4224re: historical metrology
- Nov 2, 2003Barry,
First, I found this page on historical metrology I thought you might like.
At the bottom, the author lists a few books he recommends.
The reason I am sending the whole page and not a link is because it is no
longer hosted. If your browser accepts the following link, you may view
If not, here it is, minus whatever images once were shown.
Second, I am looking into Greek measure, particularly in relation to the
Parthenon. I was reading what Livio Stecchini has to say about the
dimensions of the Parthenon, and encountered his idea of "trimmed lesser
feet". Where most people agree that the stylobate of the Parthenon measures
100 x 225 Greek feet, Stecchini prefers to say it measures 112 1/2 x 250
trimmed lesser feet. These are his theoretical measures. The actual ones,
according to him are 111 1/2 x 251.
He says that 1 Greek foot is 25/24 of 1 Roman foot. And that 1 trimmed
lesser foot is 15/16 of 1 Roman foot. So that 1 trimmed lesser foot is 10/9
of 1 Greek foot.
So I have no trouble converting from unit to unit. But *why* does he
introduce this unit? Is it accepted among metrologists? Or is it
Stecchini's own term for a more widespread one (he calls Greek feet
"geographical feet", for example).
I think I encountered the term "trimmed lesser feet" in the Tompkins
appendix, but this is the first time I would really like a solid context for
his claims. Stecchini's math is first-grade, and his insights are keen. I
dislike his writing style but, worse, have so little knowledge of metrology
it takes me a long time to assess what he says.
Anyway, here is the page I mentioned. Enjoy.
Historical metrology, the study of ancient weights and measures, is an
auxiliary science of history. Its aim is to expose the "wit" of our old
measuring systems, to understand structure and to draw comparison. Of
special interest to metrologists are the linear systems of antiquity, those
"representational" measures of man that by reason of their own
anthropomorphic terminology are thought to be self evident, magnitudes which
to the ancients represented fingers, palms, arms and feet. By analogy the
cubits of old are arm's-length measures, from elbow to fingertip. Variations
are to be expected but of those known most concede to be fitting for the
human frame giving cubits that range either side of an acceptable 18 inches.
In some instances however variations occur which considerably stretch the
imagination, the most notable being the world's oldest arm's-length, the
Egyptian royal cubit of 20.590 - 20.625 inches. This particular length is
considered excessive for the human arm and it remains a mystery as to why it
survived throughout the greater part of Egyptian history. The Biblical
reference to the Old world cubits suggests the larger exceeded the smaller
by "an handsbreadth".
Regardless of what took place in those times there seems little doubt that
the Egyptian measure once hid some sort of inner structural legitimacy
because to acknowledge the royal cubit's published scientific figures is to
state that the measure seldom varied more than 20 thousandth of an inch from
the mean. Of its equal nothing was to be found, that is, until in Britain
during the 1960's, an engineer named Alexander Thom alerted the world to the
existence of the megalithic yard. Both the Royal cubit measure and the
Megalithic yard are thought to date back at least 5000 years.
British megalithic yard ... 32.640 ± .036 inches (A.Thom.)
Egyptian royal cubit ... 20.590 inches (Alan Gardiner.)
20.620 inches (I.E.S.Edwards)
20.620 inches (R.W.Stoley.)
20.620 ± .005 inches (W.M.F.Petrie; 4th dynasty)
In 1967 the Scottish engineer A.Thom published his statistics for the
megalithic yard, an ancient unit of length that was found to be mysteriously
interwoven with the pre-historic menhirs and stone monuments of megalithic
Britain and Brittany. Thom proclaimed that in the age of stone megalithic
man had already mastered the complicated mathematical concepts of laying
down accurately the multiples of a standard linear magnitude. For Britain in
general Thom gave the figure 2.72 ± .003 feet, in other words a consistent
magnitude displaying no more deviation from the mean than today would be
tolerated by any modern surveyor using modern techniques or modern
equipment. For the experts who previously supported the notion that the
older measuring systems were probably pacing systems or at best those that
may have been influenced by various bodily parts, Thom's statistics proved a
slap in the face. Accuracy of this order was unheard-of.
Encountering fierce criticism Thom had to review his statistics. He also
found it necessary to return to the better known sites of Stonehenge and
Avebury to refine his figures, this time to an uncompromising 2.72 ± .002
feet. Thom defiantly stood his ground and to confuse the issue further made
an alarming statement "the megalithic yard is a measure too good for the
The implications of finding a magnitude consistently too accurate to have
been laid down in reference to any man made object is a little frightening.
What possible means available to megalithic man could have improved upon the
measuring rod ?
If Thom's megalithic yard is as real as Petrie's royal cubit, then the
latter, being the more precise measure of the two, must have been far too
good for the measuring rod. But in Egypt where the rod and the measure are
believed to be one and the same, how can that be ?
It doesn't matter whether Thom was aware of the contradiction or not because
the megalithic problem as he saw it was one of impracticality, in the sense
that man made objects (measuring rods) subject to wear and tear could never
under any circumstances be relied upon to maintain the agreed standard, let
alone continually transmit it. Of course the same "retention of accuracy"
problem manifests itself when any man made measuring device is either
replaced or copied from a master rod, which in itself is a doubtful
proposition if there is nothing to master the master against. Add to this
the reality of exposing the rods to the elements and it becomes obvious that
something is terribly wrong, at least in Egypt where both the rod and the
measure are to be found. For example; how did the Egyptians maintain an
accuracy (as the published scientific figures would have us believe) of ± 20
thousandths of an inch, or as Petrie insisted for the 4th dynasty, ± 5
thousandths of an inch ?
Rather than take Thom to task over the issue the inquiry should be directed
towards the uncertainties of Egyptian metrology, in particular to the
metrological judgments made by the Egyptologists of the 19th century.
In 1822 the French mathematician Edme-Francois Jomard, while stationed in
Egypt, procured and examined the artifact known as the Amenemipt rod of the
The unusual design of the rod intrigued him. Jomard found that it consisted
of three distinctly separate groups of divisions. Of the twenty eight
divisions counted, the first group of four were slightly larger than the
following group of twenty three, while the last unit was considerably larger
than any from the two previous groups. Not wishing to over speculate on the
peculiarities of the rod Jomard measured each of the three groups to the
nearest millimetre and published the following results:
77mm/4 =19.25mm or 58mm/3 = 19.33mm for the first 4 divisions.
422mm/23 = 18.35mm for the 2nd group of 23 divisions
and 21mm for the last division.
In 1865 the German archaeologist Reichard Lepsius examined a number of
extant Egyptian measuring rods. In his opinion all were of similar length
and design. From the many examples Lepsius examined an overall length of
525mm was thought appropriate.
One particular fine example, the Amenemipt rod (previously examined by
Jomard) convinced Lepsius that Egyptian metrology was structured on the
width of the abstract finger, the evidence of which in hieroglyphs :
was clearly to be seen marking the first four divisions of the rod. As there
were twenty-eight divisions to the rod it seemed reasonable to Lepsius that
28/4 fingers represented seven handwidths or palms. Lepsius measured the rod
523.5mm, filled in some divisional lines and labeled it the royal cubit of
twenty-eight "more or less" equal divisions.
The 3.5mm discrepancy between Jomard and Lepsius is discussed later in
In 1882 Sir Flinders Petrie published the results of his 4th dynasty survey
at Giza. By far his greatest metrological contribution came with the
successful identification of the 4th dynasty value for the Egyptian royal
cubit. From the Great pyramid of Khufu Petrie extracted the figure 20.620 ±
5 thousandths of an inch (mainly from the chambers ... particularly King's).
Working with the finest surveying instruments of the 19th century (including
the theodolite) Petrie was stunned by the superior workmanship he
encountered, so much so that he officially recognised the architects of
Khufu as the best that Egypt had ever produced.
If Petrie was correct to acknowledge Egypt's climax in the building of Khufu
then some credit must also be given to the metrological language by which it
was achieved, namely the "standards" agreed upon for the royal measures.
note : If the royal cubit and its divisions were not fully understood by
Petrie then it is possible that at Giza many of Petrie's measurements were
wrongfully acknowledged as being in royal cubits or royal cubit multiples.
Other measures in service at that time were the common cubit and the
geodetic digit. In the King's chamber (Khufu) however where he identified
the royal cubit in two multiples, 10 and 20 (even if the walls were shaken
about due to earthquake) judgement can probably be reserved. When applying a
standard to the 4th dynasty it is the royal cubit in the 'King's chamber
alone" that is most often referred to.
Petrie's data Chapter 7 section 52 ... Pyramid in general
Petrie's data Chapter 20, section 136 ... King's chamber alone.
By the end of the 19th century metrologists were in general agreement that
the royal cubit measures uncovered by survey were all between 523-524mm (
20.590-20.630 inches ). As few wished to pass judgment on Jomard's
controversial measurements which suggested a more complicated metrology,
preference was given to Lepsius for his 525mm rods of twenty-eight "more or
less" equal divisions.
The division into fingers and handwidths (palms) was also supposedly
supported by the Greek historian Herodotus who in the 5th century BC
compared his own Greek metrios to the larger cubits of the Near East.
Herodotus judged these differences in finger-breadths.
Confident that the world's ancient linear systems were no more complicated
than the agreed standard for the abstract finger (at any onetime)
metrologists turned to Herodotus for clarification.
Unfortunately for metrology Herodotus is somewhat vague with his
anthropometric measures and when comparisons are made with the
archaeological evidence of the 5th century BC it becomes evident that there
are a number of discrepancies. Herodotus always gives the impression that he
struggled with two different measuring systems, one of which to measure the
dimensions of monuments such as the Egyptian pyramids (base of Khufu 800
Greek feet), the other based on the common metrios he used more widely to
measure people and statues.
The Greek practice of expressing the dimensions of buildings in feet was
well known to ancient Athens because metrologists have confirmed the foot
value from a number of important sites including the Parthenon and the
Olympian temple of Zeus.
The foot's accepted value of 12.15 inches (.308m) not only establishes it as
the one surety of Greek metrology but it also entitles it to become the only
other measure from antiquity (along with the more ancient royal cubit and
megalithic yard) to be judged a standard. The foot's fine tuning suggests a
measure of architectural importance, one which probably belonged to some
exclusive canon of either the Athenian builders or those of whom they
contracted. Was Herodotus aware of the foot's true value? Probably not !
But Herodotus did use the foot and he also theorized about its common
metrios relationship, ie; 6 ft = 4 cubits?
The Greek custom of distinguishing between the foot and the cubit by using
each in its proper context does not support Herodotus' claim that both
measures were commonly linked, just as the simple ratio 1:112 would have
been a poor deterrent for the two systems to remain apart. Under the
circumstances Herodotus' grasp of his own metrology is in some doubt. But
the metrologists of the 19th century proved Herodotus correct and from the
following statements of his (with a little help from Lepsius) they
re-constructed both Greek and Egyptian metrology:
1. The Babylonian cubit is three finger-breadths longer than the Greek.
2. The Egyptian cubit is the same as the Samian.
3. ... because a fathom is six feet or four cubits, if a foot is four palms
and a cubit six.
Since archaeologists have confirmed that the excavated rods of Babylon are
all very similar to the Egyptian it follows that the Babylonian, the Samian
and the Egyptian all exceeded the Greek by three finger-breadths, in other
words there were two mainstream cubits. The Samian cubit was always in some
doubt until a broken relief featuring the outstretched arms of a "larger
than life " Greek was found along the west coast of Asia Minor (dated to the
first half of the 5th century BC). The Arundel marble or the Ashmolean
relief as it is known illustrates a fully extended armspan of between 81.25
and 81.50 inches, and if the arm's-length/cubit analogy is to be believed,
an arm's-length from finger tip to elbow of about 20.6 inches which is the
approximate length of the Egyptian royal cubit. although the ratio 1 : 3.96
is confusing Herodotus is not to be discredited.
Also featured on the relief is the outline of a Samian foot which bares no
understandable relationship with either the Samian armspan, the Samian
cubit, or for that matter, Herodotus' metrological beliefs. And so it is
that the last statement of Herodotus above creates many more problems than
it solves. Taken to be correct it influences the following calculations:
A Egyptian royal cubit = 7 palms = 28 fingers (Lepsius) ....
Greek cubit = 6 palms (Herodotus) = 24 fingers
B Egyptian royal cubit = 7 palms = 28 fingers (Lepsius)
Greek cubit = 3 finger-breadths less (Herodotus) = 25
Option A has been adopted in conjunction with the published figures for the
Athenian foot; 12.15 - 12.16 inches (.308mm) to form the abstract finger
foundation on which Greek metrology is built. If Herodotus believed 4 cubits
to equal 6 feet then :
The Greek cubit = 24 fingers
The Greek foot = 16 fingers
The Greek finger = 12.15 - 12.16 inches / 16 = approx .759 - .760
But option B suggests that Herodotus's finger - breadths were somewhat
Egyptian royal cubit 20.590 - 20.625 / 28 = approx .735 - .737 inches
With no archaeological evidence to support either of the finger options the
contradiction loses some of its impact, so too the genuineness of what is
now tentatively accepted to be the Greek metrological system. Earlier,
Petrie looked for divisions of the royal cubit at Giza. Instead he found the
artist's module, a .727 ± .002 inch multiple that formed the grid patterns
on the walls of many 4th dynasty tombs. Petrie was observant enough to admit
that it had no meaningful role to play in the known metrology of Egypt and
so cautiously named it the digit. It will be shown later that he identified
the unit correctly, but as a double unit ( 2 x .363 inches).
But to solve the ancient finger problem without discrediting Herodotus a
third option must be introduced, one which rejects Lepsius and his rods of
twenty eight "more or less" equal divisions and returns to the observations
of Jomard. Option C adopts a practical approach to the three finger-breadth
account given by Herodotus by admitting that the width of a practical finger
is more befitting to the last division of the Amenemipt rod according to
Jomard. Before option C is introduced however it is necessary to first
challenge the long standing belief that the royal cubit measure is in fact
an accurate reflection of the Egyptian measuring rod, namely the Amenemipt
rod of the 18th dynasty. Obviously if Jomard is correct, it is not. To a
lesser extent the same holds true for Lepsius.
Published figures for the royal cubit measure 20.590 - 20.625
Amenemipt rod according to Jomard 520mm.
extant rods according to Lepsius 525mm. Amenemipt rod 523.5mm.
Since Jomard and Lepsius both had access to the best measuring devices of
their era the 3.5mm discrepancy for the Amenemipt rod is not easily
accounted for. There is however some concern for the stability of ancient
excavated wood when exposed to moist air. In other words the Amenemipt rod
could have expanded or contracted by either absorbing or expiring moisture.
It is highly unlikely that Jomard was caught out because his examination of
the rod occurred shortly after excavation and the rod at that time would
have been stable. Lepsius on the other hand gathered information on a rod
that had been exposed to the elements for 43 years, by all accounts to a
Mediterranean climate. Was there sufficient absorption to account for the
3.5mm expansion ?
In 1961 the rod was again re-measured at the Turin museum in Italy where it
had been stored for some time under more sober conditions. The new reading
of 523.6mm (Senigalliesi) suggests that Lepsius read this rod correctly in
1865 but it also becomes apparent that it must have absorbed considerable
moisture sometime after Jomard's initial examination in 1822. To ignore the
rod's history or to act indifferently to the 3.5mm discrepancy is not in the
interests of Historical metrology. It may be ironic that the rod now agrees
with the published figures for the royal cubit measure but there were
obviously times when it did not.
Subtract Herodotus' 3 finger-breadths (according to the last division of the
rod measured by Jomard) from the published figures for the royal cubit
(523- 524mm) - 3(21mm) = 460- 461mm = Herodotus' common metrios
Is it possible that when Herodotus visited Egypt in the 5th century BC he
discussed the Greek metrios, were comparisons made with Egyptian measures ?
To illustrate the possibility we turn to Jomard and add two of his leading
divisions to the 23 divisions following:
2(19.25-19.33mm) + 422mm (23 divisions) = 460.5- 460.66mm.
The short Egyptian cubit (common) has always been claimed the equivalent of
the Greek metrios. It would follow that the royal cubit measure is an
addition of 3 finger-breadths:
(460.5- 460.66mm) + 3(21mm) = 523.5- 523.66mm = pub fig's for the royal
In the next illustration the last division of the rod has been labeled the
practical finger. Three of them together thus describe what Herodotus
referred to as the difference between the Old world cubits. Herodotus used
the term finger-breadth.
Selecting a division appropriate to the width of a human finger.
This simple 'three finger breadth' interpretation is central to Foundation
metrology. Its origin can best be viewed in the anthropometric geometry.
origin 1. hand geometry
origin 2. anthropometric model
In the absents of any real documentary evidence for Egyptian mathematics
prior to the Middle Kingdom scholars have argued for a long developmental
period where all knowledge was passed down orally from adept to pupil. Since
in Pre dynastic times a decimal system existed, the slow development is
thought to have extended over a period of about 1500 years ( 3600- 2040 BC)
Some Historians however are inclined to disagree by pointing out that long
periods of slow development are not consistent with the "history of
mathematics". Rather, they say, the emphasis should be on relatively short
periods of rapid development where the practical methods of the common man
are seized upon by an inquisitive and sometimes leisurely few, for further
inquiry. Furthermore, that if the newly acquired knowledge is not eventually
returned to the source from whence it came, stagnation is likely to result,
at worst deterioration or even complete loss.
An example of this rapid development, deterioration and susequent loss might
be seen in the extraordinary Greek artifact known as the Antikythera geared
object which was first brought to light in 1900 and then dated to the first
century BC. Also, recently, the realisation that Archimedes may have been
smarter than we think.
It is probable therefore that a solid body of arithmetic was used in Egypt
from the first half of the 4th millennium at a time when it is believed that
many of the ancient settlements along the Nile permanently linked with the
vast trade networks of the Near East. But it is not from amongst the
agriculturists or traders that any superior strain of mathematics is likely
to have evolved but rather from the powerful priesthoods which grew out of
their affluence. It is suspected that Egyptian mathematics from the very
earliest times existed on two levels, one in everyday use, the other
supported by an ever expanding and experimenting priesthood who from time to
time would have found it advantageous to co-operate with king and state.
For Sir Flinders Petrie, Egypt's missing mathematics turned up in the most
unexpected places, largely indecipherable and certainly at odds with
anything that was previously understood from Middle Kingdom papyri or New
Kingdom measuring rods. Petrie's 4th dynasty digit is a very good example,
yet in the Amenemipt rod interpretation just discussed where the
measurements of Jomard apparently confirm the beliefs of Herodotus, the
value of Petrie's digit appears to have some metrological significance:
Greek common metrios 460 - 461mm / 25 = 18.40 - 18.44mm
Petrie's Giza digit .727 ± .002 inches = 18.41 - 18.51mm
The coincidence which appears to span two millennia of Egyptian metrology is
not to be dismissed lightly. Petrie's insistence that the Giza digit was
consistently laid down with no more error than about 4 thousandths of an
inch demands an explanation, especially when it is at odds with his equally
consistent royal cubit measure of the same period.
One would think that such accuracy came from a common source.
and so to the geometry
Foundation metrology illustrates how Petrie's Giza digit is an integral part
of the Egyptian measuring rod. It also confirms that Petrie correctly
identified the 20.620 ± .005 inch royal cubit measure and that Jomard only
slightly misread the oddities of the Amenemipt rod. The rod and the measure
however are not the same thing and it is through this particular recognition
that the ancient metrological dilemma can be resolved. Petrie's Giza digit
from the 4th dynasty becomes the measuring rod's .726 inch double post
division, so named "double" because of the evidence for its half value off
the much earlier Saqqara flake of the 3rd dynasty. Both the measuring rod
and the flake reflect this rather forcefully.
To follow the geometry in any logical sequence it is first necessary to
contemplate the hand framework geometry of HANDLENGTH, HANDWIDTH, in
particular the scaled handwidth which has been taken straight off the
Amenemipt rod according to Jomard, and which by interpretation has been
given the number value 55©. The additional interpretation that the number
55© and its complimentary Fibonacci framework can only be understood under
an Imperial Linear system of measure (55© = 3025 = 3.025 inches) is a
formidable barrier for any metrologist yet on this basis has the entire work
been expanded. The mystery of the British connection is somewhat justified
however by thefact that out of the central anthropomorphic geometry is
extracted the megalithic yard. This ancient British measure and its Egyptian
counterpart, the elbowspan, are identical and to the credit of Alexander
Thom the former can still be identified in its megalithic setting.
The anthropometric geometry and its projected measuring rod which are at the
center of this Egyptian metrological science are truly unique for this
period of history. It could be said that they are truly unique for any
period in history.
Wading through the history of ancient metrology, particular linear measure,
is a totally frustrating exercise. It is mainly due to the fact that errors
have been built upon errors and that 'standards' having been found, have not
A few surveys stand out from the crowd, such as those done by Alexander Thom
and Sir Flinders Petrie because both of these men found 'standards',
recognised them and published the results. More importantly they firmly
stood by them. There are hundreds of books that touch on or deal with
ancient metrology, the vast majority of them only adding to the confusion
that had already been created by their predecessors. Listed below are a
few books that I found informative for not being repetitive.
Alexander Thom. Megalithic Sites In Britain. Clarendon Press, Oxford, 1967
... the megalithic yard.
W.M.F.Petrie. The Pyramids And Temples Of Gizeh. Field and Tuer, London,
1883 ... Almost a Bible (online this site).
Otto Neugebauer The Exact Sciences In Antiquity. Brown University Press.
1957 ... Reference to the rapid rather than slow developmental periods of
Algernon.E.Berriman. Historical Metrology. J.M.Dent and Sons, London, 1953
... classic metrology.
Herodotus.Histories. ... hints and hazy metrology.
Eivind Lorenzen. Technological studies in Ancient Metrology. Copenhagen,
Nyt Nordisk Forlag, 1966 ... The 7.68cm handwidth module of scale A or the
preferential scale (3rd - 19th dyn)
Francis Hitching. Earth Magic. Picador. Excellent section on Alexander
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