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.

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.

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.

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

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 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		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
Top horizontal	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
Bottom horizontal	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 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 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
Bottom horizontal	109.39	109
Right vertical	24.479	24.5
Right vertical	64.189	64
Left vertical	24.816	24.5
Left vertical	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.

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

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.

Borchardt gave a mean value of 0.75 cm, 1.905 inches, 0.094 cubits. This shows the error in his approximations.

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 italics.

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, x_l 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 xD_2.)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.)

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.

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.

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.)

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 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.

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%.

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.

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.

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, xD₂and 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.

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:

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.

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	0.0	0
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
	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

Location	Degrees
Upper left	89.69
Upper right	89.72
Lower left	90.39
Lower right	90.23

	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

	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

	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