Recently I read a very interesting article on the irregular divisions on Egyptian cubit rods, by someone called George Sarton. By coincidence, a day or two afterwards, I heard that name in an episode of Stranger Things, so I looked him up.

George Sarton was a historian of science, in fact a founder of the discipline. From Belgium, he and his family escaped during the first world War, initially to England, and then to America. His 1936 article "On a Curious Subdivision of the Egyptian Cubit" is short and discusses the possible reasons for the irregular divisions on various Egyptian cubit rods.

He begins by writing:

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. (1)

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:

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. (2)

It is true that the digits on an Egyptian cubit rod are always subdivided, from left to right, into 2, 3, 4, 5, 6, 7, 8 parts, and so forth, until you really couldn't fit any smaller subdivisions in. Sarton is probably right to say it relates to the Egyptian way of using number. Yet. as he points out, the quality of the workmanship seems to undermine the attempt at precision. Indeed, in 1822, Jomard, who wrote a short book on this cubit rod, was intrigued by the irregularity of the ruler too, writing:

A la première case répondent deux espaces ou divisions , marquées d'un trait ; à la seconde , trois divisions ; à la 3° , quatre ; à la 4 ° , 6 ; à la 5 ° , 7 ; à la 6° , 8 ; à la 7°, 9 , à la 8°, 10 ; à la 9°, 10 ; à la 10° , 12 ; à la 11°, 12 ; à la 12° , 13 ; à la 13° , 13 ; à la 14°, 14 ; et à la 15°, 16. (6)

The first box corresponds to two spaces or divisions, marked with a line; in the second, three divisions; at the 3rd, four; at 4°, 6; at 5°, 7; at 6°, 8; at 7°, 9, at 8°, 10; at 9°, 10; at 10°, 12; at 11°, 12; at 12°, 13; at 13°, 13; at 14°, 14; and at 15°, 16.

In a more recent article entitled "The use of the ‘ceremonial’ cubit rod as a measuring tool. An explanation", by Fr. Monnier, J.-P. Petit & Chr. Tardy, the authors come to a similar conclusion, about the purpose of the gradually smaller subdivisions on the first sixteen digits, and their seemingly imprecise appearance.

The graduations and associated metrical nomenclature are the most regularly reproduced information on all of the cubit rods. These rods adopt a digital system which consists of dividing the royal cubit into 28 fingers and multiples of fingers.9 The multiples include the palm (4 fingers), the hand’s breath (5 fingers), the fist (6 fingers), the double palm (8 fingers), the small span (12 fingers), the great span (14 fingers), the sacred cubit (16 fingers), the remen cubit (20 fingers), the small cubit (24 fingers) and the royal or pharaonic cubit (28 fingers). Finally, the last fifteen fingers of the graduated part are further subdivided successively into 2, 3, 4, 5, ..., 14, and 16 equal parts. All the subdivisions are finely cut and emphasized with white paint, and are superscripted by their unit fractions written in hieroglyphs.

The submultiples of a finger given in the last fifteen sections are all displayed with their measurements expressed as parts of a finger : r(A)-2, r(A)-3, r(A)-4, r(A)-5, ..., r(A)-15, r(A)-16, which are usually translated in our modern language into fractions : 1/2, 1/3, 1/4, 1/5, ..., 1/15, 1/16.( 3)

The authors go on to explain:

The Egyptian numerical system was fundamentally different in its treatment of numbers less than one, as it used unit fractions to decompose single units into equal parts. A measurement less than one finger was then expressed as 1/2, 1/3, 1/4, 1/5, ... down to 1/16th of a finger, which means in fact that the finger was divided into 2, 3, 4, 5, ... or 16 equal parts. As it was materially impossible to graduate all these measurements in one single section, the Egyptians wrote the different subdivisions on subsequent divisions, one after the other in decreasing order. (4)

They have come up with a practical explanation, involving a second ruler, for these increasingly small subdivisions:

If this cubit rod is used in conjunction with another, or with a simpler ruler subdivided only into whole fingers, the related graduations reveal a noteworthy property. The user first has to position the cubit rod alongside the object to be measured, then hold one side of the ruler against the rest of the cubit. The whole digit lines on this same edge then act as cursors that align against the cubit, either at an existing graduation, or between two graduations (fig. 4). In this last case the periodic offset of the ‘cursor’ from one finger to another on the ruler means that it eventually reaches a location where it coincides exactly with one of the fine cubit’s subdivisions. A reading has to be taken at this coincidence and added to the number of whole digits measured alongside the object.

Practical experimentation shows that this technique is undoubtedly effective, and this can explain the presence and arrangement of the subdivisions. According to our reconstruction, accurate measurement would have certainly required the use of the additional element that we suppose to be a ruler or a second cubit rod, but we can also imagine that a stem or a simple annotated papyrus could serve equally as well, with the benefit that they could be made and marked out by the scribes or artisans using the cubit rod which was available to them. Several similar and plausible scenarios can be envisaged. (5)

This explanation ties in well with the mathematics of Ancient Egypt, as we know them, whereby the equivalent of our decimal notation would have been expressed as 1/2, plus 1/3, plus 1/4, plus 1/5, etc. It also ties in with George Sarton's idea that "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. ", as quoted above. Each digit can be used independently from the others. As to the lack of precision in the execution of the rod markings, both articles conclude that the purpose for the actual rods that have survived may only have been symbolic or ceremonial. While the idea of a ceremonial ruler is intriguing, perhaps even a bit baffling, the idea is put forward to explain one aspect of the rods, but not really discussed further. Were the rods used in ceremonies? Or were they simply designed to combine measure with the divine, in order to overcome the problem of incommensurability in geometry? The irregularity present in the cubit rods, together with the names of gods, could incline us to believe it was all religious nonsense.

We could consider the possibility that our rulers today, with their single tier of regularly spaced markings, or sometimes two tiers, where in the world Metric and Imperial coincide, would seem ridiculously simplistic to an ancient Egyptian. Such a visitor from the past might believe that a measuring system that uses the same units for the diameter and the circumference of a circle, when clearly the two sections cannot belong to the same type of unit, either one being relegated to irrationality by the other, absurd.

George Sarton also refers to a different type of inequality in the digits, not just in the number of their subdivisions but in their length, which seems to be by design not by mistake: "the subdivisions of the cubit are sometimes unequal and the digits have not all the same length". In fact, there is a pattern in the irregularity, 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." 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"). Are these divisions supposed to be used independently from the others?

While Lepsius's book on the subject, Die alt-aegyptische Elle und ihre Eintheilung, has been digitised by Google and is free to read online, it is neither translated into English, nor is his plate 1, which is mentioned by Sarton, available in its entirety. It is meant to be unfolded in the physical book, but we can only see one fold online, showing the first few digits (as the run from right to left).

Jomard drew it (partially), also (he omitted the religious hieroglyphs): (7)

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. Using the figures given by George Sarton in mm, (16 x 18.75) + (8 x 17.19) + (4 x 21.87) = 525. The middle, pink section is (525 x 6/5) - (525 x 4/7), the green section is ((525 x 6/5) - (525 x 4/7)) x 2/π = 87.5352187, and the blue section is ((525 x 6/5) - (525 x 4/7)) x (√5 + 3)/ 2 x 5/6 = 299.98306, which is not quite 300 but very close.

The total length of the rod is close to 525 mm. This could be connected to the number 7 in that 7!, so 7x6x5x4x3x2x1 = 5040, a number referred to by Plato, and 7x6x5x4x3 = 5250.

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.

Lepsius describes the gods that each digit seems to be dedicated to:

Auf der Oberseite der Ellen no 1, 2 5 laufft zu oberst eine Reihe von Gotternamen hin, 28 an Zahl, den 28 Fingerbreiten der Elle entsprechund und wie diese durch Linien von einander getrennt. Es beginnen auf der Turiner Elle (no.1) 10 Gotter der ersten Ordnung, von denem jedoch der dritte (Ka oder Xent) fast unbekannt ist. Dir ordnung ist, bezeichnend fur den Fundort, die Memphitische, nicht die Thebanische, indem sie mit Ra, nich mit Mantu und Atmu beginnt, namlich: Ra, Su, Ka, Seb, Nut, Osiris, Isis, Set, Nephthys und Horus. Darauf folgen die 4 dem Osiriskreise zugehorigen Gotter Amset, Hapi, Kebhsenuf undTumutef.

Die andere Halfte der Elle beginnt mit Thoth, dem ersten Gott der zweiten Gotterordnung und dann ffolgt eine Reihe von Gotternm, die grossstentheils sehr wenig bekannt sind, asl vorlexter aber noch der Pan von Oberagypten. Auff der Elle des Muia ist, offenbar durch eine Unauffmerksamkeit des Schreibers, der Gott Seb ausgelassen, so dass jetzt die Nut mit dem Gott Ka verbunden erscheint. Di dadurch enstandene Lucke hat eine Verschibung der ffolgenden Namen veranlasst, bis zum siebzehnten Gott, hinter welchem des zwanzigste Gott unrichtig eingeschoben ist, obgleich er an seiner richtigen Stelle noch einmal erscheint. Asserdem ist der Name des 17. Gottes unrichtig gescreiben, statt wie aus der Schreibung der elle no 1 und der Elle no. 5 hervorgeht; und der des 22ten statt.

On the upper side of the cubits no. 1, 2, 5 runs a row of god names, 28 in number, corresponding to the 28 finger widths of the cubit and like these separated from each other by lines. 10 gods of the first order begin on the Turin cubit (no.1), of which the third (Ka or Xent) is almost unknown. The order is, indicative of the place where it was found, the Memphite, not the Theban, in that it begins with Ra, not with Mantu and Atmu, namely: Ra, Su, Ka, Seb, Nut, Osiris, Isis, Set, Nephthys and Horus. This is followed by the 4 gods Amset, Hapi, Kebhsenuf and Tumutef, who belong to the circle of Osiris.

The other half of the cubit begins with Thoth, the first god of the second order of gods, and then follows a series of gods, most of whom are very little known, but as an example, the Pan of Upper Egypt. On the cubit of Muia, the god Seb is left out, apparently due to the scribe's inattentiveness, so that the Nut now appears connected with the god Ka. The resulting gap has caused the following names to be shifted, up to the seventeenth god, after which the twentieth god is incorrectly inserted, although he appears again in his correct place. In addition, the name of the 17th god is spelled incorrectly instead of cubit no. 1 and cubit no. 5, as can be seen from the spelling; and that of the 22nd.

Jomard, in his analysis of this cubit rod, also reflects on the presence of gods, one for each digit, and writes:

Il ne serait pas difficile de montrer dans les hiéroglyphes qui remplissent les cases de la première bande , des caractères qui correspondent aux divers dieux du pays, au Soleil , à Osiris , Isis , Horus , Thoth , etc. , ce qui porterait à croire que les 28 doigts de la coudée répondaient eux - mêmes à autant de personnages religieux ou mythologiques, dont les noms avaient été imposés aux vingt - huit jours du mois lunaire. (...)

On ne doit pas être étonné de trouver ici les images des dieux , appliquées à la fois aux jours du mois , et aux divisions de la mesure : chez un peuple aussi attaché à son culte , la religion se mêlait à tout . Ici on voit le dieu ou roi Soleil répondant à la première case , Osiris à la septième , Isis à la huitième , Horus à la dixième , Thoth à la quinzième , etc .; mais il faut se souvenir que les nombres gravés au - dessous sont plus forts d'une unité que rang de la case correspondante . (8)

It would not be difficult to show in the hieroglyphics which fill the boxes of the first band, characters which correspond to the various gods of the country, to the Sun, to Osiris, Isis, Horus, Thoth, etc. , which would lead one to believe that the 28 fingers of the cubit corresponded themselves to as many religious or mythological characters, whose names had been imposed on the twenty-eight days of the lunar month. (...)

One should not be surprised to find here the images of the gods, applied at the same time to the days of the month, and to the divisions of the measure: among a people so attached to their worship, religion mingled with everything. Here we see the Sun God or King responding to the first square, Osiris to the seventh, Isis to the eighth, Horus to the tenth, Thoth to the fifteenth, etc.; but it must be remembered that the numbers engraved below are higher by one unit than the rank of the corresponding box.

The lunar connection, by way of the number 28, seems very plausible. However, the 28 digits on this cubit rod seem to be superimposed on another division of 24 digits, although of course 28 is what the markings show. Perhaps we should think of the divisions as both pertaining to the ordinary and royal Egyptian cubits, so both 24 and 28 digits. This is what the smallest section on the diagram tells us, it is one sixth of the total length. the blue section is four palms of the Royal cubit, while the pink section is neither a palm of the royal nor the ordinary cubit. It is something else altogether, neither royal nor ordinary. Together with the green section, yes, it becomes three palms of the royal cubit, but together with the blue section, it becomes five palms on the ordinary cubit. It's role is like a mathematical machine through which various irrational ratios can be processed, the section of a shape fed into it, and the irrational other section obtained, without having recourse to notions of irrational lengths, only irrational ratios. In this way, perhaps it was envisaged that the problem of incommensurability in geometry might be overcome, albeit in an applied way, overseen by deities in charge of the theoretical problems. Perhaps I might measure the radius or diameter of a circle in the units provided by the green section, or an ordinary palm (one sixth of the rod), and the circumference in the units provided by the pink section. this could be termed diametric and perimetric units, two terms coined by Geoff Bath. Or the pink section could be used to measure the sides of a square, and the blue section its diagonal, or the pink section one part of a pentagon, and the blue section the other. The square root of three doesn't seem to be covered here. Also, the role of the millimetre itsel is curious, but it does seem relevant in that the pink section is related to a length of 7,000 mm (the pink section measuring 7,000 / (36 x √2) = 137.492985 mm, and the blue section measuring 300 mm. The presence of the metre potentially links us back to the Egyptian digit of 0.7291666667 inches, which can be thought of as 150 / 8,100 = 0.0185185185 metres using the 39.375 inch conversion, a figure put forward by Letronne, who also put forward a possible stade of 185.185 metres (9). There are 378 digits of 0.0185185 metres in 7 metres.

I have reasoned that a possible explanation for the presence of irregular markings on the cubit rod, at least on this example, is to keep seperate the various sections of the ruler which belong to the various sections of a geometric shape which are in irrational ratio with each other. Even though when we measure things in practice, it doesn't much matter if we don't give much heed to the problem of incommensuarility in geometry, the theory behind the measure is still problematic. It would make sense to measure the diameter of the circle in a unit that is quite seperate from the unit which is used to measure its circumference, for example, and to transfer the problem of irrationality from the act of measuring to the measuring system itsel, and its design. But then why not just have a ruler that looks more like this?

You could just stick to the most commonly found irrationals, and subdivide them or multiply them as you like. So then why doesn't the cubit rod look more like this? It consists of 6 units of irregular length, one of which overlaps two others, corresponding to the three most commonly found irrational square roots, pi and Phi, and a base unit against which all the othrs can be used. Curiously, the number 7 is present here: the total length, in whatever unit the base unit is measured in, is close to 200 / (7 x 3). Also worth mentioning is that (1 + √2 + √3 + √5 + π) x 39.375 =375.0045, which is the sum of the first three irrational square roots, pi and 1, taken as a length in inches, and converted to metres using 39.375".

What the ruler above doesn't have is the symbolically important numbers for the moon: 7, 4, 28, and the eye of Horus numbers such as 4, 8, 16, 30, etc. Nor do we have the nifty little 7 palm laid over 6 palm divisions, with the middle section (in pink in my diagram) to bridge the two. Could the reason for these superimposed divisions into 6 and 7 sections be connected to Phi squared? Taking the length of the cubit as simply 10 units, the pink section is 10 x (5/6 - 4/7) = 55/21 , or a Fibonacci number Phi squared approximation. And this is confirmed by 525 x 55/21 x 1/10 = 137.5, the lengths in millimetres of the cubit and the pink section. So these superimposed divisions by 6 and 7 along the same length seem to produce the middle (pink) section being 55/21 in relation to the total length of the cubit, as 10.

The middle, pink section of the rod is (525 x 5/6) - (525 x 4/7), the green section is ((525 x 5/6) - (525 x 4/7)) x 2/π = 87.5352187, and the blue section is ((525 x 5/6) - (525 x 4/7)) x (√5 + 3)/ 2 x 5/6 = 299.98306. The superimposition of the divisions of the same length by 6 and 7 equal parts creates intriguing results: for example ((5 / 6) - (4 / 7)) multiplied by 10 is Phi squared, or at least the approximation of it produces by 55/21, or multiplied by 2 gives the Egyptian Royal cubit in metres, or multiplied by 12 gives the 22/7 pi approximation, or divided by 10 and the 99/70 approximation of the square root of two gives the digit in cm, or multiplied by 54/10 gives 99/70, or multiplied by 56 gives David Kenworthy's eclipse unit 14.6666667.

I think the divisions by 6 and 7 superimposed on each other on each other offer a practical way of producing Phi squared, and all sorts of things derived from that such as approximations of pi, the square root of two, and of course the royal cubit of 20.625".

If you were to double to pink boxes representing Phi² and then remove three pink boxes, you would obtain the square root of five, or an approximation of it, as Phi² - 3 = √5. Or if you were to take this 2 Phi² length and turn it into a circle, the diameter would be 100 shusi or 90 digits long.

Taking the length of the cubit as simply 10 units, the pink section is 10 x (5/6 - 4/7) = 55/21 , or a Fibonacci number Phi squared approximation. So these superimposed divisions by 6 and 7 along the same length seem to amount to the middle (pink) section being 55/21 in relation to the total length of the cubit, as 10.

It's amazing how many familiar metrological numbers can be produced by combining sixths and sevenths.

((1/6) + (1/7)) x 2 = 0.619047619, close to phi

((1/7)+2(1/6)) x 21 = 10

((1/7)+3(1/6)) x 14 = 9

((1/7)+6(1/6)) x 7 = 8

((1/7)-(1/6)) x 7 = 36

((1/7)+4(1/6)) x 42 = 34

((1/7)+7(1/6)) x 42 = 55

((1/7)+8(1/6)) x 20 = 29.5238095

((1/7)+9(1/6)) x 2 = (8(1/7)+(1/6)) x 12 = 3.142857 = 22 / 7

(9(1/7)+(1/6)) x 2 = 1.6190476 = 34/21

((1/6)-(1/7)) x 3 = -0.261904762

((1/7)-(1/6)) x 4 = - 0.5238095

0.5238095 x 39.375 = 20.625

1 - (1/6) = 0.833333 Megalithic yard in metres

(1 - (1/6)) x 39.375 = 32.8125 Megalithic yard in inches

(1 - (1/6)) / 45 = 0.0185185185 Digit in metres

0.0185185185 x 39.375 = 0.72916667

20 (1 - (1/6) / 45 = 14.583333

1 - (1/6) = 5 / 6

12 (1/6) = 10

14 (1 - (1/6)) = 11.666667 Roman foot

16 x 3 ((1 - (1/6) = 40

9/10 x (1 - (1/6)) x 39.375 = 29.53125 Lunation in days approx

18 x (1 - 3(1/6)) = 354.375 lunar year approx

4/10 x (1 - (1/6) x 39.375 = 13.125 = 1/2 x (1 - (1/6) x 39.375 Northern / Sumerian foot in inches

2/100 x (1 - (1/6) x 39.375 = 0.65625 = 25/1000 x ( 1 - 2(1/6) x 39.375) Shusi in inches

22/700 x (1 - 2(1/6)) x 39.375 = 0.825

22/10 x (1 - 2(1/6)) x 39.375 = 57.75 Mayan zapal, and height of GP in inches

24 x (1 - (1/6)) x 39.375 = 630

6 x (1 - 2(1/6)) x 39.375 = 157.5

5 x (1 - 5(1/6) x 39.375 = 32.8125

22/7 x (1- 5(1/6)) x 39.375 = 20.625

Today we value user-friendliness, and we don't tend to assert the divine nature of things in our every day life, as the ancient Egyptians did. Incorporating these symbolic numbers may have been very important. Besides, there are probably many more elements woven into the irregular markings of the cubit rod which combine to make something quite complex.

Together with the presence of the gods above the digits, the cubit perhaps offers a practical but also theoretically far superior method of measuring the irrational sections of geometrical shapes than we can manage today. Is the presence of gods on each division some kind of code for what we call irrational number? The presence of pi and Phi squared, and the square roots of two and five, as ratios in the cubit rod markings is very curious, especially when so many ancient and less ancient units of measure relate to each other by irrational ratios. Is one aspect of this rod about converting irrational ratios, which pertain to another world, into simple workable fractions that can be used, in a process overseen by the gods, like a filter through which the divine flows, and from which emanates something humans can understand and use. If you understand magic not just as spectacle and trickery, but as the way the divine might mingle with the everyday, then this cubit rod is a magic wand, in that spirits or deities are overseeing the transformation of irrational ratios, which cannot be expressed adequately in any mathematical language, into simple workable fractions, ready to be used by mere humans, in this world. Furthermore, Phi, and Phi squared, are present, in applied and approximate ways, in many life forms, be they flora or fauna. The transition from the pure, abstract, irrational essence of the golden ratio, that we can never know, to an expression into the world we live in, in trees, in animals, and in ourselves, is almost as mysterious as the perpetual appearance of life itself, even to the secular mind. This Egyptian cubit rod is therefore caught up in the question of existence, how and why there is something rather than nothing.

Cubit rods are often described as ceremonial, presumably as an attempt to explain their use, even though they do not seem very useful, or even precisely because they seem impractical. Some contain mistakes, slight inaccuracies in the markings, or they are simply to heavy. Though they seem to have no practical purpose, the interpretation of these rods still focuses on use. While cubit rods may well have been given as a gift in a ceremony, or displayed at important events, perhaps we should move away from the concept of ceremonial to explain them, and think in terms of symbolism instead, the symbolism of life in particular, and the forces that bestow it. We can also think in terms of magic, as an attempt by human minds to process what they have identified as necessarily incomprehensible, such as how it is life appears spontaneously all around us, or such as the true essence of the golden ratio, or the ratio between the diameter and circumference of a circle, or the ratio between the sides and diagonal and a square. Metric and geometric concerns are all bound up with metaphysical and epistemological questions. So measure, as a tool to study the world, and by extension, this cubit rod, and all examples of applied measure, are inextricably bound up with these questions. Because it links elements of measure and metaphysics, geometry and life ,the cubit rod is in many aspects a sacred object. In what way is it magic?

When magic is used in children's stories, it has none of the negative connotations the word 'magic' carries by serious people who believe in religion or science, and who may be prone to expressions of sarcasm, cynicism, or warnings to others against credulity. Magic has many meanings. Sometimes it describes a wonderful experience, such as a day in Venice. Or it is about the supernatural, and efforts to manipulate it, such as we see in stories, or in the way charlatans might want to trick others. Or simply it is about what we can't explain, when something happens 'as if by magic'. I think if we can accept the concept of magic in this last sense, simply to describe something that is difficult for us to grasp, we can use it to describe life, or even replace the word 'irrational' in relation to ratios. The word 'magic' can effectively be used to describe things that are beyond rational understanding, perhaps even beyond the reach of human knowledge.

The small but insurmountable difference between approximations of Phi² such as 55/21 and Phi² itself is like the gap between the hands of Adam and God, on the ceiling of the Sistine Chapel.

.

## Notes

1. Sarton, George. “On a Curious Subdivision of the Egyptian Cubit.” Isis, vol. 25, no. 2, 1936, pp. 399–402. JSTOR, http://www.jstor.org/stable/225377. Accessed 19 Jul. 2022.

2. Ibid

3. Monnier, Fr., Petit, J.-P., & Tardy, Chr., 2016, "The use of the ‘ceremonial’ cubit rod as a measuring tool. An explanation.", The Journal of Ancient Egyptian Architecture vol. 1, 2016, JAEA 1 - Monnier, Petit & Tardy - The Journal of Ancient Egyptian Architecture (egyptian-architecture.com)

4. Ibid

5. Ibid

6. Jomard, E.f., 1822, Description d'un étalon métrique, orné d'hiéroglyphes, découvert dans les ruines de Memphis

7. Ibid

8. Ibid

9. See Letronne, Antoine Jean, 1851, Recherches critiques, historiques et géographiques sur les fragments d'Héron d'Alexandrie, ou du système métrique égyptien, https://play.google.com/books/reader?id=8OPfAAAAMAAJ&pg=GBS.PR2&hl=en

## Bibliography

Bath, Geoff, 2021, Stone Circle Design and Measurement: Standard Units and Complex Geometries: 2: Stylised Plans and Analysis of over 300 Rings

Jomard, E.f., 1822, Description d'un étalon métrique, orné d'hiéroglyphes, découvert dans les ruines de Memphis

Google Books, from the collections of Oxford University, pp. 20 - 21

Letronne, Antoine Jean, 1851, Recherches critiques, historiques et géographiques sur les fragments d'Héron d'Alexandrie, ou du système métrique égyptien, https://play.google.com/books/reader?id=8OPfAAAAMAAJ&pg=GBS.PR2&hl=en

Monnier, Fr., Petit, J.-P., & Tardy, Chr., 2016, "The use of the ‘ceremonial’ cubit rod as a measuring tool. An explanation.", The Journal of Ancient Egyptian Architecture vol. 1, 2016, JAEA 1 - Monnier, Petit & Tardy - The Journal of Ancient Egyptian Architecture (egyptian-architecture.com)

Sarton, George. “On a Curious Subdivision of the Egyptian Cubit.” Isis, vol. 25, no. 2, 1936, pp. 399–402. JSTOR, http://www.jstor.org/stable/225377. Accessed 19 Jul. 2022.

Scott, Nora E. “Egyptian Cubit Rods.” The Metropolitan Museum of Art Bulletin, vol. 1, no. 1, 1942, pp. 70–75. JSTOR, https://doi.org/10.2307/3257092. Accessed 13 Jul. 2022.

Hi David,

Thanks. Just trying to find out what the divisions on a cubit are! The square root of five link to Phi might be relevant, or maybe not, I'm just trying different things out... Also thanks for the megalithic yard and digit numbers, it's worth trying all the connections we can think of!

Melissa

I like the way you have found Phi as 1 plus root 5/2, this is a great article, thanks

dave

I suppose a square digit divided into 16 would be 236 square units and divided into 9 would be 81 square units .

When i tried to reconcile the digit to the megalithic yard I found the digit could be divided into 80 units and the megalithic inch was 90 of these units.

This meant that the megalithic digit was 8/9 of 0.816 in. = 0.725333... in

To determine the unit that worked at SH using Thom it turned out to be 81/80ths of a digit.

This is 0.7344 making the Aubrey circle of 10771.2 in 14666.666 of these units.

https://www.tapatalk.com/groups/newmetrology/understanding-the-stonehenge-cipher-of-0-7344-impe-t48.html