Squaring the Circle 

.. God the Geometer, Manuscript illustration. Clark, Kenneth. Civilization. NY: Harper, 1969. p. 52
According to Cowan, churches had been built on geometric principles since early Christian times. Geometry was the basis of all Gothic cathedrals, everything being created from basic relationships. We've seen that the ground plan was always cruciform, the baptism font always octagonal, and the baptistry itself often was, and the circle was everywhere. This was symbolized in art by God holding a pair of compasses, a common motif in the Middle Ages. The art historian Ernst Gombrich credits a passage from the Old Testament as the inspiration for these portrayals. In Proverbs, Chapter 8 par. 27,
Wisdom put forth her
voice;


The Golden Ratio & Squaring the Circle in the Great Pyramid "Twenty years were
spent in erecting the pyramid itself: of this,
which is square, each face is eight plethra, and
the height is the same;
it is composed of polished stones, and jointed
with the greatest exactness;
none of the stones are less than thirty feet."
Heroditus, Chap. II,
para. 124.
In his History Heroditus says that the pyramids, already ancient, were covered with a mantle of highly polished stones joined with the greatest exactness. Definition of the Golden Ratio The golden ratio is also called extreme and mean ratio. According to Euclid, A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. Rest is here  LESSON 2 

Let's now return to the pyramids. If we take a crosssection through a pyramid we get a triangle. If the pyramid is the Great Pyramid, we get the socalled Egyptian Triangle. It is also called the Triangle of Price, and the Kepler triangle. This triangle is special because it supposedly contains the golden ratio. In particular,the ratio of the slant height s to half the base b is said to be the golden ratio. To verify this we have to find the slant height. Rest is here  LESSON 2 

A British railway engineer, Robert Ballard, saw the pyramids on his way to Australia to become chief engineer of the Australian railways. He watched from a moving train how the relative appearance of the three pyramids on the Giza plateau changed. He concluded that they were used as sighting devices, and wrote a book with the grand title of The Solution of the Pyramid Problem in 1882. He also noted that the crosssection of the Great Pyramid is two of what we have called Egyptian triangles. He then constructs what he called a Star Cheops, which, he says, "... is the geometric emblem of extreme and mean ratio and the symbol of the Egyptian Pyramid Cheops." To draw a star Cheops: * Draw vertical
and horizontal axes.
Project Draw a star Cheops. Fold it to quickly make a model pyramid. Rest is here  LESSON 2 

..
Now we'll look at his other claim, that the Great Pyramid's dimensions also show squaring of the circle. But just what is that? The problem of squaring the circle is one of constructing, using only compass and straightedge; (a) a square whose perimeter is exactly equal to the perimeter of a given circle, or (b) a square whose area is exactly equal to the area of a given circle. There were many attempts to square the circle over the centuries, and many approximate solutions, some of which we'll cover. However it was proved in the ninteenth century that an exact solution was impossible. Squaring of the Circle in the Great Pyramid The claim is: The perimeter of the base of the Great Pyramid equals the circumference of a circle whose radius equal to the height of the pyramid. Does it? Recall from the last unit
that if we let the
base of the Great pyramid be 2 units in length, then
pyramid height =
Then for a circle with radius equal
to pyramid height
So the perimeter of the square and the circumference of the circle agree to less than 0.1%. Editors Note: So what was "impossible" in the ninteenth century, the Egytians managed to accomplish on a grand scale to within less than .1% accuracy The Value of Pi in the Pyramid An Approximate Value for in Terms of Since the circumference of the circle (2 ) nearly equals the perimeter of the square (8) 2
8 we can get an approximate value for ,
4 /
= 3.1446
Area Squaring of the Circle The claim here is: The area of that same circle, with radius equal to the pyramid height equals that of a rectangle whose length is twice the pyramid height() and whose width is the width (2) of the pyramid. Area of rectangle = 2 () ( 2 ) = 5.088 Area of circle of radius = r 2 () 2= 5.083 An agreement within 0.01% 

We just saw that the circle is the ultimate geometric figure, perfect, infinite, representing the divine, and we had seen that the square often represented mankind. Combining the two figures had special significance, the reconciliation of the heavenly and infinite with the earthly and manmade. Recall from our unit on Egypt we said that the problem of squaring the circle is one of constructing, using only compass and straightedge; (a) a square whose perimeter is exactly equal to the perimeter of a given circle, or (b) a square whose area is exactly equal to the area of a given circle. In that same unit we also saw that a circle whose radius is the pyramid height 1. has the same perimeter of the base of the Great Pyramid 2. and the same area as the rectangle whose width equals the pyramid's base and whose length is twice the pyramid height 

A Note on
Dartmouth University
"Welcome to Dartmouth, a private, fouryear liberal arts institution that has been at the forefront of American higher education since 1769. A member of the Ivy League, Dartmouth is a superb undergraduate residential college with the intellectual character of a university, featuring thriving research and firstrate graduate and professional programs."  James Wright, President Editors Note: This is a mainstream Ivy League University studying Ancient Science and Sacred Geometry 

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