ ◦ m−3 ◦ ◦ m ◦ m−1 ◦ m−2 ◦ ◦ + m−1 ◦ + m−2 ◦ + ... R sinm−2 + AR sin (45 ++(45 ϕ) ++ BR sin (45 + 2ϕ) m ◦m m−1 ◦sin m−2 ◦ m ◦ sin m−1 ◦ sin m−2 ◦(45 + sin (45 + 3ϕ) ++ LR sin (4 CR sin (45 3ϕ) + ... + LR sin (45 + − m−3 ◦ϕ) ◦(m R45 45 + AR ϕ) + BR sin (45 + 2ϕ) CR cos (45 + 3ϕ) ... + LR cos (45 + (m − 1)ϕ) + M cos (45 +cos = T, R 45 AR sin (45 + ϕ) + sin1)ϕ) (451) + CR cos (45 3ϕ) + ... LR cos + (m − 1)ϕ) M cos (45 + mϕ) =mϕ) T, R sin 45 + AR sin (45 +CR BR sin (45 + 2ϕ) m−1 cos 45 ++ AR cos (45 + ϕ)cos ++ BR cos (45 + 2ϕ) m−3 R3)ϕ cos 45 + AR cos (45 +R ϕ) + BR cos (45 + 2ϕ) m−3 ◦(45 ◦BR + CR ++ 3ϕ) ++ ... + LR (45 (m − m−3 m mϕ + m−1 m−2 m−3 rm3)ϕ sin Ar sin (m − 1)ϕ + Br sin (m − 2)ϕ Cr ... + Lr sin ϕ = u cos (m − 3)ϕ + ... + Lr cos ϕ + M = t m−2 + (m − 3)CR sin (45 + 3ϕ) + ... LR sin (45 + (m − 1)ϕ) = m−3 + Cr sin (m − 3)ϕ + ... + Lr sin ϕ = u m m−1 m−2 m − + ... + Lr cos ϕ + M = t r sin mϕ + Ar sin (m − 1)ϕ + Br sin (m − 2)ϕ m−3 ◦◦2)ϕ ◦ ◦ m − 1)ϕ + Br m ◦ ◦+ Mm−3 ◦◦ m ◦ (45 m−1 ◦ (45◦m−1 m−2 m−3 ◦ ◦ (45 Cr2)ϕ sin (m 3)ϕ ...CR +m−3 sin ϕ(45 =− u r sinsin mϕ(m +− Ar Br (m m+ ◦ Lr m−1 m−2 ◦◦ m− 1)ϕ ◦+ m−1 m−2 ◦ ◦mϕ) m−3 m−2 ◦m−1 sin + 3ϕ) + ... + LR sin + (m − 1)ϕ) sin ++ U, m−3 ◦(m ◦ sin ◦CR mR cos 45 +CR (m − cos + ϕ) +−cos (m − mR 45 +sin (m − 1)AR cos (45 + ϕ) −= 2)BR (45 m ◦ ◦3ϕ) m−2 ◦ + sin CR 3ϕ) +cos ...cos + LR sin (45 +1)ϕ) (m − 1)ϕ) +1)AR M sin (45 + mϕ) U, R45 sin 45 AR (45 ϕ) BR + 2ϕ) R sin + AR sin (45 +3ϕ) ϕ)◦+ ++ BR sin (45 + 2ϕ) sin (45 + 3ϕ) + + LR sin (42 + sin (45+ + + ...◦BR + LR sin (45 += (m 1)ϕ) m−3 cos cos (45 +... ϕ) BR 2ϕ) + CR cos (45 + 3ϕ) + ... + LR cos (45 + (m − ... 1)ϕ) + M cos m m−1 m−2 m m m−1 m−2 + CR cos (45 + +−+ LR (45 +(45 (m − + M cos (45 + mϕ) = T, R sin 45 + AR sin (45 + ϕ) + sin (45 + 2ϕ) m−1 m−2 m−3 m−2 mmϕ m−1 m−2 + Cr sin (m − 3)ϕ +m ... + Lr sin ϕ = u mr cos (m − 1)Ar cos (m − 1)ϕ + (m − 2)Br cos (m − 2)ϕ r sin Ar 1)ϕ Br sin (m − 2)ϕ m m−1 m−2 m−3 mr cos mϕ + (m − 1)Ar cos (m − 1)ϕ + (m − 2)Br cos (m 2)ϕ m − 1)ϕ + Br sin (m − 2)ϕ + Cr sin (m − 3)ϕ + ... + Lr sin ϕ = u ◦+− m−1 ◦ m−2 ◦ m − 3)ϕ +mr ... + cos Lr sin u − 1)Ar m−3 ◦ m−3 ◦ ◦ m ◦ m−1 ◦ m−2 ◦ mϕϕ+=Cr (m cos (m 1)ϕ + (m − 2)Br cos (m − 2)ϕ sin (m − 3)ϕ ... + Lr sin ϕ = u m−3 ◦◦cos ◦◦cosm−1 ◦◦ + m−3 ◦cosm ◦ −− ◦ ◦ +cos m ◦(45 m−1 ◦ + m+ ◦+◦+ ◦++2ϕ) mR 45 + (m − 1)AR (45 + (m 2)BR + 2ϕ) ◦ϕ) m−1 ◦1)AR m−2 m cos ◦ cos m−1 ◦ sin m−2 + (m −sin 3)CR (45 3ϕ) + + (m 3)CR cos (45 3ϕ) ... LR (45 + (m m−3 ◦M ◦m−2 mR 45 + (m − 1)AR (45 +... ϕ) (m 2)BR cos (45 CR + 3ϕ) + LR sin (45 ++ (m − 1)ϕ) + M sin + = U, + CR sin (45 + 3ϕ) ... ++ LR sin (45 + (m 1)ϕ) (45 mϕ) =mϕ) U, mR cos 45 (m − 1)AR cos (45 +cos ϕ) + (m 2 mR cos 45 + − cos (45 + ϕ) +(45 (m − 2)BR cos... (45 m m m−2 cos (45 + 3ϕ) + ... +− LR cos (45 (m −+ 1)ϕ) + M cos (45 + mϕ) = T,(m R(45 sin 45 AR sin (45 +− ϕ) ++(45 BR sin (45 + 2ϕ) m−3 m−3 m−1 R sin 45 +Lr AR sin (45 + + sin (45 +(m 2ϕ) + CR sin + 3ϕ) + ...+ + LR sin (45 + −− 1) m−3 BR m−1 m−2 m−3 cos+ mϕ + Lr (m −+1)Ar cos (m −+1)ϕ + (m −++ 2)Br cos (m − 2)ϕ + (m (m − 3)Cr cos (m − 3)ϕ + sin ... Lr ϕcos = tϕ) Cr sin (m − 3)ϕ ...1)ϕ + ϕ = ucos m−1 m−2 (m − 3)Cr cos (m − 3)ϕ ... + Lr ϕ = t m−3 2)ϕ m m−1 m−2 mmr −cos 3)ϕ + sin ϕ = u− mr + − 1)Ar cos (m − + (m − 2)Br cos (m − m−3 ◦ ◦ (mcos −...mϕ 1)ϕ + (m − 2)Br cos (m − 2)ϕ m ◦ m−1 ◦ m ◦ m−1 ◦ m−2 m−3 ◦ ◦ 3)Cr 3)ϕ ... + Lr cos ϕ = t mr cos mϕ + (m 1)Ar cos (m − 1)ϕ + (m − 2)Br cos (m − 2)ϕ m ◦− m−1 m−2 ◦1)AR m ◦ + (m m−1 ◦mR m−2 ◦sin m−3 ◦ + ϕ) + m−3 ◦= ◦ (45 m m−1 ◦ m−2 ◦ 3)CR cos (45 + 3ϕ) + ... + LR cos (45 + (m − 1)ϕ) T , m−3 ◦ m−3 ◦ ◦ ◦ mR sin 45 + (m − sin (45 (m − 2(◦ sin 45 (m − 1)AR (45 + ϕ) + (m − 2)BR sin m m−1 m−2 + (m − 3)CR cos (45 + 3ϕ) + ... + LR cos (45 + (m − 1)ϕ) = T , mR cos 45 1)AR ϕ)(45 + (m − +sin 2ϕ) mR cos + (mLr −cos 1)AR cos (45 ϕ) + (m − 2)BR cos 2ϕ) + (m − 3)CR (45 + 3ϕ)(45 ...+cos +(m L (m −2)BR 3)CR cos 3ϕ) +cos ...= +U, LR cos m−3 m−2 R45+ sin AR sin +...+ ϕ) + BR sin +1)ϕ) 2ϕ) CR 3ϕ)(45 ... + LR 1)ϕ) ++M sin m m m−1 m−2 m m−2 + CR sin (45 + 3ϕ) +− +− LR sin +(45 (m +cos M+(45 sin (45 +(45 mϕ) mR cos 45 − 1)AR cos ϕ) + (m − 2)BR m−1 m−1 mr m−2 m−3 (45 mmϕ m−1 m−2 + (m − 3)Cr cos (m − 3)ϕ ... + ϕ = t mr sin + (m − 1)Ar sin (m − 1)ϕ + (m − 2)Br sin (m 2)ϕ cos cos (m − 1)ϕ + (m − 2)Br cos (m − 2)ϕ m−3 mr sin mϕ + (m − 1)Ar sin (m − 1)ϕ + (m − 2)Br sin (m 2)ϕ m m−1 m−2 m−3 3)Cr ...◦++Lr ϕ2)Br = t m−1 cos (m (m − − 3)ϕ 1)ϕsin + mϕ (m+− 2)Br cos (mcos − 3)ϕ 2)ϕ ◦ − 1)ϕ ◦t◦ (m ◦ m−2 ◦◦ cos + ... =mR tm m m−3 m−3 ◦ ◦ ◦◦ mr 1)Ar sin (m (m −m−1 2)ϕ +Lr (mcos −ϕ3)Cr 3)ϕ ...cos + Lr cos ϕ =◦◦sin m−3 ◦+ + m ◦ ◦+ m−1 ◦◦ + m−3 ◦m−2 + sin m− ◦++ m−1 m−2 m−3 ◦ sin ◦m−2 sin 45 + (m − 1)AR sin (45 + ϕ) (m 2)BR (45 + 2ϕ) m ◦3ϕ) m−1 ◦+ ◦m sin m−1 m−2 (m − 3)CR 3ϕ) ... ++ (m − 3)CR sin (45 + 3ϕ) LR sin (45 +(45 ◦ϕ) ◦(m mR 45 + (m − 1)AR sin (45 + ϕ) (m − 2)BR sin (45 + 3)CR cos (45 + 3ϕ) ... + LR cos (45 + (m − 1)ϕ) = T... ,(45 + (m − 3)CR cos (45 + + ... + LR cos (45 + (m − 1)ϕ) =2ϕ) Tsin ,+ mR sin 45 +m−3 (m − sin +LR ϕ) + (m − 2◦L mR 45 + (m − 1)AR sin (45 + (m − 2)BR sin m m m−1 m−2 CR sin 3ϕ) +sin ... +− LR sin (45 + (m − 1)ϕ) +(45 M (45 +(45 mϕ) = U, mR cos 45 1)AR cos (45 + ϕ) +1)AR (m − 2)BR cos (45 + 2ϕ) m−3 (45 mR cos 45 + (m − 1)AR cos + ϕ) + (m − 2)BR 2ϕ) + (m − 3)CR cos + 3ϕ) + ... + cos (45 m−3 2)ϕ m−3 (m m−1 m−2 m−3 mrcos sin mϕ + (m − 1)Ar sin (m − 1)ϕ + (m − 2)Br sin (m − + (m − 3)Cr sin − 3)ϕ + ... + Lr sin ϕ = u cos (m − 3)ϕ + ... + Lr cos ϕ = t m−1 m−2 + (m − 3)Cr sin (m − 3)ϕ + ... + Lr sin ϕ = u m−3 m + m−1 m−2 (msin − 3)ϕ + ... + Lr cos ϕ = t mr(m mϕ 1)Ar 1)ϕ (m − 2)Br sin (m − 2)ϕ m−3 ◦ ◦ sin − 1)ϕ (m − 2)Br sin (m − 2)ϕ m−3 ◦ ◦ 3)Cr m sin 3)ϕ ... + sin ϕ(45 =◦sin u◦ + mr sin mϕ + (m − 1)Ar sin 45 (m − 1)ϕ +3)CR (m −Lr 2)Br (m −++ 2)ϕ ◦− m−1 m−2 m−1 m−2 ◦− m−3 ◦◦ m−3 ◦=1)ϕ) ◦ m ◦sin m−2 ◦− + (m sin 3ϕ) + ... + sin (45 + (m 1)ϕ) U3ϕ) , =+sin m−3 ◦m−1 m−3 ◦sin + m ◦... ◦2ϕ) m−2 + (m 3)CR 3ϕ) +LR + LR (45 + (m ,(45 45 1)AR sin (45 ϕ) + (m − 2)BR sin (45 + + (m − 1)AR (45 ϕ) (m 2)BR sin (45 + 2ϕ) + (m −◦3)CR 3ϕ) + + L (m −(45 3)CR sin (45 ...(m +(45 LR sin (45 +sin (m m−3 sin mR cos cos (45 +... ϕ) + (m − 2)BR cos (45 ++ +◦+ (m − 3)CR cos + 3ϕ) + ... cos + (m − ... 1)ϕ) 2 m 2 m−1 2− m m m−1 m−2 + (m − 3)CR (45 + + 3ϕ) +− LR cos +(45 (m − 1)ϕ) = T2ϕ) , LR mR sin 45 + (m − 1)AR sin (45 + ϕ) U + −+ 2)BR (= 22sinmR m−1 m−1mmr m−2 m−3 m−2 m−1 2cos m−2 + (m −+3)Cr (m −(m 3)ϕ + ... Lr sin ϕ− = u2m−2 cos mϕ + ... (m − 1) Ar cos 1)ϕ ++ (m − 2) Br cos (m − 2)ϕ sin 1)Ar (m −− 1)ϕ + (m − 2)Br sin (m − 2)ϕ m−3 m−3 m r− cos mϕ (m − 1) Ar cos (m − 1)ϕ + (m 2) Br cos (m − 2)ϕ m 2 sin m−1 2 m−2 sinr(m (m22m −2mr3)ϕ 1)ϕ 2)Br 2)ϕ 3)Cr 3)ϕ ... + Lr sin ϕ = u sin + + Lr sin ϕ = u cos mϕ + (m − 3)Cr 1) Arm sin cos(m (m− − 1)ϕ + (m − 2) Br (m 3)ϕ + ... + Lr sin ϕ = u m−3 ◦◦ − 2)ϕ ◦◦ m−2 m−3 ◦ cos ◦ m ◦ m−1 ◦ ◦ ◦ m−1 ◦ m−2 ◦ m−3 ◦ ◦ + (m 3)CR sin (45 + + ... LR sin + (m = + (m − 3)CR sin (45 + 3ϕ) ... sin (45 +(45 (m+− U+2)BR ,3ϕ) m 2 22m−1 2 2− m−2 cos (45 + 3ϕ) 3ϕ) +LR ... + LR cos (45 + (m −=1)ϕ) 1)ϕ) =UT+,,... + mR sin + (m (m −+ 1)AR ϕ)1)ϕ) +− (m − sinLR (45sin+(45 2ϕ)+ m−3 m−3 mR sin 45 + (m − sin (45 + ϕ) + − 2)BR sin (45 2ϕ) ++ (m − 3)CR sin (45 22m−2 m−3 m−3 (m 2 cos m−2 m−3 45 m2m rsin mϕ + (m −++ 1)(m Ar cos (m −− 1)ϕ + (m −+ 2) Br cos (m 2)ϕ + ... (m 3) Cr cos (m − 3)ϕ +...(m ... + cos = tcos 3)Cr 3)ϕ Lr sin ϕϕ = m−1 2 m−3 − 3) Cr cos (m −+ 3)ϕ +1)AR ...Lr + Lr cos = t− 2−mϕ m 2 sin m−1 2uϕ m−2 (m − Lr sin ϕm−1 = u rm cos 1) Ar 1)ϕ − 2) Br (m − 2)ϕ cos (m 1)ϕ (m− 2) Br cos (m −m 2)ϕ 3) Cr 3)ϕ ... + cos ϕ = tcos m r3)ϕ cos+mϕ +− (m − 1) Ar cos (m − 1)ϕ −Lr 2) Br (m ◦ + (m m−1 ◦ − 2)ϕ m−2◦ ◦ m−3 ◦ m−3 ◦ ◦ 2 22 m−3 m−2 mR sin...45 + (m 1)AR sin (45++(m +...ϕ) (m 2)BR sin (45 + −+ 3)CR sin +1)ϕ) 3ϕ) + m 22 m m−1 22− m−2 22 m + −+ 3)CR sin (45 +− 3ϕ) + LR sin−(45 +(45 (m − =+... U2ϕ) , LR sin (45 + (m − 1)ϕ) = 22 m−2 m−1 2 (m m−1 2 m−3 mmϕ 2m−2 + (m −(m Cr cos (m −(m 3)ϕ + Lr cos ϕ t= sin ++ (m − 1) Ar sin (m − + (m 2) Br m rrm cos cos (m − 1)ϕ ++ (m − 2)− Br cos (m −2)ϕ 2)ϕ m−3 mcos m m−1 2 m−2 m−2 2=m−1 m−3 (m r 2− mϕ +3) (m −Cr 1) sin (m −(m 1)ϕ (m 2)= Br sin −− 2)ϕ cos (m −rsin 1)ϕ +mϕ −3) 2) Br −1)ϕ 2)ϕ 3)ϕ ... +− Lr ϕ tsin (m 3)ϕ ... + Lr cos ϕAr m sin 1) Artcos sin (m − 1)ϕ + (m 2) Br (m 2)ϕ + (m − 3) Cr cos 3)ϕ ...cos +− Lr cos ϕ = tsin m−3 ◦ ◦ 2 22m−1 2 m−2 + (m − 3)CR sin (45 + 3ϕ) + ... + LR sin (45 + (m − 1)ϕ) = U , m−3 − 22m−2 m−3 m−3 (m 2 sin m−1 2sin m−2 ,2)ϕ m2m rm mϕ + (m −... 1) Ar sin (m − 1)ϕ + (m −++ 2) Br sin ++ (m − 3) Cr sin − 3)ϕ + ... Lr ϕϕ = usin , (m cos (m − 3)ϕ +(m ... + Lr cos = t(m m−1 2Ar + (m − 3) Cr sin (m − 3)ϕ ... + Lr sin = u m−3 2)ϕ 2− m 2=m−3 m−1 2ϕ m−2 cos (m 3)ϕ + + Lr cos ϕ t rm sin mϕ + (m − 1) sin (m − 1)ϕ + − 2) Br − Δυναμις Vol. 3 No. 3 December 2008 sin (m − 1)ϕ (m − 2) Br sin (m − 2)ϕ 3) Ar Cr 3)ϕ + (m ... +−Lr m r sin mϕ + (m − 1) sin (m − 1)ϕ 2) sin Brϕ = usin, (m − 2)ϕ m−3 2 − 3) 22 Cr m−3 (m + 2+ Lr 2ϕ =m−2 2 , 2 m 2222 m−1 22− ... 22−sin 22 (m − 2)ϕ m−1 m−2 2 + (m sin − 3)ϕ u + u )(tt + uu ) + (tu − ut ) − (tt + uu ) (t msin r sin mϕ + (m 1) Ar sin (m − 1)ϕ + (m 2) Br sin m−3 +2Lr u sin )(tt uu ) + (tu ut )++ uu (t + 2 ϕ 2= u 2 (tt , 2 m−3 3) Cr 3)ϕ ...ut Lr ϕ + =sin u)ϕy. sin (m (m − − 3)ϕ 1)ϕ + + ... (m 2) Br sin (m 2)ϕ ,uu + u )(tt + )− + (tu− )− −sin (tt+ uu )= (t +− (m − 3) + Cr sin (m − − 3)ϕ ... + Lr = y. u , = 2 + 2 )2 222 22 2 2 2 2 2 m−3 r(t u 2 2 22 ut = y. m−3 (m )− 2 (tt u )(tt + uu ) + (tu − ) + uu ) (t r(t + u 2 2 2 + (m − 3) Cr sin − 3)ϕ + ... + Lr sin ϕ = u , 2 2 +)(tt utu) −)−ut(tt 2 r(t− + 2 + uu )= y. 2 ... Lr ϕuu u+uu , (tu + uu sin ) +(m (tu−−3)ϕ ut(t+ )+ −u+ (tt2)(tt +sin ) =2)+ + uuu ) + (tu ) − (tt + = uuy.) (t 2 )22 2 2 y. r(t + = y. 2u 2 2 2 (t 22+ u2 )(tt + uu 2 = r(t + u ) ) + (tu − ut ) − (tt + uu )2 2 2 2 + uu r(t )++ (tuu − ) ut ) − (tt + uu ) r(t + u ) = y. = y. r(t2 + u2 )2 r(t2 + u2 )2 Let the following function of the indeterminate x be cosBR 45 + AR (45 + ϕ) BR + 3ϕ) cos+(45 2ϕ) +Rϕ) + cos (45 cos + 2ϕ) CR cos+(45 ... ++LR cos (45 + (m − 1)ϕ) + M cos (45 + mϕ) = T, ◦ m−2◦ ◦ ◦ m−3 ◦ ◦ ◦ m−2 m m−1 ◦ cos (45 ++...ϕ) BR cos (45 + 1)ϕ) 2ϕ) CR (45 + 3ϕ) ++...M+cos LR(45 (45mϕ) +sin (m − ◦1)ϕ) + M cos (45◦ + mϕ) = T, + 3ϕ) ++ LR cos45 (45cos (m − + = (45 T, R sin ++AR sin (45 ϕ) +cos BR + 2ϕ) ◦ ◦ ◦ m ◦ ◦ ◦m−3 m−2 ◦ m−2 ◦ cos (45 +453ϕ) ... m−1 + LR (45 + (m − ◦1)ϕ) + cos (45 + mϕ) sinBR ++AR sin (45 + ϕ) +(45 BR sinM 2ϕ) +R ϕ) + sin (45 ++cos 2ϕ) CR sin + 3ϕ) +(45 ... ++ LR sin (45◦= +T, (m − 1)ϕ) + M sin (45◦ + mϕ) = U, m−2 m−3◦◦ ◦ m m−1 ◦◦ + sin (45 ++...ϕ)++ BR 2ϕ) + CR + 1)ϕ) 3ϕ) LR sin (45 + + 3ϕ) LR sin (45sin+(45 (m◦ − +... M+sin (45 mϕ) ◦ Third Demonstration m−2 39 (m −2)BR 1)ϕ)Gauss + M cos (45 + + 2ϕ) mϕ) = U, = U, mR cos 45 + (m − 1)AR cos (45 + ϕ) + (m − ◦m−2 ◦ m−1 ◦m−3 ◦ m−2 ◦ ◦ ◦ ◦ sin 3ϕ) ... − + LR sin(m (45 +(45 (m − M− sin (45+ + =cos U,+ mR + ϕ) (m 2)BR (45 2ϕ)+ (m − 1)ϕ) = T , cos(45 (45cos+ +45 ϕ) + ++(m (m − 1)AR 2)BR cos (45 +1)ϕ) 2ϕ) + −cos 3)CR cos++ (45 + 3ϕ) ...mϕ) +cos LR (45 ◦ ◦ m−3 ◦ m−2 ◦ m ◦(m AR (45 +... ϕ) + − (45 2)BR cos (45(45 +◦+ (m − 3)CR + 1)ϕ) 3ϕ) + ...T2ϕ) LR+cos + (mm−2 − 1)ϕ) = ◦T + , 2ϕ) cosm−1 (45◦cos + 3ϕ) +45 LR +(45 (m◦m−1 − =+ mR sin +cos (m −cos 1)AR sin +, ϕ) (m(45 − 2)BR sin (45 extending from r = 0 to r = R and from ϕ = 0 to ϕ = 360°, m−3 ◦ m ◦ ◦ ◦ m−1 ◦◦m−3 m−2 ◦ m−2 ◦ ◦ CR (45 + 3ϕ) ++ ... (m + LR cos (m − 1)ϕ) =T mR(45sincos + (m − 1)AR (45(45+ + ϕ) +(45 (m − (45 2ϕ)+ (m − 1)ϕ) = U , sin +45 ϕ) 2)BR 2ϕ) −sin 3)CR sin +2)BR 3ϕ) + ,... +sin LR sin+ (45 which thus acquires a finite, completely determined value. This m−3 ◦ m−2 AR (45+◦+(m +...ϕ) (msin − 2)BR sin (45◦+=+...U2ϕ) − 3)CR sin+(45 + 1)ϕ) 3ϕ) + = which U , we will signify by Ω , must be obtained regardless sin m−1 (45◦ sin + 3ϕ) ++ LR (45 (m◦ − , LR sin (45◦ + (m − 1)ϕ) value, of whether the integration is performed first with respect to ϕ and CRm−3 sin (45◦ + 3ϕ) + ... + LR sin (45◦ + (m − 1)ϕ) = U , then with respect to r, or the inverse order. We have, however, the √ Rm−1 if we consider √indefinite, (R + mA 2 cos (45◦ +r as ϕ))a constant, ydrdϕ I. t is composed of the terms m 2 √ tu − ut Rm−2 2 √ Rm−1 , y dϕ = √ (R + mB 2 cos (45◦ + 2ϕ)) + ◦ 2 + u2 ) √ (R + mA 2 cos (45 + ϕ)) m−1m 2 r(t √ R ◦ m 2 ydϕ √ (R + mA 2 cos (45 + ϕ)) m−3 ydrdϕ √ Rm−2 tu − ut m +2R √ (R3 + mC √2 cos (45◦ + 3ϕ)) + √ (R2 + mB 2 cos (45◦ + 2ϕ)) ydϕ = , ϕ is easily confirmed by differentiating to . A r(t2with + u2respect ) m 2 Rm−2 m2 2 as √ ◦ 2 cos (45 + 2ϕ)) + √ (R ydrdϕ √ constant is not added, if we presume that the integration begins m−4+ mB Rm−3 3 m +2R √ (R4 +ϕmD√2 cos (45◦ϕ+ 4ϕ))..., + √ (R + mC 2 cos (45◦ + 3ϕ)) tu −ut = 0 at = 0, since for = 0 one obtains . Now, since 2 2 m 2 r(t +u ) √ Rm−3m 32 tu −ut ◦ tu − ut √ Rm−4 4 ϕ 0 + 3ϕ)) √ 2 cos (45 evidently also vanishes for ydϕ = 360°, the ,integral 2) = = then r(t2 +u + √ (R + mD 2 cos (45◦ + 4ϕ))...,+ m 2 (R + mC 2 + u2 ) ydϕ= r(t y dϕ from ϕ = 0 to ϕ = 360° becomes 0, while r remains m 2 √ ydϕ Rm−4 4 tuΩ −= ut ◦But from here follows indefinite. 0. , T T + U U+ √ (R + mD 2 cos (45 + 4ϕ))..., ydϕ = 2 + u2 ) integral, in which we 2 seen, Likewise we have the r(t indefinite which, for each determined real value of ϕ , as is m easily tu −ut ) =0 ydr as2constant, considerr(tϕ2 +u has a single positive value; consequently, must be taken as a r =t R tt + uu tt +=uu positive value. In a similar manner it will be proven that u, t’, u’ , ydr m−1 √ y dr = 2 R 2, 2 tu −ut ydϕ t + u2t + u + mA cosbecome (45◦ + ϕ)) UU also become positive, such that T T√ +(R must 2also a 2 +u2 ) = 0 r(t m 2 positive magnitude. T T +UU m−2 √ R T 2 +U 2 ◦ ◦ ydϕ 2 t’, u’ change over◦into r = R II. For r = R, the functions t, u, T cos +Tismϕ) + U sin (45 confirmed + mϕ), by differentiating likewise easily with to T +UU 2ϕ))(45 as + √ (R + mB 2 cos (45 + tt + uurespect 2 +U 2 , ydr = m 2 ◦ r:T again ◦ 2+ 2 we begin no constant is needed here, supposing that + mϕ) − U cos (45 + mϕ), T sin (45 t u m−3 T cos (45◦ + mϕ)R+ U sin3(45◦ + mϕ), √ TT to the proofs in the according ◦ ◦ the integration with r ◦ = 0. Hence, tt + uu ◦ ◦ √ + mC 2 cos (45 + 3ϕ)) (R + cos (45 + mϕ) + U sin (45 + mϕ), T T sin (45 + mϕ)m− U2 cos (45 +1mϕ), , ydr = preceding paragraphs, the integral,T2extending T T +2 U U from r = 0 to r = R, ◦ ◦ cos +(45 mϕ) U sin− + mϕ), T cos (45◦ + mϕ) + U sin (45◦ √ + mϕ),T cos (45 ++mϕ) (45◦ + mϕ), t +2 u 2 dϕ T ◦sin T (45 TU +UU m−4 , and will, consequently, according to the theoequals T + U 1 R 1 2 2 4 (45◦ + mϕ), ◦ T +U ◦ T sin (45◦ ++mϕ) U cos √− (R + mD 2 cos (45 4ϕ))..., + mϕ) rems − U cos (45◦preceding + mϕ), paragraphs, always be a positive magniT sin+(45 of the m 2 1 U ◦ T T +UU ◦ ◦ Ω will cos (45 + mϕ) + U for sineach (45 real + mϕ), − arctan = mϕ + 45also tude value of ϕ . Therefore, necessarily 2 2 respectively, as will be easily shown by the actualTdevelopment. T +U T U U T T + Uintegral ◦ be apositive − =Umϕ +(45 45◦◦magnitude, + mϕ) − cos + arctan mϕ), that is, the value of2the dϕ Consequently, the value of the function tt + uu , T forsin r =(45 R, will 2 U T T +U arctan equal T T + U U and thus will be a positive magnitude. t2 + u 2 T U T T + UU arctan dϕ t2 + u2 Incidentally we conclude from the same formula that 2 + U2 T T U r = of R the function t2 + u2 , for r = R, equals T 2 + U 2 the value = mϕ + 45◦ − arctan tt + uu 2 2 T and T +will U thus be positive, and thence it follows that for √ no √ ϕ √ 3 0 to ϕ = 360°.1 This is absurd, since just before we had T 2 +than U 2 the individual magnitudes mA 2, mBfrom 2,◦ =mC 2UU value of r which isgreater √ √ 3 √ −arctan arctan = mϕ + 45 the √ found sameTmagnitude = 0. Our presumption, thus, can not be t =√0, u mA 2, mB 2, mC 2 √ ,..., ◦can = 0 at√the t + uT −1 ◦ same 3 √ TmA cos (45 + mϕ) + U sin (45 + mϕ), true, and with it the validity of the theorem is proven. 2, mB 2, mC 2 time. t + u −1 U ◦ ◦ √ arctan T sin (45 + mϕ) − U cos (45 + mϕ), x= T √ r(cos ϕ + sin ϕ 3. −1) √ 2. ◦ + mϕ) + U sin (45◦ + mϕ), t + u −1 T cos (45 x√= r(cos ϕ + sin ϕ −1) √ ◦ ◦ Thet function into t + u −1 by the sub− u −1X transforms + mϕ) − U 0cos (45 T sin √ Theorem. Within the(45 boundaries r= and r =+R,mϕ), as √ −1) x = r(cos ϕ + sin ϕ stitution and into t − u −1 by the √ well as ϕ = 0 and ϕ = 360°, there exists such values of the √ = r(cos ϕ + sin ϕ −1) substitution x = determined values √ r(cos ϕ − sin ϕ −1). Ifxfor indeterminates r, ϕ , for which t = 0 and u = 0 at the same √ 1 t−u ϕ − sin ϕ −1) of r, ϕ , 1say for −1 r = g, ϕ =G, results that t = x√ 0,=ur(cos = 0 simultatime. t − u −1 tt Demonstration. + uu √ We will suppose that the proposition is √ √ 1 x = r(cos ϕ − sin ϕ −1) 2 2 x =ofg(cos + sin G one −1), = g(cos G − sin G √ not true; it is evident that the value of t + u for all values of 1 This is clear in and itself.GMoreover, willx easily ascerx = r(cos ϕ − sin −1)G + si x =ϕg(cos U the indeterminates within the assigned limits must be a positive tain the indefinite integral = m ϕ 1 + 45° – arctan T , and can be prov2 2 1 1 en in different ways (indeed in itself it√is not yet obvious which √ T + finite. U magnitude, such that the value of y remains Let us conx = g(cos G + sin G −1), x = g(cos G − U sin G −√ 1 sider the double integral of the infinitely many values of the multi-valued function arctan T , √ √ √ 3 x = g(cos G + sin G mA 2, mB 2, mC 2 which correspond to ϕ =360°, one must adopt), that the value which √ y dr dϕ one obtains for the integration for set =Gm+• 360° x −be g(cos sin G −1) 1 ϕ = 360°, must √ or = 2 m π . However, this is not necessary for our purpose. and similarly x − g(cos G − sin G −1). Δυναμις Vol. 3 No. 3 December 2008 √and simil 1 1 x − g(cos G + sin G −1) √ and similarly 1 − sin G1 −1). 1 x − g(cos G and similarly 1 2 x − 2g cos Gx + g2; 1 1 1 1 m ◦◦ sin (45◦◦ √ √−1 t + u√ +u u −1 −1 tt + √ √−1) x = r(cos ϕ + sin ϕ√ 40 x= = r(cos r(cos ϕ ϕ+ + sin sin ϕ ϕ −1) −1) Third Demonstration x √ Gauss √−1 t − u√ −u u −1 −1 tt − √ The integration neously, (and that √ x = r(cos ϕ − sin ϕ√ −1)there are such values became verified in the x= = r(cos r(cos ϕ− − sin sin ϕ −1) −1) then X obtains the value 0 for each of the ydrdϕ previous paragraphs), x ϕ ϕ tu − ut substitutions y dϕ = √ √ r(t2 + u2 ) √−1), x = g(cos G − sin G√ √−1), x = g(cos G + sin G√ x= = g(cos g(cos G G+ + sin sin G G −1), −1), x x= = g(cos g(cos G G− − sin sin G G −1), −1), ξ = 0, η = x is only an actual integration---that is, summation---so long as y is everywhere a finite magnitude between the boundaries in which and will consequently be divisible by the indefinite one integrates; on the contrary, it will be absurd if y istuinfinite − ut ydϕ = anywhere between those bounds. If we specify such an integral √ r(t2 + u2 ) √−1) x − g(cos G + sin G√ η dξ , which generally indicates the surfacebetween the axis of x − g(cos G + sin G −1) x − g(cos G + sin G√−1) the abscissa and the curve developed accordingηdξ to the customary √−1). and similarly x − g(cos G − sin G√ and similarly similarly x x− − g(cos g(cos G G− − sin sin G G −1). −1). rule in which the ordinate η corresponds to the abscissa ξ , and and 1 of the continuity, then we are withal negligent very often η =could 2 So long as G is not = 0, nor g = 0, these divisors are unydrdϕ ξydrdϕ find ourselves entangled in contradictions. If we set, for example, equal, and X will consequently also be divisible by their product 1 1 C − η = ξ2 , then analysis will yield the integral ydrdϕ ξ , which will x22 − 2g cos Gx + g 22; correctly indicate the surface so long as the curve maintains its ξ = 0,◦ η ξ = 0, η = 0, x2 − − 2g 2g cos cos Gx Gx + + gg2 ;; u ζ = 0, ξ = R, η = 360 , x continuity; since this is interrupted at ξ = 0,t then, if anyone of unreasonableξ manner should theR, magnitude = 0, η = 0, ζ ask = 0,for ξ= η = 360◦of , the surface for would r = e,deliver ϕ = E by however, if either sin G = 0 and thus cos G = ±1 or g = 0, each from a negative abscissa to a positive, the formula tu −ϕut= F by sin G = 0 tu − ut r = e, ∓ of the factors will be identical, namely, = x g. It thus holds ydϕ = sin G G= = 00 the absurd replyydϕ that = it would be negative. However, what 2these 2 sin r(t2 + u2 ) r =r(t f , up+ϕinu=)E by true that the function X possesses a real divisor of the second or and similar analytical paradoxes mean shall be followed tu − ut cos G = ±1 first degree, and since the same conclusion will again hold for the ydϕ more = detail ηdξ r = f , ϕ = F by cos G G= = ±1 ηdξ occasion. on 2 + u2 cos ±1 r(t ) another quotient, X is thus completely dissolvable into such factors. Here it is only possible to affix but a single remark. =x∓g 1 1 ηdξ Were questions =x x∓ ∓ gg η which = 2 in certain arctan Θ − arc proposed without restriction, = η = 4. ξ ξ 2 absurd, then analysis thereupon cases could become aids itself 1 1 1 η = 2very frequentlyC by C − delivering a response in part variable. If we, Although we have brought the task we had undertaken − ξ ξ ξ y dr dϕ extending from r = for example, signify the integral completely to an end with the preceding, still it would not be suC − 1ξ e to r = f and ϕu = E to ϕ = F and the value of ut perfluous to add something further concerning the inferences of t § 2. By the assumption that t and u vanish at the same time for for r = e, ϕ = E by Θfor r = e, ϕ = E b no values of the variables r, ϕ within the boundaries given there,ut r = e, ϕ = F b r = e, ϕ = F by Θ we happen upon an inevitable contradiction, whereupon we have for r = e, ϕ = E by Θ r = f, ϕ = E b r = f , ϕ = E by Θ concluded the incorrectness of the assumption itself. This con r = e, ϕ = F by Θ r = f, ϕ = F b ϕ r = f , ϕ = F by Θ tradiction must therefore cease, if there is an actual value of r, r = f , ϕ = E by Θ for which t and u become = 0 at the same time. In order to further r = f , ϕ = F by Θ integral arctan clarify this, we observe that for such values t2 + u2 will = 0 and Θ−a then by analytical operationsarctan one easily obtainsΘthe valΘ − arctan − arctan Θ + arctan 1 thus y become infinite, so that it no longer is permitted to treat the 1 ue 1 y dr dϕ as an assignable T 2magnitude. + U2 double integral Given arctan Θ − arctan Θ − arctan Θ + arctan Θ . if ξ , η , ζ signify the coordinates of spatial in general, √ points, √the √ 3 y dr dϕ signifies the volume mA ydrdϕ 2, mB integral of a body, which2,is mCThe2 integral in actuality can then only have a determined value, bounded by the five planes whose equations are if y remains finite between the specified boundaries. This value is itself contained in the specified formula, however it is not fully ξ = 0, η = 0, ζ = 0, ξ = R, η = 360◦ , determined by the same, since indeed the arctan is a multivalued 1 function, and it must further become decided which functional and by a surface whose equation is ζ = y, if one considers those values are preferable in a determined case, by means of other partstuof −the as negative whose coordinates ζ are negative. utbody not-so-difficult considerations. If, on the contrary, y is infinite ydϕ However, = anywhere between the specified boundaries, then the question rer(t2 + here u2 ) it will be tacitly assumed that the sixth surface will be continuous; if the latter condition comes to pass in such a way specting the value of the integral y dr dϕ is absurd. This ηdξ that y will be infinite, it can very well happen that the former condoes not hinder, that, if one will obstinately extort a response from analysis, this might soon give different methods, whereby 1ception no longer makes sense. Inthis case, there can be no talk η = 2of the evaluation of the integral y dr dϕ , and for that reason the individual values of the foregoing1 general formula are obξ it is not incomprehensible that analytical operations applied in tained. 1 C − 1ξ blind reckoning to meaningless things lead to nonsense. 1 Δυναμις Vol. 3 No. 3 u t for r r r r = e, = e, = f, = f, ϕ=E ϕ=F ϕ=E ϕ=F by by by by 1 Θ Θ Θ Θ 1 1 December 2008 41 Gauss to Bessel December 18, 1811 Correspondence Carl Gauss to Wilhelm Bessel December 18, 1811 In this letter, Gauss responds to an assertion made by his student Bessel about the theory of numbers. Gauss used this opportunity to intervene on the way mathematicians had been thinking about their science, making clear that mathematics must represent an investigation of physics, not merely mathematical theorems. A few days ago, I finally recieved the Königsberger Archiv that I had ordered. With great interest, dear Bessel, I read through your first treatise,1 and for now, have at least skimmed the other. That and the wonderful assistance, which it offers in the determination of li for large numbers, is now all the more agreeable to me, since I have recently obtained a beautiful table of factors from Chernac up to 1,020,000, appearing in this year of Deventer, with which I want to count up, little by little, the prime numbers from myriad to myriad, in order to compare them with the value of the integral ex−1 dx x √reckoned from x = 0 on. You have made known to me the desire, a + b that −1 I=report a + bithis theorem in our GGA: before I do it, dear Bessel, our friendship obligates me to converse in writing with you li(a + about bi) one or another point, where my outlook is not entirely in accord with yours. Therefore, kindly accept the following re ϕ(x)marks · dx and, in the meantime, impart to me your thoughts just as frankly and candidly, as I mine. I think I can infer from one of α + βiyour casual remarks, that a basic principle is common to both of us: “In mathematics, there are no controversial truths,” and thus I x = a do + bi not doubt, that through the mutual exchange of our ideas, we will already be in agreement. ϕ(x) · dx First and foremost, from someone who wants to introduce a new function into analysis, I would ask for an explanation, a + bi whether he will understand them as applied to merely real magnitudes (real values of the arguments of the function), and regard ϕ · dx the imaginary values of the arguments as if they were only quasioutgrowths; or whether he would accede to my principle, that, in √ ϕ(x) the realm of magnitudes, the imaginaries a + b −1 = a + bi must enjoy equal rights with the reals. The discussion here is not ϕ(x) =of∞ + bi) practical utility, rather, to me, analysis isli(a an independent science, which, by the neglect of those imaginary magnitudes, loses ϕ(x) · dx 1 Investigation of expressible transcendental functions by the α +Science βi integral ldxx , Königsberger Archive of Natural and Medicine 1, 1812, S.1, FR. W. Bessel’s Werke II, Leipzig 1876, S.330 Δυναμις Vol. 3 No. 3 x = a + bi x−1 ex−1 √ e = a + bi a + bx−1−1 dx x e x √dx enormously in beauty and roundness and, in a moment aall+xtruths, bbi)−1 = a + b li(a x−1 e which otherwise would be universally valid, are obliged to add dx x−1 √ e√ xbiex−1 a−1 += bemploy =the a+ the most weighty limitations. assumption, that ϕ(x) · li(a + bi) a√+ bI must a−1 +dx bi √ dx dx x −1 = + b point, a + bi you are essentially ain+accordance since b −1 = awith + bixme aboutathis li(a + bi) that you by your elucidation2 in art. 18 already indicates, noϕ(x) means α+ βi ·dx x−1 li(a + bi) √ e li(a + bi) intend to block the way toabi) investigations about . What a + b −1 = + bi li(a + dx √ √ √ ϕ(x) · dx ,awhere x= aa+ α=+ βi a + b be −1imagined = aa+ +b −1 +bibi? x should ϕ(x) dx +bi b then −1 with = a +· bi ϕ(x) · dx li(a + bi) Evidently, if one wants from clear notions, it must ϕ(x)to· proceed dx α + βiinfinitely x = a + bi ϕ(x) · dx li(aaccepted + bi) that li(a + bi) be x proceeds, through small increments li(a + bi)α + βi √ α which + βi the inteϕ(x) · dx (each of the form ), from that value for a + b −1 = a + bi α + βi x bi =, aand + bi bi · dx ϕ(x) ϕ(x) · dx be 0,ϕ(x) · adx+ then ϕ(x) add all gral should up to· dx x=a+ x =however, a + bi the ex−1 + βi x = a bili(aestablished. . Thus the α significance is + fully Now, + bi) dx ϕ(x) · dxmany ϕ a ·+dx bi the x α + βi can beα constructed α +ways: βi thus pathway as dx + βi ϕ(x)in · infinitely ϕ(x) a+ bi magnitudes ϕ(x) · dx ϕ(x) entire realmxof=all real can· dx be thought of as· dx an infinite ϕ imag· dx x = a +line, bi soxthe x = a real +ϕ(x) biand straight of+allbimagnitudes, + bi a = entire a + biarealm √ ϕ(x) · dx aphysical + bi aαwith inary, can be made ++ −1infinite = a + plane, bia + biwhere each b βian · dx =ϕ(x) ϕ(x) ϕ(x) =∞ ϕ(x) ·determined dx · dx point, by b, can represent the dxa,ϕordinate ϕ(x) dx ϕ · = · abscissa ϕ ·from dx a value x =the + bi ϕ · dx li(a magnitude a + bi. Accordingly, continuous path +abi) ϕ(x)by a line,aand ϕ(x) = ∞ a+ bi another + biis consequently of x to ϕ(x) a + bi is represented ϕ(x) ϕ · dx ϕ(x) ϕ(x) · dx possible in infinitely many ways. ϕ(x) · dx ϕ(x) ϕ · dx I maintain ϕ · dx always ob∞integral ϕ · dx then,ϕ(x) that=the = ∞ x−1 e pathways, ϕ(x)value bidifferent ϕ(x) = to ∞ tains the same along αatwo ++βi dxϕ(x) =as∞long as x ϕ(x) never = 0ϕ(x) ϕ(x) the two lines within the area x−1 enclosed between ∞ representingϕ(x) the = pathways. This whose e x =ϕa·isdx +abibeautiful theorem,3 √ dx ϕ(x)difficult = ∞ proof ϕ(x) occasion. = a∞+ b −1 ϕ(x)I = ∞give on an appropriate not will It is= a x + bi √ ϕ(x) a + b with −1 =other a + beautiful bi connected truths ϕ(x) · dxrelating to the expansion of series. A pathway to each point can √ always be found, li(a which never + bi) ∞. = li(a +such bi) a place whereaϕ(x) a++bib= −1 a + bi contacts Hence I insist, that one must avoid such points, where the original, fundamental principle ϕ(x) · dx li(a +clarity bi) and easily leads to inϕ its · dx of ϕ(x) · dx evidently loses consistencies. Furthermore, itis at the same time clear α +from βi this, βi ϕ(x) · dx can always have many howα a+function produced byϕ(x) x = acan + bigo values for same value of x, when namely, the paths the x = such a +ex−1 bia point where ϕ(x) α += βi∞, either never, or once, or around dx ϕ(x) · dx multiple times. x 4 If one defines, e.g., log x by ϕ(x) · dx x = a + bi a + bi a + bi ϕ(x) · dx 1 1 ϕ · dx 2 See Bessel’s Werke II, Leipzig 1876, S. 341, 1. Spalte.1 ϕ · dx a + bi 3 Strictly speaking, it is still yet supposed, that 1 ϕ(x) is itself a single-valued function of x, or least1 only one system ϕ(x) ϕ at · dx 1 of values, without interruption in continuity, is assumed for whoseϕ(x) value=within ∞ each whole ϕ(x) = ∞ surface-space. ϕ(x) 4 It appears in the manuscript as x = ∞ ϕ(x) = ∞ 1 December 2008 ϕ(x) · dx a + bi 1 ϕ · dx ϕ(x) · dx α + βi x = a + bi +2πi −2πi +2πi −2πi −2πi +2πi +2πi −2πi −2πi 42 December 18, 1811 1 · dx, infinite magnitude. ex dx negative, 1 x , · dx, ϕ(x) · dx I have mentioned all of xthis before hand, in order to es x 1 1 · dx, tablish my point of view, that I must join Euler, when he said,5 · dx, x a + bi 1 x · dx, that where x starts off = 1, then logx x+2πi is arrived at either without in dx or more times; cluding the point x = 0 or by rotating aroundϕit·one dx each time, the constants +2πi 1or −2πi are added; thus the multilogx dx, If ϕ(x) can never become ple logarithms of each number xis ·clear. infinite for a finite value−2πi of x, 1then the integral is always only a xx< 1, and for values > 1, nec· dx, ex − 1as real for the case where x is taken e dx x the case single valued function. This isxe.g. for= ∞ , ϕ(x) = e dx e − 1 ϕ(x) , x essarily imaginary values. Its difference from the real, x obtains , , ϕ(x) = 0.001i), x li(0.7 + 0.001i), li(0.8 + 0.001i), x e dx li(0.5 + li(0.6 + 0.001i), x x dx e e − 1which Mascheroni, Soldner and you add to it, is =πi or 3πi or 5πi x x , e −1 , li(0.9 + 0.001i), li(1.0x+ ,0.001i), li(1.1 + 0.001i), etc., until x ϕ(x) = ϕ(x) = ex − 1 , the path through x = 1; one would beli(1.5 able to+ 0.001i) x etc. I do not examine x , ϕ(x) = in an entirely similar way, that log – x = log + x (a theorem prove x x so that − 1 can be accepted, if limited ewhich x todx real magnitudes, but which dx dx e −1 x must be immediately ommitted, when my above principle of two dx logx x dx x logx x xe − 1 e − 1 dimensions is dx bestowed upon the realm of all magnitudes.) Make , ϕ(x) e= dx logx x −x1xdx logx x li x real for any single value of x between 0 and 1! But which e x−e 1− 1 , whose value is always ϕ(x) = dx is certainly a single-valued function of x, value will then be assigned? x x 1 3 1one 1 4 converging, and will1always 1be represented by one and only li(0.6 + 0.001i), + 0.001i), + 0.001 x +li(0.5 xx + x+ x + +0.001i), etc. 1 4 li(0.5 +li(0.7 0.001i), li(0.6 +li(0.8 0.001i), li( 3 18 4 96 xx + xx + x + etc. sensible series x + li(0.5 li(0.9 + 0.001i), li(1.0 + 0.001i), li(1.1 + 0.001i), etc., until li(1.5 +0.001 0.00 +18 li(0.7 + 0.001i), li(0.8 + 0.001i), − 1 96li(0.6 + 0.001i), e0.001i), li(0.9 + 0.001i), li(1.0 + 0.001i), li(1.1 + 4 1 li(0.5 1+ 0.001i), 1 li(0.6 + 0.001i), li(0.7 + 0.001i), li(0.8 + 0.001i), 1 3 +dx 1 1 4 xx0.001i), + x3 etc., x4 + etc. x+ + + until li(1.0 li(0.9x++0.001i), li(1.1 li(1.5 + 0.001i) xx + x + etc. xxx + 0.001i), li(0.9 + 0.001i), li(1.0 + 0.001i), li(1.1 + 0.001i), etc., until li(1.5 + 0.001i) 18 4 96 1e − 1 1 4 18 4 3 1 96 dx x + xx + x + x + etc. Without doubt imaginary, but it should follow the law of the con 4 18 x 96 dx nowhere a break ex abrupto? If you then go from lix −tinuity, being , dx logx lix − , li(1.5+0.001i), while allowing the imaginary part 0.001i to de 1 logx 1 chosen this one, since he dx I would that1 Herr Soldner had x + xx + x3 +dx x4 + etc. crease to 0, [to li 1.5], it absolutely does not come to a real value lix − , would introduce a simple new function, 18 41 lix 96 instead of his − logx of li 1.5, 1but to one which depends upon -\pi i. With what you 1 3 logx dx1 , 4 x + ,x + etc. + − x + xx lix furnish 18 logx 4 for proof against Euler and myself, I find to criticize 1) dx96 ex − 1 , li x − x that you dx say, if li x must become real in the whole circumference, 1 x e − log x one must etc., but of course the onus lies not on us, if the dx x then x x dx − 1 e lix −e − 1 , dx continuity should not be lifted without cause, and it should even logx dx regarded as simpler and x since a single valued functionexisx−always 1dx lix − , dx especially since log x is be proved, of course, that it should be taken as real in the whole more classical than a multi-valued one, x logx circumference. 2) To be sure, ex dx itself already a multi-valued function. ex dx It would perhaps also be x x advantageous, for x dy x1 e − e dx x dx dx e y xxx x 1 e e−xdx dx xx is log y as well as log –y, but never both simultaneously, rather the or at least for former, if the integral can begin from y – 1, the latter, if it begins x dx 1 y = –1; the second e dx from integral just as+understood as the first, = C +1is l(±lx) etc., logx 1xx in general log y + C, if the first time C is set = 0, the second time e dx 1 C is set =±πi or ±3πi etc. However, it is very true that E uler ’s 1x 1 remarks require a correction in so far as, if the integral should to introduce a suitable symbol and name, so much the more since, will start from z = 0 , in no case C 1be infinite. – Thus, according with the problems from physics derived from li x, x itself is comex − dx to my opinion, one may not set monly an exponential magnitude. If the truths for Soldner’s li x are carried over to my 1 dx = C + l(±lx) + etc., 1x logx e dx , 1 1 x dx 1 1 logx which for brevity I will signify with Ei x, Exponential Logarithm, 5 Institutiones calculi 1 I, 1768, Sections 228, L. Euler 1 integralis xx S.128 + x3 + etc. x + then the integration is taken, so that Ei disappears for any real, Opera omnia, ser. I, vol. 11, 18 4 dx Δυναμις Vol. 3 No. 3 logx December 2008 1 1 x + xx + x3 + etc. 18 4 +2πi Gauss to Bessel li(0.5 + 0.001i), li(0.6 + 0.001i), li(0.7 + 0.001i), li(0.8 + 0.001i), (1 + 2αx + βxx) li(0.9 + 0.001i), li(1.0 + 0.001i), li(1.1 + 0.001i), etc., until li(1.5 + 0.001i) dy y dy y Gauss to Bessel 43 December 18, 1811 rather must be determined either with l(+lx) or with l(–lx), but dx only one decided on. – = C + l(±lx) + etc., dy of the investigation to Moreover, Ilogx believe the extension dx C + l(±lx)y+ etc., imaginary argumentslogx will=provide grounds for supremely interesting results. Nevertheless, from the foundations derived above, x I would rather chose the function e −1 x x dx dxe − 1 = Cdx + l(±lx) + etc., logx x than dx x logx dx e − 1 dx dx x + etc., = C +logx l(±lx) because I surmise, that the first will give a coinciding result. Thus logx for example, I would love1to know,1 whether this function, or what x + xx + x3 + etc. is the same, the series 18 4 dx 1 1 x + 1 xx + 1 x3 + etc. 3 logx 18 4 x + xx + x + etc. 4 18 (1 + 2αx + βxx) can+become x of the form a + bi. I (1 + 2αx βxx) 0, for certain finite1values of 1 3 it is even very probcan not yet maintain with certainty, whether x + xx + x + etc. 18most certainly infinite), 4 (then able to me. If there is such a value these magnitudes will be very remarkable decomposed and the entire series can be decomposed into infinite factors of the form (1 + 2αx + βxx) . From a few other details, I can only add a few words at this time... 1 1 1 Δυναμις Vol. 3 No. 3 1 December 2008 44 The First Integral Calculus Bernoulli The First Integral Calculus A Selection from Johann Bernoulli’s Mathematical Lectures on the Method of Integrals and Other Matters Johann Bernoulli Translated by William A. Ferguson, Jr., from the German translation of Dr. Gerhardt Kowalewski. It should be noted, that sometimes quantities present themselves, whose integrals at first glance, it seems, cannot be √ √ 4 found Nonetheless, the may dxintegral a2 x2 + x4easily be found dx a2by x2this + xrule. after a certain transformation, as in the following cases. 1 2 e have seen previously how to find the differentials √ √ 3 2x √ x dx√a21.+ xIf 2 x + 3ax 2 +a 2 + 3a 2 instead of2 + x3 one writes dx x dx a x 1 2 of quantities. Now we will, inversely, show how the dx a3 + √ 3a22x + 3ax42 + x3 2 2x 1 2 1 2 integrals of differentials are to be found, i.e., x2those dx √ dx a x + x√, then one finds the integral of the √ latter,√namely 2x 2x 1 2 (a 1 dx 1 2 2 √1 2 2 +x 2++xx 3 (+ 2 3 2 . And if one(writes 1 2 dx) a + x ) a2 + x2 a + x a + x ) a dx a 3a x + 3ax 2 quantities, from which the differentials are derived. Now it is al√√3a + x 3 3 3 dx) (a dx + x x dx 2x 2 2 x dx 1 3 for dx a√a3++x3a2 x + 3ax2 + x3 , then one finds the integral ready known x2 dx x2from dx previous statements, that dx is the differential 3x 1 2 1 2 2 (a dx + x dx) a +√ x √ 1 3 ofx2 x or 2 x + or – a constant quantity, x dx the differential 2 2 1 2 1 2 2 3 √ 2 ax+ 1 4 ( a + x ) a2 + (a + 3 (a dx + x dx) 1 1 3 3 2 x 2 3 3 of 3 x or 3 x + or – etc. and x dx the differential of 4 x or (a + 2ax + x ) a + x . 5 1 3 2 5 1 x 4 2 dx x dx √ 3x 2 2 4 x + or – a constant quantity, likewise also 1 4 1 4 (a + 2ax + x2 ) a + x √ x x 2. Also conversely it2can2 occur, that 2one must pull one is the differential of 1ax4 etc. 5 4 1 3 14 3 a dx (a + 2ax + x ) a + x 2 " 41x 3x 3 x ax dx or more variables under the root " ax 5 sign, before the integral can be 2 1 3 " 2 taken, as with the following example " (3ax3 dx 1 4 1 4 ax dx 3 ax (3ax3 dx + 4x4 dx) ax + x2 1 4 4x 4 x ax3 dx " " 4 ax etc. (3ax3 dx + 4x4 dx) ax + x2 . 3 4 p dx + 4x dx) ax + x2 (3ax ax dx From this the following general rule can be formed: The integral of this cannot be taken by our rule as it a p+1 p x appears. However, if one pulls in an x (under the root sign), the 2 is the differential of . ax dx p+1 (3ax dx + 4x result is a p+1 x quantity is Therefore if the integral of any differential 3 4 2 3 2 dx p+1 (3ax + 4x3 3 dx) (3ax x4 ax + x 2 dx + 4x3 dx) ax3 + to be taken, then one must first of all consider, whether the given 24x dx) 3ax + x4 ,3 (3ax dx + 4 (3ax dx + 4x dx) ax + x quantity is the product of any differential quantity 2 3 √ multiplied by (3ax dx + 4x dx) ax3 + x4 2 dy This a + is y then an its “absolute quantity” raised to a certain power. whose integral one finds by the rule = (ax3 + 3 indication that one can find the integral by the rule given above. √ 3 2 a + ay + y is to be 4 2 If, for example, the integral of the quantity dy =4 (ax xx44). ax3 + x4 (ax33 + = 2 (ax + ax343+ + 2 xx4 )3)3 ax = + xax3 + x4 found, then I see first, that dy is multiplied by a1 +1 multiple of its 1 3 (ax = + x ) 32 1 2 (a +ay) 3 1 absolute quantity a + y, raised to the power ; next I seek its + y 3 2 = (ax + x4 ) ax3 + x4 2 +1 3. If a fraction 3 presents itself, whose denominator is a integral by the above rule, namely, 1 square, cube, or other power, then one must choose its root as the a4 + 2 2 x dx √ absolute quantity. Therefore, for 1 2 dx 1 xx dx (a + y) a + y (a + y) 2 +1 4 x dx 1 +4 2a2 x2 + x4 4 + 2a2 x2 a , i.e. 3 . + xx 2 x2 + 4 2 +1 aa4 + 2a 4 2 2+ 4 2 xadx+ 2a a x + x2 x √ 3 4 2 2 4 a + 2a x + x Likewise one finds the integral of x dx a2 + x2 , √ 2 2 2 2 which is the2following −1 : (2a2 + 2x2 ) a + x (a + y) a + y a2 + + xx2 a 2 2 3 a + x is to be chosen as the absolute quantity, and one obtains 2 2 2 −1 2: (2a2 + 2x2 ) . If one chose a4 + 2a2 x2 + x4 as the ab√dy : √a +12 y 2 3 1 then a x 2 + 2x2 ) 3 −1+ (2a dy : a + y 1 (a + x2 ) 3 +1 = (a2 + x2 ) √a2 + x2 ; −1 :: (2a 2x 2) −1+: (2a + 2x2then ) solute quantity, the integral of this fraction would not be dy : a + y 8 + 1 3 √ 4 2 2 4 2 2 √ 4 2 2 4 + 2a x + x a obtainable by this rule. −1+: (2a 2xx4) 2a2 xx2++ + √ 2 a + 2y a + y √ 2a aa4 + x2If+the x4integrals of two quantities cannot be found india4 + 2ax24. the integral of dy : a + y equals 2 a + y , the integral of 2 2 x + x4then sometimes it will be the case, that one can find the a4 + 2avidually, dx : xdx : x equals √ integral of their combination. Example: dx : x 2 a+y 1 0 11x0 = 1 × 1 = ∞ 1 0 1 x = 0 × 1 =0 ∞ . x = ×1=∞ 0 dx : x0 0 0 1 0 1 Δυναμις Vol. 3 No. 3 December 2008 x = ×1=∞ 0 0 W 1 1 1 1 1 √ 2 2 √√ 2 a(a + x) √ 2 a(a + x)√a(a++xx)2 a + x=−2 (a + x) aa++xx 5 22 √ a dx 2 a +3x √ 53 2√ √ 2 2 (a ++x) x)x) aaa+ = + √ + x)2 a + x= (a a(a+ +xxx − 55(a 45 = x Calculus dx 5 a+x 3 The First Integral √ 2 √ Bernoulli √ a(a + x) a + x x dx a + x a dx a + x 3√ 2 √ √ 2√ 2 √ 2 x) a + x − a(a + x) ax+ dx x a+x √ dx (a + xx aa dx aa + 5 which is 3 √ x dx a dx a dx a + x x dx a dx √ √ 2 √ +√ + √2 √2 x2 dx a + x 2 2ax + x22ax +2ax + x a(a + x) a + x x dx a dx 2ax x x+ dxx a + x √ 22 √ √ 23 √ +√ 2 2 + a(a +ax) x) + xx , + x)√a + x a(a 2ax + x2 a dx 2ax + x2 (a +33x) a(a + xaa−+ x dx √ The integral of either aquantity is not x known. However, + the√inte√5 (a2 + 2ax + 3x2 )dx a + x dx √2ax dx 2 2 2 √ 2ax + x therefore x dx a + x +x +√ gral of their sum, √ √ x dx a + x 2 2 + x dx 2ax + xx2dx 2ax +a xdx a dx + √ (a + 2ax + x2 )dx2 a + x 2 √ 2 √ √2 (a + x) a + √ x − a(a + x) a + x 2ax++x xdx2ax + x2 √ a dx √ √ 2 2 2 2 5 3 √ + √ (a + + x) x)22 aa + a(a + x) aa + + xx − − a(a + xx 2 (a √ x2 dx a√+x) x 5 3 (a + x)2 2ax + x2 3 5 2 a dx + x dx √ x dx a + x a dx a + x 5 2 2 √ √ √ + 2ax + x )dx a dx + x dx √ √ √ √ a+x √ dx aa + +(a xx 2ax + x2 xx dx √ 2 2ax + x22ax + x2 . dx aa++xx. In this remains as the integral ofa the a dx + xgiven quantity xx3dx 2 is 2ax + x √ √ 2 same manner one finds the integral of x √dx a√+ x . In other 2ax + x2 5. Sometimes, a fraction may seem not to have an in√ √ √ 4 3 2 2 2 2 dxx ais + x x dx the a +integral x dx +xxx√ + 2axaa+ )dx xa + tegral, but if one √ multiplies words, of (axx +dx known, as x adxdx + xand 2 its+numerator dx denominator by √ 2ax +ax√dx 2ax dx a + x √ 2 √ be obtained easily. So it the same quantity, then its integral can 2 a dx a + x √ immediately , and afterwards was pfound is4that√ of 2ax + x 3a++x2x a dx dx3a + 2x x dx a + x x dx a + x 2ax dx√a + x √ √ is with . above, that of 3 √ x dx a + x 3a + 2x √ 2 a dx + x dx (a2 + 2ax + xremaining )dx a +term, x √aTherefore one has the integral of √ the xp3 dx√ +x √ √ √ √ x dx a + x a dx + x dx √ 3a + 2x 2 22 + 2ax + 2x22 )dxx3adx (a + x a + x x dx a + x . 4 (a + 2ax + x )dx a + x a dx a + x √ 2 √ √ (a(2a + 2 . x 2 dx a + x √ 4 3a+ +xax 2x ax dx dx In this same manner one will find the integrals of dx a + x x dx + x dx 2ax dx a + x 2 √ √ √ x dx a + x √ 4 √ 3 √ 2 3 2 p 3 x dx a + x and even x 3 or the quantities x dx a + ax3ax dx 2++x2x dx + 3ax x 2x dx a + x + x41)dx a √ +x (2ax √ Multiply the numerator and p √ denominator by x. Then one obtains 2 x dx a + x . Thus also, if a quantity consisting of several terms 2 3 a dx √ √ 2 3ax + 2x √ √xp dx a 3+a x+√x √ 4 ax dx + x dx 2+ dx a x x 2 is given, its integral will be found by parts. One such 2ax dx quantity a + x is 1 aa dx √ √ dx + xxa + x + 2axaadx √ ax dx + x2 dx √ 2a + √ 3 3 4 x2 dx + 2x x3 3ax 1 2 √ + x )dx a + x (2ax 1 . First I seek the integral of the first part √ 2 3 2x √ 3 4 √ √ 3 3ax + p 3 3ax + 2x 3ax2 + 2x3 , 4 +x + x )dx a (2ax 3 √ x dx a + x dx a + x x 2ax dx a + x , then that of the second, . Their 1 2 3 (2ax3 + √ 3 3ax + 2x sum√gives the integral of the whole. √ 1 2ax dx a + x √ 1 4 √ √ x2xdx x x √ dxa + a+ √ whose integral is 3 3ax2 + 2x3 . √ 2ax dx aa + + xx 1 3 2ax dx 2 3 (2ax3 + x4 )dx a + x Admonition 2ax dx +x √ andadenomina3 3ax + 2x 6. Conversely, sometimes the numerator 2ax3 dx a + x √ tor are to be divided by the same quantity, in order to obtain its 3 √ These cases √are the most important dxin awhich + x integrals 1 2ax 2 x4 dx a + x √ integral. Example: x dx a + x 4 can be formed. Indeed several, even infinitely many others yet re√x dx a + x √ √ √ √ dx main, + xxwith the help 4 are possible. xx22 dx aa + 3 which integrations of 2 + x a +However x dx x adx 2ax dx a + x x ax2 dx they do not all come to mind, and furthermore, most of them can √ 2 1 √ 2 x2dx 2 aax + x4 . axbe dx reduced to those dx ahere, + x1so x4 cited √ 1 that with the help of these, the √ 2 2 4 desired ones can be achieved; ultimately a thousand methods of a2 x2 + x4 a x +x Divide every term by x. Then one obtains solution and manifold cases according to the nature of the given ax dx √ quantities present themselves to the attentive observer. For this ax a2 dx + x2 axreason dx it were no less impossible than useless, were we to provide √ √ √ a2 + x2 . a2yet + several x2 others aside from those offered here. a a2 + x2 √ √ Let the one remark suffice, that important mathematical √ a a2 +Its x2integral follows by the rule: a a2 + x2 . x dx and a +theorems x problems directly depend on the finding of integrals, 7. It also occurs sometimes, that the integral of a given √ both those already found as well as such as are yet desired to be dxthearule. + x If one however adds anquantity is√not obtainablexby found, as for example the quadrature of plane surfaces, the rectidx a + otherxquantity toxit, whose integral one knows,√something may be fication of curves, the cubature of bodies, the method of inverse √ a dx one a +subtracts x produced whose integral can be taken. Next, from tangents, or the finding of the nature of a curve from given propx dx a + x √ that integral the added quantity, therefore the remainder is the √ a dx a +x erties of its tangents, as well as that which belongs to mechanics, √ desired integral. Example: x dx a + x . of xfinding the center of mass, of impulses, of os(alike dx the + xmethods dx) a + Because √ its integral cannot √ be taken by a simple cillations, and so forth. Through the finding of integrals, one also dx)given a + quantity. x method, add a√dx a(a+dx The result is x +toxthe obtains the involutions of curves, and the method by which to 1 √ (a dx + x dx) a + x , 1 determine their nature, and with the help of the involute to rectify a dx a + x √ whose integral is found by the rule 1 to be equal to √ the2 curves themselves, as Tschirnhaus did with his caustics. (a dx + x dx) a + x = (a + x)2 a + x The ease of finding the differential of any given quantity 5 1 √√ 2 dx + x 2dx) (a a + x is matched, conversely, by the difficulty of finding the integral = (a + 1x) a + x . √ 2 5 2 of any given differential, so that we at times cannot even confi= (a + x) a + x √ 5 dently assert, whether the integral of the given quantity can be If one subtracts from this the integral of a dx 2 a + x , 2 √ (a + x) = a + x √ 5 Δυναμις Vol. 3 No. 32 December 2008 √ a dx a +x √ = (a + x)2 a + x 2 5 a(a + x) a + x 3 √ 2 √ a dx a + a(a x + x) a + x √ √ 2 a(a + x) a + x 3 a dx a3 + x √ 2 √ √ a(a + x) a +2x(a + x)2 a + x − 2 a(a + x) a + x 1 3 √ √ (ax + x2 )dx a + (ax + x2 )dx a + x √ 46 (ax + x2 )dx a + x The First Integral Calculus ( Bernoulli √ √ a + x = y √ a + x = y√ (ax + x2 )dx a + x formed or not. I venture to assert at least, that every whole and sumed avariable. +x= y √ = y2 − a p p will2 be better 2 − x2 2 x clarified through ana example. +√ x = y Let the a− x2 This x, =xpy 2 − x a ax , x + x a rational quantity, which is multiplied or divided by 2 2 (ax + x )dx a + x. quantity whose integral is desired be p p − a x = y √ dx = 2y dy a2 − x2 , x ax − x2 , x a2 + x2 , is either integrable or reducible to a + x = y . Then x = y 2 − a and furdx = 2y dy For that purpose, I set the quadrature of the circle or the hyperbola. This we will show ther dx = 2y dy . Therefore in the whole quantity in what follows. Therefore, above all, it is to be carefully consid √ x = y2 − a dx = 2y2dy √ 6 ered whether the given quantity, which one would integrate,3can 2 a + x =2y √ 6 4 x )dx a + x = 2y d (ax 3 (ax + x )dx a + x = 2y dy − 2ay+ dy . (a + ax − x )dx √ be reduced by multiplication, division, or through the extraction x a+x 2 dx = dy a + x = 2y 6 dy − 2ay 4 dy (ax + 2y x2 )dx 2 (a3 + ax − x3 )dx of roots to a quantity which has one of these root terms, multi− a x = y x (ax + x2 )dx The integral of this is found immediately, easily, without plied by a whole and rational quantity. If it is possible, then it is 2 7 2 5 2 5ado, further and inserts the dx = 2yindeed dy is = 7 y − 5 ay . Now if2 one√ immediatelyp a serious proposition, that thep given quantity can be = 27 y 7 − 5 ay p (ax + x )dx a + x = 2y 6 dy − 2 2 2 2 2 2 2 x − x , x , x + x a ax − x a 7 5 value of y, one obtains integrated; if not, it is dependent on and reducible to the quadra=7 y − 5 ay 5 a+x ture of the circle or the hyperbola. 3 2 3 =2 27 y 7 − 25 ay √ 2 = (a + ax − x )dx √ (ax + 2x2 )dx√a + √x(x=+2ya) 63 x + a − 2 3 2 x dy − 2ay 4 dya(x + If, fora example, the following quantity is given, +x (x +2a) x2 + a −(a√ 2a(x + 2 2a) 72x + a . 5 = (a3 + ax2 − x3 )dx √ +)dx 2x2 )dx 2 7 2 2 )dx (a + 2x 74 )dx 5ay+5(a2x 2 (a4 + a3 x + a2 x2 − x x √ = (x a(x + a) a)35√ x2+√a 2− x + a 7 y+ − √ √ 2 a + ax2 +5x2 a2 +7x2 2 a+x (x + a)3 ax + x (a4 + a3 x + a2 x2 − x4(a )dx 3 2 3 In the same manner the integral of the quantity + ax − x )dx 7 √ x , = 27 y 7 − 25 ay 5 ax + x2 √ 2 2 3 √ √ √ 2 2 (x + a) x + a − a(x + a) (a + 2x )dx 2 2 2 2 2 2 a + ax + =xy = y 5 7 a neither +x =y √ and its integral is to be taken, then it appears at first glance √ 2 + x2 = y 2 + x2 a a to be integrable nor to have a relationship to the quadrature of the √ √ p 2 2 2 −2x2 , pxp 2ax −2 x2 , xp x = + a)2 x + a (x + a)3 x + a − a(x − a2 x = x y=2 − ya22 − a2 yx22be a2 +to xp In − x2 ,words, xpx ax , x athe+absolute x a2 other − xassumes √ circle. if aone quantity 5 7 2 2 2 2 by setting (a+x) √ will be found a + x, then its differential y = a + x . Then will x = y − a that under the root 2 2 +2 x2 =2 y x = (a3 sign, + axthe − fraction x3 )dx 2 2 a 2 2 y dy = y: dyy: − ya − a . Further we get for the quantity and itdx = y dy −=adx x can be concluded will also be a fraction, so from that nothing by: ydx y 4 − a2 y 2 a2 ya22 self dx = y dy : y 4 y−2 − 4 3 2 2 4 4 − a22 y 2 2 the rule. Therefore I(amultiply the and denominator of + a x + anumerator x − x )dx y x = 4 The y a− 2integral of this is y 4 − a2 y 2 .Likewise, if one has ya2 this irrational fraction by the√numerator, uniteathe product of y − +x a+ +x2x2 and 3 y4 − a2 y2 3 2 3 (a3ax ax − component x )dx (a +with ax itself − x to )dxthe+rational the numerator of the quan(a − x)dx 2 2 2 2 2 2 2 2 2 2 2 x x dx = y dy : (a y − =x2√ (a2x = +a2x )dx : (aa ++ x +)dx =2x :)dxa: + ax22 + (a − x)dx tity, so that a fraction results whose numerator is purely rational 4 2 2 2ax − x2 y − a y (a − x)dx √ (a − x)dx 3 2 3 2 3 2 4(2y −−aa−2y)dy : y:4 − ya42 y−2 a2√ − y2 y 4 − a2 y 2 (2y − a y)dy : (2yy√ yx22a y)dy and whose denominator is irrational. Namely, (a − 2ax 2ax − x2 2ax − x2 √ ( This quantity now shows, that either it has an integral, 2ax √ √ − x2: = ay2 + x2 (a 2ax − x2 = (a2 + 2ax 2x2 )dx = − x)dx √ √ (a − x)dx √ √ √ a+x 2ax − x2 2= y a+x 3 2ax 2 −x 2: =√2y4 − a2 y 22ax − x2 = (a3 + ax2 −=x3 )dx 2ax − x = y (a3 + ax2 − x3 )dx − a y)dy (2y 2 2 2ax − x x , y − x2 = y x = a ± a −2ax x x = a ± a2 x=a± a2 − y 2 2 +4 )dx a3 x + a2 x2 − x4 )dx (a4 + a3 x + a2 x2(a−4 x y 2 y x = a ± a2 −√ √ x = a ± a2 − √ √ 2 2 dx √ ∓y dy x : =aa2 ± − y 2a2 − y 2 , dx = ∓y dy : == yThen y = 2ax a2 +−xx ax + x2 . set ax + x2 2 − xdy=: y 2 dx2ax = ∓y a2 − y dx = ∓y dy : a2 − y 2 dx = ∓y dy : a2 − y 2 2 and or is reducible to the quadrature of the hyperbola. Rules will be dx = ∓y dy : a − y 2 x2 = a ± a2 − y 2 2 − y x = a ± a (a − x)dx given below, as to how this can be recognized and done. √ = dy (a − x)dx 2 2 There remains yet something else, before we come to 2ax − x2 dx = 2∓y dy√: (a a− x)dx − y = dy (a − x)dx 2 dx = ∓y dy : a − y √ √ =(ady− x) 2ax − x2 2 = dy. the use and application of the integral calculus. To wit, we will 2ax − x 2ax − x2 √ explain another procedure for the formation of integrals, which √ √2ax − The integral of this is = y = 2ax − x2 .(a − x)dx = y = 2ax − condenses the general method by more than a little. Because √ √ (a − x)dx = dy √ be =y= √ 2ax − x2 2 rule can √ henceforth many = applied dy2 √ sometimes, due to the complexity of the given quantity, it is not 2axin−infinitely x2 = y = 2ax −This x = y2ax = −2ax 2 −x x 2 cases, even in those which seem almost hopeless because of their = y = 2ax − x immediately clear, whether it is of a kind which is reducible to 1 the fact that this rule sometimes 1 complexity. Because, aside from one of the cases that we have presented before, and even, whether √ 1 √ makes the2= quantity in question y = 2ax − x2 much briefer, it also offers the it has an integral or not. This procedure, however, reduces the = y = advantage, 2ax − x that it stands immediately before one’s eyes, whether quantity to fewer terms, so that one may find the desired integral the transformed quantity can be integrated. without difficulty. This is done however, by taking the quantity 1 To all these methods of finding integrals one can add the under the root sign as the absolute quantity and setting it equal following, which because of its usefulness and easiness, is almost to some variable, and transforming the quantity to be integrated 1 preferable to all the rest. This method is however 1 only relevant accordingly into another consisting only of terms of the assumed 1 to those quantities which are combined with irrationals. Its whole helping variable. One takes the integral of this quantity, which for application, accordingly, consists of transforming irrational quanthe most part appears much simpler, which can be transformed tities into rational ones, so that the given quantity assumes a comback to the desired integral, by reinserting the value of the as- Δυναμις Vol. 3 No. 3 December 2008 1 The First Integral Calculus 47 Bernoulli pletely rational character, after which, when it is possible, its integral is easy to form. This leads therefore into the Diophantine a3 dx √ problem, which provides excellent assistance on such occasions, x ax − x2 as will become more clearly evident by examples. a3 dx √ For example, let a3 dx x√ ax − x2 √ a3 dx 2 x ax − x2 √ 3 dx ax 3− x x √a ax −√ xa2 dx 3 √ a2 dx 2 − 2x xaxxax x ax − − x√ ax − x2 √ x ax − x2 ax − x2 √ 2the quantity be to be integrated, which is not feasible by the preax − x √ √ ax2 − x2 2 methods, but which can be done in the following manner. x ax −vious xax − 2 a2 x2 √ ax − x 2 2 ax − x2 is an irrational quantity, then to make it rational,ax − x = m2 ax − xSince 2 − x2 must become a square. Let therefore ax − xax a2 x2 ax − x2 = ax − x2 a2 x2 m2 2 2 ax − x2 = 2 x a=xam2 : (a2 + m2 ) m2 ax − x = a2 x22 a2 x2 2 . m 2 − x 2= 2 2 ax − xax=√ x 2 2 a m ax−−xx2m ==a2 m2 : (a2 + m2 ) ax x = am2 : (a2 + m2 ) m x = am2 : (a2 + m2 ) and consequently From this assumption 2 x = am√2ax : (a−2 x+2 m ) 2 dx = 2a32a m3dm dm: (m2 + a2 )2 , = a m : (a2 + m2 )a, 3 dx 2 2 : (a22 + m2 ) √ x2=: (a am √ x = am + m ) 23 = 2 2a3 2dx 2 3 2a3 dm3 : (a m+ m ) − x = ax2m √ aax dx dm therefore whole given 22 2 2 3 quantity 3 2=the 2 am 2 =x √ = 2a3 dm a dx am :+ (am + )x2 ax=−2a √ 2a dx dm a m : (a ) ax − x 3 2 2 2√ √ 2 √ 2 dx2 = 2a2 2m dm22 : (mx √ + a ) m = 2 x ax − x 3 m = 2ax − 2x 2 2a3 dm a m+:m (a2dx )32m dm3m 2 2 axm=: (a )ax ax − xax=−√ 2 2 x ax −√xa2 dx m= 3+=m x − x 2a : (m + a ) 2a a dx dm 2 2 2 3 2m : (a + m ) ax − x2 2+=a2a)√ m2 = dx = 2a m3dm : (m x ax − x2 2 −2a :m 3 22 ax − 2 2x2 m 3 = 2a 2 : (m 2 x , dx m dm + a ) dx = 2a m dm : (m + −2a a )3:m −2a33 : m 3 m dm : (m2 + a2 )2 −2a3 : m −2a :m m= dxa2=x 2a : (a − x) whose integral is easy to find, and indeed is −2a3 : m . m = a2 x : (a − x) 2 x) now m =3 a22x : (aIf−one −2a :m inserts the value m = a x : (a − x) , the m = a x : (a − x) a2 x : (a − x) result is m = a−x m = a2 x : (a − x) − 4a5 − 4a4 x = 2 −2a 5 − 4a4 x 4a5 − 4a4 x x a−x 4a a − x 2 x = −2a4a5 a − x − 4a4 x − 4a5 − 4a4 x = −2a22− a − x x 2 x 5 = −2a − x 4x = −2a x − a−x − 4a 4a = −2ax2 − x a −x x 4a5 −x4a4 x 2 √ x x = −2a a3√dx : x ax− − x2 x x √ 3 3 x+a=y the a33 dx :as x√ axintegral − x22 of a dx : x ax − x2a.3 dx : x√ax − x2 a dx : x ax −Likewise x √ to integrate √ 3 2 √ dx −ax= 3 x+ y3 2 + 2ax a 2 : x ax x dx + a √ a3 dx : x ax − x2 √ 3 2 2 √ 3 dx 3 dx √ x2 + 2ax +xa2 ax3 + 2ax + a dx 3 x2 + 2ax + a2 dx 3 x2 + 2ax + a2x, x=+ya − =y x √ x √ dxx3 x2 + 2ax + a2 3 x 3 2 2 2 dx x =xy +−2ax a + a dx = 3y dy x x2 + 2ax + a2 must become axcube. Let therefore x + a = y 3 . √ 2 2 3 x a+ 2ax 2 2 x22 + 2ax + will a2 x = y 3 − =+3ya2 dy and then , dx x2 + x2 + 2ax + 2ax a2 + a = y x + 2ax + a2 Therefore the whole quantity √ x22 + 2ax + a2 3 x x2 + 2ax + a2 dx = 3y 2√ 3 2 + 2ax + a2 = y dy √ x=y −a 3y 4 dy dx 3 x2 + 2ax + a2 3y 4 dy dx 3 x2 + 2ax + a2 √ . = 3 = 3 2 2 2 2 y x + 2ax y −s x x= y3 − a √ dx + =a3y dy 3y 4 dy dx 3 x2 + 2ax + a2 If one can obtain the√integral of this, then one also has the integral = 3 3 y −s x a2 √ = y2 2 3 of the given quantity. x2 + 2ax +dx 3y 4 dy x + 2ax + a2 = 3 y −s x √ 1 3y 4 dy dx 3 x2 + 2ax + a2 = 3 y −s x 1 1 1 1 1 Δυναμις Vol. 3 No. 3 1 December 2008 48 What’s the Matter with Descartes Vance Exclusive Interview: René Descartes What’s the Matter with Descartes? Timothy Vance T here’s no denying it, folks. No matter how you slice it, there’s just no amount of geometric extension in the world that could bail out Descartes’ utterly bankrupt notion of moving bodies. You may keep trying to extend this matter of his if you like, but it still won’t move. “It’s impenetrable,” the experts may say. I don’t care, my friend. It’ll take a miracle from God to get those bodies of his moving again. Why? Because Descartes’ argument for motion lacks force ... and not just the kind necessary for convincing you he’s right, which, by the way, he would enjoy very much. This may seem a little suprising at first because René, as I like to call him, is a man whose books are still purchased every year by undergraduates enrolled in philosophy coursTim es they can ill afford to take, and shouldn’t. He is a man who is credited with making mathematics “modern” by freeing Geometry from the tyranny of ... well, Geometry; And finally, Descartes is a man whom my loving friend Wilhelm Gottfried Leibniz would go out of his way on just about every occasion to refute. Naturally, such a reputation led me to expect more gravitas from Monsieur Descartes; instead, I found a man led astray (from making any damn sense) by too great a faith in his own genius and inflated sense of self–importance. What follows is a hastily arranged interview I conducted with Monsieur Descartes himself on Sepember 28 of this year: VANCE – Monsieur Descartes, thank you so much for taking the time to be with us today. DESCARTES – It really is my pleasure, and I’m very lucky to have actually made it here this evening. I had some trouble earlier today ... Δυναμις Vol. 3 No. 3 Vance interviews René Descartes V – Trouble with your flight over the Atlantic? D – No, trouble getting out of bed. It’s where I draw the vast majority of my philosophical conclusions, you know; and it’s so very hard to find a nice quiet place these days for meditative contemplation (much more so if you have friends like mine). I’ve also discovered that one’s bed provides the perfect environment for thinking through life’s most challenging questions like “What can I know?” and “How can I know it?” – especially if doe in the late morning and early afternoon when everybody else is at work. V – You’ve been described by leading academics as “one of the most influential thinkers in human history.” That’s quite a remarkable statement! How did you come by all that knowledge which made you so famous? D – That’s a great question Tim, thanks so much for asking me. It’s really simple. “All that I have, up to this moment, ac- December 2008 49 What’s the Matter with Descartes Vance cepted as possessed of the highest truth and certainty, I received either from or through the senses.”1 The idea for it came to me in a dream a few winters back. It was then that I discovered something very important which “got the ball rolling” so to speak– V – What did you discover? V – But that would imply that– D – “[M]otion and rest ... are nothing other than two different modes–”4 V – Of the same thing? D – MYSELF! You know: Cogito ... D – Yes, all corporeal substance. V – Ergo sum? V – Substance made of what? D – Yeah! Isn’t that great? V – Excusez–moi, Monsieur. Would you mind if I were to ask you a few simple questions concerning motion? It appears that my friend Leibniz could not have disagreed more with you on this very subject. He accuses you of preferring applause over certainty, or rather a fan–base over fact; and attributes your failure to properly describe the laws of nature– D – I FAILED? V – Well, before I answer you, René, let me ask: What is motion when it’s properly understood? D – Well, I am most sure of myself when I say “it is the translation of one part of matter, or of one body, from the vicinity of those bodies that are in direct contact with it and are viewed as at rest to the vicinity of others. Where by ‘one body’ or ‘one part of matter’ I understand everything that is transferred at the same time, even if this itself might again consist of many parts which have other motions in themselves. And I say that translation is not the force or action that transfers, as I shall show that this [motion] is always in the mobile, not in the mover.”2 V – Très bien, Monsieur. However, I became a little confused when you assigned the name “motion” to what is in fact only an effect of motion (namely “translation,” or a change from one place to another in a rectilinear fashion I presume), only to separate that effect immediately afterwards from its cause (that is, the “force or action that transfers”). You make little distinction between something which is moved, i.e. the mobile as you say, and that which moves, i.e. the mover. D – “It is not a matter of that action which is understood to be in the mover, or in that which arrests motion, but of translation alone and of the absence of translation, or rest.”3 1 D – Extension which is impenetrable. V – But how are things extended? D – How else are things extended through Euclidean space? By merely adding or subtracting individual points in the case of a line, or by adding and subtracting a series of points in the case of a surface, and so on. V – Euclid tells me a point has no parts. So what part of a point would make it impenetrable? D – I don’t know. V – I think you’re making it a lot harder for yourself by applying the same generic metric of “translation,” or the lack thereof, to both a body in motion and one at rest, and similarly applying it to any moving body regardless of whether it moves or is moved. Once again, why lump the two together? D – “[Because] no more action is required for motion than for rest.” V – But René, if no more action is required for motion than for rest, what has a body formerly at rest ever done in order to move? And if a moving object isn’t doing anything either, who or what has moved it? D – “God.”5 V – I see, très bien! And let me guess: “God is the primary cause of motion and always conserves the same quantity of motion in the universe.” D – “Mais oui.” René Descartes Meditations – No. 1 V – Uh–huh ... Right. 2 René Descartes Principles of Philosophy Part II – ‘On Motion’ Translated by M.S. Mahoney, 1977 4 Mahoney, ibid. 3 5 Mahoney, ibid. Mahoney, ibid. Δυναμις Vol. 3 No. 3 December 2008 50 What’s the Matter with Descartes Vance D – “[Well], it seems clear to me that it is nothing other than God Himself, who in the beginning created matter together with motion and rest and now conserves just as much motion and rest as a whole as He then posited. Now, although this motion in moved matter is nothing other than its mode, nevertheless it has a certain and determinate quantity, which we easily understand to be able to be always the same in the whole universe of things, even though it be changed in its individual parts.”6 V – For example ...? D – “[I]t is evident [...] when one part of matter is moved twice as fast as another, and this second [part of matter] is twice as large as the first, there is as much motion in the smaller as in the larger; and by as much as the motion of one part is made slower, the motion of some other equal to it is made faster. We also understand perfection to be in God, not only that He is immutable in Himself, but that he works in a most constant and immutable way, such that, save those changes that clear experience or divine revelation renders certain and that we believe or perceive to be made without any change in the Creator, we should suppose no other [changes] in His works, lest one then argue an inconstancy in Him. Whence it follows that it is most wholly in accord with reason that we think on this basis alone that God moved the parts of matter in various ways when He first created them and that He now conserves all of this matter clearly in the same way and for the same reason that He formerly created, and that He also conserves the same amount [tantundem] of motion in it always.”7 V – C’est magnifique! Except ... D – Except what? V – Except that doesn’t happen. D – Whatever ... C’est la vie. V – Descartes! D – “I will continue resolutely fixed in this belief, and if indeed by this means it be not in my power to arrive at the knowledge of truth, I shall at least do what is in my power, viz., [suspend my judgment.]” Explain!8 even works if we limit our investigation of motion to the common machines of antiquity, e.g. the lever, windlass, pulley, wedge, and screw – what is often referred to as “mechanics,” or “statics” to be more precise. But, as the great Monsieur Leibniz would point out, in all of those machines “there exists an equilibrium, since the mass of one body is compensated for by the velocity of the other” and that “the nature of the machine here makes the magnitudes of the bodies – assuming they are of the same kind – reciprocally proportional to their velocities, so that the same quantity of motion is produced on either side.”9 Therefore, your law of motion only works by accident. D – It’s no accident. I didn’t look into any other kind of machine! V – You know, René, it wasn’t a pulley that put man on the moon ... D – The moon? V – Anyways, shall I now proceed to show an exception to your rule using a demonstration from Leibniz? D – I suppose if you must. V – Thank you ... First, assume that a body falling from a given height acquires the same force which is necessary to lift it back to its original height if its direction were to allow it, and there was no interference with its motion through friction, air– resistance, and the like. D – As for example with an ideal pendulum? V – Yes, exactly. Second, assume that the same amount of effort is consumed in raising a four–pound body to a height of one foot, as is consumed in raising a one–pound body to a height of four feet. D – I will generously grant Monsieur Leibniz these two assumptions, as will any of my pupils. V – Wonderful! Then a four–pound body falling one foot has acquired the same amount of force as a one–pound body falling four feet. V – Your straightforward example of two bodies which possess the same “quantity of motion” – or what you define as the product of mass times velocity – certainly pleases the senses. It D – Mais oui, for what you say is to me both “clear and distinct.” 6 Mahoney, ibid. 7 Mahoney, ibid. 8 Meditations – “I. Of the Things of Which We May Doubt” 9 Gottfried Leibniz A Brief Demonstration of a Notable Error of Descartes and Others Concerning a Natural Law, Philosophical Papers and Letters Translated by Leroy E. Loemker, Kluwer Academic Publishers, Boston, 1956. Δυναμις Vol. 3 No. 3 December 2008 51 What’s the Matter with Descartes Vance V – Except, the one–pound body acquires only two units of velocity. D – What of it? V – Descartes! Do the math ... shouldn’t it have acquired by your reasoning four units of velocity? Remember how evidently, you implied, a body twice as large as another would travel half as fast, thus always conserving the same quantity of motion?10 D – Aidez–moi! Do I have to show my work? V – Don’t worry about it. I’ll just assume you’ve actually calculated the results of this or any other physical experiment for that matter, ok? D – What a relief! Merci. V – But remember: Galileo’s proposition, and subsequent demonstrations show that a height is proportional to the square of the velocity of an object having fallen from it. Which means that, by your definition of ‘quantity of motion’, were we to construct a machine which would allow us to transfer all of the motion of the four–pound body falling one foot (previously in motion but now at rest) to the one–pound body (previously at rest but now set into motion), we would miraculously discover that such a one–pound body would be able to raise itself to a height of sixteen feet! D – What’s wrong with that? V – Well, earlier you had agreed with Leibniz when he asserted that as much effort was consumed in raising a four– pound body to a height of one foot, as was consumed in raising a one–pound body to height of four feet. Thus, with as much effort as was consumed in raising a one–pound body to a height of sixteen feet, we could have raised the four–pound body to a height of at least four feet. But we previously stated that the four–pound body in question originally fell from a height of one foot (before transferring its motion to the one–pound body), which should have only been enough to raise itself to a similar foot. Where did all that extra motion come from? doesn’t exist. This is because, as Leibniz would say, there would be no reason for it to exist. For if effects could at any time exceed their causes, of what use would those causes be if not for producing all of their subsequent effects! D – How was I supposed to see that?! V – You weren’t! You don’t see causes, you know them by their power. And so the very things by which you correctly understand motion or change (powers), happens to be the name of a new science of motion founded by Leibniz11 in opposition to yours and Newton’s. And so, while you sought to preserve the same quantity of motion, Monsieur Leibniz sought to preserve the same quantity of power, i.e. the equality of cause and effect. D – I see. V – René, all this “seeing” with your eyes got you into this trouble in the first place. You unfortunately assumed that there was nothing to be found in motion, or in corporeal bodies for that matter, which could not be measured through your senses. If indeed there was nothing more to it, then quantity of motion would consist of the only two things one could sense about a moving body, namely mass and velocity ... I mean, have you ever seen with your own two eyes actual squares on a velocity? D – [Non cogito–] <POOF!> [Note: Having over–extended himself at this point in the interview, Descartes was no more.] V – Ergo, it appears you no longer exist, and neither does the credibility of your methods. D – Je ne sais pas. But I do know that such results could be used to produce some pretty interesting machines– V – Don’t even say it! Perpetual mechanical motion 10 If Descartes’ total “quantity of motion” (mass times velocity) is to be preserved in this example, then m1 · v1 = m2 · v2 (here: 4 times 1 = 1 times 4), i.e. a body of four pounds with one unit of velocity would impart four units of velocity to a one pound body. Δυναμις Vol. 3 No. 3 11 i.e. Dynamics December 2008