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4224re: historical metrology

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  • Chris
    Nov 2, 2003
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      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
      google's cache:

      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
        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)

      megalithic yard

      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
      measuring rod."

      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 ?

      the inquiry

      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
      18th dynasty.

      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 1:11Ž2 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
      smaller :
      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
      inches 523-524mm.
      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.

      option C

      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

      egyptian mathematics
      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.

      the unknown

      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.

      further reading
      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
      been recognised.
      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
      ancient mathematics.
      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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