Here is an unexpected philosophical solution to the central mathematical riddle of the ancient world: Is there a way to express the transcendental - Pi - relationship between the radius and circumference of a circle ("squaring the circle") using only a straight-edge and a compass? Mathematicians are certain this is not possible, because they've proven there can't be an exact geometric construction for any transcendental number.
Scholars are not certain how the riddle of squaring the circle was understood in the ancient world. Most scholars, who read the riddle in an exclusively mathematical context, believe that the Greek mathematicians did not know that the problem wasn't possible to solve exactly. These scholars believe that ancient mathematicians were speaking plainly about a real problem in geometry whose solution they were actually seeking.
Other scholars are not so sure. They believe that the riddle of squaring the circle was a way of teaching about important philosophical concepts that were based on geometric thinking. For these scholars the riddle is not seeking an exact geometric construction for its resolution. Unfortunately, even scholars open to a philosophical solution have not been able to propose one. [For more information on how the ancient world may have viewed this problem, see Background Information on Quadrature.]
Tenen reminds us that the letters of the traditional western alphabets, Hebrew, Greek, and Arabic, are claimed to be sacred, and that name implies an explicit way in which these alphabets are also, like the Pi relationship, transcendental.
Draw a Circular Ring with a compass and extend a Tangent Line from it with a straight edge. Here is the result:
The construction turns the Circular Ring and Tangent Line into a model human hand.
We express our will to others by both gestures and words. When we wear this transcendental hand and make meaningful gestures, we see outlines of the model hand that match both the shapes and the meanings of the names of the Hebrew, Greek, and Arabic letters. For example, pointing to one's mouth displays the letter Pe, which means "mouth." Because root meanings can be built from these universally recognized letter-gestures, a naive person can read the meanings of Hebrew words spelled out in gestures of this special hand. Thus this natural hand-gesture language, based on a philosophical interpretation of the ancient circle-squaring riddle, also explains biblical claims of a universal language lost at the Tower of Babel.
Taken together, Tenen believes this demonstrates that the Western sacred alphabets are the desired philosophical solution to the ancient circle-squaring riddle. Posing the riddle of squaring the circle resolves the riddle of the Bible's lost universal language.
SqCirc B2 21jan9 Graphics ©'99 S. Tenen / MERU
From Isaac Newton: The Last Sorcerer ©1967 Michael White. Reading, Mass.: Perseus Books, p. 89
A thought experiment familiar to natural philosophers was that of the stone on a string. This may be visualized by imagining a stone attached to a string being whirled around the experimenter's head. In this model, one force draws the stone towards the centre of the circular path and another pulls the stone away. The Dutchman Christiaan Huygens called the first of these forces the centripetal force and the other the centrifugal force. The stone continues to travel in a circle around the experimenter's head because the two forces cancel out. If the string is cut the stone will fly off in a straight line at a tangent to the circle.
Using this as a basis, Newton created a thought experiment to determine a way of calculating the outward or centrifugal force experienced by an object travelling in circular motion. To begin with, he imagined a ball travelling along the four sides of a square inscribed in a circle.
From "Torah Metaphysics versus Newtonian Empiricism," ©Rabbi Dr. Shimon Dovid Cowen, article published in the journal B'Or Ha'Torah: Science, Art, and Modern Life in the Light of the Torah, Number 11 - English 5759/1999, pp. 107-108. [Note: Italics are sic; boldface is added.]
Part Three: Torah Cosmology and the Metaphysics of Newtonian Science
The Physical World as an Analogue of a Divine Metaphysic
Hasidism and Kabbala explain that the concept of enlivening ex nihilo all entities in Creation, from the most sophisticated to the simplest object, is associated with a transcendent level of G-dliness. This level of G-dliness is called sovev kol almin (enveloping all worlds), which also pervades all levels and aspects-'worlds'-of creation equally. The metaphor used for transcendent encompassing G-dliness is that of a circle, which has no beginning and no end, symbolizing its relevance to all within it and its indifference to the distinctions of the entities encompassed by it. The concept of encompassing G-dliness in its pristine sense is comprehended by the metaphor of the great circle, which encompasses all levels of Creation.26
The graded descent of worlds, the realm of immanent G-dliness, termed memaleh kol almin(filling all worlds), within this great circle, on the other hand, is comprehended metaphorically as a downward directed 'line.' It measures out an ordered and differentiated Creation, in which 'upper' and 'lower' are significant distinctions. The 'line'-representing the overall descent of immanent spiritual levels of Creation-however, also incorporates within it the transcendent aspect of G-dliness. At a given juncture it forms a lesser circle27 and then descends as a line, forming a circle and then again proceeds as a downward line. Thus, although the circles found in the line are themselves stages in the descent of seder hishtalshalut, they nevertheless derive from and have something of the encompassing character of the great circle.28
All the foregoing, as mentioned above, is a metaphor describing a spiritual realm. It is explained, however, in discourses29 of the first Rebbe of Habad, Rabbi Schneur Zalman of Liadi, that the physical heavens (the nine spheres of physical Creation) model the nine circles of their spiritual essence in the seder hishtalshalut in which they are enlivened.
26 See Schneur Zalman of Liadi, Likkutey Torah, Korah 52a-d.
27 See Dov Ber, Shaat Ha'Yihud, chapters 16 and 17.
28 Tsemah Tsedek, Mitsvat tsitsit, chapter 2 in Derekh Mitsvotekha.
29 See Sefer Maamarim, (5562) pages 475-479.
Kabbala was a species of symbolic writing among the initiated, setting forth the secret teachings of the Bible; and the key of Kabbala is thought to be the geometrical relation of the area of the circle inscribed in the square, or of the cube to the sphere, giving rise to the relation of diameter to circumference of a circle, with the numerical value of this relation expressed in integrals. The relation of diameter to circum-ference being a supreme one connected with the god-names Elo[-]him and Jehovah (which terms are expressions numerically of these relations, respectively - the first being of circumference, the latter of diameter), embraces all other subordinations under it.
From: A History of Greek Mathematics,
Vol. 1, From Thales to Euclid, by Sir Thomas Heath, 1921.
Reprinted by Dover Publications, 1981, ISBN 0-486-24073.
The Squaring of the Circle.
There is presumably no problem which has exercised such a fascination throughout the ages as that of rectifying or squaring the circle; and it is a curious fact that its attraction has been no less (perhaps even greater) for the non-mathematician than for the mathematician. It was naturally the kind of problem which the Greeks, of all people, would take up with zest the moment that its difficulty was realized. The first name connected with the problem is Anaxagoras, who is said to have occupied himself with it when in prison.1 The Pythagoreans claimed that it was solved in their school, 'as is clear from the demonstrations of Sextus the Pythagorean, who got his method of demonstration from early tradition'2; but Sextus, or rather Sextius, lived in the reign of Augustus or Tiberius, and, for the usual reasons, no value can be attached to the statement.
The first serious attempts to solve the problem belong to the second half of the fifth century B.C.E. A passage of Aristophanes's Birds is quoted as evidence of the popularity of the problem at the time (414 B.C.E.) of its first representation. Aristophanes introduces Meton, the astronomer and discoverer of the Metonic cycle of 19 years, who brings with him a ruler and compasses, and makes a certain construction 'in order that your circle may become square'.3
1 Plutarch, De exil. 17, p.607 F.
2 Iambi. ap. Simpl. in Categ., p.192, 16-19 K., 64 b 11 Brandis.
3 Aristophanes, Birds 1005.
We must conclude that the so-called Arabic numerals and the alphabet originated from the ten digits and the zero, or rather from two symbols, 1 0, the stroke and the circle. L. D. Nelme35, in his essay on the origin of letters, shows that all elementary characters, or letters, derive their forms from the line and the circle. As I understand the Sefer Yetzirah, it also holds that all written characters originated from a line and a circle, but from a line that was originally a symbol for unity, and a circle that was originally the symbol for zero. Similarly, all cuneiform characters originated from two symbols - those for one and ten. L.L. Conant) says: "Two centuries ago the distinguish-ed philosopher and mathematician Leibnitz proposed a binary system of numeration, the only symbols needed in such a system ,would be 0 and | . . . Leibnitz found in the represen-tation of all numbers by means of two digits 0 and | a fitting symbolization of the Creation out of chaos or nothing, of the Universe by the power of the Deity." We have seen that not only a binary system of numeration, but even the decimal system may be expressed by a stroke and a zero. Moreover, it has been pointed out that the alphabet and the so-called Arabic numerals originated from these two symbols. There-fore, the author of the Sefer Yetzirah may have meant by two --, with which God created void and chaos, a digit and a zero; for as the ten digits may be expressed by nine digits and a zero, so may two digits. be represented by a digit and a zero. Thus, the Sefer Yetzirah may have believed two di-gits, 0 and |, a fitting symbolization of the creation, out of chaos or nothing, of the universe, by the power of the Deity.35Comp. "An Essay towards an Investigation or the Origin and Elements of Language and Letters" by L D. Nelme, London 1762. On page 16 we read as follows: "All his (God's) creation, and every minutest part thereof, participates of two most essential forms; the line I the symbol of the altitude, and the circle 0 the symbol of the horizon. These symbols contain in them the first elements, the forms of all crea-ted nature. There doth not exist in the whole creation any being, or thing, that doth not partake of the first principles; nor can the human mind conceive of any existence, without ideas that include these first elements; which are not only forms essential to all matter but also to every idea of matter that arises in the human mind: they contain in them the elements of every art, and of every science known to man; and they are the radix of letters also, which we have already considered as symbols expressive of ideas."
The "one" in the Pythagorean dualism is the symbol, I. Contrary to the prevailing opinion, I believe that the Pythagoreans regarded the Zero, 0, as the second element which was called the infinite, indeterminate duality, infinite binary, etc. In a binary system of notation the Zero is the second Symbol. We know now that even the decimal system of notation originated from the two symbols the one, I, and the Zero, 0. This is in perfect harmony with the Pythagorean formula that all numbers originated from two elements, the limited (the one, 1,) and the unlimited (the Zero, 0). Therefore, all things according to the Pythagoreans originated from two elements One, 1, and Zero, 0.4 Since One, 1 is the finite, the Zero, 0, is the true infinite of the Pythagoreans. The One, 1 was considered the Good, for it represents that which exists, but the Zero, 0 was called the evil, for it represents non-existence.4The Chinese Philosophers even actually said that the circle 0 and the line - are the first elements from which all writing and everything originated. (Thimus Harmonikale Symbolik Koeln 1876, vol. 1, pp. 79-83). By the "bounded line" and "unbounded line" from which according to the Pythagoreans everything originated (Diels H. Die Fragmente der Volsokratiker p. 250) they surely meant the line and the circle the symbols for one and zero.
It was the view of Hermippus that mystical numerology originated with the Jews from which Pythagoras copied it. Origen writes: It is said, moreover, that Hermippus has recorded in his first book, On Lawgivers, that it was from the Jewish people that Pythagoras derived the philosophy which he introduced among the Greeks.17
17Against Celsus, Book I, Chap XV, Alexander Roberts and James Donaldson, eds., The Ante-Nicene Fathers, p. 402
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