HOW THE GREEKS MEASURED THE INVISIBLE.pdf

8/11/2019 HOW THE GREEKS MEASURED THE INVISIBLE.pdf

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HOW THE GREEKS MEASURED THE INVISIBLE, Part 1

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 senseperception, 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 theUniversal 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) itmust 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.


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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 theplane 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 whodon'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.


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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, thediscovery 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 couldnever 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 thewhole 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 itto 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


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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 propereducation 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 thedistance 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 impliesthe 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 fitperfectly 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.)


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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 andproportionality, 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 ofgenerosity, 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 theuniverse, 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


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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 ismade, 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 downonto 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


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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 shadowof 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 shadowon 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 theMoon 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 thetapering 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.


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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 principlewhich, 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.

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HOW THE GREEKS MEASURED THE INVISIBLE.pdf