The Luxor Ellipse

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 75-76.

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.

My Procedure

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 Measured Values

Borchardt's measurements are shown below in centimeters, then in calculated cubits.

Vertical center of the drawing
from the right end proceeding left:

Horizontal center of the drawing
from the top proceeding down.

  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
intersection with the ellipse
across bottom from left to right

Left vertical



Left bottom intersection

36.0 0.687

Center bottom intersection of
ellipse with vertical center line

72.0 1.374

Right bottom intersection

109.0 2.081

Right vertical

145.0 2.768


Rectangle vertical distances and
Intersection with ellipse transverse lines

rectangle line
rectangle line
  cm. cubits cm. cubits


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.

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My Graphical Measures

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 photo-image-to-wall-drawing 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
from inches
V = 6.624
H = 6.55

Horizontal Lines

Top most (ellipse)






Top rectangle line

6.82 to 6.83





Bottom rectangle line

1.54 to 1.55





Bottom most (ellipse)






Vertical Lines

Right most (ellipse)






Right vertical line

9.89 to 9.91





Center vertical line

5.56 to 5.57





Left vertical line

1.17 to 1.20





Left most (ellipse)






Transverse intersections with respective rectangle lines

(Intersections with drawn ellipse are virtually the same.)

Top horizontal


H - 3.390

V - 6.830








H - 7.800

V - 6.820







Bottom horizontal


H - 3.350

V - 1.550








H - 7.760

V - 1.550







Left vertical


V - 5.350

H - 1.180








V - 3.030

H - 1.190







Right vertical


V - 5.370

H - 9.900








V - 3.010

H - 9.890








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.

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Comparison of Measured Values

Borchardt used four reference points. The first was the vertical center of the ellipse, starting on the right-most end and proceeding left. The second was the bottom horizontal for the rectangle, starting at the left-most 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
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.

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Major Drawing Dimensions

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.

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Borchardt's First Geometric Solution

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 work-drawing. (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 eye-level.

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 AA1, BB1, CC1, and DD1 = 1/4 AB = a/2, and on the short sides, the sections AA2, BB2, CC2 and DD2 = 1/6 AB = a/3. The centers of the oval curved line going through A2A1B1B2C2C1D1D2 lie firstly on the middle/center of the long sides in x, xl and secondly on the intersection of the lines xC2 and x1B2 or x1A2 and xD2.

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  x1 and x2 denote his location of the centers of the large circles, while y1 and y2 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 x1 and x2. In fact, the amount of displacement appears to be half the span distance. (The radii of his large circles are shown by xC2 and x1B2 or x1A2 and xD2.) Thus it would seem that Borchardt "forced" the drawing to cause x and x1 to fall directly on the rectangle horizontal lines. (Or he may not have had drawing resolution that would permit him to see the differences.)

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Radii of the Two Large Circles

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 half-span 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 one-half 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.

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Radii of the Two Small Circles

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.

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The Transverse Lines

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 one-half 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
Bottom Horizontal 36.02 36.85 36.52 35.60 36.25

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 AA1, BB1, CC1, and DD1 = 1/4 AB = a/2 = 36.25 cm.

and on the short sides, the sections AA2, BB2, CC2 and DD2 = 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%.

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Comparison of Areas

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.

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Borchardt's Other Solutions

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) d2 instead of (Pi/4) d2]. (64/81 = 82/92)

In our ellipse-problem, 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 [(2-2/7) X (3-2/7) = 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 xC2, xD2 and x1A2, x1B2 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.

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The Description of Sylvia Cuchoud

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 237-41] 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.

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I shall now go on to a more rigorous analytical discussion of this construction.

Refer to following Paper.