THE DISCOVERY OF PRINCIPLE OF THE THALES THEOREM

By Pierre Beaudry

"The heavens are filled with gods." -Thales of Miletus

Let me start with a provocative question: what do the height of the Egyptian pyramids and the distance to the Moon have in common? How did Thales of Miletus (600 B.C.) discover a principle by means of which one is able to determine the height of the Egyptian pyramids (without the use of the Pythagorean theorem), as well as the distance to the Moon, given a knowledge of its size? If you can answer those questions, then you have made what Lyndon LaRouche calls a "discovery of principle." That is, what you have discovered is not some THING, some physical reality, which can be acquired through sense perception, and be measured with a ruler, in some way. No! You have discovered an idea, a Platonic idea, in the form of a principle of measure, a principle of congruence, or concordance, between ideas and actions on the universe, which can only be developed by the human mind. Recall in this connection, what Plato said about the teaching of astronomy, and the ordering principle of the heavens in the {Republic}, VII, 529, d,e, and {Laws}, X, 899 b.

What is required for the discovery of such causality is a higher cardinality; and that cardinality is no less than addressing the subjective power of the human mind to become the causal agency, which commands the ordering of the Universal Blazonry of the Heavens. This is the specific task that Thales, Kepler, and Gauss, especially, all three challenged themselves with, each in his own way: to develop a truly scientific notion of universal congruence beyond the grasp of sense perception. This also means, that each of us must muster the courage to do exactly the opposite that the evil Francis Bacon proposed when he said that man should not "give out a dream of his own imagination for a pattern of the world."

But this principle, in one form or another, implies the resolution of Plato's ontological paradox of the One and the Many. This means that whatever the nature of the discovery, it must imply three conditions of axiomatic change: 1) it must involve a multiplicity, a Many of some sort, 2) it must imply an axiomatic break with a previous set of assumptions, 3) it must be determined by a One that bounds the process of discovery from the outside. To put it in a nutshell, you need a Many, a Discontinuity of perception, and a One. From the advanced standpoint of Lyndon LaRouche, those are the three necessary conditions that must make up the Platonic Idea of any discovery of principle.

Thales of Miletus lived 600 years before Christ and was recognized as one of the Seven Sages of Greece during Solon's archonship of Athens. It was Thales who forecast the solar eclipse of May 28, 585 B.C. which put an end to the protracted war between the Lydians and the Medes, and ultimately settled a lasting peace between them. Not only did Thales know when eclipses would occur, but he also knew that the cycles of the Sun, the Earth, and the Moon had to concur in the plane of the ecliptic in order to cause such eclipses. This knowledge was outstanding, since no one at that time understood what eclipses were all about. Furthermore, Thales is reputed to have created the first almanac, giving the solstices, the equinoxes, the phases of the moon, and a long-range calendar with eclipse and weather prediction. He invented a means of steering the course of ships on the sea, and and a way to determine their distances from shore by sighting them from a tower.

Thales developed a very elegant and simple theorem which is so elementary that its simple beauty and generality leave blind -- mentally blind that is -- those who don't investigate its purpose. His idea resembles a changing geometrical figure, which is never the same: sometimes it is a triangle, sometimes a line, sometimes overlapping shadows and, more often than not, it also takes the form of an array of spheres circumscribed by cones. Like the principle of water, which he took as the basis of his philosophy, it is forever changing; however, in all cases and everywhere, it remains the same. As his follower Heraclitus said: "You never bathe in the same river twice," and yet, it is always the same river. Underlying all of the changes, there remains a mental congruence which is not apprehensible by sense perception, a principle of similarity and proportion, which enabled him to establish certain astronomical measurements involving the determinations of lunar and solar eclipses. The Thales Theorem can be stated as follows:

ANY LINE PARALLEL TO ONE OF THE SIDES OF A TRIANGLE WILL DIVIDE THE TWO OTHER SIDES IN PROPORTIONAL SEGMENTS, AND WILL DETERMINE ANOTHER TRIANGLE SIMILAR TO THE FIRST.

First draw a triangle, preferably an elongated irregular triangle, and draw across it, near the apex, a line parallel to the shorter side. This will immediately show two similar triangles.

Second, consider that since the similarity of triangles is derived from the similarity of cones with the same apex angle, the Thales Theorem can be easily conceived as the extension of a conic projection. Indeed, any triangle can be conceived as a perpendicular cut down the axis of a cone.

This is the method that the Thales Theorem implies for the purpose of determining the Moon's distance from the Earth, given its size. However, there exist no claim that Thales did make such a discovery. We are merely stating that the method of the Thales Theorem is sufficient to make such discoveries.

If you know that the diameter of the Moon is approximately 2160 miles, the discovery of the distance to the Moon requires an experiment which is relatively simple. It involves a conical projection and implies an important discovery which is fundamental in projective geometry: the discovery of self-similarity.

The idea is to create an imaginary cone which begins with your eye at one end and circumscribes the Moon at the other end. You must then conceive that any portion of that cone, starting from your eye, and cut anywhere between you and the Moon, will be self-similar to the entire Moon-cone, and will reflect the same proportionality everywhere. As perceived from your eye, that circular cut will always be the same size; which is another way of saying that your eye could never give you any knowledge of the size of the Moon.

It is fundamental to grasp this concept because, without this projective property of self-similarity, you cannot even begin to think how to solve the problem of measuring the distance to the Moon, unless you use another method. This is the bounding principle, so to speak, of the whole discovery. It is important that you illustrate this for yourself with an elongated triangle of the Thales Theorem.

This concept is absolutely necessary, because it means that if you can construct a small portion of that cone (a similar triangle), you will be able to construct the whole cone. An answer will be given in Tuesday's briefing.

THE DISTANCES TO THE MOON AND SUN

All you need, to make this exciting discovery, is a full Moon, a small piece of cardboard, a long stick, and the Thales Theorem in the back of your mind.

Cut a square of 1/2 inch in the center of a small piece of cardboard and attach it to the end of a long stick. Line up the apparatus, so that the Moon is circumscribed perfectly by the square, and mark the distance to your eye on the stick. This distance should be approximately 55.5 inches.

Since the ratio of the distance of the "perceived" image of the Moon to your eye, and the 1/2 inch square, is 55.5/.5 = 111, then, by self-similarity, the diameter of the Moon, is 2160 miles x 111 = 240,000 miles, which is the approximate distance between the Earth and the Moon.

The exciting idea in all of this is the realization that knowledge simply cannot be acquired through sense perception. Knowledge can only be acquired through the apprehension of congruence of physical-space-time, which requires a proper education of one's own visual imagination with respect to the axiomatics of the creative process of discovery.

Following the same kind of discovery of principle that we have already employed, with the Thales Theorem, we can also determine approximately the proportion between the distance of the Sun, and its diameter. However, this experiment calls for caution with respect to an observation of the Sun: NEVER LOOK AT THE SUN DIRECTLY.

So, as opposed to the direct method of projection that we used to discover the distance to the Moon, you may use a projective derivation of the Thales Theorem, an indirect method, which is called the HOMOTHETIC PROJECTION (from "homos," similar, and "thesis," position [Chasles]).

A HOMOTHETIC PROJECTION is the projective property of two figures whereby all the points of one figure can be projected onto the other, one on one, by means of straight lines, the which all pass through a homothetic center between them, a point of infinite similarity. As a result, all homothetic lines (or curves) are parallel, all homothetic angles are self-similar, and the two homothetic figures are proportional to one another. Again, this discovery implies the Thales principle of self-similarity and proportionality.

Now, imagine that the homothetic figure of a square ABCD, that you draw on a piece of cardboard, is another square A'B'C'D' which hypothetically circumscribes the Sun. Take a long wooden stick, and tape a piece of cardboard to its end, showing the drawing of a square whose side is 1/4 of an inch. Slide another piece of cardboard, with a pinhole in it (which is the homothetic center), along the stick at the level of your eye, at about 27 inches from the square at the end of the stick. Line up the whole thing toward the ground, in such a way that the sun is projected from behind you through the pinhole, and is made to fit perfectly into the 1/4 inch square. You have now discovered the ratio of the distance to the diameter of the Sun, 27 /.25 = 108. If we already know the Sun's diameter to be 866,000 miles, this corresponds in miles to the approximate average distances of 93,600,00 miles. (93,600,000 / 866,000 = 108.)

This is yet another demonstration of the very powerful Thales Theorem, which has been the germ, the "Motiv," for many other discoveries in the history of astronomy and geometry, from Aristarchus, to Pappus of Alexandria, Pascal, Monge, Poncelet, and Steiner, to identify but a few of the main successors of Thales.

How did Thales use this principle to measure the height of the Egyptian pyramids? The answer will appear in the final installment.

PROPORTIONS BETWEEN SINGULARITIES OF THE SAME TYPE -

In the three previous discoveries, the reader will have noticed that the singularities are of the same type, because they all contain an explicit discovery of a One with reference to a Many, in the form of self-similarity and proportionality, and they explicitly contain a discontinuity, an explicit rejection of sense perception as a source of knowledge. These are the conditions that Lyndon LaRouche has established to "test," so to speak, the validity of a hypothesis; the resolution of the ontological paradox of the One and the Many is the litmus test that bears upon all of the crucial discoveries in history.

The very joy and excitement of such discoveries are that they involve a congruence of higher proportionality, which is properly located in the subjective will of the creative individual. Indeed, in that location, the distance between the perceptible and the intelligible, is proportional to the willful enthusiasm of generosity, of Agape, which is the generative principle for such discoveries. It is this higher hypothesis level of discovery, which will provide more truth in science than any particular "objective" measure of a particular body. There is no such thing as an "objective measure," no such thing as a true measure of anything for that matter; there exists only true congruence of proportionality and self-similarity in the image of God. That is how the Greeks measured the invisible, and that is what the validity of these Thales discoveries is based on.

The point to be made with Thales is that man is capable of discovering that he is not simply a speck of star-dust in the universe; he is capable of mastering the universe, as a whole, and putting it under his command, for the singular purpose of increasing the powers of reason of the next generations of human beings. Such is the purpose of the Thales' Theorem: give mankind the sentiment of elevation and scope, to become more and more in the image of God [Imago Dei] and participating in God [Capax Dei]. That is how the discovery of principle of the Thales Theorem is achieved. Man is larger than nature; he can stand outside of the event of his own discovery, as an outside observer looking in from a distance, into the particular discovery that he has made, and compare it to other discoveries. That is the standpoint of the higher hypothesis, looking over its shoulder at the different hypotheses. That is how the congruence of self- similarity of proportion is understood as a type of higher hypothesis.

Man is the Measure

Thales discovered that if he stood next to an Egyptian Pyramid, at precisely the time of day when he cast a shadow equal to his own height, he could use that measure of "one Thales" to determine the height of the pyramid. Indeed, to discover the height of the pyramid, all you need do, is measure its shadow, at the precise time of day when your own shadow is exactly equal to your height. The length of that shadow is its height. Simple, isn't it? This is the discovery of the principle of measuring, the subjective principle of the ruler. You see by this that the ruler is not an "objective" instrument of measure. And again, the discovery is made, not by the senses, but by a proportion of self-similarity.

Another extraordinary discovery that Thales made was his determination of the distance of a ship at sea. He hypothesized that you could tell such a distance by observing the ship from the top of a tower, using only the congruence of two self- similar right-angled triangles, one small and the other large. By standing at the top of the tower with only two legs of a small right-angled triangle, the vertical and the horizontal sides, all you need to do is line up the open side of the triangle from your eye to the ship, and mark the projected position of this hypotenuse on the base of the small triangle.

The proportions of this small triangle will be similar to those of the larger triangle formed by 1) the height of the observation tower above sea-level, 2) the distance from the foot of the tower to the ship, and 3) the distance from the ship to your eye. Once you are able to determine the ratio of the small triangle, you are able to determine the distance to the ship.

For instance, if you could stick a long measuring-rod horizontally out your window at the top of the tower, exactly one foot below your eye, you could line up a distant ship with a mark on the rod, by sighting the ship while looking down onto it past the rod. Let us say the ship lines up with the ten-foot mark, ten feet out the window. Now, the height of the small triangle is one foot, and its base is ten feet. If we know that our tower window is 500 feet above sea level, then by similarity, the ship is 10 X 500, or 5,000 feet away. Take the ratio of the two right-angled legs of the small triangle and multiply it by the height of your observation level from the surface of the sea. That will give you the distance to the ship.

This singularity is of the same type as that of the discovery of the height of the pyramid, and of the diameter of the Moon. These are ideas of self-similarity and proportionality of singularities belonging to the same type, such that the shadow of Thales is to himself as the shadow of the pyramid is also to itself. The higher proportionality is the higher congruence that exists between perception and the higher principles of self-government of the soul.

The following are only a few of the discoveries that were inspired by the Thales Theorem, before the Christian Era: a series of discoveries which provide macrophysical readings of astrophysical phenomena.

1) Eratosthenes (276-194 BC) discovered the circumference of the Earth, by applying the principle of similarity and proportionality, to the ratio of a shadow on a sundial, and the distance along the meridian between Syene and Alexandria. At noon on the longest day of the year, a radius-stick set in a hemispherical sundial at Syene (Aswan) cast no shadow, while a parallel stick at Alexandria cast a shadow 1/50 of a great circle (i.e., about seven degrees). Eratosthenes knew that Syene and Alexandria were on the same meridian, and concluded from the above measurement, that their distance, which he knew, was 1/50 of the length of the Great Circle, around the Earth, on which they lay.

2) Aristarchus (310-230 BC) discovered the distance of the Sun and the Moon, initially by applying proportionality and similarity to the movement which the Moon makes during one hour, in its circling of the Earth. He observed the Moon to move one Moon-diameter in one hour, and concluded that the circle of the Moon's orbit is 720 Moon-diameters, since the Moon's period is one month, or about 720 hours. He confirmed the same thing, by measuring the visual size of the Moon at one-half degree, or 1/720 of a 360-degree circle.

3) Hipparchus, born around 190 BC, discovered that, during an eclipse of the Moon, if the Moon takes one hour to enter the Earth's shadow, and 2.5 hours to completely cross it, this gives a width of the shadow of about 2.5 Moon diameters. By adding the value of one more Moon diameter, to account for the tapering of the conical shadow at the distance of the Moon's orbit, Hipparchus discovered that the Earth's diameter is about 3.5 Moon diameters. The correction of one Moon-diameter for the tapering of the shadow, is based on the similarity between the Moon's umbra (conical shadow) and that of the Earth, coupled with the observation, during eclipses of the Sun, that the length of the Moon's umbra is about equal to the Moon's distance from the Earth.

It is the ordering principle of such subjective discoveries, which determines the movements of the stars in {imago Dei,} making man truly in the image of God; a characteristic also applicable to other fields of knowledge, and especially to music.

It is the effective historical-scientific impact of such a discovery of principle which, yesterday, today, and tomorrow, has the effect of increasing the power of mankind over the universe, increasing the relative population density of our planet, as Lyndon LaRouche has shown, and forcing this universe to obey mankind. Thus, as Plato has indicated in his dialogues, the soul regulates the movements of the stars, through this superior subjective congruence of the creative process; such that the proportional distance between perception and reason, this higher divine proportion, becomes a transfinite more real than the physical measures of the Moon and the Earth themselves. So much for astrologers.