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Today, a tourist guide to the Old House of Congress in Washington, D.C. could easily make you discover the reason why John Quincy Adams, when he served as a Congressman, was often ``sleeping'' during the sessions that were held in that room. In point of fact, he was not really sleeping; he was actually listening to the secret talks that his political opponents were quietly having at the other end of the room. Often, they were discussing how to thwart Adams's attempts to bring the issue of slavery (through the fight over the right to petition) or of scientific progress (everything from observatories--what Adams called the ``lighthouses of the sky'' to the U.S. Coastal Geodetic Survey headed by Alexander Dallas Bache) before the Congress; sometimes, they were discussing the fact that it might be better for all concerned if Adams weren't around at all. Indeed, Adams reports in his diaries overhearing conversations which were in fact out-and-out death threats against him. The question is, how was he able to hear such whispering conversations from this listening post at a distance of more than 50 feet away?
The answer can easily be found once one understands the basic harmonic proportions that are embodied in the Old House's ceiling. Although very ordinary in appearance, and not at all noticeable to anyone who is unaware of the properties of certain geometric shapes, the ceiling of the Old House held a very dear secret for John Quincy Adams, who had learned the science of optics and acoustics from the experiments that were being conducted at the French Ecole Polytechnique of Gaspard Monge, and Lazare Carnot, and that were later introduced at the West Point Military Academy by the Polytechnique geometer, Claude Crozet, under Superintendent Thayer. Adams himself replicated a number of these experiments with other members of a science study group he had created in Massachusetts in the early part of the 19th century. From that vantage point, John Quincy Adams knew that a very special conception of amplification of light and sound was involved with respect to the geometric shape of the ceiling of the Old House of Congress.
Every sound uttered in the vicinity of one of the two foci of the ellipsoid can be heard very clearly and distinctly at the location of the other focus! A simple look at the geometric property of the ellipse will show the reflexive principle involved here, which is applicable for sound, as well as for light.[fn1]
Let the seat of John Quincy Adams be in the position of the focus A. (See Figure 1).
All of the utterances coming from the leader of the opposition, who is sitting at the focus B, will be heard very clearly at the focus A, even though the distance between the two points is too great for the two people to normally hear one another. How do you explain such an anomaly? What is the principle of reason which underlies such an exceptional phenomenon?
Let me point out, first of all, that the usual geometrical explanation for determining the property of the ellipse is not adequate, nor sufficient, to answer this question. In other words, it is not enough to simply relate to the usual textbook definition of the ellipse as ``The curve with the property that the sum of the distances from two given points to any point on the curve is constant.''[fn2] This simple definition of the ellipse is not appropriate to explain the anomaly.
What is required is a higher conception, a determination which is defined from a higher m-manifold of culture and truthfulness, in which Leonardo da Vinci, for example, had initially developed his investigations in painting touching on questions of application of the reason for the inverse square of distances between the source of light and the illuminated body, and questions of the significance of the catenoid-caustic and related topics in optics.[fn3] Indeed, we must here follow the Leibniz precept whereby there is always more in nature and art than can be determined by geometry.
How, then, can we explain the anomaly from this higher vantage point?
The first thing is to realize that the constancy of the sum of the two radii of an ellipse is merely a footprint, a reflection, which is not determinant enough to explain, by itself, the anomaly which must serve as a sort of crucial physical experiment. However, it is potent enough to lead an inquisitive mind to the discovery of the relationship between the constant equality of the two radii, and the constant equality of angles between the same radii, and a tangent to the ellipse, at their point of intersection. Indeed, this small discovery is enough to suggest the question of the sufficient reason that should bring together the harmonic constancy of lengths between the two foci, and the constancy of angles between the two focal radii, at the point of curvature of the ellipse. Such endeavor of bringing those two different viewpoints together should serve as a metaphor for the act of integrating two different levels of geometry.
But, to do this, an epistemological leap must be made. The common constancy of the sum of the two radii, and of the equality of the two angles, must be discovered by way of passing from a lower to a higher level; in a sense, by adding a new dimension, a new hypothesis. In point of fact, the question is no longer what is the characteristic feature of the curve that we call the ellipse, but rather: What is the property of the generative principle that determines the ellipse?
There are generally three ways of constructing an ellipse; the first is the pipefitter's method, which is obtained by making a diagonal cut through a cylinder or a cone; the second is the gardener's method, which is obtained by stretching a closed string around two fixed points, and the third method is Nature's method, devised by Christiaan Huyghens and Leibniz, and which consists in the evolution of a tightly held string that one can unravel slowly from an evolute curve, and thus, trace another curve called an involute.
I will discard the first method for the time being, simply because it does not give us any immediate information concerning the properties of the two focal radii of the ellipse. The second method is the textbook construction expressed by Figure 1, whereby the sum of the distances from the two foci to any point on the curve is always the same; that is to say, AC + BC = AD + BD. This first property, however, can be related to a second property which can lead indirectly to the discovery of the third method; that is, the fact that two rays projected from the two foci onto any point of the ellipse will form constantly equal angles with a tangent at that piont, such that: [angle sign] = [angle sign]BC(E).
However, if we treat these angles as Leibniz did, that is, as transcendental osculating angles, as opposed to simple tangent angles, we will be able to see how, from that advanced standpoint, the principle of sufficient reason informs the discovery of a new physical principle, a new dimension.
Thus, the question now becomes: How do you find the curve F(F) generating another curve DC, and which has the property of ordering in position, and amplifying, the reflecting self-propagation of incidence and reflective rays in a least action form by directing them from one focus B to another focus A, or vice-versa? (See Figure 2.)
Such a reflective phenomenon of self-propagation, of curving the curvature, should lead us to discover the true reason behind the anomaly we are trying to resolve; but only under the condition that we follow the least action principle of Fermat, Huyghens, and Leibniz, whereby light travels in the least possible time, and along the shortest possible pathway. It is from this principle that can be deduced the constant ratio of equal angles between the radii of an ellipsoid; that is, the angle that an incidence ray forms with the curve is always equal to the angle that the reflexive ray makes at that point with the same curve.
As shown in Figure 2, if you take two infinitely close perpendiculars DF and CF, well ordered in position, their intersection point F will determine the center of an osculating circle, or osculating sphere, whose circumference embraces the inside of an involute curve DC, and whose osculating radii touches the evolute curve F(F) by inverse tangency. This astroid-evolute is the sought-for generative curve F(F), and it represents the measure of curvature of the ellipse-involute GDCE. As one can observe, this curvature of intersections, from which the evolution action proceeds, is what contains most directly the reason for the constancy of the sum of the two raii of an ellipse, as well as the constancy of the equality of angles between the incidence and the reflexive rays coming and going from the two focal radii, and touching every point of that ellipse.
However, even though it is the evolute which is the generative principle of the involute, when they are taken together as one harmonic whole, both the astroid-evolute and the ellipse-involute form a self-evolving, self-propagating process, whose harmonic proportion is such that EO is to BO as (F)G is to (F)O; that is to say, together, they express a minimum-maximum principle where the minimum curvature of the ellipse corresponds to a maximum extension of the osculating radius, while the minimum extension of the osculating radius corresponds to the maximum curvature of the ellipse. Furthrmoe, it is the sum total of all of those light (or sound) carrying osculating radii, FC and FD, which direct light (or sound) from one focus to the other, that is, acting as an unassignable One of the Many, and where each point of each ray is itself proportionately radiating.
John Quincy Adams also applied this sentiment of proportionality to statecraft, as he stated in his first Annual Message to Congress: ``The great object of the institution of civil government is the improvement of the condition of those who are parties to the social compact, and no government ... can accomplish the lawful ends of its institution but in proportionas it improves the conditions of those over whom it is established.''
Thus, the two focal radii of an ellipse are so harmonically conjugated that, because of their constant proportionality with respect to the osculating measure of curvature of the ellipse by the astroid, anything that passes through one focal point must always be reflected as well through the other focal point, and that isochronically, with the least possible resistance, and in the least possible time. If you want to imagine that situation in the simplest ideal way, create the the following mental image: imagine if you were standing at the focal point B, and were to shine a light beam everywhere on the reflecting ellipsoid ceiling of the Old House of Congress, the totality of all of the rays of incidence, coming from that point B, and striking the curvature of the ceiling, as well as the totality of all of the secondary rays coming from each luminous point of each of those rays also touching the ceiling, would all be changed into as many reflexive rays, and would all be directed back down in the region of point A, where the seat of John Quincy Adams is located; and the invisible surface-envelope that would contain, and direct, all of those rays, in ordered position of equal angles, would be an osculating-astroid-evolute-envelope.
This is how John Quincy Adams was able to know in advance, all of the plans of his political enemies; that is, by ``going to sleep,'' head resting sideways on the table, with one ear cocked, at the right place, and at the right time.
Hilbert, Geometry and the Imagination, Chelsea Publishing, New York, 1983.
This amplification exists because of this lawful property of optical mirrors which have the shape of an ellipse; and it is for that reason, that the Latin origin of the English word focus, signifies ``fireplace,'' which in French is translated by ``foyer,'' and in German by the term ``brennpunkt,'' meaning ``burning point.''
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