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Delian Problem

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  • mikebispham@aol.com
    Does he mention the Delian Problem, that of doubling the cube Dominick? I ve pasted an (pretty poor) outline below Mike PS I don t get Livio Stecchini. He
    Message 1 of 10 , Oct 21, 2006
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      Does he mention the Delian Problem, that of doubling the cube Dominick?
       
      I've pasted an (pretty poor) outline below
       
      Mike
       
      PS I don't get Livio Stecchini.  He can obviously do sums and has great ideas, but his work - at least that I've seen - is so sloppy - mistakes everywhere, no references, castles of speculation built on sand...  Does anyone agree?
       
       
       
       
      In a message dated 21/10/06 03:42:28 GMT Daylight Time, scorpio_eagle2002@... writes:
      Mike, et al:

      Still, IMHO, the best work ever done on the subject of the so-called
      Cheops Pyramid is embodied in "Secrets of the Great Pyramid" by
      Peter Tompkins, which contains a fascinating appendix on ancient
      weights & measuring systems by Livio Stecchini (and Stecchini's
      conclusions therein re: Akhenaten and his "religious revolution"
      should make all sit up and take notice!)
       Geometry > Solid Geometry > Polyhedra > Cubes
      Geometry > Geometric Construction
       

      Cube Duplication
       
       
       
      Cube duplication, also called the Delian problem, is one of the geometric problems of antiquity which asks, given the length of an edge of a cube, that a second cube be constructed having double the volume of the first. The only tools allowed for the construction are the classic (unmarked) straightedge and compass.
       
      The problem appears in a Greek legend which tells how the Athenians, suffering under a plague, sought guidance from the Oracle at Delos as to how the gods could be appeased and the plague ended. The Oracle advised doubling the size of the altar to the god Apollo. The Athenians therefore built a new alter twice as big as the original in each direction and, like the original, cubical in shape (Wells, 1986, p. 33). However, as the Oracle (notorious for ambiguity and double-speaking in his prophecies) had advised doubling the size (i.e., volume), not linear dimension (i.e., scale), the new altar was actually eight times as big as the old one. As a result, the gods remained unappeased and the plague continued to spread unabated. The reasons for the dissatisfaction of the gods under these circumstances is not entirely clear, especially since eight times the volume of original altar was a factor of four greater than actually requesting. It can therefore only be assumed that Greek gods were unusually ticklish on the subject of "altar"-ations being performed to their exact specifications.
       
      Under these restrictions, the problem cannot be solved because the Delian constant  (the required ratio of sides of the original cube and that to be constructed) is not a Euclidean number. However, the impossibility of the construction required nearly 2000 years, with the first proof constructed by Descartes in 1637. The problem can be solved, however, using a Neusis construction.
       
       

      Equally informative and packed with excellent material are his
      companion volumes:

      Mysteries of the Mexican Pyramids

      and

      The Magic of Obelisks

      Respectfully,
      Dominick
       
    • Chris
      I didn t find any reference by him. What would you like to know? An excellent starting point is:
      Message 2 of 10 , Oct 21, 2006
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        I didn't find any reference by him.

        What would you like to know?

        An excellent starting point is:

        http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Doubling_the_cube.html

        If link doesn't work it, try google's cache:

        http://72.14.205.104/search?q=cache:57LLG2iNNSwJ:www-groups.dcs.st-and.ac.uk/
        ~history/HistTopics/Doubling_the_cube.html+doubling+the+cube
        +math&hl=en&ct=clnk&cd=1

        Only story it does not relate is that this was not the only one: also story of doubling the
        king's throne (cube).

        -Chris

        --- In sacredlandscapelist@yahoogroups.com, mikebispham@... wrote:
        >
        >
        > Does he mention the Delian Problem, that of doubling the cube Dominick?
        >
        > I've pasted an (pretty poor) outline below
        >
        > Mike
        >
        > PS I don't get Livio Stecchini. He can obviously do sums and has great
        > ideas, but his work - at least that I've seen - is so sloppy - mistakes
        > everywhere, no references, castles of speculation built on sand... Does anyone agree?
        >
        >
        >
        >
        > In a message dated 21/10/06 03:42:28 GMT Daylight Time,
        > scorpio_eagle2002@... writes:
        >
        > Mike, et al:
        >
        > Still, IMHO, the best work ever done on the subject of the so-called
        > Cheops Pyramid is embodied in "Secrets of the Great Pyramid" by
        > Peter Tompkins, which contains a fascinating appendix on ancient
        > weights & measuring systems by Livio Stecchini (and Stecchini's
        > conclusions therein re: Akhenaten and his "religious revolution"
        > should make all sit up and take notice!)
        >
        >
        >
        > Geometry > Solid Geometry > Polyhedra > Cubes
        > Geometry > Geometric Construction
        >
        >
        > Cube Duplication
        >
        >
        >
        > Cube duplication, also called the Delian problem, is one of the geometric
        > problems of antiquity which asks, given the length of an edge of a cube, that a
        > second cube be constructed having double the volume of the first. The only
        > tools allowed for the construction are the classic (unmarked) straightedge and
        > compass.
        >
        > The problem appears in a Greek legend which tells how the Athenians,
        > suffering under a plague, sought guidance from the Oracle at Delos as to how the
        > gods could be appeased and the plague ended. The Oracle advised doubling the
        > size of the altar to the god Apollo. The Athenians therefore built a new alter
        > twice as big as the original in each direction and, like the original, cubical
        > in shape (Wells, 1986, p. 33). However, as the Oracle (notorious for
        > ambiguity and double-speaking in his prophecies) had advised doubling the size
        > (i.e., volume), not linear dimension (i.e., scale), the new altar was actually
        > eight times as big as the old one. As a result, the gods remained unappeased and
        > the plague continued to spread unabated. The reasons for the dissatisfaction
        > of the gods under these circumstances is not entirely clear, especially
        > since eight times the volume of original altar was a factor of four greater than
        > actually requesting. It can therefore only be assumed that Greek gods were
        > unusually ticklish on the subject of "altar"-ations being performed to their
        > exact specifications.
        >
        > Under these restrictions, the problem cannot be solved because the Delian
        > constant (the required ratio of sides of the original cube and that to be
        > constructed) is not a Euclidean number. However, the impossibility of the
        > construction required nearly 2000 years, with the first proof constructed by
        > Descartes in 1637. The problem can be solved, however, using a Neusis construction.
        >
        >
        >
        > Equally informative and packed with excellent material are his
        > companion volumes:
        >
        > Mysteries of the Mexican Pyramids
        >
        > and
        >
        > The Magic of Obelisks
        >
        > Respectfully,
        > Dominick
        >
      • jjdepompeo
        ... At the end of this post I touch upon a possible variant of the Delian problem in relationship to the King Chamber ... Here the Delian constant is not the
        Message 3 of 10 , Oct 28, 2006
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          --- In sacredlandscapelist Dominick wrote:

          > Mike, et al:
          >
          > Still, IMHO, the best work ever done on the subject of the so-
          > called Cheops Pyramid is embodied in "Secrets of the Great
          > Pyramid" by Peter Tompkins, which contains a fascinating appendix
          > on ancient weights & measuring systems by Livio Stecchini (and
          > Stecchini's conclusions therein re: Akhenaten and his "religious
          > revolution" should make all sit up and take notice!)

          --- In sacredlandscapelist Mike wrote:

          > Does he mention the Delian Problem, that of doubling the cube
          > Dominick?
          >
          > I've pasted an (pretty poor) outline below

          At the end of this post I touch upon a possible variant of the
          Delian problem in relationship to the King Chamber

          > Under these restrictions, the problem cannot be solved because
          > the Delian constant (the required ratio of sides of the original
          > cube and that to be constructed) is not a Euclidean number.
          > However, the impossibility of the construction required nearly
          > 2000 years, with the first proof constructed by Descartes in
          > 1637. The problem can be solved, however, using a Neusis
          > construction.

          Here the Delian constant is not the cube root of 2 or 2^(1/3) -
          the inverse ratio of the edge of a cube to the edge of the cube
          double its volume, it is the cube root of the square of 2 or 2^(2/3)
          - the inverse ratio of the area of the face of a cube to the area of
          the face of the cube double its volume. Hence, the area of the
          square which equals 440 square cubits to the area of the square
          which equals 280 square cubits equals 1.57142 = 22 / 14 or around
          one half Pi which is very roughly equal to 2 ^ ( 2 / 3 ) the Delian
          Constant. The square root of 2 is more directly incorporated in
          the monument since the area of the horizontal cross section of the
          pyramid at the height of the floor of the king chamber equals half
          the area of the square base of the pyramid, and the distance from
          the apex of the monument to the level of the floor of the king cham-
          ber is 198 cubits, so that the ratio of the height of the monument
          to the distance from the height to the floor of the king chamber
          equals 280 / 198 = 1.41414... ~ 1.41421... = Square Root of 2. And
          of course it is one of those interesting things that if your divide
          280 + 280 / 198 king chamber cubits into the mean volumetric radius
          of the earth you get 43200.1 - the number of seconds in one half day
          accurate to .1 seconds.

          6370973.27862 /.5240528 cubit in meters /(280+280/198) = 43200.1

          http://www.jqjacobs.net/astro/cosmo.html

          > PS I don't get Livio Stecchini. He can obviously do sums and has
          > great ideas, but his work - at least that I've seen - is so
          > sloppy - mistakes everywhere, no references, castles of
          > speculation built on sand... Does anyone agree?


          I would not be quite so severe in judgment Mike. At times it does
          appear that he reasons from an a priori conviction that the ancient
          Egyptians, and the ancient world in general, knew the dimensions of
          the Earth, including its flattening, to a high degree of accuracy,
          and incorporated that knowledge into the lengths of their basic
          units of measure, and in the case of the ancient Egyptians, into the
          dimensions of their greatest monument. Stecchini is not unique. It
          is said that Newton thought the ancient Egyptians knew the dimensions
          of the earth, and asked the astronomer John Greaves to obtain the
          length of the cubit of the Great Pyramid so that Newton might
          evaluate his theory of gravitation using the data of the dimensions
          of the earth he thought in that cubit. I think I remember that
          Greaves even included the essay of Newton on the cubit of the He-
          brews and the cubit of Memphis as an appendix to his work Pyramido-
          graphia. Newton, however, was more a bibliomaniac than a pyramido-
          maniac, and in that work, more concerned with determining the cubit
          of the temple in Jerusalem. For a brief, and misleading overview see

          http://en.wikipedia.org/wiki/Pseudoscientific_metrology

          Then compare the essay of Stecchini on the Roman and Egyptian foot:

          http://www.metrum.org/measures/romegfoot.htm

          It is important to distinguish his meticulous historical research
          from some of the hypotheses and theories he based on it. None-
          theless, they do not diminish the value of his work, if it is a
          field one is interested in. One can always refer to his sources
          and judge for oneself. The last quarter of the link above con-
          tains an illuminating discussion of pyramidites and their errors.
          The first link above on pseudoscientic metrology is so misleading,
          it characterizes Petrie as a practitioner of it, when he is in fact
          one of the fathers of scientific archaeology, metrology, Egyptology,
          and surveying. His father was a pyramidite, and friends with Smyth,
          then astronomer royal of Scotland, and one of the foremost pyramid-
          ites of his time, being the author of Our Inheritance in the Great
          Pyramid.

          The current Director of the Gizeh Plateau, who along with Lerner
          has very little patience for modern pyamidites like Hancock and
          Graham, writes in his introduction to a 1990 abridgement of the
          Pyramids and Temples of Gizeh by Petrie

          "Petrie's work inside the Great Pyramid and at the other pyramids at
          Giza, concentrated on the taking of measurements of the structures.
          These measurements were so accurate that even today they are still
          used by many Egyptologists. At the same time with the advance of
          scientific practices more detailed information has been extracted.
          The base of the Great Pyramid was resurveyed by Cole in 1926, by J.
          Dorner and by Lehner and Goodman in 1984. Recently, Maragioglio and
          Rinaldi did a visual survey of the pyramids of Giza and took measure-
          ments of many elements inside the Great Pyramid." p.98 ( 1990 )

          Maragioglio and Rinaldi, note in their work L'Architettura Delle
          Piramidi Menfite, Parte IV, La Grande Piramide di Cheope, --
          ( Tipagrafia Canessa - Rapallo, 1965 ), page 5,

          "...in caso di quasi identità...fra le nostre misure e quelle date
          dal Petrie nella sua opera « The Pyramids and Temples of Giza »
          abbiamo spesso data la preferenza a quelle dell'archeologo inglese
          in quanto ottenute con strumenti di alta precisione."

          "in case of small differences...in the measurements we took and
          those given by Petrie in his volume « The Pyramids and Temples
          of Giza » we have often given preference to his measurements
          as they were obtained with very high precision instruments"

          Notwithstanding the remarks of Hawass, no one has measured the
          interior of the monument to the degree of precision of Petrie. I
          would be happy if a team of surveyors went in with the best laser
          instruments and measured the Grand Gallery, the Passages, the
          Chambers, the Niche, and the Sarcophagus. But you work with what
          is available. The measures of Petrie work almost too perfectly.
          It is difficult to think he did not apply the highest standards
          in his work, but in science, unless there are at least two sets of
          data which corroborate each other, one may be erecting a theory
          on the basis of erroneous data. The external survey of Petrie
          has been corroborated by Cole, and no doubt by Lerner, although
          I have yet to review his GPS data, and the measures of Maragioglio
          and Rinaldi of the internal features of the monument corroborate
          Petrie to the lesser degree of precision that they measured. His
          measures work so precisely that it is necessary to measure at least
          to his degree of precision or greater(measures taken to 1 hundredth
          of an inch accuracy, and the divisions of the rulers machined to one
          thousandth of a inch accuracy ) to verify the precision geometry of
          the monument that I have discovered through his measures.


          However, it is just as unscientific to dismiss out of hand a know-
          ledge of geodesy and/or astronomy on the part of our ancestors
          without examining all available evidence, as it is to assert such a
          knowledge on the basis of faulty survey data, complex geometrical
          speculation, and a priori historical knowledge, e.g. - the testa-
          mony of Greece and Israel affirms the wisdom of the Egyptians, there-
          fore they must have known how to measure the beetle ball of Kephra,
          by Ra Heru! The academy tends to swing like a pendulum at times, con-
          servatively retreating from hypotheses it embraced uncritically on
          the basis of little or no evidence at one time, and then tentatively
          re-examining earlier hypotheses when it gets over its knee jerk re-
          actions. Egypt is a case in point. All the fanfare of the ages led
          historians to believe the ancient Egyptians possessed advanced math-
          ematics, astronomy, and geodesy, at least to the level of the civil-
          ization of the Babylonians. However, as I mentioned in an earlier
          post, when at last the few crumbling remains of the hieroglyphic
          record of the knowledge of the ancient Egyptians in those areas began
          to be discovered and translated after Champollion, scholars were
          gravely disappointed. For Mathematics, the most extensive record
          we have is that of the Rhind Papyrus, and another few of similiar
          content. Although it demonstrates the ancient Egyptians knew how
          to reckon well with fractions, and calculate volumes of truncated
          pyramids, it does not demonstrate an advanced knowledge of algebra,
          or geometrical reasoning ( a la Euclid ). An argument has been made
          that their method of reckoning fractions hindered their advances in
          mathematics. The Rhind Papyrus has been speculated to be a col-
          lection of schoolboy mathematical exercises, so there is not alot
          of justification for concluding that it represents the limit of math-
          ematical knowledge of the Egyptians. In Essay 24, Non-literary
          Texts, of his Middle Egyptian, An Introduction to the Language and
          Culture of Hieroglyphs, James P. Allen, Curator of Egyptian Art at
          the Metropolitan Museum of Art in New York, writes

          Middle Egyptian mathematical treatises are represented by four
          papyri and two wooden tablets. Of the most important is
          the Rhind Mathematical Papyrus, which contains a table of division
          of 2 by odd numbers from 3 to 101 and a series of 84 problems in
          arithmetic and plane and solid geometry. Page 360

          The Papyrus dates from the reign of the Hyksos Pharoah Apophis, ca.
          1560 bce, 1000 year after Khufu, although apparently copied from an
          older writing dating to the reign of Amenemhat III, ca. 1844-1797
          bce. The Hyksos Pharoahs were invaders. Three times as many major
          medical papyri have come down to us as major mathematical papyri.
          Egypt is known to have been a profoundly conservative culture, and
          at times it gives the appearance that its earlier dynasties
          surpassed its later dynasties in knowledge. Their are several
          difficulties over and above the paucity of textual evidence in
          determining the scientific knowledge of the ancient Egyptians:

          1) The bulk of the written testamony of their civilization has sur-
          vived through the fact that it was engraved or painted on material
          able to withstand the decay of time: monuments, and stone and
          wooden stele. However, it is almost invariably the case that these
          monuments and stele are either commemorative of acts of the Pharoah
          accomplished during his reign, or of the deeds of a deceased person
          in life, or are religious in nature - spells to guard the soul of
          the deceased in the afterlife, or accounts of cosmogony and the life
          of the gods. So although there are references to a more extensive
          body of scientific, linguistic ( i.e. hieroglyphic ), geometical,
          mathematical, astronomical, and medical knowledge ( for instance the
          42 Books of Thoth ), very few scientific papyri have survived the
          debris of their civilization, and the accidental burning of the
          Library of Alexandria by the Romans did not help. A fair number of
          literary papyri primarily from the Middle Kingdom, in the genre of
          wisdom literature and stories, have also come down to us, but it is
          the architectual art of their monuments, for which Egypt is argu-
          ably most celebrated, so much so that it caused Hegel to write
          in his Philosophy of History:

          It is the distinguishing feature of the Egyptian spirit, that it
          stands before us as this mighty taskmaster. It is not splendour,
          amusement, pleasure, or the like that it seeks. The force which
          urges it is the impulse of self-comprehension; and it has no other
          material to work on, in order to teach itself what it is, to realize
          itself for itself, than this working out its thoughts in stone;
          and what it engraves on stone are its enigmas--these hieroglyphs.
          They are of two kinds: hieroglyphs proper, designed rather to ex-
          press language, and having reference to subjective conception; and
          a class of hieroglyphs of a different kind, viz., those enormous
          masses of architecture and sculpture, with which Egypt is covered.
          While among other nations history consists of a series of events--
          as, e.g., that of the Romans, who century after century, lived only
          with a view to conquest, and accomplished the subjugation of the
          world--the Egyptians raised an empire equally mighty--of achievements
          in works of art, whose ruins prove their indestructibility, and which
          are greater and more worthy of astonishment than all other works of
          ancient or modern time.

          ( Great Books of the Western World, Volume 46, page 255 )

          One need not agree with the dialectical method of Hegel and the
          subject object dyad ceaselessly sublimating itself which lies at
          the foundation of his philosophy, to agree that when it comes to
          the art of the monument the Egptians exerted themselves to the limit.
          Also contrary to his assessment it is not true the Egyptians were not
          a fun loving people. The question Dan raises of enduring art versus
          fashionable art finds one index of an answer in the perennial popu-
          larity of ancient Egypt in the imagination of mankind.

          2) Another difficulty in determining the level of scientific know-
          ledge of the ancient Egyptians is related to their identification of
          divine science and physical science, Thoth being the patron God of
          both sciences. In our culture certain kinds of knowledge is class-
          fied, knowledge related to state security, for instance, or know-
          ledge which gives a business a competitive edge in the marketplace.
          Did the priests of Egypt jealously guard their knowledge? What gave
          a priest power in ancient Egypt is the idea that they embodied the
          knowledge of the gods of their country, & were the actual represent-
          atives of the gods, and at least in the case of the Pharoah, embodied
          the god itself. It is striking that Khufu is considered to be the
          first Pharoah who explictly identified himself with the sun Ra in his
          life, and his sons are the first Pharoahs whose cartouches carry the
          additional title Son of Ra. Whereas all other pyramids place the
          crypt either at, below, or a little above the base of the pyramid,
          his crypt is placed high up in the structure of the monument, which
          from one perspective is an earlier form of the heresy of Akhenaton,
          the absolute identification of the Pharoah with Ra in life, rather
          than merely upon death, as the pyramid is a form of the ancient
          Ben Ben stone of Heliopolis, and symbolizes Ra himself. Now if it
          is true that the entire Horizon of the earth, and the celestial
          bodies which move through the Horizon, were conceived to be divine
          in the mind of ancient Egypt, the actual motion of the Horizon, and
          the measure of that motion, will measure the motion of divinity, &
          the embodying of the measured motion in a monument will embody
          heaven on earth, & divinize Pharoah through his entombment in that
          Horizon, and its perfect reflection of the movements of Heaven and
          Earth. When we remember than Khufu is celebrated in the historical
          imagination of ancient Egypt as having sought out the knowledge of
          the secret passageways of Tahuti, to incorporate it into his Akhet,
          or Horizon, and we reflect that Tahuti is the scribe of the gods,
          and the measurer & weigher of Heaven and Earth, it is not too great
          of a leap of the imagination to think that Khufu, and his priests
          and architects incorporated their astronomical and mathematical
          knowledge in the monument. The question arises, why is it not
          incorporated through direct lineal measurements, or if one is able
          to demonstrate that some of it is incorporated directly, why is it
          that a knowledge of elementary solid geometry is necessary to dem-
          onstrate other of this knowledge? First is should be remarked that
          even it one is able to demonstrate a very precise measure of the
          movements of the heavens incorporated in the monument, that fact
          of being able to precisely measure the time and space of these
          movements does not demonstrate a knowledge of heliocentric theory,
          or celestial mechanics. The knowledge of the actual fact that the
          heavens precess through time is a case in point. Given the length
          of the Egyptian civilization and the period of time that they ob-
          served the heavens, it is impossible that they did not notice that
          the heavens precess, as they do so at around one minute a year, or
          1.39686 degrees per century at our era, and using some recent pre-
          cessional formulae, at around 1.369 degrees per century at the era
          of Khufu. However, to measure the rate of precession is extremely
          difficult. One can measure it in a given time span, and say that
          the rate of precession changes at such a rate, but a true under-
          standing of the rate of precession requires an understanding of
          the non-linear dynamics of the solar system, and the mathematics
          of a non-rigid earth. Of all local astronomic constants the rate
          of precession is one subject to the greatest uncertainty. However,
          it is extremely easy to measure the actual fact of precession if
          you are a regular observer of the heavens, as each spring equinox
          the stars which arise before the sun will be different, & will be the
          stars of the zodiac which are one arc minute behind the stars which
          arose in the previous year. The Egyptians were fully capable of ob-
          serving the heavens to a minute degree of accuracy. The Great Pyramid
          itself is oriented to true north to that degree of accuracy, & Petrie
          argues that its foursquare form demonstrates an ability to measure to
          12 seconds of accuracy. To know why the heavens precess requires a
          knowledge not only of the actual rotation and revolution of the earth
          around the sun, it also requires a knowledge of the revolution of the
          moon around the earth, and a knowledge of the gravitational effects
          of the sun and moon on the oblate spin of the earth, meaning, at the
          least it requires a knowledge of elementary Newtonian mechanics. Re-
          cent research suggests that one possible way that ancient Egyptians
          oriented the Great Pyramid is through using a plumb line centered on
          two polar stars circling the center of the Heavens. If indeed that
          was how they went about orienting their monuments, it would soon be-
          come apparent to them as they built through the decades, and even
          centuries, that the center of the Heavens did not remain fixed. In
          fact, using an extremely simple formula, it is possible to derive
          centennial precession at the era of Khufu, through the slope of his
          monument. The 9 / 10 rise run on the edge works best, although the
          14 / 11 rise run on the face is almost as accurate. Using the date
          2528 bc as the date of the death of Khufu cited most often in the
          literature, current precessional formulae calculate centennial pre-
          cession for that era at 1.369 degrees. If one uses a face slope of
          the monument as if it was built using a 9 / 10 rise run on its as-
          cending edge, then its slope in radians raised to the inverse power
          of Pi equals centennial precession of that era accurate to the time
          of the building of the monument

          .9048518739 ^ - PI = Radian Slope Khufu ^ - PI = 1.36904 ~

          Number of Degrees Centennial Precession circa Khufu 2528 bce.

          If you would like to calculate it yourself, first determine the
          radian slope of Khufu. Using a perimeter inradius of 220 cubits,
          the perimeter circumradius equals Square Root ( 220² + 220² ), the
          height will equal 9 / 10 * Square Root ( 220² + 220² ), the ratio
          of the height and the perimeter inradius will equal the tangent of
          the radian slope, and the radian slope equals the Inverse Tangent
          of 9 / 5 / Square Root 2 = 0.90485187394...

          To calculate precession circa era Khufu use the precessional formula

          (1.396291666... + 0.0006180555...T)°

          given at http://www.jqjacobs.net/astro/astrofor.html#periodicity

          where T is the number of tropical centuries from 1900.00, although
          in our case we will be subtracting .0006180555 times the number
          of centuries before 1900.00. Notice that the constant of precession
          .0006180555 is quite close to 1 / GoldenRatio / 1000. Solving for T
          the number of centuries prior to 1900 at which precession equals
          .9048518739 ^ - PI° = 1.36904°:

          (1.396291666... - 0.0006180555 * T)° = .9048518739 ^ - PI°

          and T = 44.09411418... centuries or 4409.411418 years. Subtracting
          this number of years from 1900 to determine what year centennial pre-
          cession equaled Radian Slope Khufu ^ - PI° we get

          1900 ce - 4409.411418 = 2509.411418... bce

          Now that is pretty darn remarkable. The date of the death of Khufu
          is usually given as 2528 bce, it is often stated that it took 20
          years to build the Great Pyramid, so that by these figures, the
          radian slope of Khufu raised to the inverse power of PI equals
          centennial precession in degrees at the year the architects and
          masons were laying the foundations of the monument. Compare this
          method of deriving precession from Khufu with that of Stecchini. Of
          course, there are many uncertainties. The two main king lists from
          which the dynasties of Egypt are dated have yielded at least two
          major competing dating schemes. The date I give is the one most
          often given, for instance in the Oxford Atlas of Egypt, and in the
          Complete Pyramids of Lerner. The precessional formula is open to
          uncertainty. In fact it contradicts the division of the perimeter
          into celestial sphere obliquity of the current era accurate to 1/3
          arc second. Why does the face slope and the circumference / diameter
          ratio yield centennial precession era Khufu, and the celestial
          sphere / perimeter ratio yield obliquity ecliptic our era? It would
          appear that the perimeter of the Great Pyramid is predictive of the
          tilt of the axis of the earth - i.e. that it predicts a given time.
          It is true that we live in a time that is absolutely critical, and
          the decisions we make in this era in regards to carbon dioxide
          emissions, deforestation, depletion of aquifiers, soil conservation,
          active forestation, equilibrium economics, species extinction,
          overfishing of the oceans, ozone holes, et cetera will determine
          specific consequences, so it is absolutely appropriate that the
          celestial sphere / perimeter ratio of the Great Pyramid, the
          greatest monument and tomb in the solar system, and perhaps the
          universe, a monument signifying death and resurrection, is the
          perfect analogy for the creation of a new thinking and new logic, a
          logic of life and not death, a logic of the living and not the dead,
          a logic of peace and not war, a logic of the actual existence of
          heaven on earth, of earth in heaven, a logic of transcendental
          consciousness identical matter, a logic of absolute exteriority and
          the surface of existence, the infinitely flat surface of the four
          dimensional hypersphere of universal expansion, a logic of the
          surface of the earth, a logic of the word of God made flesh.

          But given the current imprecision in our ability to actually cal-
          culate the rate of precession through time, limited by both a small
          empirical set of temporal data, and the complexity of the equations,
          we really have no way of knowing if that was the actual rate of pre-
          cession of that era, let alone, given the other fundamental geodetic
          and astronomic features of the monument more directly related to its
          mathematical structure, why such a simple equation should allow its
          slope - apparently what it is for other reasons - to be so easily
          related to the period of precession - after all this would appear to
          suggest that the slope which relates the length of the polar axis
          and a circle of equatorial latitude mediated through the period of
          the rotation of the earth - meaning the height and perimeter slope
          of the Great Pyramid - possesses a real mathematical relationship
          to the period of the precession of the earth. That would need
          to be demonstrated through celestial mechanics for it to be anything
          other than a coincidence, which brings us to the third point of why
          it is difficult to know what if any knowledge of astronomy et cetera
          is incorporated in the Great Pyramid: coincidental occurances of
          numbers, and jumping to conclusions from these coincidences, all of
          which has contributed to the general disrepute any research into
          the subject has fallen these days.

          3) When one reads the works of Smyth, Edwards, and the Edgar brothers
          and the crazy theories, and wild speculations they contain, they make
          folk like Stecchini and Schwaller De Lubicz look like sober scholars.
          Nothing has done more damage than a faithful attempt to identify the
          length of the passages of the pyramid measured in an imaginary pyra-
          mid inch derived from an equally imaginary sacred cubit of the He-
          brews with the timeline of major events in the Bible ( whether real
          or imaginary events ) & human history. You do not even need to re-
          sort to this imaginary pyramid inch to engage in predictive pyramid-
          ology. Petrie measured the edge of the base of the north face to the
          center axis of the pyramid - meaning the point below the apex, which
          is also the distance of the point below the apex of the roof of the
          Queen Chamber, & the distance of the point below the step of the
          Grand Gallery, from the northermost edge of the Great Pyramid - to
          be 4534 British Imperial Inches. Well, if you take the commonly
          accepted date of the death of Khufu - 2528 bce - and subtract it from
          4534, you get 2006 the current year. Anything world shaking occur
          in this year? The most historical event that I perceive which has
          occurred this year is that for the first time in history the fact
          that global warming is actually occurring has finally begun to be
          admitted by the American power structure. Biblically, 2006 is the
          year of Creation that Noah died. Remember him? The guy that rode
          out the flood and preserved species on the universal ark? Now these
          numbers may be able to be used to catalyse the popular religious
          imagination to act if embodied in a creative work of sufficient
          appeal, but otherwise, they have little or no value outside of the
          role myth can play in cultivating unscientific and irrational
          understanding.

          There is alot more than can be said on these matters, but I am a
          little tired, and I will save it for another occasion. I will sim-
          ply conclude by noting that the value of folk like Stechinni and
          Tompkins, is they awaken the imagination to the possibility of a
          specific fact of history. The truth or falsity of that fact, if it
          exists, and it is not merely a nebulous fancy and wishful think-
          ing, can only be examined on the basis of extremely accurate scient-
          ific measurements. That is the first conditio sine qua non. Once
          one has these measurements one can examine the basic geometrical
          facts of the monument and compare them to the facts of the science
          of geometry, astronomy, and geodesy, and see if they warrant the
          conclusion that the authors of the monument incorporated any such
          knowledge in it. The analogy of the blind watchmaker is not exact.
          The geometry ought to be as evident as if one were examining geome-
          trical figures drawn on paper. The difference is this: on paper
          one is able to explicitly see the geometrical problem and its sol-
          ution, and one is often given a few words to help demonstrate the
          proof of its solution. In a monument like the Great Pyramid, the
          proof is often not quite as explicit. Yet though implicit in the
          geometry of its architecture, it ought to be as transparent as the
          geometry drawn on paper. After all, what is the difference between
          seeing a geometrical problem and its solution in two dimensions, and
          seeing one built in stone in three dimensions? Let us take a very
          simple example - one of the basic geometrical facts of the King
          Chamber. I have already pointed out that it exactly doubles the
          volume of the Queen Chamber - accurate to 1/2 millimeter. I think
          it is more improbable that the volume of the King Chamber is double
          the volume of the Queen Chamber by accident than that it is by de-
          sign. One could perchance calculate the odds. It by itself demon-
          strates no great measure of intentional design in the monument. How-
          ever, let us look a little closer at the geometry of the King Cham-
          ber. The King Chamber cubit is derived from the hypothesis of a 10
          by 20 cubit width and length of the base of its walls. This is a
          hypothesis proven by the fact that the cubit so derived is within
          .002 inches of the mean cubit of ancient Egypt derived from a variety
          of its monuments - 20.632 versus 20.63 inches. Knowing then that the
          architects designed the chamber to be 10 by 20 cubits at its base, we
          ask ourselves, given the height of the walls of the King Chamber -
          235.2 inches, or in king chamber cubits - 11.399767... cubits - why,
          if at all, did the architects design the height to be that many cu-
          bits? The easy answer, since the King Chamber was built after the
          Queen Chamber, is that number of cubits allows the volume of the
          chamber to equal 2280 cubic cubits, or twice the volume of the 1140
          cubic cubits of the Queen Chamber. However, to settle on just this
          does not do justice to the possibility that the architect may have
          had a level of intelligence above the average bear. And indeed,
          the simplest volumetric analysis of the chamber ( & one need not in-
          voke the idea of De Lubicz, correct as it may be, that the ancient
          Egyptians were fond of volumetric thinking ) demonstrates the King
          Chamber solves not the Delian Problem, but a two dimensional variant
          of it: given two squares equal in area to each other - here the
          actual dimensions of the floor of the King Chamber - 10 cubits in
          width by 20 cubits in length - two squares each 10 by 10 - how high
          must the chamber be so that its volume equals that of a cube whose
          circumradius is that height? It turns out that there is one height
          which perfectly cubes the volume of a 10 by 20 hexahedron so that
          the volume of the cube equals the volume of the hexahedron, and the
          height of the hexahedron equals the circumradius of the cube, and
          that height is 11.397535...

          Since the actual height of the walls of the King Chamber equals
          11.399767 cubits, it is one millimeter more than the height the
          chamber would need to be for its volume to equal the volume of
          a cube whose circumradius ( half its space or body diagonal ) is
          the height of the King Chamber. Now again, it is not this actual
          fact itself of the King Chamber which by itself demonstrates the
          intentional incorporation of elementary solid geometry into its
          design. It is only through the demonstration of the existence
          of a series of such simple geometrical facts in the monument that
          the existence of intelligence designing it there acquires the force
          of a demonstration, as it outweighs the improbability of such beau-
          tiful and simple geometrical facts existing there by accident.

          - John
        • jjdepompeo
          One way to get the cube root of 2 Delian Constant http://en.wikipedia.org/wiki/Doubling_the_cube from the volume of the King Chamber ( KC ) is as follows: Let
          Message 4 of 10 , Oct 30, 2006
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            One way to get the cube root of 2 Delian Constant

            http://en.wikipedia.org/wiki/Doubling_the_cube

            from the volume of the King Chamber ( KC ) is as follows:

            Let the Volume of the Chamber equal 10 by 20 by 11.4 cubic cubits or
            2280 cubic king chamber cubits. The area of a sphere of volume 2280
            cubic cubits is 837.736341, and if we divide it by half the area of
            the floor of the king chamber - 100 square cubits - we get a unit
            free number

            Area of Sphere of Volume of KC / Area of 1/2 Floor of KC = 8.37736341

            8.37736341 is the Volume of a Sphere of Radius 1.259910...
            which is quite close to the Delian Constant of 1.2599210...

            Or if we want to get a little ridiculous, let the height of the GP
            equal 280 cubits, and its base equal 440, then its Apothem will equal
            356.0898..., and the Inverse of the Cosine of its Face Slope in rad-
            ian or the Secant of its Slope will equal

            1 / Cosine .904827089... = 1.618590346... = 356.0898... / 220

            and instead of letting the sphere be equal in volume to that of the
            King Chamber - 2280 cubic cubits - let the sphere be equal in volume
            to a cube of edge length 13 + one tenth of the Secant of the Radian
            Slope of Khufu:

            13 + Secant .904827089... / 10 = 13 + .1618590346...

            If you divide the length of the apothem of a pyramid of height 280 &
            base 440 by the sum of the length of its perimeter and the length of
            the indiameter of its perimeter, you get .1618590346...

            So let the Sphere be equal in volume to

            ( 13 + .1618590346... )³ cubic king chamber cubits

            then the king chamber cubit surface area of that sphere divided by
            half the king chamber cubit area of the floor of the king chamber -
            100 square cubits - equals 8.37758021264, and the radius of a sphere
            of volume 8.37758021264 equals

            1.2599210400 which equals the cube root of 2 or

            1.2599210498 and the Delian constant accurate to nine digits!


            - John
          • jjdepompeo
            A generalization of the geometry of the king chamber from end of post 7000 and beginning of post 7007 : The area of the sphere equal in volume to a cuboid of
            Message 5 of 10 , Oct 30, 2006
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              A generalization of the geometry of the king chamber from end of post
              7000 and beginning of post 7007 :

              The area of the sphere equal in volume to a cuboid of width 1, length 2,
              and height equal to the circumradius of a cube equal in volume to the
              cuboid, equals the volume of a sphere of radius 1.2598 ~ 1.2599 = 2 ^ (
              1 / 3 ) = Delian constant

              Another way to the Delian constant through the GP:

              The length of the ascending edge of a pyramid of height 280 cubits and
              base 440 cubits is 4.18569 egyptian rods, and the length of the radius
              of a sphere whose volume equals the number of rods in the 4 ascending
              edges of the pyramid is

              1.58700.. ~ cube root of 4 = 1.58740... =

              the ratio of the area of the face of a cube to the area of the face of
              the cube half its volume, and the square of the Delian constant
            • jjdepompeo
              ... 100 square cubits equals 1 land cubit in ancient Egypt. Mike - notice that the commonly agreed height of the GP - 280 cubits - and its inradius - 220
              Message 6 of 10 , Jan 20, 2007
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                --- In sacredlandscapelist
                >
                >
                > One way to get the cube root of 2 Delian Constant

                ...

                >
                > So let the Sphere be equal in volume to
                >
                > ( 13 + .1618590346... )³ cubic units
                >
                > then surface area of that sphere divided by 100 square units -
                > equals 8.37758021264, and the radius of a sphere of volume
                > 8.37758021264 equals
                >
                > 1.2599210400 which equals the cube root of 2 or
                >
                > 1.2599210498 and the Delian constant accurate to nine digits!

                100 square cubits equals 1 land cubit in ancient Egypt.

                Mike - notice that the commonly agreed height of the GP - 280 cubits -
                and its inradius - 220 cubits - yields an apothem of

                Square Root ( 280² + 220² ) = 356.0898762952... cubits

                and hence its apothem divided by the length of a side of its square
                base equals

                356.0898762... / 440 = .809295173....

                This ratio is close to .809295992... =

                Square Root ( 2 × 2 ^ ( 2 / 3 ) - 2 × 2 ^ ( 1 / 3 ) )

                the square root of ( twice the area of the face of a cube double the
                volume of a unit cube minus twice the length of the edge of the cube
                double the volume of the unit cube ) a number which figures promin-
                ently in one of the most celebrated solutions to the doubling of the
                cube - Archytas' Duplication of the Cube

                http://plato.stanford.edu/entries/archytas/
                http://www.ms.uky.edu/~carl/ma330/projects/dupcubfin1.html

                Archytas determined the length of the Delian constant from the inter-
                section of a cone, a cylinder, and torus. Using modern notation the
                Delian constant equals the solution of three simultaneous equations:

                Cone: x² = y² + z²

                Cylinder: 2x = x² + y²

                Torus: ( x² + y² + z² )² = 4x² + 4y²

                Here the length of the edge of the cube to be duplicated equals unity
                ( See the second link for details ) When x = the cube root of two or
                the Delian constant 1.259921... in the above equations z equals

                Square Root ( 2 × 2 ^ ( 2 / 3 ) - 2 × 2 ^ ( 1 / 3 ) )

                = .80929599291... ~ = 0.809295173... 356.089876295... / 440 =

                Apothem of Great Pyramid / Base of Perimeter of Great Pyramid

                As the second link notes "The human eye has a difficult time reading
                a 3-D figure, so we will take a slice of this figure to make it easy
                to see...You need to slice it at the appropriate z = constant or y =
                constant plane in order to "see" that the x-coordinate of the inter-
                section is 2 ^( 1 /3 ). Of course you can only approximately see
                that ...By playing with the solutions to the equations, you can work
                out that (.80929599291... ) is the exact value for z."

                - J
              • Daniel N. Washburn
                Hi, John You have a note to Mike in this post. Because of other priorities, Mike has signed off the list for the time being. I have forwarded your post on
                Message 7 of 10 , Jan 23, 2007
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                  Hi, John

                  You have a note to Mike in this post. Because of other priorities,
                  Mike has signed off the list for the time being. I have forwarded your
                  post on to him. If you want to contact him directly his e-mail is Mike
                  Bispham@....

                  Dan

                  jjdepompeo wrote:

                  >--- In sacredlandscapelist
                  >
                  >
                  >>One way to get the cube root of 2 Delian Constant
                  >>
                  >>
                  >
                  >...
                  >
                  >
                  >
                  >>So let the Sphere be equal in volume to
                  >>
                  >>( 13 + .1618590346... )³ cubic units
                  >>
                  >>then surface area of that sphere divided by 100 square units -
                  >>equals 8.37758021264, and the radius of a sphere of volume
                  >>8.37758021264 equals
                  >>
                  >>1.2599210400 which equals the cube root of 2 or
                  >>
                  >>1.2599210498 and the Delian constant accurate to nine digits!
                  >>
                  >>
                  >
                  >100 square cubits equals 1 land cubit in ancient Egypt.
                  >
                  >Mike - notice that the commonly agreed height of the GP - 280 cubits -
                  >and its inradius - 220 cubits - yields an apothem of
                  >
                  >Square Root ( 280² + 220² ) = 356.0898762952... cubits
                  >
                  >and hence its apothem divided by the length of a side of its square
                  >base equals
                  >
                  >356.0898762... / 440 = .809295173....
                  >
                  >This ratio is close to .809295992... =
                  >
                  >Square Root ( 2 × 2 ^ ( 2 / 3 ) - 2 × 2 ^ ( 1 / 3 ) )
                  >
                  >the square root of ( twice the area of the face of a cube double the
                  >volume of a unit cube minus twice the length of the edge of the cube
                  >double the volume of the unit cube ) a number which figures promin-
                  >ently in one of the most celebrated solutions to the doubling of the
                  >cube - Archytas' Duplication of the Cube
                  >
                  >http://plato.stanford.edu/entries/archytas/
                  >http://www.ms.uky.edu/~carl/ma330/projects/dupcubfin1.html
                  >
                  >Archytas determined the length of the Delian constant from the inter-
                  >section of a cone, a cylinder, and torus. Using modern notation the
                  >Delian constant equals the solution of three simultaneous equations:
                  >
                  >Cone: x² = y² + z²
                  >
                  >Cylinder: 2x = x² + y²
                  >
                  >Torus: ( x² + y² + z² )² = 4x² + 4y²
                  >
                  >Here the length of the edge of the cube to be duplicated equals unity
                  >( See the second link for details ) When x = the cube root of two or
                  >the Delian constant 1.259921... in the above equations z equals
                  >
                  >Square Root ( 2 × 2 ^ ( 2 / 3 ) - 2 × 2 ^ ( 1 / 3 ) )
                  >
                  >= .80929599291... ~ = 0.809295173... 356.089876295... / 440 =
                  >
                  >Apothem of Great Pyramid / Base of Perimeter of Great Pyramid
                  >
                  >As the second link notes "The human eye has a difficult time reading
                  >a 3-D figure, so we will take a slice of this figure to make it easy
                  >to see...You need to slice it at the appropriate z = constant or y =
                  >constant plane in order to "see" that the x-coordinate of the inter-
                  >section is 2 ^( 1 /3 ). Of course you can only approximately see
                  >that ...By playing with the solutions to the equations, you can work
                  >out that (.80929599291... ) is the exact value for z."
                  >
                  >- J
                  >
                  >
                  >
                  >
                  >
                  >Topics suitable for discussion in this e-list can be found at:
                  >http://www.luckymojo.com/sacredland.html
                  >
                  >To UNsubscribe, send email to:
                  >sacredlandscapelist-unsubscribe@yahoogroups.com
                  >
                  >Yahoo! Groups Links
                  >
                  >
                  >
                  >
                  >
                  >
                  >
                • jjdepompeo
                  Thanks Dan. I mentioned his name in passing as he indicated his researches led him to think that the proportional external geometry of the GP indicates a
                  Message 8 of 10 , Jan 23, 2007
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                    Thanks Dan.   I mentioned his name in passing as he indicated his researches led him to think that the proportional external geometry of the GP indicates a concern with the Delian problem on the part of its architect.  I stumbled on the Archytas z coordinate and thought to share it.  It still may be that it is possible to directly derive a precise Delian constant from the simplicity of the proportional geometry of the GP ...  BTW if anyone followed up on any of those links in my last post, the geometrical drawing in the first link http://plato.stanford.edu/entries/archytas/#Cube has a 3-D animation analogue of it at ( of all places! )

                    http://www.larouchepac.com/pages/economy_files/2004/041014_anim_eco_graphics.htm

                    Click on the control and drag it to rotate it about any axis.   It is easy to see how the cylinder and torus are generated ( respectively ) from the original horizontal circle ABDZ at the stanford link, and the rotation of the semi-circle erected perpendicular to it on the diameter AED.  The center 'hole' of the torus is the point A.   A question:  is an infintesimal point a hole, and is a torus with such a center actually a torus?  :)   It is more difficult to see why the curve of the cone is necessarily generated from the rotating triangle.  Also I did not review or check the math at the second site ( the one that uses the maple program ) so I am not able to vouch for it.  

                    John

                     

                    > Hi, John
                    >
                    > You have a note to Mike in this post. Because of other priorities,
                    > Mike has signed off the list for the time being. I have forwarded your
                    > post on to him. If you want to contact him directly his e-mail is Mike
                    > Bispham@...
                    >
                    > Dan

                  • jdepompeo
                    Or more simply and accurately where Pyramid Apothem = 356.089876... cubits = Square Root ( 280² + 220² ) = Square Root ( Height² + Inradius² ) Pyramid Side
                    Message 9 of 10 , Jan 25, 2007
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                      Or more simply and accurately where

                      Pyramid Apothem = 356.089876... cubits = Square Root ( 280² + 220² ) = Square Root ( Height² + Inradius² ) 

                      Pyramid Side = 440 Cubits

                      Delian constant = 1.25992104... = cube root of 2 = Edge of Cube of Volume Two = D

                      Edge of Unit Square of Volume One = 1

                      ( Apothem ÷ Side )² ÷ 2 = D² - D  = ( D - 1 ) × D = ( D - 1 ) ÷ ( 1 ÷ D )

                      the D variable on the right side of the equation equals 1.259921 ~ 1.25992104... = Delian constant accurate to 7 places

                      Apothem² ÷ 2  / Area of Square Base of Pyramid = 1.259921² - 1.259921 = ( 1.259921 - 1 ) ÷ ( 1 ÷ 1.259921 )

                      1/2 the area of the square of the altitude of each face triangle of the Great Pyramid is in the same proportion to the area of the square of its base as the difference between the delian constant and unity is to the quotient of unity and the delian constant

                      Thanks to Mike for suggesting the idea!

                       

                      - John

                    • jdepompeo
                      ... 220² ) = Square Root ( Height² + Inradius² ) ... Volume Two = D ... - 1 ) ÷ ( 1 ÷ D ) ... 1.25992104... = Delian constant accurate to 7 places ...
                      Message 10 of 10 , Jan 31, 2007
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                        In sacredlandscape list jdepompeo wrote:

                        > Or more simply and accurately where

                        > Pyramid Apothem = 356.089876... cubits = Square Root ( 280² + 220² ) = Square Root ( Height² + Inradius² ) 

                        > Pyramid Side = 440 Cubits

                        > Delian constant = 1.25992104... = cube root of 2 = Edge of Cube of Volume Two = D

                        > Edge of Unit Square of Volume One = 1

                        > ( Apothem ÷ Side )² ÷ 2 = D² - D  = ( D - 1 ) × D = ( D - 1 ) ÷ ( 1 ÷ D )

                        > the D variable on the right side of the equation equals 1.259921 ~ 1.25992104... = Delian constant accurate to 7 places

                        > Apothem² ÷ 2  / Area of Square Base of Pyramid = 1.259921² - 1.259921 = ( 1.259921 - 1 ) ÷ ( 1 ÷ 1.259921 )

                        > 1/2 the area of the square of the altitude of each face triangle of the Great Pyramid is in the same proportion to the area of the square of its base as the

                        >  difference between the delian constant and unity is to the quotient of unity and the delian constant

                        >Thanks to Mike for suggesting the idea!

                        Even more simply, where the rise / run of Khufu = 280 / 220 = 14 / 11,

                        ( 1 + 142 / 112 ) / 23 = D ( D - 1 )

                        the actual value of D = 1.2599206... = ( 22 + 11181/2 ) / 44 = 1 + ( 1 + rise2 / run) / 23 / 21/3 =  21/3

                         

                        - John

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