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The Transverse Lines offer a most intriguing insight into the construction of the ellipsoid.

Consider how one might proceed to construct an ellipsoid by the use of circles.

Why use circles?  If one knows the theoretical construction of an ellipse why not use it? Like a cord pegged at two points? Or is it easier in construction, but not ignorant of the theoretical? What would be involved in constructing circles? How to know where to place them?

Remarks on Geometric Construction

Borchardt does not discuss how the Geometrician knew that his transverse intersections would define a rectangle of area equal to the ellipse. He merely reports the fact that they do.

How did the ancient Geometrician know how to construct a Figure that would make the ellipsoid and rectangle equal in area, or nearly equal? He had to have mathematical knowledge and experience. He had to know how to calculate the area of an ellipse, as well as that of a rectangle.

The remarkable aspect of the drawing is the fact that the tangent points of the large and small circles occur at the intersection of the rectangle vertical lines with the dawn ellipsoid, while those vertical lines define an area for the rectangle to be equal in area to the ellipsoid.

Subject: The Conceptual Curiosity of the Luxor Ellipse
Date: Sun, 3 Aug 2003 09:29:11 -0400
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The mathematical ellipse inscribed on a wall of the Temple of Luxor,
reported by Borchardt, is more than a random construct. It has important
mathematical properties. See my report of July 14.

To understand this claim one must examine the usual discussion of the
"squared circle." In mathematical discourses on the subject two aspects are
presented. The first is the determination of a square equal in area to a
given circle. There is now universal agreement that this cannot be done with
the simple tools of straight edge and compass first prescribed by the
ancient Greeks because the properties of the circle include Pi, a
transcendental number. The second is the common practice of placing the
square superimposed and centered over the circle.

This produces a space between the edges of the square and the circle.
Borchardt named this a "span." I shall use his terminology.

If one constructs a circle of unit radius, (feet, meters, or cubits) the
area of the circle is equal to Pi X radius squared. If the radius is 1.0
the area is equal to Pi.

This circle may be described as an ellipse with the respective radii of "a"
equal to "b," where "a" and "b" are the respective short and long radii,
both equal to one unit.

If one constructs an ellipse with "a" equal to a unit distance (feet,
meters, or cubits) and "b" equal to 1.5 units one would obtain the Luxor
ellipse (short axis = 2, and long axis = 3). This theoretical ellipse has an
area of 1.5 Pi.

Returning to the numbers reported by Borchardt, the area of the Luxor
ellipse calculated from his measured dimensions is 4.725 sq cubits. This
gives a value for the area of the ellipse of 1.505 Pi.

The area of the rectangle calculated from Borchardt's measurements is 4.717
sq cubits. This gives a value of 1.501 Pi.

From the Expanded Drawing, the ellipsoid measures 3.046 cubits in length and 1.978 cubits in height, (ideally 3:2). If we use the theoretical formula of Pi ∙ a · b, where a and b are the half lengths for a true ellipse, this would produce an area of 4.732 sq cub. (1.506 Pi) Compare to 4.71 sq cub calculated by Borchardt.

The rectangle measured 2.768 cubits wide by 1.694 cubits high. This gives an area of 4.689 square cubits. Compare to the 4.58 sq cub calculated by Borchardt.

Compare measured ellipsoid area of 4.732 sq cub to rectangle area of 4.689 sq cub. They differ from one another by 0.043 sq cub. Borchardt's reported difference is 0.13 sq cub.

For comparison, a theoretical ellipse of "a" = 1 and "b" = 2 would give an
area of 2 Pi. If ellipses are constructed with "a" always equal to 1.0 unit,
the area of the ellipses will always be some value of Pi equal to the "b"
dimension.

The span distance for the squared circle is 0.1138 units.

When constructing a rectangle of equal area to an ellipse one has a choice
of how to represent the rectangle superimposed over the ellipse. One can
maintain the same ratio of width-to-length as the ellipse, or one can
maintain the same span distances on all four sides.

If one holds the same width-to-length ratio for the rectangle as the
ellipse, with "a" = 1.0, the span distance between the rectangle and the
ellipse in the direction of the short dimension, "a," is always 0.1138, the
same as the square. The span distance for the long dimension, "b," then
becomes proportionately larger with increasing "b."

The Egyptian geometricians elected for the second choice. They maintained
the span distances equal on all four sides, within construction error. These
calculate to 0.137 for the right and the left, 0.141 for the top, and 0.147
for the bottom. (Since the Luxor ellipse uses royal Egyptian cubits for its
measurements, the values are in cubits.)

Other mathematical curiosities exist about the Luxor ellipse.

Please note that I do not infer any knowledge of the ancient Egyptians on
the concept or value of Pi. I am merely reporting the facts.

Ernest

From: "Ernest Moyer" <epmoyer@suscom.net>
To: "EEF" <eef@lists.yale.edu>
Sent: Friday, July 11, 2003 5:00 PM
Subject: EEF: An Inquiry on Geometry

Dieter Arnold, in his Building in Egypt, pg 22, shows a mathematical ellipse
inscribed near the entrance to the tomb of Ramesses VI. This was a
full-scale drawing to guide the stone cutters for shaping the roof of the
tomb. Arnold offered a few brief remarks concerning the properties of the
ellipse.

Ludwig Borchardt also published an ellipse in ZAS, Vol 34, 1896, Plate 4,
Figure 7, that was scratched on a wall in the Temple of Luxor. Borchardt
attempted to provide an interpretive explanation. I do not know if it had a
practical application.

Both of these examples predate Greek mathematical developments by a 1000
years, and offer support for ancient traditions that the Greeks obtained
their mathematical knowledge from the Egyptians.

Do list members know of any other mathematical forms that exceed the
examples illustrated in the several Egyptian mathematical papyri?

Ernest Moyer

From: "Ernest Moyer" <epmoyer@suscom.net>
To: "EEF" <eef@lists.yale.edu>
Sent: Monday, July 14, 2003 10:26 PM
Subject: EEF Re: An Inquiry on Geometry

This is to clarify remarks I made on the mathematical ellipse reported
by Borchardt, July 11, 2003.

> Ludwig Borchardt also published an ellipse in ZAS,
> Vol 34, 1896, Plate 4, Figure 7, that was scratched
> on a wall in the Temple of Luxor. Borchardt attempted
> to provide an interpretive explanation. I do not know
> if it had a practical application.

The total figure included
One: the ellipse, 2 X 3 cubits on the "a" and "b" dimensions, 103.5 cm X
159.5 cm by Borchardt's measure.
(Assuming 52.39 cm per cubit the measurements calculate to 1.976 X 3.045
cubits.)
Two: a rectangle centered on the ellipse and apparently intended to be
equivalent in area to the area of the ellipse, measured at 89.25 cm (mean) X
145 cm, (1.704 X 2.768 cubits).
Three: sloping transverse lines at the four corners, constructed by the
Egyptian architect or mathematician apparently to help define the equal-area
rectangle. The intersection of the transverse lines with the ellipse was at
the points where the rectangle also intersected the ellipse.

Borchardt measured 7.5 cm (X 2) difference between the two small extremes of
the ellipse and the short dimension of the rectangle, and 7.0 cm (X 2)
difference between the two large extremes of the ellipse and the long
dimension of the rectangle.

The calculated area of a theoretical ellipse measuring 2 X 3 cubits is 4.712
sq cubits.
The area for the ellipse taken from Borchardt's measurements is 4.727 sq
cubits.
The calculated area of a rectangle measuring 89.25 cm X 145 cm (1.704 X
2.768 cubits) is 4.717 sq cubits.

Obviously, the designer intended for the rectangle to be the same area as
the ellipse. The difference in area between the theoretical ellipse and the
measured rectangle is 0.005 sq cubits. The difference in area between the
actual values measured by Borchardt is 0.010 sq cubits, or about 0.2%.

I do not analyze here the methods employed by the ancient architect or
mathematician to obtain these equal areas.

In the ancient literature this is known as the infamous "squaring of the
circle" problem.

Ernest Moyer