Talk:Ancient Roman units of measurement/Archive 2

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Impossible precision

Initially topo

Paul Martin,

On what basis do you hold that the Roman foot is exactly 296 1/3 milimetres? The metre is based on the speed of light in a vaccuum and the period of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. Is this what the Romans used to define their foot?

Also, I not convinced of the following ...

Since the Roman cubit measures 17.5 English inches exactly one,
the English foot is sixteen twenty-eighth of the Mesopotamien cubit.
Thus the Roman foot measures 35 lengths of the English barleycorn. 
This table retains the value of the English barleycorn issue of the 
foot of the compromise 1959,
even if a modern value, increased by 0.533 µm, would be better than
the current  8,466.667 µm. 

... not convinced ... well it doesn't actually make any sense. Jimp 07:05, 1 December 2005 (UTC)

Hi Jimp,

Like you can see it with the five different English pounds, the simple prime number ratio has always be applied by good metrologists.
The five pounds: the "London pound", the "avoirdupois pound", the "merchant pound", the "troy pound" and the "tower pound" measures respectively: 7200, 7000, 6750, 5760 and 5400 grains, i.e. (7^0 x 5^2 x 3^2 x 2^5), (7^1 x 5^3 x 3^0 x 2^3), (7^0 x 5^3 x 3^3 x 2^1), (7^0 x 5^1 x 3^2 x 2^7) and (7^0 x 5^2 x 3^3 x 2^3).

Also the metrologists in ancient times always respected these simple primary number ratios. They "took references" to already known standards applied by neighbour nations. (Mediaeval metrologists, less scientific than their colleges of ancient times didn't do this every time, but although often. So it is not a coincidence, that the English foot is 15 sixteenth of the French foot. 0.086 %, i.e. nothing for Middle-Ages needs of accuracy.)

If it is proofed, that 20 Mesopotamian cubits equal 24 Roman cubits equal 35 English feet, we can apply the modern value of the English compromise foot of 1959 to know the proper value of the historical measures. Scientific researches with real existing archaeological standard sticks confirm this with a precision of ± 0.17 %.

17.5 inches for the Roman cubit, with 25.4 mm per inch, gives 444.5 mm exactly one for the Roman cubit.

But you are right Romans didn't know definitions like "the transition between [...] hyperfine levels [...] of the caesium-133 atom." But never the less their precision with about ± 0.17 % was not too bad. But we, today, knowing that one Roman foot is 35 thirty-sixteenth of the English foot, we can say Roman foot should be something like 296 1/3 mm. This is confirmed by the archaeological founds with ± 0.17 %.

The modern defined value of the Roman foot is even 296,352 (= 7^3 x 5^0 x 3^3 x 2^5) nanometres !

Then the English inch should be redefined officially to be hence 25.4016 mm, because a inch of  7² x 5² x 3⁴ x 2⁸ nanometres better matches with the omnipresent simple primary factors of the ancient system of measure than the current 127 x 5⁵ x 2⁶ nanometres. Just 1.6 µm more per inch, that is 8/127,000 or about 0.0063 %. That is nothing even for high-level technical precision. In praxis nothing changes, but this would simplify all the theoretical values for all the ancient measures in a considerable manner.

Best regards, Paul Martin 18:12, 1 December 2005 (UTC)

Paul,

There is a vast difference between those different pounds and these different feet. All those pounds are based on the same base unit: the grain. The English foot is based on the metre which in turn is based on the speed of light and the second with the second being based on the frequency of the aforementioned radiation. This is not the basis of the Roman foot. It seems that you are redefining the Roman foot to your liking. What people want to know is what the actual historical Roman foot was they aren't going to be interested in what you call the "modern defined value of the Roman foot".

I don't doubt that the ancients "respected this simple primary number ratios." and "'took references' to already known standards applied by neighbour nations." I'd not be surprised to find simple interger relationships between the units of various ancient systems. However, what you're suggesting is something more. You're suggesting simple interger relationships between ancient units and modern ones. The Romans didn't have the benifit of the modern SI definitions to base their units on. We can't just go around and redefine their units for them.

A side point: in my opinion, the inch would be better redefined as exactly 25 mm. Why redifine something to fit in nicely with a long obsolete ancient system instead of the current international standard? Jimp 2 December 2005

Jimp,

The example of the English pounds was given to show that the "principle of easy prime number relationships" exists in the (old and newer) metrology. This is not a new principle of Renaissance times, but this is attested from ancient times. Heron and many, many ancient writers discoursed precisely about this topo.

And that's "normal". Ancient merchant had to know that one Roman pound was exactly 3/4 Greek mine. (International business dealings were common and very developed during the ancient times, but broke down, like the sciences, in the European Middles-Ages.)  Also it is attested that in ancient times it was a general knowledge that 625 Roman feet equal 600 Greek feet, i.e. one (Roman–Greek shared) stadion. Because, one of the main Greek feet measured indeed 25/24 Pes Romanus, quite 308.7 mm.

You wrote: "I'd not be surprised to find simple interger relationships... " ??? I don't understand you. The ratio 2:3 for example is a relationship of simple integers. Even if the result, expressed in decimal (like in hexadecimal, but not in duodecimal base) is recurrent.^[1] Serious scientists like Lelgemann or Knobloch, both Berlin, and the pioneer Prof. Rottländer, Tübingen (Germany) who spent his life, in the second half of the 20th century, to collect or examine hundreds of archaeological measure sticks, graduated rulers and other standards. He found not one not related one. In this or that case you can say: coincidence! But every time? I remember you that from Middle-Ages, we know some not related measure systems.

That's not surprising. Because European Middle-Ages were, beyond doubt, much less scientific than the ancient times, international trade relationships were rather rare in Middle-Ages. Thus some egomaniac Prince, without good metrological consulters, took the liberty to officialise a doubtful, obscure and corrupt local standard, mandatory for the whole realm. Such things never occurred in ancient times. Good scientific metrologists watched over. Utterly, this wouldn't be accepted neither by dealer nor by the international working high-quality craft.

If in ancient times a replacement part of an engine (of war) was commanded to be fabricated by the adroit hand workers of an ally country: The ordering party must be sure that the consignee uses the same ruler or a compatible one. If not, it's not operating. (This is a technical need, just like, in our times for the Airbus, fabricated in four or five different countries.)

But the needs of accuracy in ancient times were very less exigent than in modern times. Therefore you can find later called Roman inches (since about BC 2800) and than Roman foot sticks (up to AD 476, and even from later, if you count the various realms conserving the Roman foot during all the Middles-Ages) from about 295.7 mm to about 296.7 mm (= 296.2 mm ± 0.17 %, like Rottländer calculates with modern scientific statistical methods. If you want to give the value of the Roman line (used at least at the beginning of the Middle-Ages) you obtain by the Rottländer value 296.2 mm / 144 = 2.05694 mm. Suppose, you round this to 2.1 mm. Then 2.1 mm x 144. This gives you a Roman foot of 302.4 mm ???  That's not a Roman foot !!  An error exceeding the two percents !

You can take 296 mm, but that's not better. Furthermore a Roman foot of 296.0 mm is not more correct than a Roman foot of 295.9 mm or 296.1 mm for example. Because all the graduated sticks beginning by 295.7 mm up to 296.7 mm can be considered to be Roman feet, are Roman foot rulers.

You wrote: "We can't just go around and redefine their units for them." That's very obvious. A friend of mine, a great metrologist, certainly the greatest living, contemporary metrologist, proposes since 2003 – for the reasons described above – a conventional, modern defined Roman foot of 296.352 mm exactly one.

But his definition (which me, for my part, I recognise, because it's the best one, the most adapted one. Not in contradiction with the values obtained by the modern science) is nowadays still not well-known and therefore "not widely accepted". For that reason I prefer to use for conversion the 1959 compromise English foot standard of 304.8, respectively its barleycorn of 8.46 mm. (This one who has forgotten what's a barleycorn, he follows the given link to remember and to retain it: "One English barleycorn is equal to 1/36 English foot exactly one."

So long.

Summary

I summarise:
Because the English system of measurement is now the last surviving variant of the 5000 years old ancient system of measurement, we have to calculate the Roman foot in the English standards. Since the English foot is identified to be 36/35 Roman foot, we know that the Roman foot is 35 English barleycorns. Then we have to convert these values faithfully into decimal SI, since 36 English barleycorns equal 0.3048 meter.

Even if – that's clear – Romans never heard of "speed of light" nor had a NIST with measure knowledge and know-how up to nanometers and above.
Despite this fact, the chosen procedure is – in my humble opinion – the only accurately one.

Paul Martin 15:09, 2 December 2005 (UTC)

The new Roman foot

Paul,

You note that "the needs of accuracy in ancient times were very less exigent than in modern times." You point out that Roman foot sticks are found to range in length "from about 295.7 mm to about 296.7 mm". "Furthermore", as you also note, "a Roman foot of 296.0 mm is not more correct than a Roman foot of 295.9 mm or 296.1 mm for example."

Yes, I agree with what you're saying here. What I'm trying to bring to your attention to is the fact that what you've put into the article is rather at odds with this. The value you give in the article is 296.3 mm. By your own very logic this cannot be more correct than 296.3 mm or 296.4 mm.

Indeed since the Roman foot ranged from about 295.7 mm to about 296.7 mm precission to the nearest tenth of a milimetre is plain nonsense. What belongs in an encyclopædia are the facts. The fact, as you've explained here, is that the Roman foot is 296.2 mm ± 0.17 %. This is the fact that belongs in the article.

You agree that "We can't just go around and redefine their units for them." What I'm putting to you, Paul, is that this is exactly what you're doing. If the Roman foot ranged from about 295.7 mm to about 296.7 mm, you can't just pick a value within that range and call this the true Roman foot (at least not for the purpose of writing an encyclopædia). By choosing 296 1/3 mm you are redifining the Roman foot.

I appreciate your motives for choosing this value. Yes, a Roman foot of exactly 35 barleycorns is a nice round foot but it's your Roman foot not the Roman foot. Similarly 296.352 mm exactly may be nice too but this is your friend's Roman foot. I've got an idea: how about we make the Roman foot 296.32 mm exactly then ten Roman stadions will be excatly one international nautical mile (our friend Rktect would love me)? Or how about making it 2^-1.75 m (approximately 297.301779 mm): the perfect height for an A4 sheet? Surely you'll like my Roman feet ... well the second is a little large ... but they are my Roman feet.

What readers want to know is the length of the true historical Roman foot. In short, the article must be editted to reflect the facts rather than your preference. This article is Ancient Roman units of measurement not Paul Martin's modernised Roman units of measurement. I'm reverting your edits to the length table.

Jimp 5 December 2005

Hi Jimp, I saw your modifications in this article and this reply. I'll answer you within the next few days. So long, Paul Martin 09:55, 7 December 2005 (UTC)

I await your reply. You will forgive my haste in reverting your changes to the table but for the reasons above I felt it necessary and saw little reason to hold out any longer. Also you will have noticed that I have left a mention of the fact that one Roman foot is about 35 English barleycorns in the paragraph at the bottom of the section. I hope that this kind of compromise might be acceptable to you. I did also note, however, that one Roman foot is approximately ³⁴/₃₅ of an English one as well and that this approximation was just as valid. Surely you'd agree. This is certainly interesting trivia but trivia should be presented as such and not masquerade as a defintion. Jimp 8 December 2005

Barleycorns

What is the rational for putting equivalence in English barleycorns? This is nothing short of absurd. How many people use barleycorns as a unit of measurement today? How many even know what one is? Jimp 07:23, 1 December 2005 (UTC)

You are right, this can be worthy a discussion. I only wanted to exprime the Roman values into the values of the English system of measure.

If we express the Roman foot into English foot, we have to give 35/36. Then the palm = 35/144 EF, the digit = 35/576 EF.

For the larger measures, this done:
Cubit = 35/24 (or 1 11/24) EF, step = 175/72 (or 2 31/72) EF, pace = 175/36 (or 4 31/36) EF, perch = 175/18 (or 9 13/18) EF, arpent = 350/3 (or 116 2/3) English feet a.s.o.

If you prefer, why not?!  But I wonder if these values (even expressed in decimals) would be quite easier than the barleycorn reference. It's an occasion to remember that a foot is actually 36 barleycorns. It's not so difficult. No?

Paul Martin 00:03, 2 December 2005 (UTC)

Given that the Roman foot is exactly 35 English barleycorns then, yes, this reference is useful ... to some. However, most of those who still use feet & inches wouldn't have a clue what a barleycorn is supposed to be. Now, instead of simply replacing barleycorns with more familiar units, why not have both? Of course, this is all supposing that the Roman foot is exactly 35 English barleycorns which is the point we're debating above. If, as I contend, the Roman foot is not exactly 35 barleycorns, then this will still be an interesting but rough conversion and still worth a mention. Jimp 2 December 2005

Suite (Paul)

Hi Jimp, I saw your modifications in this article and this reply. I'll answer you within the next few days. So long, Paul Martin 09:55, 7 December 2005 (UTC)

I await your reply. You will forgive my haste in reverting your changes to the table but for the reasons above I felt it necessary and saw little reason to hold out any longer. Also you will have noticed that I have left a mention of the fact that one Roman foot is about 35 English barleycorns in the paragraph at the bottom of the section. I hope that this kind of compromise might be acceptable to you. I did also note, however, that one Roman foot is approximately ³⁴/₃₅ of an English one as well and that this approximation was just as valid. Surely you'd agree. This is certainly interesting trivia but trivia should be presented as such and not masquerade as a defintion. Jimp 8 December 2005

I'll try to reply you in a new chapter, in which I moved our last two messages just here, because all above too related to split it.

Article concerning talk

For your described ratio of 34 to 35? 34/35 x 304.8 ≤ 296.09 mm. A small Roman foot, but indeed: It's a Roman foot!

But what's about 33:34? 296.835 mm. Acceptable. Though, why not the bonny ratios 71:73 or 853:859, which give the very good values of 296.449 and 296.224 mm respectively? No, peremptorily no! The aim is not to find any ratio, but the plainest, most simple one ! Therefore your insertion in the article 34/35 is not consistent and only confuses the reading users. Or do you want to vindicate a new – never heard – theory in metrology, pursuant to the English foot is derived from the other ancient measures by a 34-digit-cubit ??

Casually, also your A4 mention is not useful: The DIN A4 is an approximation of fourth root of two divided by four i.e. more accurately about 297,302 mm (like you mentionned rightly yourself) and so there is a difference of about a third of percent. Too much ! Then: This superficial thought, which everyone, including me, at first can have, should not be specified in a serious encyclopaedia, because there is no cause and effect. In this manner, we also could make researches, whose prominent feet size about 44½ and therefore incidentally one approximate Roman foot. Should we add a long listing of these personnalities?

Before continue, I take it for useful to remember you several attested and well-known main measures of the ancient metrology:
Between the foot, always shared into 16 digits in ancient times, and its double, the 32-digit-cubit named nibw, there are five other very important measures:
the pygme = 18 digits, the pygon = 20 digits, the pechys = 24 digits, the neilos = 28 digits and lastly the kus, which is the sumerian dividing of the cubit into 30 digits.
Because in ancient times the 21-digit-measure, like the 25-digit-cubit and the 27-digit-cubit are used rather seldom, also we, now today, we don't have to occupy with all too often.

Except in the English 66-feet-chain and in the French arpent length, the prime factor 11 is rather rare in the old systems. The factors 13 or 17, I never encountered. Did you? So, we can neglect the 22-digit-measure, just like the cubits of 26-digits and 34-digits. (You see, no place for an improbable 34:35 ratio theory.)

Let's not speak about other prime factors like: 19, 23, 29 etc. and its absolutely inexistent respective measures.

By this we can retrace, for instance, the way from the Nippur cubit to the French and English feet: (Other paths are imaginable, but the result would be identical.)

Let's going out – for simplifying – from a Nippur cubit specimen which measures, incidentally, 6223/3 mm exactly one.

Since the Drusian pygme – equal 18 Roman digits – is 18/28 of the Nippur cubit :	(518.583 / 28) x 18	=	333.375 mm
The Babylonian cubit is a nibw of this attested Drusian pygme seen as a pygon :	(333.375 / 20) x 32	=	533.400 mm
The Hashimi kus is a 30-digit-measure of 32/28 Babylonian cubit (seen as neilos) :	(533.400 / 28) x 32	=	609.600 mm
The Hashimi nibw is the 32-digit-measure of precisely this Hashimi kus measure :	(609.600 / 30) x 32	=	650.240 mm

As easy like that, in metrology, we can retrace millenia of history. When both, the Hashimi kus and the Hashimi nibw came to northern Europe in the Middles-Ages, they each chose their respective measures, the Norman kingdom (after 1066) in England and the late Carolingians (751-987) or the early Capetians (987-1328) in France. Since these times :

• The English foot is the half of the Hashimi kus.
• The French foot is the half of the Hashimi nibw.

Even if it's quite a little harm that you deleted the barleycorn conversion in the table, the fact that you also eliminated the good, not-controversial definition of the Roman foot is so much the worse. That the Roman foot is 16/20 of the Egyptian Remen and thus 16/28 Nippur cubit (This means the Nippur cubit measure, practiced over millenia, and not means one of the several Nippur cubit specimen found by archaeologist.) is one of the most important informations in this article. Above all, not-controversial in science !

Beyond the 35:36 ratio is not "a convienient rough conversion" since it is proved that both English foot and Roman foot are historically related. (I don't know any metrologist who would contest that.) Quite another question is the exact values expressed in millimetres, a mean rivalry of authority by the different standard institutes. It's obvious that even before 1959, the Imperial foot as well as the American foot was essentially the same English foot. You approve? It's rather anecdotic that the first one measured 0.304 799 472 mm (official in 1928) and the second one is about 304.800 609 601 219 2 mm (I saved you from the 210 recurrent decimals of the exact value obtained by the 1200/3937 ratio of the US definition.) All this is either to laugh or to weep.

In the same manner, the Austrian foot used until about 130 years is the same foot than the at least 2500 years old Greek "pous metrios". Also here, the official value for conversion, chosen by some bureaucratic functionaries: 3793/12000 is only one of other possible (not less accurate) and gives 0.316 083 mm exactly one.

Happily in 1959 the anglo-saxon metrologists and legislators abolished these doubtful, but hitherto official definitions like 1200/3937 and set a common rational standard:
"One English foot equal 381/1250 metre." This is a very good definition. This is the legal one.
Therefore the Roman foot is exactly 296 1/3 mm. Defined neither by me nor another individual, but by the Commonwealth legislators.

However – and here I apologise – without doubt, you are fully in right, this value for the Roman foot can't give without an important information. The ascertained margin of ± 0.17 % relative to a statistical value of 296.2 mm. But we should not confound a mean value neither with originate value nor with best value nor with legal value.
Even if the Austrian definition continue to have a local juridical status (for instance of old surveying conflict), without doubt, the English foot and its legal definition, as nowadays the only one of the ancient measures still in use, and so makes also authority for the Roman foot, failing with a serviceable value definition of an inexistent Roman standard institute.

But, thanks to you, we found the mayor lack of my former version: The absence of any mention to the ascertained spread in the length of the historical Roman foot.

Here ends, the reply concerning directly the article.
But, as above, we also discusses some connected topos like "new definitions for feet and inches" (not the same as "new feet and new inches") I'll answer to your suggestions.

Reply form Jimp

Paul,

"The aim is not to find any ratio," you insist "but the plainest, most simple one !" This is where we rather disagree. What I would say is that aim is neither to find just any ratio nor the plainest, most simple one but to find the most accurate ratio. "34 Eng. ft. ≈ 35 Roman ft." is no less accurate than "35 Eng. ft. ≈ 36 Roman ft." Noting this in no way vindicates the theory that "the English foot is derived from the other ancient measures by a 34-digit-cubit". All it is is a rough approximation it does not imply anything about the derivation of the English foot.

In just the same way noting that the height of an A4 sheet is about one Roman foot (ignore the 0.3% error for a second) doesn't imply any historical link between them. Is mentioning this unuseful? A4 sheets are quite a common sight these days (I happen to have a bunch sitting in front of me). Perhaps it would help readers visualise the length of a Roman foot. How may of us have rulers with barleycorns marked off lying about these days? "we also could make researches, whose prominent feet size about 44½ and therefore incidentally one approximate Roman foot." Yes, we could but nor do we have these in the drawer of the desk.

Okay, 36 is three dozen is 2²×3² and 35 is five sevens whereas 34 is twice seventeen. As bizzarre as the factors between units in traditional systems got you never had 17, sure. And it is a well attested fact the the English units derived from the Roman ones, indeed. And from Nippur cubits to Hashimi nibw then the French feet and the like and all down throught the ages till we land in England. Let's assume that the English foot had been defined as 36/35 Roman foot. It is now redefined as 304.8 millimetres.

We can't pretend that in 1959 the Commonwealth & U.S. legislators redefined the Roman foot. What they did was to redefine the English foot. Certainly the English foot may have once been defined such that it were 36/35 of a Roman one but it had since been redefined and more than once. Changing the definition of a unit doesn't redefine the units it was once based on. The older definition is abandoned and a new one adopted or are you going to insist that the length of the earth's meridian along a quadrant is exactly the distance travelled by light in vacuum during a time interval of 10 000 000/299 792 458 of a second?

I'm glad you agree that we should note "The ascertained margin of ± 0.17 % relative to a statistical value of 296.2 mm." But you also note "we should not confound a mean value neither with originate value nor with best value nor with legal value." Indeed not. However, our ideas of what constitutes original, best and legal vales are leagues apart.

"English foot and its legal definition, as nowadays the only one of the ancient measures still in use, and so makes also authority for the Roman foot, failing with a serviceable value definition of an inexistent Roman standard institute." you write. I cannot agree with this. There is no extant organisation with the authority to define the length of the Roman foot. Roman units of measurement are obsolete. Neither you nor I can redefine them: you agree that would be absurd. Nor can the U.S. government in conjunction with the Commonwealth governments do so. Not even the U.N. can claim the authority to do this. The Roman foot was what it was and cannot be redefined (except, perhaps, by a god but even this would be to change history).

Anyway, you also note my deletion of the Nippur cubit reference. Yes, if it is valid then it belongs here. However, I'd ask you to cite some verifiable sources before we put it back.

Jimp 12Dec05

Paul's new reply

Thanks for your answer Jimp.
Even if you are an agreeable knowledgeable talk partner, I think we should not eternalise our discussion.

There are points in your version – even if I see it differently – however I can live with.

• Your millimetre rounded mean value is one of these points. We can let it in the table.
  Nevertheless I consider, giving this value you redefine not only the Roman foot to 296 mm, but also the English foot to be 304 16/35 mm, i.e. – 0.1135 % of the 1959 value.
  A Nippur cubit of 518 mm exactly one, seems me to be a short Nippur cubit. However it's a Nippur cubit!
  
• The barleycorn conversions of each Roman unit of length in table is not necessary.
  You are right, to read that the correct length of the Roman stadium is 21,875 English barleycorns don't give anyone a vision of the length of this stadium.
• Some minor format modifications are also acceptable.

Concerning some other points we are still in disaccord.

• For the A4 mention you write: "Perhaps it would help readers visualise the length of a Roman foot." This is not consistent. The Roman foot is about 29.6 cm. Everybody can imagine a 30 cm ruler. Subtracting in mind four millimetres (or surely some decimals less) than you have a very good visualisation of the Roman foot length.
• I persist: If you give a 34:35 ratio, you also suggest that the English foot historically derived from a 34-digit-cubit. I'll come back below.
  But it's with pleasure that I can confirm you that a hypothetical 34:35 ratio would be "no less accurate than" the well-known and accepted 35:36 ratio.

Both the A4 mention and the 34:35 ratio – like it seems me – have been introduced by you to relativise the pertinency of the 35:36 relationship.
Therefore I'm not mad at you. I understand. 200 years of a hegemonic decimal SI system, unlearnt, broke the habit to reason in "ratio relationships", once the only possible one.

• Concerning the Nippur cubit reference: I propose you to mention it briefly like before in the length table, and to add a new, not all too long chapter at the end of the article.
  I propose me to prepare a version for example at Discussion version: Chapter: History of the Roman foot which we can review together before integrate it into the article.
   
  If you ask for references, I can give you for example this document of Dieter Lelgemann, Acting Director of the Institute for Geodesy and Geo-Information Technology,
  Technical University Berlin, who works amongst others for the European GPS project called Galileo. This doubtless, serious and eminent scientist recognises, of course like all the knowledgeable colleagues, the real existing historical ratios inside the ancient measure system. Before invention of the decimal digits, this was the only possible manner to evaluate the different measures. (Sadly he even integrates the doubtful "megalithic yard" researches of Thom, even without commentaries. I never studied these in details, but like others, here, I'm very sceptical. In the opposite, since the well-attested, graduate Nippur cubit measure found in several specimens, the relationships are clear.)

The plainest ratio is the accurate one.

To close this topo of that reply. You wrote: "What I would say is that aim is neither to find just any ratio nor the plainest, most simple one but to find the most accurate ratio."
I'll try to illustrate you how the old measure system in ancient times developed by dispersing.

Imagine a smaller, (nowadays we would say) "satellite state" standing former, for example, under influence of the Babylonian Empire, now came under influence of another regional power, say: the Egypt. Several years later, in consequence of the close commercial relationships with the new friend and protector, they decided to change their system of measurements by abandoning the Babylonian main measure and for adopting the Egyptian Royal Cubit of 529.2 mm.
But they would say, in this hypothetical illustration example: Okay, our new main measure will be hence the Royal Cubit. However because it's in our babylonian tradition, we'll continue to share the cubit into 30 equal parts. That will better match with our well-known sexagesimal system. So they created a digit of 28/30 Egyptian digit equal 17.64 mm. Some generations later exactly this digit can give a (not attested) foot main measure of 282.24 mm equal 8/15 Royal Cubit.
Those exchanges of main measures could happen even between great powers, for similar reasons or for example by the arriving of an intellectual elite from a declining former important region to the new dominant metropolis. (See role played of slaved Greek teachers at Rome.)
Another example of such exchanges can be studied in the case of the Russian cubit called "arshin" equal 28 English inches.

So, you see, it's well our duty to rediscover "the plainest, most simple ratios" because they are identical to the "most accurate ones". In your logic the excellent ratio of the two prime factors 829 and 853 would have prevalence. (304.8 / 853) x 829 ≥296.224 mm. If you want or not this suggest that the English foot are deduced from the Roman one via fantastic 829- and 853-digit-arpent lengths. Also hypothetical pseudo-ratios of 1481:1524 or 370:381 are only another way to give respectively 296.2 and 296.0 mm exactly one. But that's nothing to understand of the essence of the ancient ratio values. These ones ever also suggested a real historical deduction. Since the 34:35 ratio is not notably better than the widely accepted 35:36 ratio, the second, simpler one is prevalent, because more probable. That's also a principle in modern science.

Conclusion concerning these points

In conclusion: Let's concentrate, in our further talk to these six or five proposals expressed just above, for obtaining a solution.
Other exchanges can be interesting, but they are less urgent.

Paul Martin 18:36, 13 December 2005 (UTC)

Jimp's next few words

And thank you too, Paul, for your responses.

• I'm glad that you can live with the mean value rounded to the nearest millimetre.
  • However, I don't agree that this is in anyway a redefinition of the Roman foot. If I were to note that Mount Everest is 8844 metres high, this in no way constitutes a definition of the height of the mountain. When giving the length of an object it is normal to assume that 296 mm to imply 296±0.5 mm.
  • Furthermore, I don't accept that this is a redefinition of the English foot either. It is not I but you who are insisting that an English barleycorn is exactly ¹/₃₅ of a Roman foot. What I'm saying is that even if it were the case up to 1959 that the English foot had been defined in terms of the Roman one, it is no longer so defined. The English foot is defined in terms of the metre. If it were possible to redefine the Roman foot, then the only way to end up with the reationship 1 Roman foot equals 35 English barleycorns exactly would be to choose the definition 296¹/₃ mm.
  • "A Nippur cubit of 518 mm exactly one, seems me to be a short Nippur cubit." you write "However it's a Nippur cubit!" Nor am I trying to redefine the Nippur cubit. The same arguement applies but there's more involved here which I'll come to later.
• I'm also glad that you agree that "The barleycorn conversions of each Roman unit of length in table is not necessary."
• "Some minor format modifications are also acceptable." of course.
• With respect to my A4 mention you write "Everybody can imagine a 30 cm ruler. Subtracting in mind four millimetres (or surely some decimals less) than you have a very good visualisation of the Roman foot length." Yes, this is true. However 297 mm is still closer to the Roman foot than 300 mm is. What's going to be easier: subtracting four millimetres in the mind or subtracting one? However, this is, as I say, nothing more than a little trivia and so I can easily live without its being mentioned.
• "If you give a 34:35 ratio," you persist "you also suggest that the English foot, historically derived from a 34-digit-cubit." I continue to counter that I do not. What I have written in the article makes it absolutely clear that this is a rough approximation.
  • Yes, I acknowledge that I had intended to relativise the pertinency of the 35:36. This was the purpose of my introduction of the 34:35 ratio and my choice of wording for the A4 mention. Note, however, that I refer to the mention's wording. It was for the purpose of helping people visualise the Roman foot's length that I mentioned A4 at all.
  • I can accept a change of wording here. I can also accept that we drop the mention of 34:35.
  • "200 years of a hegemonic decimal SI system, unlearnt, broke the habit – once the only possible one – to reason in 'ratio relationsips'." Yes but I'm not the only one with this broken habit. Other readers will be in the same habit as I. Decimals are what readers will be expecting these days.
• Thank you for the link to the document of Dieter Lelgemann's. I have read it.
  • I too was dismayed that he unblinkingly incorporates the megalithic yard. On the one hand: who am I to refute the research of such an eminent and serious scientist? On the other: how can I take this seriously when he also says the the megalithic yard was based on the same Nipur cubit?
  • Another point: perhaps I've overlooked something but I don't quite see anywhere where he states that the Roman foot was defined as ¹⁶/₂₈ of a Nippur cubit.
  • However, I'm sure that you'd be able to find plenty of other sources if it is as you say such a "good, not-controversial definition of the Roman foot"
  • Then, on the other hand, you write "Roman foot is 16/20 of the Egyptian Remen and thus 16/28 Nippur cubit". So which is the definition of the Roman foot. For comparison consider the English foot. It is exactly 304.8 mm but this is not its definition. The definition of the English foot is one third of a yard which is, in turn, defined as 0.9144 metres.
  • I look forward to reading your "new, not alltoo long chapter" regarding the definition of the Roman foot.

Now I must go. I shall continue my response later. Jimp 14Dec05

Is the plainest ratio the accurate one?

Paul,

You write "it's well our duty to rediscover 'the plainest, most simple ratios' because they are identical to the 'most accurate ones'." The logic behind this is that because of the influence of neighbouring states, one ancient society should be expected to have adjusted their system of measurement to be compatible with that of their neighbours.

I can certainly understand the validity of the arguement that the values of the units of measurement used in contemporary societies may be expected to have existed in simple ratios. Now, if you tell me that the Roman foot was defined such that 24 of them equalled exactly 25 Greek ones, then I'll happily accept this. However what you're saying is something entirely different.

The arguement you present applies only to contemporary societies. It could well be extended so as to include the possibility that a unit be based on an older one. However you cannot base your measurement standard on something that is two millenia yet to come ... at least not unless the causal theory of time is wrong.

Were the English yard defined as ^3×36/₃₅ Roman feet, then you could say that one Roman foot was exactly 35 English barleycorns. The English yard is not so defined. It is defined as 0.9144 metres. The simple ratio of 35:36 may once have been valid but the English yard now has a new definition. This does not force a redefinition of the old Roman foot: it is an abandonment of the ratio.

Take, for example, the nautical mile. Until 1970 the British nautical mile was defined as 6080 feet. In 1970 the international definition of 1852 metres was adopted. It would seem to me, by your way of thinking, then that this would be to redefine the Imperial foot to be ¹⁸⁵²/₆₀₈₀ (⁴⁶³/₁₅₂₀ or ~0.304605263) metres. Of course this is not the case. The old definition was abandoned for the new one.

"In your logic the excellent ratio of the two prime factors 829 and 853 would have prevalence. ... If you want or not this suggest that the English foot are deduced from the Roman one via fantastic 829- and 853-digit-arpent lengths." If I don't want to make this suggestion then I have merely to note that this is a rough approximation. However, no, I wouldn't want to use such mind-boggling ratios ... not when readers will be expecting decimals.

"Also hypothetical pseudo-ratios of 1481:1524 or 370:381 are only another way to give respectively 296.2 and 296.0 mm exactly one." Okay, but I do not intend to suggest an exact relationship. What I'm saying is that this kind of precission is not valid. Giving "296 mm" is not to suggest any exact relationship. "296 mm" is normally taken to mean "296±0.5 mm" (unless some other range is specified).

You write of ratios suggesting real historical deductions. I'm afraid, my friend, that in your hands they seem to imply a backwards historical deduction, at least to me they do. You write of principles in modern science. One such principle would be that the cause comes before the effect.

No organisation today has the authority to redefine the ancient Roman foot. If you have a unit which is based on the Roman foot and you redefine that unit based on the metre, then you've abandoned the old definition in favour of a new one. The two different definitions don't just continue to coexist thence redefining the Roman foot back through time.

If the ratio 36:35 is historically valid, then it deserves a mention. However, the place for such a mention will be English unit not here. Such a ratio may once have been valid: it no longer remains so. Today it is nothing but a rough approximation. It would certainly be an approximation with historical importance but no less an approximation.

Jimp 15Dec05

I've disentangled these two different topics. You'll notice links between this page & the new one (the last of these is right above this paragraph). By clicking on each of these as you come to them you can preserve the flow of this chat of ours. Paul, I shall respond again when I find time but now I must go. Jimp 16Dec05

You made it very well. Me too, I'll reply soon. Paul Martin 09:55, 16 December 2005 (UTC)

Excuse for not having replied in our topo since a long time. This will be come soon. Promised.
Hovever I didn't stay lazy. I prepared you an overview for the Hexadecimal metric system. So we'll know exactly what we are talking about.

So long!  Paul Martin 14:18, 20 December 2005 (UTC)