Hi,
George Sarton wrote:
This is quite true. The subdivisions, as they appear on cubit rods, are often glossed over, and don't necessarily correspond to the length of an ideal digit as it is understood generally, usually 0.729 inches or 0.7291666667 inches, sixteen of which make up an Egyptian or Roman foot, eighteen of which make up a Northern or Saxon foot. In fact this digit is usually understood as a twentieth part of a remen, and not as a regular subdivision of a cubit at all. The remen itself is understood to be related to the Egyptian royal cubit by way of the square root of two. So a square with a side of a remen would have a diagonal measuring one Egyptian royal cubit.
George Sarton continues:
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There seems to be a pattern in the irregularity of the markings, as Lepsius, quoted by Sarton, observes: On his plate I, "the first sixteen digits are 18.75 mm. long, the eight following (17th to 24th), 17.19 mm long, the last four, 21.87 mm long."
Lepsius, Richard, 1865, Druckerei der Königlichen Akademie der Wissenschaften,
Die alt-aegyptische Elle und ihre Eintheilung - Google Play Books
Jomard also observes this: "On observe que les 28 divisions ne sont pas égales entr'elles. Du côté gauche, les quatre premières sont plus grandes ; celles qui suivent sont plus petites." ("We observe that the 28 divisions are not equal to each other. On the left side, the first four are larger; those that follow are smaller").
Jomard, E.f., 1822, Description d'un étalon métrique, orné d'hiéroglyphes, découvert dans les ruines de Memphis
Are these divisions supposed to be used independently from the others, or in various combinations? If these irregular lengths are deliberate and reasonably precise, what do they mean? I had a look for irrational ratios that might link them, following a hunch that the names of the gods above each digit represented something mathematically ineffable, or irrational, hence belonging to the realm of the divine. It is possible that various irrational square roots, pi and Phi squared might explain the divisions, as per the diagram below:
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The longest, blue, section is the one with the gradually smaller subdivisions on each digit, from right to left, (these subdivisions are not represented in this diagram). It is 300 mm long. This is exactly 7/4 of the total length of the cubit. If the total length of the cubit were divided into 28 equal parts, the digits from this section would be very close to that average, in length. The green and pink sections combine to make the remainder a third of the cubit. The question is: why are the digits from the green and pink sections bigger and smaller than an average 28th part of the total length? The green section on its own corresponds to a sixth of the cubit length. This is surprising since the cubit is in fact divided into not six but seven palms of four digits. The remaining six palms (the total length of the cubit minus the green section) make up five sixths of the cubit. This rod seems to be both an ordinary Egyptian cubit of 6 palms, or 24 digits, with one palm equal to the green section, and a Royal cubit, of 7 palms, or 28 digits, with 3 palms equal to the green and pink sections combined, and the remaining 4 palms, equal to the blue section.
The accuracy of these divisions came as a surprise, and suggests these divisions are deliberate. Furthermore, the accuracy of a pi ratio between the green and pink sections was also surprising, and suggests deliberate intent. If half the green section were the diameter of a circle (i.e. two green digits of 21.89 mm), the pink section would correspond to the circle's circumference. This is allowed by the juxtaposition of ordinary 6 palm cubit with royal 7 palm cubit. Usually, the 7 palm royal cubit is considered a palm longer than the ordinary 6 palm cubit, but maybe here we can consider the ordinary and royal cubits simply two different ways of dividing the same cubit length, in this case of very nearly 525 mm. Seven may seem a surprising number of divisions for a measuring system, when six or ten or twelve would make more sense. But perhaps the reason there are seven here, is that, combined with the six divisions on the exact same length, this pi connection can be made between one regular Egyptian cubit palm (one sixth of the total length) and 3 palms of the Royal Egyptian cubit, divided into two sections, one being the ordinary palm, and the remaining two being the royal palms. Seven is an important lunar number of course, there being four seven day weeks in a month. Also 12 lunations are a lunar year, and minus the five holiday days of the Egyptian year, gives 350 days. Two of these cycles make 700 days. On the other hand, the number six is more easily associated with the sun, there being 6 x 6 x 10 = 360 days, plus the five holiday days, in a solar year. Juxtaposing seven divisions onto six divisions may be about balancing solar and lunar cycles.
On the diagram, I've tried to show that if half the green palm, i.e. ordinary palm, corresponded to the diameter of a circle, then the pink section, i.e. two palms of the royal cubit, would correspond to the circle's circumference. I was quite surprised to find that there should be such a connection allowed by the juxtaposition of ordinary and royal cubits, specifically one ordinary palm and two royal palms, as part of the same total length. Take the pink section as 137.492985 mm, divided by pi, this gives 43.765376 mm, which is half the green area, or half an ordinary palm. Is the green section's purpose to measure the diameter or radius of a circle, and the pink section to measure the circumference? The 21.88262 mm length, a quarter of the green section, one ordinary palm, can also be linked to the pink section via the square root of two (approximately): 21.885 x √2 x 10 x 4/9 = 137.556
Furthermore, the pink section relates to the blue via 5/6 and Phi squared, approximately. The presence of 5/6 with Phi squared and pi points to the equivalence of these two irrational ratios via the fraction 5/6. Phi is derived from the square root of five, as is Phi squared, so I have written Phi squared as (√5 + 3)/2 on the diagram above. Phi squared can be approximated as π x 5/6, but also as 100 / (√2 x 27). the square root of two is the ratio between the sides and diagonals of a square. Perhaps it acts as a single key to move between measures on a geometrical shape that are incommensurable, alternately bridging the measure of the diameter and the circumference on a circle, or the side and the diameter of a square, or parts of a pentagon. Used in the right way, perhaps the sections of the rod which correspond to Phi squared can be used interchangeably to measure the incommensurable parts of a geometrical figure.
I wondered if the green digits given as 21.89 mm, might in fact more accurately be thought of as 21.885048627 mm, as 21.885048627 x π x 5/6 x (√5 + 3) = 300.0001133, and 300 mm represent the pink section. The total length of the cubit then becomes closer to 525 mm, and a 28th part of this becomes close to a regular digit in the pink section. 525 millimetres convert today using 39.3700787402 metres to the inch, to 20.66929 inches.
The values given for these lengths by Lepsius are naturally in millimetres, and I was not expecting the values in this unit to correspond to anything interesting in and of themselves. I was really just looking for ratios to link the palms and digits. The theoretical link between the metre and the cubit is well known though, as a cubit is often held to be π /6 = 0.523599 metres, or you could call it 20 x √2 / 54 = 0.523783 metres, (i.e. 20.614125 and 20.62137 inches respectively). The total length of this particular cubit is given as 524.98 mm, which is quite close to these values, but if the theory of the total length of the Egyptian royal cubit relating to pi and the square root of two is going to convince in this particular case, then we would have to assume that the metre that the ancient Egyptians were using, if they did in fact use it, was very slightly shorter, maybe around the 39.265 inch mark.
In any case, I was not expecting to find that a value close to the one Lepsius gives for what I have called the pink section, i.e. two royal palms, to be 7,000 / (√2 x 36) = 137.49299 millimetres. Lepsius gives 137.52 mm. This 36 may well correspond to the 300 mm of the longest, blue section, multiplied by 6/5, or to the degrees of a circle. The 21.88262 mm , very close to the length Lepsius gives for the green section, one ordinary palm, is 7,000 / (√2 x π x 72). consequently the 300 mm of the longest section, four royal palms, are Phi² x 5/6 x 7,000 / (√2 x 360). Furthermore, the number seven itself, which is the number of palms in the royal Egyptian cubit, can be approximated by: (21.883864 x 2 x π )² x Phi² x √2 / 10,000. If the pink section (two royal palms), is the circumference of the circle of which the diameter is half the green section, then this same circumference can also be thought of as equal to two sides of a square, of which the diagonal measures 7,000 / 36 mm. And so the pink section could be thought of, in millimetres, as 21.883864² x π ² x Phi² x 4 / 360 = 137.49012, or half the green section multiplied by pi and Phi divided by 6, the result of which is squared and divided by 10.
Thanks,
Melissa
George Sarton wrote:
Sarton, George. “On a Curious Subdivision of the Egyptian Cubit.” Isis, vol. 25, no. 2, 1936, pp. 399–402. JSTORQuote
There are many publications on the Egyptian cubit but most of them are naturally devoted to the determination of its length length and its possible relationships with other standards of the ancient world.
This is quite true. The subdivisions, as they appear on cubit rods, are often glossed over, and don't necessarily correspond to the length of an ideal digit as it is understood generally, usually 0.729 inches or 0.7291666667 inches, sixteen of which make up an Egyptian or Roman foot, eighteen of which make up a Northern or Saxon foot. In fact this digit is usually understood as a twentieth part of a remen, and not as a regular subdivision of a cubit at all. The remen itself is understood to be related to the Egyptian royal cubit by way of the square root of two. So a square with a side of a remen would have a diagonal measuring one Egyptian royal cubit.
George Sarton continues:
Quote
According to other examples described by Lepsius (1866) and Schiapareli (1927), the royal Egyptian cubits were divided into seven palms of four digits each, but the fact upon which I wish to draw attention is the very curious subdivision of the digits.
The first (beginning from the right) is divided into two parts, the second into three, and so on, until the fifteenth which is divided into sixteen. There is one example mentioned by Lepsius (his no.5, p.15, 28), wherein the sixteenth digit was divided into seventeenths, but this seems to have been the result of a mistake. In other examples the thirteen remaining digits are not subdivided. It should be noted that as the length of the digit is less than 2 cm, subdivision into sixteen parts reaches almost the practical limit of visibility.
What may have been the purpose of that strange subdivision? Why was it necessary to have ready scales in fractions of digit from the half to the sixteenth? This was probably connected with the Egyptian exclusive interest in fractions of the type 1/n. Their rulers made it possible to determine the actual length indicated by such an expression as one cubit plus one fifth, one eighth and one fourteenth, yet the same purpose might have been attained in a simple manner.
The use of such rulers for the graphical solutions of arithmetical problems cannot be countenanced for the divisions were not precise enough. Indeed as Lepsius remarked (1866, 18) the subdivisions of the cubit are sometimes unequal and the digits have not all the same length. In the schematic example on his plate I, the first sixteen digits are 18.75 mm. long, the eight following (17th to 24th), 17.19 mm long, the last four, 21.87 mm long. Indeed various characteristics of the cubits preserved in our museums suggest that they were objects meant for ceremonial rather than practical use. For example, in the beautiful cubit of the museum of Torino, each digit is associated with a god whose name is written above it. Moreover, various cubits being made of stone were too heavy and fragile for convenience. The subdivision of the digits into a number of parts increasing one by one from two to sixteen, was thus more probably of theoretical than of practical importance. Some of their arithmetical problems, whether published or unpublished, may possibly throw light on this mystery.
There seems to be a pattern in the irregularity of the markings, as Lepsius, quoted by Sarton, observes: On his plate I, "the first sixteen digits are 18.75 mm. long, the eight following (17th to 24th), 17.19 mm long, the last four, 21.87 mm long."
Lepsius, Richard, 1865, Druckerei der Königlichen Akademie der Wissenschaften,
Die alt-aegyptische Elle und ihre Eintheilung - Google Play Books
Jomard also observes this: "On observe que les 28 divisions ne sont pas égales entr'elles. Du côté gauche, les quatre premières sont plus grandes ; celles qui suivent sont plus petites." ("We observe that the 28 divisions are not equal to each other. On the left side, the first four are larger; those that follow are smaller").
Jomard, E.f., 1822, Description d'un étalon métrique, orné d'hiéroglyphes, découvert dans les ruines de Memphis
Are these divisions supposed to be used independently from the others, or in various combinations? If these irregular lengths are deliberate and reasonably precise, what do they mean? I had a look for irrational ratios that might link them, following a hunch that the names of the gods above each digit represented something mathematically ineffable, or irrational, hence belonging to the realm of the divine. It is possible that various irrational square roots, pi and Phi squared might explain the divisions, as per the diagram below:
The longest, blue, section is the one with the gradually smaller subdivisions on each digit, from right to left, (these subdivisions are not represented in this diagram). It is 300 mm long. This is exactly 7/4 of the total length of the cubit. If the total length of the cubit were divided into 28 equal parts, the digits from this section would be very close to that average, in length. The green and pink sections combine to make the remainder a third of the cubit. The question is: why are the digits from the green and pink sections bigger and smaller than an average 28th part of the total length? The green section on its own corresponds to a sixth of the cubit length. This is surprising since the cubit is in fact divided into not six but seven palms of four digits. The remaining six palms (the total length of the cubit minus the green section) make up five sixths of the cubit. This rod seems to be both an ordinary Egyptian cubit of 6 palms, or 24 digits, with one palm equal to the green section, and a Royal cubit, of 7 palms, or 28 digits, with 3 palms equal to the green and pink sections combined, and the remaining 4 palms, equal to the blue section.
The accuracy of these divisions came as a surprise, and suggests these divisions are deliberate. Furthermore, the accuracy of a pi ratio between the green and pink sections was also surprising, and suggests deliberate intent. If half the green section were the diameter of a circle (i.e. two green digits of 21.89 mm), the pink section would correspond to the circle's circumference. This is allowed by the juxtaposition of ordinary 6 palm cubit with royal 7 palm cubit. Usually, the 7 palm royal cubit is considered a palm longer than the ordinary 6 palm cubit, but maybe here we can consider the ordinary and royal cubits simply two different ways of dividing the same cubit length, in this case of very nearly 525 mm. Seven may seem a surprising number of divisions for a measuring system, when six or ten or twelve would make more sense. But perhaps the reason there are seven here, is that, combined with the six divisions on the exact same length, this pi connection can be made between one regular Egyptian cubit palm (one sixth of the total length) and 3 palms of the Royal Egyptian cubit, divided into two sections, one being the ordinary palm, and the remaining two being the royal palms. Seven is an important lunar number of course, there being four seven day weeks in a month. Also 12 lunations are a lunar year, and minus the five holiday days of the Egyptian year, gives 350 days. Two of these cycles make 700 days. On the other hand, the number six is more easily associated with the sun, there being 6 x 6 x 10 = 360 days, plus the five holiday days, in a solar year. Juxtaposing seven divisions onto six divisions may be about balancing solar and lunar cycles.
On the diagram, I've tried to show that if half the green palm, i.e. ordinary palm, corresponded to the diameter of a circle, then the pink section, i.e. two palms of the royal cubit, would correspond to the circle's circumference. I was quite surprised to find that there should be such a connection allowed by the juxtaposition of ordinary and royal cubits, specifically one ordinary palm and two royal palms, as part of the same total length. Take the pink section as 137.492985 mm, divided by pi, this gives 43.765376 mm, which is half the green area, or half an ordinary palm. Is the green section's purpose to measure the diameter or radius of a circle, and the pink section to measure the circumference? The 21.88262 mm length, a quarter of the green section, one ordinary palm, can also be linked to the pink section via the square root of two (approximately): 21.885 x √2 x 10 x 4/9 = 137.556
Furthermore, the pink section relates to the blue via 5/6 and Phi squared, approximately. The presence of 5/6 with Phi squared and pi points to the equivalence of these two irrational ratios via the fraction 5/6. Phi is derived from the square root of five, as is Phi squared, so I have written Phi squared as (√5 + 3)/2 on the diagram above. Phi squared can be approximated as π x 5/6, but also as 100 / (√2 x 27). the square root of two is the ratio between the sides and diagonals of a square. Perhaps it acts as a single key to move between measures on a geometrical shape that are incommensurable, alternately bridging the measure of the diameter and the circumference on a circle, or the side and the diameter of a square, or parts of a pentagon. Used in the right way, perhaps the sections of the rod which correspond to Phi squared can be used interchangeably to measure the incommensurable parts of a geometrical figure.
I wondered if the green digits given as 21.89 mm, might in fact more accurately be thought of as 21.885048627 mm, as 21.885048627 x π x 5/6 x (√5 + 3) = 300.0001133, and 300 mm represent the pink section. The total length of the cubit then becomes closer to 525 mm, and a 28th part of this becomes close to a regular digit in the pink section. 525 millimetres convert today using 39.3700787402 metres to the inch, to 20.66929 inches.
The values given for these lengths by Lepsius are naturally in millimetres, and I was not expecting the values in this unit to correspond to anything interesting in and of themselves. I was really just looking for ratios to link the palms and digits. The theoretical link between the metre and the cubit is well known though, as a cubit is often held to be π /6 = 0.523599 metres, or you could call it 20 x √2 / 54 = 0.523783 metres, (i.e. 20.614125 and 20.62137 inches respectively). The total length of this particular cubit is given as 524.98 mm, which is quite close to these values, but if the theory of the total length of the Egyptian royal cubit relating to pi and the square root of two is going to convince in this particular case, then we would have to assume that the metre that the ancient Egyptians were using, if they did in fact use it, was very slightly shorter, maybe around the 39.265 inch mark.
In any case, I was not expecting to find that a value close to the one Lepsius gives for what I have called the pink section, i.e. two royal palms, to be 7,000 / (√2 x 36) = 137.49299 millimetres. Lepsius gives 137.52 mm. This 36 may well correspond to the 300 mm of the longest, blue section, multiplied by 6/5, or to the degrees of a circle. The 21.88262 mm , very close to the length Lepsius gives for the green section, one ordinary palm, is 7,000 / (√2 x π x 72). consequently the 300 mm of the longest section, four royal palms, are Phi² x 5/6 x 7,000 / (√2 x 360). Furthermore, the number seven itself, which is the number of palms in the royal Egyptian cubit, can be approximated by: (21.883864 x 2 x π )² x Phi² x √2 / 10,000. If the pink section (two royal palms), is the circumference of the circle of which the diameter is half the green section, then this same circumference can also be thought of as equal to two sides of a square, of which the diagonal measures 7,000 / 36 mm. And so the pink section could be thought of, in millimetres, as 21.883864² x π ² x Phi² x 4 / 360 = 137.49012, or half the green section multiplied by pi and Phi divided by 6, the result of which is squared and divided by 10.
Thanks,
Melissa