In 1896 Ludwig Borchardt, a famous Egyptologist from Germany, published his discovery of a drawing of a mathematical ellipse on a wall in the Temple of Luxor. See Zeitschrift für ägyptische Sprache und Altertumskunde (Berlin/Leipzig), Journal of the Egyptian Language and Archeology, Volume 34, 1896, pgs 7576.
Borchardt provided two drawings. First, the ellipse as he measured it on the Temple wall, and second, his interpretation of construction. I reproduce his published drawing by photocopy. (Click on the drawing below for a larger image.)
Borchardt offered three solutions for the manner in which the Geometrician created the drawing. Curiously, he did not offer a theoretical discussion of how the Geometrician may have arrived at this goal. This step is important to understand the level of knowledge and mathematical skill of the ancient Egyptians. Perhaps he felt limited by journal space in reporting his discovery, and did not wish to carry examination beyond the level he shows in his report. I am unaware that anyone has published a more theoretical examination.
First, I report Borchardt's measured values, in centimeters, as he shows them on his drawing. I then calculate the values in Royal Egyptian Cubits. I use a conversion of 52.388 cm/cub. (Borchardt reported measurement resolution within 0.1 cm = 1 mm.)
Second, I report my measures from an Expanded Drawing. I copied Borchardt's published drawing into QuickCad software. This provided a scale expansion that permitted coordinates to be measured to within +/ 0.01 English inches. For comparing results I used a conversion factor of 2.54 cm/inch. I then report those values. I use a conversion factor of 20.625 inches/cubit to calculate my measured results in cubits.
Third, I compare Borchardt's values with those I measured from the Expanded Drawing.
Fourth, I then provide a table showing major values of the dimensions from the Expanded Drawing. I examine the angles, and horizontal and vertical accuracy of the lines.
Fifth, I then turn to Borchardt's first solution. I show how his idealized suggestion deviates from a more accurate rendering of the drawing as created by the ancient Geometrician. Such close examination is necessary to more concretely judge the theoretical aspects of the drawing. I include sections on"
Sixth, I compare areas to determine how well the ancient Geometrician reproduced equal elliptical and rectangular areas.
Seventh, I examine Borchardt's other proposed solutions.
Eighth, I examine remarks published by Sylvia Cuchod on how the drawing was constructed.
Ninth, I then enter into a discussion of the theoretical aspects of the drawing, and what this implies for the level of mathematical knowledge of the ancient Egyptians.
Borchardt's measurements are shown below in centimeters, then in calculated cubits.
Vertical center of the drawing 
Horizontal center of the drawing 


cm.  cubits  cm.  cubits  
Right ellipse end  0.0  0  Top ellipse  0.0  0 
Right rectangle vertical  7.0  0.134  Top rectangle  7.5  0.143 
Vertical Center Line  79.5  1.158  Bottom rectangle  96.0  1.832 
Left rectangle vertical  152.2  2.905  Bottom ellipse  103.5  1.976 
Left ellipse end  159.5  3.045 
Location of rectangle vertical lines and 


Left vertical 
0.0  0 
Left bottom intersection 
36.0  0.687 
Center bottom intersection of 
72.0  1.374 
Right bottom intersection 
109.0  2.081 
Right vertical 
145.0  2.768 
Rectangle vertical distances and 


Left rectangle line 
Right rectangle line 

cm.  cubits  cm.  cubits  
Top  0.0 
0  0.0  0 
Top intersection  24.5  0.468  24.5  0.468 
Bottom intersection  64.0  1.221  64.0  1.221 
Bottom  89.0  1.700  88.5  1.689 
The isolated value of 29.5 toward the right end of the drawing may be a reference measure for Borchardt's benefit. It does not appear to be related to the drawing dimensions.
From my Expanded Drawing I found some difference between the vertical and horizontal scales. This may have been due to the Geometrician's original drawing, to Borchardt in making his copy, in conversion by his printer, or to my photocopy. The differences are most likely from the last. From the expanded view with coordinates within +/ 0.01 inches, and multiplying the photoimagetowalldrawing ratio of approximately 6.6 permitted measure of the Borchardt drawing to within +/ 0.07 inches, +/ 0.18 cm, or about +/ 1.8 mm. (My drawing scale was 3.15 inches per cubit.)
I found the following measures, with the horizontal and vertical numbers appropriately scaled. The multipliers are shown. My measured values are compared to Borchardt's report below. Where measurable I give the range of values for the respective differences in the location of the lines in the second column. I use the means in the third column to calculate the locations. I show resolution to three decimal places.
All measures are from the lower left drawing zero point. All points were then normalized to ellipse reference lines in order to calculate distances. The reference lines are the top most ellipse at 7.25 drawing inches, and the right most ellipse at 10.34 inches.
Respective Feature 
Inch measures from QuickCad drawing 
Inches from reference line 
Drawing centimeters calculated from inches 
Actual Centimeters Multiplier V = 6.624 H = 6.55 


Horizontal Lines 

Top most (ellipse) 
7.250 
0.000 
0.000 
0.000 

Top rectangle line 
6.82 to 6.83 
6.825 
0.425 
1.080 
7.151 
Bottom rectangle line 
1.54 to 1.55 
1.550 
5.700 
14.478 
95.902 
Bottom most (ellipse) 
1.090 
6.160 
15.646 
103.642 

Vertical Lines 

Right most (ellipse) 
10.340 
0.000 
0.000 
0.000 

Right vertical line 
9.89 to 9.91 
9.900 
0.440 
1.118 
7.320 
Center vertical line 
5.56 to 5.57 
5.565 
4.775 
12.129 
79.442 
Left vertical line 
1.17 to 1.20 
1.185 
9.155 
23.254 
152.312 
Left most (ellipse) 
0.750 
9.590 
24.359 
159.549 

Transverse intersections with respective rectangle lines (Intersections with drawn ellipse are virtually the same.) 

Top horizontal 
Left 
H  3.390 V  6.830 
6.950 0.420 
17.653 1.067 
115.627 7.068 
Right 
H  7.800 V  6.820 
2.540 0.430 
6.452 1.092 
42.261 7.233 

Bottom horizontal 
Left 
H  3.350 V  1.550 
6.990 5.700 
17.755 14.478 
116.295 95.902 
Right 
H  7.760 V  1.550 
2.580 5.700 
6.553 14.478 
42.922 95.902 

Left vertical 

Top 
V  5.350 H  1.180 
1.900 9.160 
4.826 23.266 
31.967 152.392 

Bottom 
V  3.030 H  1.190 
4.220 9.150 
10.719 23.241 
71.002 152.228 

Right vertical 

Top 
V  5.370 H  9.900 
1.880 0.440 
4.775 1.118 
31.630 7.323 

Bottom 
V  3.010 H  9.890 
4.240 0.450 
10.770 1.143 
71.340 7.487 
Except for the rectangle vertical lines the several values across the drawing for each line, both horizontal and vertical, were all within my measurement error. That is, they do not show variability; they are truly horizontal and vertical. Some verticals differed from top to bottom by only 0.02 and 0.03 inches respectively. The horizontal lines were held parallel to one another, and to the published drawing edge. How much of this lack of variability and fine parallelism may be due to the original Egyptian drawing or Borchardt's copy we cannot say.
I was surprised by this remarkable accuracy. The drawing was not a sloppy rough sketch, but a careful rendering, contrary to the opinion voiced by Borchardt below. If Borchardt maintained a faithful reproduction, the original drawing was both accurate and well preserved. The Geometrician was careful to draw exact arcs and lines, with proper dimensions to simulate a 3 X 2 cubit ellipse. Of course, it is possible that Borchardt idealized the drawing but his measures shows that he was attempting to report faithfully.
Borchardt used four reference points. The first was the vertical center of the ellipse, starting on the rightmost end and proceeding left. The second was the bottom horizontal for the rectangle, starting at the leftmost end and proceeding right. The third and fourth (left and right) were vertical from the top down, using the maximum height for the ellipse, and the upper horizontal line for the rectangle. Therefore, all measurements must be adjusted to bring them into a common reference frame to make comparisons.
Respective Feature  My Calculated Values 
Borchardt's Values 

All dimensions are in centimeters  
Horizontal Lines  
Top most (ellipse)  0.000  0 
Top rectangle line  7.151  7.5 
Bottom rectangle line  95.902  96 
Bottom most (ellipse)  103.642  103.5 
Vertical Lines  
Right most (ellipse)  0.000  0 
Right vertical line  7.320  7.2* 
Center vertical line  79.442  79.5 
Left vertical line  152.312  152.2 
Left most (ellipse)  159.549  159.9 
Ellipse intersections with respective rectangle lines  
Bottom horizontal  36.017  36 
109.39  109  
Right vertical  24.479  24.5 
64.189  64  
Left vertical  24.816  24.5 
63.851  64  
*This value is calculated from Borchardt's numbers. 
Thus it would appear that Borchardt's values are very close to the measurements I made from the Expanded Drawing, agreeing within a few millimeters in all cases.
Below are the calculated results from my measured values.
Property:  Centimeters  Cubits 

Ellipse right to left  159.549  3.046 
Ellipse top to bottom  103.642  1.978 
Rectangle horizontal distance  144.992  2.768 
Rectangle vertical distance  88.751  1.694 
Right span  7.320  0.140 
Left span  7.237  0.138 
Top span  7.151  0.137 
Bottom span  7.740  0.148 
Right ellipse to center line  79.442  1.516 
Center to left ellipse  80.107  1.539 
Right rectangle to center line  72.122  1.377 
Center to left rectangle line  72.870  1.391 
I also measured the angles at each of the rectangle corners:
Location  Degrees 

Upper left  89.69 
Upper right  89.72 
Lower left  90.39 
Lower right  90.23 
These values sum to 360.03 degrees. This indicates the amount of error in my measures from the expanded drawing, 3 parts out of 36,000.
The rectangle has verticals that slope slightly inward from top to bottom on both right and left sides. The top horizontal line of the rectangle has a measured length from my reproduction of 8.74 inches (22.20 cm) while the bottom has 8.69 inches (22.07 cm). Multiplied to the actual drawing width this would be 145.41 cm (2.776 cub.) and 144.56 cm (2.759 cub.) respectively. This difference is 0.85 cm, or 8.5 mm.
I also measured the verticality of the center line. I obtained a slight slope left to right from top to bottom, with angles of 89.72 deg and 90.14 deg.
The ellipsoid is slightly off center from the vertical center dividing line. From the right end to the center line is 1.527 cubits; from the left to the center is 1.534. This is a difference of only 0.007 cubits, or about 0.14 inches, 3.6 mm.
The corresponding ellipse vertical values are 0.984 cubits from top to center and 0.987 from center to bottom. This difference is even smaller than that of the horizontal error, 0.003 cubits, or 1.5 mm.
The measured span (spanne) distances from the Figure are
right: 0.140,
left: 0.138,
top: 0.137, and
bottom: 0.148 cubits.
Borchardt gave a mean value of 0.75 cm, 1.905 inches, 0.094 cubits. This shows the error in his approximations.
We are now ready to examine Borchardt's three proposed solutions.
His solutions are not analytical in a mathematical sense, merely constructional in a geometric sense. The figure below is his first interpretation of the construction.
Borchardt then offered the following description. I have corrected obvious errors in his designations. My editorial remarks are in bold.
Finally, a further drawing deserves to be mentioned here, although this one can hardly be taken for a workdrawing. (He was referring to previous drawings not discussed here. He means an academic drawing, clearly intended for teaching, not a temple construction drawing.)
In the temple of Luxor, on the eastern wall of the eastern room that starts from the late (BC) Coptic church, opposite the door, a construction of an elliptic oval is scratched in the wall at eyelevel.
As auxiliary lines for the making of this figure, the sides of a horizontal rectangle have been used, of which the corners have been cut away by symmetrically placed transverse lines. (Thus Borchardt appears to limit the purpose of the rectangle to construction of the ellipse. He does not work out the implications of their equal areas. Refer to discussion below.)
The construction is approximately like this:
In the rectangle ABCD, of which the lengths of the sides are AB = DC = 2a = 2 + 1/2 + 1/4 cubits, and
AD = BC = 2b = 1 + 2/3 cubits (of each ca 53 cm),
have been measured off on the long side, going forth
from the corners, the sections AA_{1}, BB_{1}, CC_{1}, and
DD_{1} = 1/4 AB = a/2,
and on the short sides, the sections AA_{2}, BB_{2}, CC_{2} and
DD_{2} = 1/6 AB = a/3.
The centers of the oval curved line going through
A_{2}A_{1}B_{1}B_{2}C_{2}C_{1}D_{1}D_{2} lie firstly on the middle/center of
the long sides in x, x_{l} and secondly on the intersection of the lines
xC_{2} and x_{1}B_{2} or x_{1}A_{2} and xD_{2}.
The axes of the thus generated curve are approximately 2 and 3 cubits, the centers of the small arcs of a circle are approximately 2 cubits removed from each other. This is the first possibility to explain the construction.
I shall now more fully explain the interpretations offered by Borchardt. To make his first proposal clear I provide his drawing with the appropriate circles superimposed.
His drawing shows that the Figure is not a true ellipse, but is composed of two large circles and two small circles in tangent construction to one another. The designations x_{1} and x_{2} denote his location of the centers of the large circles, while y_{1} and y_{2} show his location of the centers of the small circles.
We can clearly see that exact circles compose the top and bottom arcs of the ellipsoid. By careful scrutiny (and expanded view) we can see that Borchardt's circle distances are not quite correct. His large circles are slightly too small. As proposed by Borchardt they fall exactly on the Figure at the maximum points but fail to do so throughout the arc. If they were made minutely larger the arcs would identically fall on the curves he copied from the Luxor wall, thus more correctly showing that the ellipse was composed of the circular construction he proposed. Then the center of the circles would move slightly downward and upward, and hence off x_{1} and x_{2}. In fact, the amount of displacement appears to be half the span distance. (The radii of his large circles are shown by xC_{2} and x_{1}B_{2} or x_{1}A_{2} and xD_{2.) }Thus it would seem that Borchardt "forced" the drawing to cause x and x_{1} to fall directly on the rectangle horizontal lines. (Or he may not have had drawing resolution that would permit him to see the differences.)
In adjustment by eye to my copy of Borchardt's published Figure, and using a multiplier of 6.6, I determined the radii of the two large circles.
Borchardt's radii values are the distance from the rectangle horizontal lines to the respective ellipse maximum vertical points. The Expanded Drawing values are from the halfspan distances to the respective ellipse maximum vertical points. I report these in actual wall drawing centimeters and cubits.
Bottom of Rectangle (or half span) to Top of Ellipse 
Top of Rectangle (or half span) to Bottom of Ellipse 


Centimeters  Cubits  Centimeters  Cubits  
Borchardt  95.902  1.831  96.491  1.842 
Expanded Drawing  99.772  1.904  99.477  1.899 
Mean of Values  Difference of Values  
Borchardt  96.197  1.837  0.589  0.011 
Expanded Drawing  99.625  1.901  0.295  0.005 
The difference between the two means is 3.43 cm, 0.064 cubits. Even though the Expanded Drawing was adjusted by eye the difference in the two radii is onehalf of that constructed by Borchardt, who assuredly also adjusted by eye. This shows the advantage of modern graphical techniques.
The distance between the two circle centers for Borchardt was the rectangle height, 88.751 cm, 1.694 cubits. The Expanded Drawing value for the two span centers was 95.399 cm, 1.821 cubits.
According to Borchardt the center of the small circles fall on the circumference of his dashed center circle that measures two cubits in diameter, the vertical distance of the ellipsoid, and hence are two cubits distant from one another. (A similar superimposition is shown by mathematicians in analytical dissertations on the construction of true ellipses.) Ideally one would like to see the small circles exactly 1/2 cubit in radius. They then would form a trio of inner circles just tangent to one another, to make up the length of the elliptical figure of three cubits, as I show with the small dashed inner circle. However, they actually are just slightly larger than 1/2 cubit in radius to create the right and left curve of the elliptical figure, as I show. This difference from an ideal three circles tangent to one another shows that the Geometrician was not forcing three tangent circles but was designing to some other criteria.
In adjustment by eye to my copy of Borchardt's published Figure, and using a multiplier of 6.6, the radii of the two small circles are 0.528 cubits. The distance between the two circle centers is 2.016 cubits.
Using the above estimates from eye fit, the following figure shows how the circles actually arrange on the published Figure. They do not quite match, in tangent or in size, to obtain Borchardt's idealistic solution. Note the slight disparity in the lower left. (Because these are eye fits, one could debate the most correct solution.)
My small circle radii are 28.88 cm, 0.551 cubits.
The evidence all points to the fact that his drawn circles do not exactly fit the theoretical drawing model he proposes. My solutions for both the large and the small circles are slightly larger than his.
Thus it would appear that the Geometrician used circle center points somewhat different from those proposed by Borchardt. A question then naturally arises as to the Geometrician's choice. Why did he not use the points proposed by Borchardt? Did he recognize subtleties in his construction to obtain a more accurate simulation of an ellipse? Or did he understand that the circle diameters proposed by Borchardt would not be exactly tangent to one another in his ellipsoid simulation?
We can carry these questions ever further. Did he know an exact mathematical ellipse? If not, why did he simulate one through a simple geometric construction? His method suggests not only that he knew an exact mathematical ellipse, but also that he was conversant in geometric construction methods to produce such simulations.
The Transverse Lines offer a most intriguing insight into the construction of the ellipsoid. Before entering into a discussion of the deeper implications I shall present the graphical data on their location.
Borchardt's defined the distance of onehalf the rectangle width as "a." He then stated that the intersection of the transverse lines with the rectangle top and bottom horizontal lines distant from each end were at a/2. Thus the rectangle horizontal lines were divided into four equal parts. The intersection of the transverse lines with the rectangle left and right edges distant from top and bottom were at a/3.
These ratios are the same as the ratio of the ideal ellipse 3:2.
He specified values
for AB = DC = 2a = 2 + 1/2 + 1/4 cubits = 2.75 cubits = 144.07 cm,
and AD = BC = 2b = 1 + 2/3 cubits = 1.67 cubits = 87.49 cm.
Measured from the Expanded Drawing these are 144.99 and 88.75 respectively.
Hence, measured a = 72.496 cm.
Measured b = 44.375 cm.
From my Expanded Drawing:
Measured Length of Segment in Centimeters  (a/2) 


Line  Left  Left to Center  Center to Right  Right  Range 
Top Horizontal  36.68  36.19  37.18  34.94  36.25 1.31 +0.93 
Bottom Horizontal  36.02  36.85  36.52  35.60 
36.25 0.65 +0.60 
Within construction and measurement tolerance these distances are nearly equal.
Measured Length of Segment in Centimeters  

Line  Top  Center  Bottom 
Left Vertical  24.82  39.03  24.90 
Right Vertical  24.48  39.71  24.56 
His AA_{1}, BB_{1}, CC_{1}, and DD_{1} = 1/4 AB = a/2 = 36.25 cm.
and on the short sides, the sections AA_{2}, BB_{2}, CC_{2} and DD_{2} = 1/6 AB = a/3.
The calculated distance of a/3 from the rectangle half width of 72.496 = 24.164 cm.
Thus we see that the measured values from the Expanded Drawing are slightly higher than Borchardt's proposal, 24.69 cm mean vs 24.16 cm, a difference of about 2%.
We can now more accurately compare the rectangular area with the elliptical area.
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. 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.
Importantly, as we saw above, the ellipsoid is not mathematically pure. It is slightly smaller in area than a true ellipse. Therefore we cannot legitimately compare the two areas according to ideal criteria.
To emphasize the fact that the Figure is not a true ellipse I show one, centered and properly scaled, superimposed upon the drawing. We can see how closely the constructed Figure agrees with a true ellipse.
Clearly, the ancient Geometrician was constructing a figure with a rectangle the same area as the theoretical ellipse. He was not merely using rectangle construction lines to produce an ellipse.
Thus the agreement between the area of the rectangle and a theoretical ellipse is closer from my measure than from Borchardt: a difference of 0.043 sq cub is 0.9 per cent. Since the constructed Figure is not a true ellipse, the actual area of the drawn ellipsoid is somewhat less than the theoretical. This brings the agreement of the areas between the drawn rectangle and the ellipsoid even closer, but we cannot be certain this was the intent of the Geometrician.
This is a representation of the famous "squaring of the circle" problem, discussed many times in the mathematical literature, including the Greek ancient past, except that here it is done with a rectangle and an ellipsoid. This representation predates historic Greek mathematical discussion by a thousand years.
We should consider that a circle is a mathematical special case of the ellipse. Hence the Figure, with expression of equal rectangular and ellipsoid areas, is strongly suggestive of mathematical sophistication that exceeded the later Greek developments.
A second possible explanation of the drawing can not however be excluded. I suggest that it is not impossible that we have here
an attempt to determine the area of an ellipse with the radii
of 1 and 1 1/2 cubit, analogous with the exercise/problem, known from
the London mathematical papyrus, concerning the area
of a circle [(64/81) d^{2} instead of (Pi/4) d^{2}]. (64/81 =
8^{2}/9^{2})
In our ellipseproblem, the area [Pi • a • b = Pi • 1 • 1 1/2 = 4.71
square cubits] would be set to be approximately equal to the area of the rectangle [1 2/3 • 2 3/4 = 4.58 square cubits]
The error could in this case be only 13/471 (0.0276), i.e. ca.1/36
(0.0277).
Borchardt was explaining that the Geometrician was simulating an approximation similar to that used for a circle by the ancient Egyptians, rather than the theoretical ellipse (or circle) formula, to determine the area. Again he does not explain how the Geometrician knew the proper construction methods for equal areas centered on one another. According to this remark Borchardt saw the rectangle constructed with the same approximation as that used for the circle, without theoretical knowledge of Pi, or theoretical knowledge of how to obtain equal areas.
It would also be possible that the area determination would be like this: the ellipse with the diameters of 2 and 3 cubits is equal to a rectangle of which the sides are at every side 1 span (0.75 cm) shorter as the diameters of the ellipse [(22/7) X (32/7) = 4.65 square cubits]. In this case the error would be only 6/471 (0.0127), i.e. ca. 1/78 (0.0128).
Which one of all these possibilities is the right one, I cannot determine
because of the inaccuracy and imperfect preservation of the
drawing. Also the time when it was done is doubtful  a post quem
is provided by the wall itself, on which the drawing is placed. It was
executed after the time of Ramesses II.
He does not explain how the ancient Geometrician would have known this span distance would produce a correct solution to the equal area problem.
He makes no further comments concerning the construction methods in any of these proposals. His xC_{2}, xD_{2 }and x_{1}A_{2}, x_{1}B_{2} define the larger circular arcs. He does not explain construction of the smaller arcs, that we now know were not exactly one cubit in diameter. Nor does he discuss the mathematical relationship of the distances of a/2 and a/3 to the ellipse theoretical ratio. He also does not discuss the significance of fact that the rectangle horizontal lines divide into four (equal) parts, and the vertical lines are spaced at the a/3 distance.
Marshall Clagett, in his comprehensive review of Ancient Egyptian Mathematics, Volume Three, American Philosophical Society, Philadelphia, 1999, gives Borchardt's description and a copy of the Figure. He also mentions the explanation of S. Cuchoud. She stated in Mathematiques egyptiennes : recherches sur les connaissances mathematiques de l'Egypte pharaonique (Paris, 1993), the following:
This one (i.e. this drawing) has been made with the help
of parts of circles with different centers and diameters.
This drawing shows the approximation of an ellipse
and a rectangle which represents, except for a small error (of
approximately 1%), the area of the ellipse.
It could have been defined by analogy with the circle
by the formula:
[ a  (1/9)a ] X [ b  (1/9)b ] = a X b X (64/81) = a X b X (Pi/4)
Whether the rectangle was used as help with construction or as
representation of the area has only little importance: the essence is
that the drawing gives a clear proof that the idea itself of an ellipse
was not foreign to the Egyptian mind. One could thus construct it and in
all probability calculate it. Let us recall moreover that Daressy [Un trace egyptien d'une voute elliptique, ASAE , Vol 8, (1908) pgs 23741]
believed to be able to recognize in the drawing of the construction of a vault,
the arc of an ellipse.
Thus Cuchoud stated what I summarized in the above detailed description.
a: Parts of circles of different centers and diameters.
b: The difference between the two areas is approximately 1%. I calculated 0.9%.
c: She more rigorously defines an equation for the approximation of the area of the ellipse, extended from the circle approximation. Here she merely grants the ancient Egyptians with knowledge of the approximation, not the theoretical understanding.
I cannot agree with her note "of little difference". Whether the rectangle was used as a help in construction of the ellipse, or if it represents the "squaring of the ellipse" is a highly significant difference. She emphasizes the fact of the knowledge of the ellipse, not its mathematical properties. That is poor scholarship.
Her reference is to the construction of the ceiling of the tomb of Ramesses VI, which I discuss in a separate paper.
I shall now go on to a more rigorous analytical discussion of this construction.