Loki 2

I've worked on the translation of the discription of Loki a bit more today and this is nearly word for word for the original Icelandic (circa 1220 A.D.):

"Loki is fine and fair to see, bad by nature, very shifty in manner. He has that wisdom beyond advanced, but [it] is called sleight, and devices for all things; he would bring the Æsir even into a complete morass and often he would free them with fabrications."

"Loki er fríður og fagur sýnum, illur í skaplyndi, mjög fjölbreytinn að háttum. Hann hafði þá speki umfram aðra menn er slægð heitir og vélar til allra hluta. Hann kom ásum jafnan í fullt vandræði, og oft leysti hann þá með vélræðum." -GYLFAGINNING 33

There seems to be some wordplay going on in the Icelandic.

So, Loki was a magician and a prankster and there is an implication that he was as changable as the weather.

Some Ambivalence on Global Warming

Some people raise concerns about global warming and others cite cycles as an explanation for changes but the historical record offers support for both points of view. It does not appear to be purely global warming or necessarily a cycle. To get an accurate reading on any long term warming we have to allow for the possibility of cycles or fluctuations and eliminate their effect on a fit to the data. The results are shown for Jul, the warmest month on average, for Mean Land Temperatures in the northern hemisphere. The only explanation for the data is that both long term effects and short term effects are present.

These fits are just empirical and not based on any particular theory as to what is happening. The fits can be extrapolated to give an estimate of future temperatures. Note that for this plot the temperatures are given in °F while in previous blogs °C was used.



Supplemental: If σ is the standard deviation for the fit one would expect the temperatures to be within 3σ (approximately 0.5 °F) of the solid curve. These are mean values for the month so one would also have to add an additional correction to estimate bounds for the daily highs and lows for a particular location.

Monthly Mean Temperatures in the Northern Hemisphere for 1880-2009*

NOAA's monthly temperature anomalies are given relative to the mean monthly temperature averages for the reference period 1901-2000. If these averages are added to the mean monthly land anomalies for the northern hemisphere one gets the mean monthly temperatures which are displayed in the following plots.

An analysis by month shows a similar pattern for the entire record and the best fits for each month displays a similar pattern. The following table gives the period, amplitude and phase for the fundamental frequency of the best fit. April appears to have the longest period and October has the largest amplitude. These months are mid spring and mid autumn although this might not necessarily be significant. One has to ask if these changes are due to oscillations or if they are just fluctuations in the temperatures which appear to be oscillations. To assert that these "fluctuations" are oscillations at this point would, in my opinion, be "deeming" them to be so. The cause of these fluctuations needs to be determined if it is shown that they are not just random fluctuations.

The coefficients for the quadratic portion of the fit are given below. The columns are the months, the mean values, the rates of change and half** the "accelerations" for each month. Note that on average the rates of change indicate both increasing and decreasing trends depending on the month.

*edit: The available data is from Jan, 1880 through Nov, 2009.

** 2nd edit: The numbers are just the quadratic coefficients and represent values at the beginning of the curves. One would have to set yr to 129 to estimate the 2009 values. The fit used 0 for the Dec, 2009 value which would add a slight error for the Dec curve.

Northern Hemisphere Land Temperature Anomaly

I downloaded some monthly temperature data from NOAA today and was able to find a fit to the data for Northern Hemisphere's Land Temperature Anomaly. An anomaly is a deviation from some reference temperature. The fit assumed a parabolic curve and 20 sinusoidal harmonics of a fundamental frequency. The fundamental frequency that gave the best fit had a period of 63.6 years. The fit in the plot below includes the fundamental frequency plus two harmonics. This fit suggests that the temperatures will soon switch to a declining trend if the pattern holds. But there is no guarantee that it will. The fit is what we would expect if the data were cyclical in nature.



(click to enclage)

A little Learning is a dang'rous Thing

.....

First follow NATURE, and your Judgment frame
By her just Standard, which is still the same:
Unerring Nature, still divinely bright,
One clear, unchang'd and Universal Light,
Life, Force, and Beauty, must to all impart,
At once the Source, and End, and Test of Art
Art from that Fund each just Supply provides,
Works without Show, and without Pomp presides:
.....

Those RULES of old discover'd, not devis'd,
Are Nature still, but Nature Methodiz'd;
Nature, like Liberty, is but restrain'd
By the same Laws which first herself ordain'd.
.....

Of all the Causes which conspire to blind
Man's erring Judgment, and misguide the Mind,
What the weak Head with strongest Byass rules,
Is Pride, the never-failing Vice of Fools.
Whatever Nature has in Worth deny'd,
She gives in large Recruits of needful Pride;
.....

A little Learning is a dang'rous Thing;
Drink deep, or taste not the Pierian Spring:
There shallow Draughts intoxicate the Brain,
And drinking largely sobers us again.
Fir'd at first Sight with what the Muse imparts,
In fearless Youth we tempt the Heights of Arts,
While from the bounded Level of our Mind,
Short Views we take, nor see the lengths behind,
But more advanc'd, behold with strange Surprize
New, distant Scenes of endless Science rise!

- Alexander Pope, An Essay on Criticism, 1709

Loki

The Nordic equivalent of Hermes and Mercury was known as Loki. His father was Fárbauti, a "hacker", and his mother was Laufey, possibly a silvan nymph. Like the Greek and Roman gods his connection with bounds is that he often ignored them. There is a discription of him given in an Icelandic saga about Gylfi, the first king of Sweden,

"Loki was cute and alluring* in appearance, bad by nature, very duplicitous in manner. He surpassed others in that wisdom which is called craftiness with devices for all things; he would even bring the Æsir into a complete quagmire and often he would free them with fabrications." - Gylfaginning, 33

Loki was a slippery character who did not engender trust and probably was responsible for the downfall of the gods. He had a fate similar to that of Prometheus.

(*edit: fríður og fagur can be translated as "beautiful" and "enchanting" but, allowing for changes over time, "plain" and "simple" might be a better fit. A modern translation for fríður is "peace." vélar and vélræðum → "devices" and "fabrications," at least that seems to be the impression.)

Problems with Analyzing Data with Cyclical Error

There may be some doubt concerning the rate of global warming which is due to the nature of the temperature measurements. For statistical data one usually encounters a normal distribution but with temperatures there are both daily and annual cycles in the data. This can bias the estimate of global warming as seen by what happens to the data in the following example. The solid line is the true curve, the dots are the data and the dashed line is the estimated trend based on the data. Notice that the slope is greater for the estimated trend. This increases the time necessary to make accurate estimates of the rate and rate of change of global warming. When analyzing the data one has to be on guard against subtle errors.

Naive Impressionism in Ancient Times?

I found some passages in the Bible which touch on naive impressionism.

and said, “Assuredly, I say to you, unless you are converted and become as little children, you will by no means enter the kingdom of heaven. (Matthew 18:3)

When I was a child, I spoke as a child, I felt as a child, I thought as a child. Now that I have become a man, I have put away childish things. (1 Corinthians 13:11)

Paul may have seen naive impressionism as somewhat lacking. Perhaps this was due to his background. The ancient Greeks also may have viewed Hermes, the younger brother of Apollo, as somewhat naive.

Christ's message was one of renewal/rebirth.

Commons vs Lords

COP15 dealing with global warming is taking place in Denmark. There are two aspects to the problem. First, what is actually happening and, secondly, what should our response be. Whatever the cause of global warming, our response will primarily be an economic and political solution and perhaps we should consider the difference between economics and politics and how they might work together. In ancient times economics dealt with "rule of the home" while politics concerned itself with management of the city-state. There is still this division today in the separation of microeconomics and macroeconomics. The first deals with local problems while the latter deals with more general problems. The difficulty that we are having with global warming might be that the two approaches are out of balance.

In nature one finds a similar division between Special Relativity and General Relativity. Special Relativity is local while General Relativity deals with weaker but more "global" forces that ultimately dominate. These are the Commons and Lords of Nature.

We can probably do something similar with economics and global warming and try to get the two systems to work together better. The question is, "Are we up to the task?"

Beyond Relativity

The weakest assumption in the derivation of the Lorentz Transformation is Einstein's assumption that the speed of light is a universal constant. This assumption is validated by the null result of Michelson-Morley experiment. The conclusion that there is no ether may not be justified. The validity of the Lorentz Transformation suggests that the observed change to the speed of light for a given relative velocity is neglible. If this were not so naive impressionism suggests that one can still find a transformation between the two reference frames which is similar to the Lorentz Transformation. An expression for A can be found by plugging the two values for the speed of light into the formula for the addition of velocities and solving for A. One then gets the following results where Δc is the change in the observed value of the speed of light,

Relativity and Naive Impressionism

How can we justify the assumption of symmetry used in the derivation of the Lorentz Transformation? It appears to be a form of naive impressionism or the belief that what is true for one is true for all. But it appears to fit the facts. The Michelson-Morley experiment gave a null result on the measurement for the velocity of the ether. The speed of light appears to be independent of the Earth's motion throughout the year at least for the value of the Earth's orbital velocity. Relativity has proven to be a useful tool for scientific research. But it seems to validate the simplistic worldview and the possibility of making a false assumption.

One could view Ockham's razor as a form of naive impressionism. But it is an economy measure. One has to seek a balance between ignoring the lack of evidence to the contrary and unnecessarily complicating an explanation of the facts. Making unjustified assumptions raises doubts.

The operational rule of the scientific community appears to be naive impressionism with doubts. So we are justified in saying, "Don't trust them." As a matter of expediency, however, this may be the best way of proceeding, but under the circumstances one needs to show that the use of Relativity is justified in a particular case and that the results are reasonable. The assumptions break down if there is an asymmetry in the point of view of the observers. This may be the explanation of the imaginary values in the transformation for velocities exceeding the speed of light. Strong gravitation may bias the transformation and Einstein attempted to address this in the General Theory of Relativity. Special Relativity may still be the best first approximation to the laws of the Universe.

Special Relativity & the Lorentz Transformation

In Special Relativity the Lorentz Transformation allows one to convert measurements such as distances and times made in one frame of reference to those of another. It is easiest to derive the transformation when the situation is symmetric, i.e., changing between the two frames of reference doesn't alter any of the parameters involved. The same transformation can then be used for both reference frames. Consider two spacecraft headed directly towards each other with a relative velocity of v. We can use unprimed variables for measurements made by the first spacecraft and primed variables for the second. In both cases the transformation is L.


The transformation is assumed to be linear and can be represented by four components of a matrix which only depend on the relative velocity.





By noting that a point in the second spacecraft doesn't move relative to itself while it appears to be moving with velocity -v to the first spacecraft, we can deduce B. The method can be generalized to find the any velocity, V', as it appears to the second spacecraft if its value for the first spacecraft, V, is known. This is the formula for the addition of velocities.



















With L the same for both reference frames we can use it twice to make a transformation from the first to the second spacecraft and then back again to the first. Since we should get the original values back the result is the identity matrix. Doing the multiplications and equating terms gives two more terms of the transformation leaving only one unknown, A.














This is as far as symmetry will take us. To go further Einstein had to assume that the speed of light was a universal constant. This is not unreasonable if space is homogeneous. We just have to be careful about directions though. A ray of light moving along the common line of the spacecraft will appear to be moving in different directions to the two observers. Let's say that it moves away from the first and towards the second. This allows us to simplify the expression for C and determine an expression for A.














We then have to use the minus sign so that the direction of time will be the same in both frames of motion.









So we have found the transformation in this particular case. We can use other transformations to convert to situations that are less symmetrical.














This derivation indicates that Relativity doesn't impose any constraints on time travel. There are transformations which will convert positive changes in time to negative ones. But at the same time they will also convert positive energies into negative ones. So it seems likely that if one could travel back in time one would find oneself in an antimatter universe which would be extremely hazardous.

Timeline of the Julian calendar reforms

239 BC Ptolemy III issues the Decree of Canopus

63 BC Caesar elected Pontifex Maximus

49 BC Civil War

48 BC Caesar meets Cleopatra

46 BC Forum of Caesar

46 BC Julian calendar

44 BC Assassination of Julius Caesar

42 BC Apotheosis of Julius Caesar

30 BC Cleopatra's death

26 BC Alexandrian calendar

After examining this timeline one has to ask if the Julian calendar reforms were incomplete due to the assassination of Julius Caesar?

Are the number of days per month rational?

During the last week I have been studying the computation of the Julian Day Number that astronomers use to keep track of time. To do this one needs to know how to convert from month and day to day of year. The number of days in a month is quite irregular and one wonders how this came about. Most of the features of our present calendar are due to the reforms of Julius Caesar in 46 BC. Caesar turned to an Egyptian astronomer, Sosigenes of Alexandria, for assistance in correcting the errors in the Roman calendar at that time. The lengths of the months and leap day date from that time. At first the length of the months seem quite arbitrary but if one considers the period from March to the following February the pattern if more regular. Placing the leap day at the end of this period is consistent with December being the "10th month."

The problem of designing a calendar with twelve month and 365 days is how to distribute the odd 5 days. The pattern seems to be consistent with using multiples of 30 7/12 rather than the more obvious 30 5/12 as seen in the calculation below. An irregularity is that the sequence is shifted by one month.

In order to do this one needs to be able to perform integer division and this can be done quite easily using multiplication tables and Sosigenes would have been quite capable of doing this. This gives us the sums of the days of the months. The formula for computing the sums turns out to be rather simple and allows us to derive a formula for January through December counting January as the first month. This simplifies converting month and day to day of year with the inclusion of a leap day in leap years.

Space Elevator Thermal Cycling

As a diversion from ancient history we might look to the future for a change.

A space elevator is a mechanism which claims to provide easy access to space. It is basically a cable with a counterweight that rotates with the Earth as it turns about its axis. The cable is heated by sunlight which varies as the elevator rotates. So there will be daily variations in the temperature of the cable also know as thermal cycling. Why study themperature variations? Most materials expand as they are heated and since the space elevator extends beyond geosynchronous orbit, 36,000 km above the Earth's surface, the change in length can be considerable.

It has been suggested that carbon nanotubes might be strong enough to create a cable that is self supporting. The carbon nanotubes are similar in structure chemically to graphite. Since the planes of carbon atoms form tubes, we would expect the density of a cable to be less than that of graphite. But one would expect the thermal properties per unit mass to be about the same.

So to approximate the thermal properties of a cable, we will assume it is made of graphite and behaves like a black body. The following calculation shows the daily temperature variation of the cable as it rotates about the Earth at the time of the equinoxes when the Sun is above the Equator. The cable absorbs energy from the sunlight which strikes it and radiates heat at a rate depending on its temperature. The method is similar to that used for simple climate models of the Earth.

The images below are a slightly condensed version of the program used to do the calculations. A simplifying assumption was that a section of the cable had a uniform temperature throughout. (For a better view of the images double click on them.)

The calculations above indicate the daily changes at the time of the equnoxes. There are two minimums because when the cable aligns with the direction of the Sun it is essentially in its own shadow and only experiences cooling. The shifts in times of the minimum and maximum temperatures for thicker cables can be attributed to thermal inertia.

The daily variations for different times of the year have to take into consideration changes in the angle of the Sun relative to the Equator. There is surprisingly little change though in the thermal cycles throughout the year. The reason is that the projection of sunlight onto the surface of the cable doesn't change that much. It is on the order of 10% as can be seen from the necessary change below. ι_S is the inclination of the rotational axis of the Earth, 23.5°. The first formula gives the declination of the Sun in terms of the angle φ which is the angle of the Sun in the ecliptic plane. θ is the angle of the cable relative to the Sun in the plane of cable's rotation.

Arrested Development?

History sometimes gives the impression that we are viewing ancient Egypt through minimalist eyes. But what we do have may be that of an archivist who long ago tried to collect what remained from former times in order to preserve it. The Rhind Papyrus is not a single document but appears to have been an ancient text with other material attached to it. So it may be just remnants pasted together and not complete texts.

In ancient times the oral transmission of information was the primary mode of instruction. There was no publishing industry and if someone wanted a text of their own they would have to make a copy of some existing text. And limits of the length of a papyrus roll and time available may have resulted in some judicious editing. Cramming may have been practiced even then.

There may not have been a organized effort to preserve and pass on knowledge in early times. What was needed tended to be passed on. What was not most likely was forgotten. The mathematical texts that we do have come from the Hyksos period when foreigners ruled Egypt. And there appears to have been an effort to recover some of the past.

Egypt may have been the victim of its own success. Its relative stability and isolation resulted in little change over long periods. The rate of change may have been too slow. Time might have passed it by and the illiterate "barbarians" were the ones who were motivated to change and ultimately ended up in control.

How a Scribe Might Have Done It

The guesses for the formulas for triangular and pyramidal numbers used rational fractions. This might have been a little advanced for the scribes of the Pyramid Age who only used unit fractions. The guesses can be rewritten using the four arithmetical operations addition, subtraction, multiplication and whole number division. The notation is a common one used for unit fractions. The Egyptian method for writing numbers was a little more cumbersome that decimal notation. There was a special glyph that represented 2/3rds. The scribes would have had the skills needed to do calculations for following problem. And the need to know the number of blocks required to build a pyramid would have suggested it.

Another Derivation of Triangular and Pyramidal Numbers

The methods previously indicated are not the only way that one could arrive at the formula for the triangular and pyramidal numbers. If one guesses that the triangular numbers are proportional to n then one finds, on dividing the triangular numbers by the corresponding value of n, that the result is linear as seen below and the unknown factor is easily guessed at.

Similarly, one might guess that the pyramidal numbers are proportional to the triangular numbers and divide a pyramidal number by its corresponding triangular number. One again gets a linear sequence of numbers and the factor can be easily determined.

So it is not always obvious how a particular formula was arrived at. The formulas found in the ancient papyri were probably intended for use by scribes functioning as clerks and probably do not comprise a mathematical treatise. The ultimate source of the procedures may have been lost with the passage of time and what we now have may have been copied and recopied over thousands of years.

The procedures given in the mathematical papyri are in the most general form. One finds both the method for finding the area of a truncated triangle (with the upper portion cut off parallel to the bottom) and the volume of a truncated pyramid.

For more information see,

The Rhind Mathematical Papyrus by Gay Robins and Charles Shute

Mathematics and Measurement by O. A. W. Dilke

Mathematics in the Time of the Pharaohs by Richard J. Gillings

Pyramidal Numbers

One can ask how many identical blocks would be needed to create a pyramid. Suppose there is one block at the very top, four block blocks beneath it, nine blocks in the third row down, etc. To get the total number one needs to know the sum of a series of square numbers. If one considers a series of sums, a particular sum is equal to the prior sum plus k².

The sum of n 1s is n. The sum of 1 through n is n(n+1)/2. So it would seem likely that the sum we are looking for is some cubic expression and by substituting k-1 for k we get the prior sum and the difference.
We can equate the coefficients of k on both sides of the second equals sign to obtain equations for the coefficients.
Note that only unit fractions are needed for the sum. Finally we deduce the formula for the total number of blocks in a pyramid of n rows.
If we assume that the blocks are square in shape with width w and height h the volume of the pyramid is

V = n(n+1)(2n+1)w²h/6.

If we ignore the additive constants and set W = nw and H = nh we get the usual formula for the volume of a pyramid,

V = W²H/3.

A method for computing the volume of a truncated pyramid if found in the Moscow Mathematical Papyrus.

It is not necessary that all the scribes knew this rudimentary level of geometry and algebra. It would take only one master of sufficient skill to deduce the methods of calculation. The builder of the first Step Pyramid was known as Imhotep. He was immortalized by the ancient Egyptians.

It could be that the pyramids contained the mathematical knowledge of the ancient Egyptians. It is not unlikely that they would set this knowledge down in stone. At least the pyramids provided an opportunity to do so.

Sum of 1, 2, 3, ... , n

The sum of a linear series is fairly easy to deduce. As seen below one writes down the series from 1 to n on the first row. The second row is the same series in reverse order. Adding each column separately one finds each sum is n+1. So twice the sum is n(n+1). Therefore, the sum is n(n+1)/2 as indicated in the last blog.

The Area of a Triangle

The ancient Egyptians knew how to compute the area of a triangle. There is a simple way that they might have arrived at a method for computing the area. Consider a stack of blocks which decreases by one block for each higher row as in the figure below. As can be seen the total number of blocks in n(n+1)/2. One then has to subtract the number of blocks outside the triangle which is n/2. So the total number of blocks is n²/2. If the width of a block is w and its height is h then the width of the triangle is W = n·w and its height is H = n·h. So the area of the triangle is A = ½·H·W.

(click on the image above to enlarge it)

Is it likely that the ancient Egyptians knew of the formula for the total number of blocks, n(n+1)/2? Well, consider doubling an odd unit fraction 1/n by finding the difference between 2/n and 2/(n+1). The difference is 2/[n(n+1)] so

2/n = 2/(n+1) + 2/[n(n+1)]

and, since n is odd, n+1 is even and divisible by 2. The second term is just the inverse of n(n+1)/2. It is likely that the ancient Egyptians could easily compute the total number of blocks and therefore determine the area of a triangle.

This sequence of "triangular numbers" was known to Pythagoras who studied under the Egyptian priests in Memphis, Egypt. And so it is likely that the sum of a linear series was known much earlier in time.

The Slope of the Sides of a Pyramid

The ancient Egyptians knew how to computed the slope of the sides of a pyramid which we know from the Rhind Mathematical Papyrus and other ancient scrolls. The value they used was known as the seked which was the base of a right triangle whose height was a unit length. It is similar to an Egyptian fraction which were the sums of unit fractions. The fractions were just a series of divisors.

The procedure for computing the seked was to take half the base of the pyramid and divide by the height. This makes computing the slope of the Great Pyramid of Giza especially simple since the ratio of the base to height was 11/7. The seked is just 11/2 or 5½ (palms) since the unit length, the cubit, is 7 palms.

The pyramid designers needed to keep track of the slope of the pyramids since they were working near the limits of the construction material. The Meidum Pyramid may have collapsed due to internal stresses which can result in cracks within the structure. A sand dune similarly has a critical slope. It is believed that the Great Pyramid was reentered for inspection after it had been sealed because there is a tunnel from beneath the pyramid to the Grand Gallery that was concealed afterwards.

It is doubtful that 11/7 had anything to with π but may have just been a convenient slope. Still the scribes were devoted to Thoth and may have shared his attitude on secrecy. Their education consisted of figuring things out for themselves. Problems in the scrolls were worked out in detail for the reader but he had to determine the general method. The problems just illustrated the procedure involved.

The Royal Cubit & Foot of the Fourth Dynasty

A significant event in the history of Ancient Egypt was the unification of Upper and Lower Egypt. This was accomplished by focussing on similarities rather than differences. The subsequent events show what happens if two people work with each other instead of against each other. The central administration of the Old Kingdom resulted in a new class of scribes and a general advance in society over previous times.

But our main focus is the royal foot (remen) whose use can be seen in the design of the pyramids of the Old Kingdom. Let's take a look at the Great Pyramid of Giza whose original dimensions are estimated to have a base of 230.5 m and a height of 146.6 m. What does this tell us? Well, the ratio of the base to the height is 1.57231 which doesn't say much by itself but we can look at its continued fraction.

Ignoring the sequence after 1,1,1,2 tells us that the ratio is nearly equal to 8/5 while ignoring that after 1,1,1,2,1 suggests a ratio of 11/7. Comparing these multiples of the royal foot and cubit with the dimensions of the pyramid gives the following information,

which shows that one gets a best fit for 11/7 with a scale factor of 40 cubits or 70 feet. This implies that in Khufu's time,

1 royal foot = 299.35 mm

1 royal cubit = 523.86 mm

which is very close to the values for the cubit stick from the 18th Dynasty.

An interesting coincidence is that,

11/7 = 1.5714

π/2 = 1.5708

π/2 is the arc of a circle corresponding to 90°.

Empowerment

There is an ancient Egyptian fable about the Book of Thoth which illustrates the sense of empowerment. It is from the Ptolemaic period when the Greeks ruled Egypt and so is several periods removed from the time of Pyramid building. It is not known how old the story is but was written on papyrus in Demotic and found in the grave of a Coptic monk at Thebes. Thoth is a god of wisdom and is attributed as the source of writing and the arts and sciences. What is interesting about the story is that Thoth goes to Ra to be empowered to seek vengeance for the theft of his book. The moral seems to be that even kings are subject to the will of the gods.

Another aspect of the story is the secrecy associated with the Book of Thoth. It was hidden away and protected by a series of defenses. But no matter how well something is protected the defenses can always be defeated.

Thoth appears in the story to have a dark side. While Ra, the Sun god, is the source of truth and light Thoth, the Moon, is more obscure. To obtain vengeance on the thief he first gets power over him by taking him and his family to the land of the dead. Thoth is jealous of his power and it is not easily taken.

Pharaoh

The title of Pharaoh was not solely a personal honorific. Literally it means "great house" so the power of the Pharaoh extended to those who served him. It is somewhat akin to the "Crown" in England and the "Presidency" in the US. He was the Egyptian Chief of State. Servants were referred to as "arms". Their power came from the Pharaoh whose power ultimately came from the gods.

An Emblem of Authority

The cubit, foot and other units of length can be viewed as a tool to extend the Pharaoh's power throughout Egypt. The sacred foot was an emblem of power as the ideographs below indicate. There is an ideograph similar to that of the tcheser (foot) in which the hand is holding what appears to be a scepter. The royal cubit may therefore be a symbol of excellence.

One of the duties of the scribes was to distribute goods. When precision was required the royal measures were beyond question.

Tcheseru

The royal cubit was considered sacred and used to determine the dimensions of temples. I found the hieroglyphics for a tcheser length in Wallis Budge's Egyptian Hieroglyphic Dictionary which confirms its special status. The name also suggests a connection with the Eye of Horus fractions mentioned earlier. It could be that tcheser conveys the meanings hallowed, exalted and royal. There is a sense of truth and justice associated with it also. Copies were made of the royal cubit for the use of workers and compared with the original monthly. The penalty for tampering with the royal cubit was death. For more information see Thoth: Architect of the Universe by Ralph Ellis.

The Royal Cubit in the 18th Dynasty

On 18th Dynasty Egyptian cubits the 20th finger is labeled a remen. The 16th finger is indicated as being a "t'eser" and the hieroglyphic symbol on the 15th and 16th fingers is similar to the remen but overlaps the border between them. The glyph appears to be that of a forearm with the hand grasping a rod which can be seen more clearly on the cubit at the Museo Egizio in Turin, Italy. There is a picture of this cubit stick in Mathematics and Measurement by O. A. W. Dilke on p. 23.

t'eser (foot) indicated as being 16 fingers in length

Mathematics in the Time of the Pharoahs by R. Gillings p.220

If one does an actual measurement one does get a value very close to a foot. If you want to check this measurement I would recommend an assistant since balancing the tape measure while taking a picture is rather difficult.

measuring the foot

Plausible Deniability

So without evidence to the contrary, i.e., no evidence of ETs or temporal agents being involved, we have plausible deniability on the ancient foot or remen being based on the nanosecond. There is indeed a natural explanation for it. But why so much misdirection in calling it a foot? Was it ancient propaganda? The conspiracy theorist might be harder to convince. Ockham's razor says that a simple explanation is to be preferred over a more elaborate one if they both fit the available evidence.

The Remen?

Just measured the distance from the lower tendon of the biceps of my right arm to the notch between my thumb and the palm of my hand and it turned out to be 300 mm. Could the remen have been the distance from the upper arm to where one grasps something? One could probably check out that measurement on the statues of the period and compare them with values for the remen. The remen glyph shows part of the upper arm and the notch between the thumb and hand. Calling this distance a foot is a misnomer.

Cubits and Palms

The hand's breadth, a 4 fingered palm, may be more fundamental than the Egyptian bd (foot). The cubit was most likely a means of counting palms. The royal cubit was 7 palms but there was a more common cubit of 6 palms. The counting scheme seems to be based on the powers of two idea. 4 fingers = 1 palm. 16 fingers = 4 palms = 1 bd (foot). But the cubit doesn't fit in with this scheme. Neither 6 or 7 palms is a power of 2. There was a cubit rod found a Memphis which had the 7 palm royal cubit on one side and the 6 palm common cubit on the other. So the simplest explanation of the cubit is that it was a measuring stick which allowed the user to measure lengths without counting them one palm at a time. Lengths could just be read off in terms of palms and fingers.

The fingers of the royal cubit were marked as follows: 1finger, 2 fingers, 3 fingers, 4 fingered palm, five fingered palm, fist, and so on up to 8 fingers. The glyph for the remen is placed above the 15th finger on the scale. This appears to be what Wikipedia calls the bw (foot) which was 15 or 16 fingers. The term remen was also used to represent half the diagonal of a cubit square which was about 20 fingers. And a remen square would have a diagonal about equal to a cubit. The remen marked on the cubit seems to be associated with the forearm and not a foot. Perhaps this is an older usage that of the diagonal of a square came later.


It you look at the side of a royal cubit you will see that there are fingers which are subdivided into parts varying from 2 to 16. The Egyptian scribes needed all these divisions since they used unit fractions. But the scale shows that with 16 subdivisions they were capable of millimeter accuracy. One can compare the royal cubit with the modern engineer's scale or architect's scale.