One of our readers, John Frewen-Lord, speculates that the metre may be the modern version of a measure that was familiar to the Pharaohs.

While we think of today’s metric system (SI) as mostly a modern invention (1960), we have been led to believe for many years now that its most fundamental base unit, the metre, originated in France in 1793, and represented one ten-millionth of the earth’s quadrant (the distance from the earth’s equator to the North Pole, as measured at sea level) . Yet just a few years ago, the late Pat Naughtin discovered that the proposal for a universal standard of length very close to the metre may in fact have originated much earlier, via Bishop John Wilkins, an English cleric and philosopher, and a member of the Royal Society, in the mid-1600s. Recent comments on Metric Views now bring even that assertion into doubt, with the discovery of a measuring device called the wand having been around much longer still.

It is known that the wand, divided into ten segments, was almost exactly, to within a few millimetres, the same length as today’s metre, and that it was used as long as 1000 years ago. But what if all these versions of the metre were simply the rediscovery (or the handing down over time) of a standard measure, equating to the metre, that was invented in Egypt over 4500 years ago?

When we think of units of measure used in Biblical times, the cubit usually springs to mind. In fact, opponents of metric conversion have often referred to the cubit, in jest at least, as having as much validity as the metre. Such people should be careful for what they wish for, for, as we shall see, the cubit and the metre may in fact be directly related – and remarkably both are directly traceable to the Great Pyramid at Giza.

At first sight, such direct relationship may not be immediately apparent. There are a number of variations of the cubit, each different in length, but it is accepted that the Egyptian royal cubit is the definitive cubit, of which a physical example is on display in the Liverpool museum. Used to set out the Great Pyramid, its length measures 524 mm, or 0.524 m. For anyone hoping to see a nice round relationship between the cubit and the metre, I’m afraid the story is much more complicated than that! But keep in mind that number of 0.524 – for it will crop up again.

Let us look briefly at some of the mathematical properties of the Great Pyramid. Apart from the fact that it is just 3/60ths of a degree off an orientation of true north (the Prime Meridian through Greenwich is 9/60ths of a degree out of such an alignment), the Great Pyramid contains some quite stunning dimensional relationships between the numerical constants of pi (?), phi (?) and Phi (?) – and those relationships involve a dimension that is exactly equal to today’s metre. Let us explore this a bit further.

We all know what pi is. It is the ratio of a circle’s circumference to its diameter, and is approximately equal to 3.1416 (another number to keep in mind). We are probably less familiar with Phi and phi. One is the reciprocal of the other, with values of 1.618 and 0.618 respectively. The value of 1.618 is known as Phi with a capital P (?), while the reciprocal 0.618 value is represented by the lower case phi (?), and the two collectively are known by many names, such as the Golden Ratio, the Golden Mean, the Golden Number, and others, but they are values that exist throughout nature. Their discovery is attributed to mathematician Fibonacci in the 13^{th} century.

Fibonacci noted that much of nature – and indeed much of Roman architecture – encompassed relationships of 1.618 and 0.618 for various aspects of design, and that these relationships relate to what is known as the Fibonacci sequence, consisting of 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on, where each number is the sum of the previous two numbers. What is not always realised is that if you take any two adjacent numbers, say 55 and 89, you can obtain two ratios – 1.618 if you divide the second by the first, and 0.618 if you divide the first by the second (the minor variations in the decimal places get smaller as the numbers get bigger, coinciding at infinity). The Golden Ratio has a few unique properties – in fact these equations work only with the Golden Ratio and nothing else:

? = 1 + ? (i.e. 1.618 = 1 + 0.618);

? = 1/? (i.e. 1.618 = 1 ÷ 0.618);

? + 1 = ?² (i.e. 1.618 + 1 = 1.618² = 2.618);

? – 1 = 1/? (i.e. 1.618 – 1 = 1 ÷ 1.618 =0.618).

If we skip alternate numbers in the Fibonacci sequence, we end up with the same result as either ?² or adding 1 to ? – e.g. 144 ÷ 55 = 2.618 = 1.618 + 1 = 1.618² (keep in mind also the number of 2.618). Now you may be saying that this is all very interesting, but what has it got to do with the Great Pyramid at Giza, let alone the origins of the metre? All will become clear!

It is well known and accepted that the Great Pyramid incorporates the value of ? in its geometry – this was discovered by Englishman John Taylor in 1859, when he found that if you divide half the length of the Pyramid’s base perimeter by its height, you end up with ?. The base length of one side is 230.3 m, while its original height is 146.6 m. Therefore (230.3 x 2) ÷ 146.6 = 3.1418 – not precisely ?, but then the height of 146.6 m is at best an estimate of just how high the Pyramid was 4500 years ago (the very top is now missing, as is part of its external cladding, and ground level has likely changed). Likewise, take a circle with the same circumference as the perimeter of the base of the Great Pyramid. Calculate the radius of this circle. It will be found to be exactly equal to the Great Pyramid’s height (230.3 x 4 = 921.2. 921.2 ÷ (2 x 3.1416) = 146.6).

We must note that these relationships, along with many other relationships embodied in the Great Pyramid, can be made using any measurement units – they are not exclusive to the metre.

The Golden Ratio ? is there as well. If we take the surface area of the four sides, and divide that by the area of the base, we come to the value of ? (4 x 0.5 x 230.3 x 186.4 ÷ 230.3² = 1.618). Again, that is purely a ratio, and is not dependent upon any particular unit of measure. But now let us do some more calculations involving the Great Pyramid’s geometry that are dependent upon the metre – and only the metre.

- If we add two of the sides of the Pyramid’s base together, then subtract the height, we end up with a rounded value of 100 x ? (230.3 x 2 – 146.6 = 314.0).
- The King’s Chamber measures 5.24 m x 10.47 m. The Chamber’s perimeter = 10 x ? (31.42 m). There are also many measurements in the King’s Chamber that relate to even multiples of ?, but only using metres.
- If we draw two circles, one circumscribing the Pyramid’s base (i.e. intersecting the four corners) and one inside (i.e. touching the mid-point of each side), then subtract, in metres, the circumference of the inner circle from that of the outer circle, you end up with a figure of 299.71. This is almost exactly one millionth of the speed of light in metres per second (299 792 458 m/s – the slight discrepancy is due to rounding at various points along the way).

Hold on – the ancient Egyptians may have known about the metre, but surely they didn’t know about the second? Perhaps they did. The length of two sides of the base of the Great Pyramid is the distance a point on the equator moves through space in exactly one second.

I’m sure if you tried hard enough, the Great Pyramid may be found to contain some mathematics that support imperial measures, even though things like the foot and inch were not anywhere near close to existence 4500 years ago, and anyway are promoted as being based on human properties, not mathematical ones. But there is one thing that really does indicate that the ancient Egyptians were very familiar with the metre. I mentioned early on in this article that the cubit, which was used to build the Great Pyramid (each side has a length of 440 cubits), was 0.524 m long, an apparently odd relationship to the metre. Let us however look at three equations:

- One sixth of ? is 0.5236 – to all intents and purposes exactly the length of the cubit in metres (to within 0.4 mm of the known physical example, and even that assumes that this example’s stated length has not been rounded to three decimal places); quite why one sixth is not clear, but the Great Pyramid is located exactly 30° above the equator – i.e. one sixth of the distance between the two poles.
- One fifth of ?² (2.618) = 0.5236 – again, exactly the length of the cubit in metres. There are five increments of 72° in a circle of 360°. It is known that the earth wobbles slightly on its axis, at the rate of 1° every 72 years.
- ? – ?² (3.1416 – 2.618) = 0.5236 – another relationship that yields the length of the cubit in metres, and ties together, by means of the cubit (and hence the metre), the two constants that are embedded in the Great Pyramid’s mathematical properties.

These equations cannot be pure chance or coincidence, but must have been created by a society that knew all about the metre 4500 years ago, and from which they derived the cubit. One thing is certain – no measurement unit can be more natural than the metre, based as it is on nature’s constants of ? and ? (not to mention the circumference of the earth). Clever people, those ancient Egyptians.

[Note: I claim little original material in this article, but have made extensive use of sources from Wikipedia, YouTube and others, all of which must be treated with the usual caution as to their absolute accuracy. J F-L]

I find this all very interesting. What I find the most interesting is finding evidence for the metre in the pyramid. The article also reminded me of a book I purchased in the '80s called: The Great Pyramid by Piazzi Smythe and Englishman. I still have it. He however claimed the opposite, that imbedded in the pyramid were all the units of the imperial system. Smythe loathed metric and even called it the French metrical system.

http://www.touregypt.net/featurestories/pyramidinch.htm

So at one end of the room, we have a person claiming that one can find the inch hidden in the pyramid and at the opposite end of the room there is the claim that one can find the metre. Can both be right or can it be possible that one can find whatever one wants to find in the pyramid?

This article reminds me of those old "think of a number" tricks, where you get the subject to do enough mathematical calculations until you end up declaring all of a sudden that they're now thinking of the number 9, regadless of what they started with.

As much as I like the metric system, I just find the claims in this article to be a bit far fetched.

I think we should take this speculation with some caution. The ratio of the kilometre to the statute mile is about 1.61. Does that mean that they are related? I rather doubt it. I think it's more likely to be a matter of coincidence that the ratio should be so close to the Golden Mean.

Human beings have a fascination with patterns and look for them everywhere. However, we can be led astray in our search.

Patterns and coincidence may be clues to further research but should not be taken as conclusive evidence in itself. To do so would be bad science.

If we take a step back from all the fascinating observations in the article we have to consider what possible reason could the ancient Egyptians have for recognizing the metre?

The eighteenth century line of thinking must have been that the figure of the Earth was one of a very few known constants (by the standards of the time) in nature that they could measure with any accuracy and so it proved a useful basis for the measurement of length. For this they had to be able to measure the size of the Earth accordingly.

If the ancient Egyptians were similarly motivated then it must mean that they too could do this (assuming that they realized that the Earth was round and not flat). From a brief research of the internet it appears that the earliest known reasonably accurate measurement of the size of the Earth was by the Greek Eratosthenes in the third century BCE - more than two thousand years after the Pyramids were built.

Typically of such speculation it is difficult (or even impossible) to prove a negative but I, frankly, remain sceptical.

Eratosthenes calculated the circumference of the earth as 700 stadia per degree of latitude. There were, unfortunately, several versions of the stadion around in the third century BC, and we don't know which one Eratosthenes used. Current thinking is that his estimate of the circumference of the earth is within 20%.

In contrast, the meridian survey carried out in the 1790s in order to define the metre came within 0.02% of what we now accept as the mean distance from pole to equator.

Moving on to the present day, we have a precise and universal standard for measurement of length, no longer relying on a timber rod, metal bar or the size of the earth. The quest for this is described in a programme broadcast last night on BBC4, repeated tonight, and available on BBC i-player: "Precision: the measure of all things".

Derek,

Unfortunately there are those who hide behind the title of "Professor" and show not only their ignorance but feed those who don't know better a handful of pseudoscience when it comes the metric system:

Oxford Professor's view of metric.

Marcus du Sautoy is the Charles Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford. He presents a new three-part series about the history of measurement on BBC Four, "Precision: The Measure of All Things"

Here’s what he said about the metric system, speaking on the BBC Radio 4 programme "Midweek":

"... I think actually the metric system is mathematically flawed ... (the originators of metric) made us all go decimal basically because we have ten fingers. But actually the system we were using before, which had units divided into twelve or sixteen, are much better units actually because they are more easily divided.”

He went on to say that an hour of sixty minutes is better than one of a hundred, and to praise the Babylonian use of the number sixty

Referring to Ray's comments, this is an old argument about divisibility. The preference for bases such as 12 or 16 is based on greater divisibility. This issue was seriously considered by the 1862 Report on Weights and Measures and they looked at reducing the imperial system to some harmonious form. Two bases that were seriously considered were decimal and duodecimal (i.e. base 12). They concluded that decimal is best for accounting and calculation purposes and because the world uses the decimal number system. Decimalisation of the currency was also considered, though that did not happen until 1971. Interestingly, the world uses decimal currencies and nobody argues for duodecimal, hexadecimal (base 16) or sexagesimal (base 60) currencies.

You can find my blog article on Metric Views about the proposed decimalisation of imperial units at http://metricviews.org.uk/2013/05/imperial-units-and-decimals-not-a-winning-combination/.

Ronnie,

That old comment about divisibility was obviously concocted to frighten the masses of people who don't know better. Whenever I encounter someone who uses it as an argument, and it is rare it happens, I often question why the 99 % of the people who use the metric system in the world aren't troubled by this. Why only the Americans and British seem to have a problem dividing by three? I never do get an answer to that question.

Of course there is the standard soft drink can of 330 mL. Of course an opposer would insist it isn't a true 1/3 litre, but who said it has to be? We are being sold 330 mL, nothing more and nothing less.

The construction industry is able to divide by 3. They use the 100 mm module and have common product sizes of 1200 mm, 2400 mm and 3600 mm. All are divisible by three such that the result is a whole number. Opposers just can't seem to comprehend that the rules of SI does not dictate the numbers to be used in usage just the grammar of the units. So if division by three is important to someone, then they can select there base amount to be a value like 1200 mm of 2400 g or 3600 L that can easily be divisible by three.

You may find this article, although dated, somewhat interesting. Even though the guy supports metrication I did notice some errors in the article.

http://howell.seattle.wa.us/essays/metric.html

If professor Sautoy thinks the metric system is flawed for the reason he stated on Radio 4 then the imperial system is even more so. It doesn't make consistent use of twelve let alone sixty. In fact the unit ratios for weight are not even divisible by 3.

I strongly agree with Ray. This argument about divisiblity is completely bogus. I am surprised frankly by the professors comment. Doesn't he understand that the metric system was based on ten because we already counted that way when it was invented?

There is a very good reason why imperial was never able to stay consistent with a common conversion factor. It has to do with the human scale of imperial the opposers love to draw attention to.

If you use your thumb for some measurements and you foot for others and the width of your outstretched arms for another, the ratios are not going to always be the same number.

Imperial was designed around numbers ignorant and illiterate people could relate to. But we don't live in ignorant and illiterate time (well some still do), so we don't need or desire to be handicapped by the pre scientific fundamentals of imperial.

The human scale is flawed because every human is different. My foot is not the same as yours and if we standardize on a foot, then whose foot and why? SI is perfect in that it is based on the invariable standards of nature (except presently for the kilogram, which will change shortly).

We can pick any series of numbers to use within SI that suits our needs. We don't need to be confined to numbers that only work with binary fractions. There is nothing more unnatural and cumbersome than having to express a dimension in a string of unit names. A foot divided into thirds still results in a non-whole number and can only be expressed in a whole number when the unit is changed to inches. A third pound even when changed to ounces is not a whole number. The third argument does not work with a conversion factor of 16. So in truth this divide by three nonsense only works with length units in imperial and only if you change the unit from the original.

I think the comments on Prof du Sautoy are unfair. He may have chosen his words carelessly but it is fairly obvious that he was referring to the decimal numbering system rather than the metric system itself - which is fair comment. In fact the programme is very supportive of SI and is well worth watching.

Last night's programme, no. 2 in the series "Precision: the measure of all things", was every bit as good as the first. Prof. du Sautoy began in France, visited Maryland to view the Watt Balance, but focussed throughout on the importance of internationally-accepted, precise standards. Only one mention of imperial, none of USC. He tactfully omitted to refer to the unfortunate fire at the Palace of Westminster in 1834 that destroyed the newly-created imperial standards of length and mass. Repeated tonight at midnight on BBC4 (the programme, not the fire).

Next week's programme is about work, energy and light. Another walkover for metric, I expect. Let's hope the series is repeated on BBC2.

Not surprisingly, these conclusions have been vehemently challenged on various grounds, including disputes about the actual measurements (which must infer the additional mass of missing face stones and capstone) and doubts about the length of cubit the builders actually had in mind. But if the above figures are accepted, a case can be made for the pyramid inch and the pyramid cubit as the ideal units of reckoning for both time and space. By basing the unit on the dimensions and orbit of the planet itself (assuming for the moment that the ancients knew these quantities), the Egyptians would have been measuring distance in fractions of the earth’s axis and time in fractions of the circumference?whose rotation equals a day?as well as fractions of the distance to the sun, which is the orbital radius on which the earth cycles once each year. Or, using the .4618 meter cubit, one cubit equals a thousandth of a second in the sense that the earth, at its equator, rotates a distance of 1,000 cubits per second.

Not fair! How do we get to see this program in the States??

Why are the measurements of Khufu's Great Pyramid such odd figures like 432 cubits for each base? The cubit, about 21 inches long, was the standard measure of the Egyptians. It seems reasonable that the architect would have used an even number like 400, 450, or 500 cubits for the bases. Imagine Khufu asking his architects to build a pyramid 300 cubits high or 525 feet, so it would be the greatest monument ever built. (I doubt that he would have said, "Build the pyramid 275.43 cubits high.") The architects would then need to know two things: the angle of the pyramid and the length of each of the four sides of the base. The most manageable size block of stone at the time was about two cubits by two cubits, so the pyramid would be 150 stones high to equal 300 cubits.

This review may be of interest:

http://www.guardian.co.uk/tv-and-radio/2013/jun/25/precision-measure-all-things-tv-review

There were different kinds of cubits. The common cubit, called the cubit of a man, was about eighteen inches ( Deut. 3:11 ). The king's cubit was about three inches longer than the common one. The holy cubit was about a yard, or two common ones. The Royal Cubit, as employed during the Old Kingdom, is generally understood to have been 524 millimeters +/- 2 mm ( 20.6 inches) in length.

Take a circle 1 metre in diameter . calculate the circumference and divide by 6 and you have a royal cubit to 4 decimal points.

pi - Phi squared = 1 royal cubit to 2 decimal places.

Could this really be coincidence????

Someone at some point is going to have to give these ancient Egyptians a little more credit with regard to their body of knowledge. After all they have built structures which we would struggle with today even allowing for our supposedly advanced technological capabilities.

The 50cm length rather than the metre was one of the measures used by Neolithic man and is found in many of the Recumbent Stone Circle radii in North East Scotland dating from 3200-3000BC. It is funny that the measurements of Alexander Thom made in feet and inches convert to precise numbers of half metres. (Examples of stone circles using the 50cm length for radii are Ninestone Rig ( Radius = 3.5m), Raedykes (5.0m), Dalcross Castle (6.0m),River Ness (10.5m),Cauldside (12.5m), Milton (14.0m), and Sheldon of Bourtie (16.5m)).The interesting thing about the measurement system used by our ancestors was the fact that they managed to combine the measurement of length and time by using a series of pendulum lengths (see Uriels Machine). A 50cm pendulum gives precisely 333 oscillations per megalithic degree (solar time at latitude 60 deg North) ( where 366 megalithic degrees equals 360 degrees). There were many different pendulum lengths that gave precise numbers of swings for different angles of separation of stars or movement of the Sun (or rather the angle of rotation of the Earth relative to either the stars or the Sun)The 50cm pendulum was a solar pendulum so to calibrate it, an isosceles triangle could be made on the ground using for example an equally spaced knotted rope of two sides of 55 and one short side aligned east-west of 28, giving an angle of 30 meg degrees at the North apex. In this way three sticks could be placed at the apices and the Sun aligned with the East stick and North stick and then by swinging the pendulum counting the swings until the Sun aligned with the West and North sticks of the triangle. Ten thousand swings would mean that the pendulum was the correct length....otherwise you would shorten it to increase the number or lengthen it if there were too many swings for that time. There are many who cannot imagine our ancestors being able to count more than 20 assuming they had all their fingers and toes but they are simply wrong. Interestingly the period of a pendulum varies as a function of latitude due to gravitational field ( because the Earth is oblate gravity is greater at the Poles than the Equator) Surprisingly perhaps the 50cm pendulum is most accurate at 60 degrees North...corresponding to the far North of Scotland/Orkney giving precisely 333.0 swings whereas in Egypt the pendulum gives only 332.5 swings per meg deg suggesting that the measurement technology was developed in the North and taken South by druid astronomers perhaps. It is also interesting that the Royal Cubit (52.36cm) is related to the 50cm length as a circle with a radius of 50cm has a circumference of precisely 6x 52.36cm...the Royal Cubit was also used in measuring the Recumbent stone circles of Scotland it was one of the star pendulums and gave 12000 swings for a rotaion of 37 megalithic degrees . In Scotland around 3000BC this pendulum was conveniently calibrated using the stars Psi and Lambda Velorum in the constellation of Vela (3000 swings separating their Hour Angles) The stone circles of Scotland predate the pyramids by over 400 years so perhaps we might consider that it is the forefathers of the Picts rather than the Egyptians who are the true mathematical genii...one only needs to look at the statue of the architect of the Great Pyramid Hemiunu to see a man who looks remarkably similar to Rab C Nesbitt..though admittedly he isn't wearing a string vest.

Suggestion for anyone interested in the metre being found in the GP is to watch the brilliant documentary directed by Patrice Pooyard. You will not be disappointed.

Marcus du Sautoy is probably quite right when he said that the metric system was flawed. That was also the view of Laplace and Legendre, two of the scientists who, in 1790, were involved with the design of the metric system. They said that both counting and measuring should be done to the same base and that ideally the decimal system of counting should be replaced with a duodecimal system. However, they did not live in an ideal world (nor, with all due respect to the Dozenal Society, do we) – replacing the decimal system of counting with a duodecimal system was not practical in 18th Century France, nor is it practical in 21st Century Britain. The second-best solution was therefore to align measurements and counting by adopting a decimal system of measurement.

"Knew about the meter" implies there is something special about that particular length--there isn't. I think the only thing that should appear in this article is the fact that a circle with a 1m diameter has a 6 cubit circumference to four decimal places. I feel that this is too much accuracy to be attributed to coincidence (consider that Michael Glass's counter example exhibits approx 100 times more absolute error and 1000 times more relative error than this [when using 1.609344498 as the value of the km/mile ratio]). Your other arguments make it look like you are resorting to some convoluted mathematical trickery to make the numbers work (which you aren't) but clearly all the pi-phi relationships you cite as "evidence of knowledge of the meter" simply follow from the fact that the unit in question (the cubit) has a numerical value of pi/6.

I think all we can say is that the Egyptians evidently had an arbitrary unit of length equal to 1 meter (perhaps by a different name), and defined a second unit of length equal to 1/6 of the circumference of a circle whose diameter was 1 meter (or perhaps, defined their "meter" as the diameter of a circle whose circumference was 6 cubits). The Egyptians wouldn't even need knowledge of pi for this to be done. Make any size circle you want and call the diameter 1, and all your other pi-phi relationships will hold true. The only amazing thing here is that 1 cubit is--to within even modern fabrication error--*exactly* 1/6 of the circumference of a circle whose diameter is 1 meter. When you consider how often pi appears in Egyptian architecture, one must conclude that they sometimes used a unit of measure equal to the meter--a unit of length that was evidently passed on to modern times because there is otherwise nothing inherently special about it. There is no means by which it could have been "rediscovered" other than by sheer coincidence.

I am inclined to agree that co-incidence cannot be discarded, unless of course the length of the earth's meridian was used to define the unit of length - after all the metre was intended to be 1/10 000 000 of the distance from the North Pole to the Equator (through Paris).

Both the inch and the meter are incorporated in the Giza Pyramid

1 cubit = pi/6 meters

1 cubit = (7*73*71)/(7*73 - 71)/4 = 36281/1760 inches

Do the math

1 inch = 2.53999... = 2.54 centimeters

Please read my views on The Egyptian Royal Cubit and it's surprising similarity with The Metre Scale in the link below...

http://horasuryachandra.blogspot.in/

Dividing the metres-per-second 'light speed' in a vacuum figure of 299,792,458 by 10,000,000 (the metre being 1/10.000,000th of the distance between the Equator and the North Pole), converts the m/s ‘light speed’ figure to 29.9792458.

The precise 'degree' latitude coordinates that the Great Pyramid of Giza is centred on are 29.9792458 N.

The Ancient Egyptian Royal Cubit is the key to history and origin of all measures. There is no such thing as the English inch, or the Greek foot. Both of these measures were known without a shadow of doubt and the key is The Ancient Egyptian Royal Cubit, and this number proves it with absolute authority; 206,264.48 also written as 206, 265

Now go do your homework.