# Talk:Ancient Roman units of measurement/Hexadecimal foot/Archive 1

The following is a discussion that Paul Martin and I had been having under the title Impossible precision on the Ancient Roman units of measurement talk page. It came about when I suggested that 25 mm exactly would be a better redefinition of the inch than Paul's suggested 25.4016 mm i.e. 72 x 52 x 34 x 28 nanometres. Jimp 16Dec05

## The new hexadecimal metric system

PS. The decimal meter is, since 200 years, the predominant system. Since many decades it is – and for good reasons – prevalent in science.  Not any longer.
Soon, the new hexadecimal metric system – I really have this conviction – will replace this "poor decimal metric system", which has not even conserved the – certainly more than 5000 years old – foot measure.
At the moment, this (not so new) proposal (it dates of the year 1989) has not the right to be presented. Not even in Wikipedia.
Because its inventor, a friend of mine, – especially in France where he lives – is dashed by the probably the most insolent censorship since Galilei. In France because since more than 200 years, the long-cherished "decimal metric system" is a kind of "state religion", inspired by the celebrated and adored "Goddess Reason" of the French Revolution times herself.
It's not fortuity, if the official devise of the decimal meter system is: "For every people and for always.(Very poor dreamers!)   However: "Vive la République!"
The foot of hexadecimal metric system has two definitions. A scientific one (since 1990) and a technical one (since 2003). Both differ by about 0.0054 %.
The scientific hexadecimal foot is defined: 19.109 257 m / 64. You can get the result by your hand-held calculator.
This, because the mayor radius of planet Earth, according to WGS84, is 6,378,137 meters. Multiplied by the number pi, then divided per 1024 twice and rounded appropriated to six right-sided decimals gives: 19.109 257 m exactly one. (19,109,257 is a prime number!) This is the modern 64-feet chain. The sixteenth part of the modern hexadecimal foot is the so-called "universal digit". The half of the circumference of Earth (little more than 20,037.5 km) divided three times by 1024: So, the "universal digit" measures about 18.6614 millimeters.
The "technical hexadecimal digit" (for uses outside of scientific and extremely high-level accuracy fields) accepts a little error of about + 0.0054 % to measure 18.6624 mm exactly one. Because a defined universal "technical foot" of 1728 x 0.1728 = 298.5984 mm better matches with the 5000 years old measuring system. Thus the "technical hexadecimal foot" is defined to be twelve power six multiplied by ten power minus seven SI meter exactly one. A hexadecimal time format also exists, for uniting finally, space–time in the same frame of reference, in the modern "base sixteen". Man has not in vain "two thumbs and eight other fingers".
I send an S.O.S. into the world entire (and especially to its the English spoken part), if there are not at least one serious media or publication, who dares to break the iniquitous censorship, to which my friend Michael and his scientific works underlies in France, the self-called "country of the Human Rights". I know, since almost 15 years Michael tempted without success to publish his researches and proposals in French medias. But in the eyes of the French establishment, "nothing can be better, superior then the French decimal meter system". So, any other serious proposal can not exist. But, I think so, the Anglo-saxon world is not so narrow-minded regarding this topo. At least I hope it.
A good referenced site in Internet since 2003, december 19th – mostly in French – clearly describes his proposals. It is visited daily by many individuals.
Soon the worldwide TABU-cork will pop up !
Excuse Jimp, to have been a little long.
[1] ^   Now I reread, more attentively and I understand. You support, that surely often there are "relationships between the units of various ancient systems".
Excuse a lot for my lake of good understanding.
[2] ^   If not – the aviation industry using inch standards – all the passengers of all the crashing aeroplanes will thank you, because once more an undercarriage or a wing has been lost, owing to a bolt, calibrated too small into "Jimp's new inch standard" ;-) Errare humanum est.
Paul Martin 15:09, 2 December 2005 (UTC)

Talk:Ancient Roman weights and measures/Archive 2#The new Roman foot

## New feet and new inches

Paul,

"Jimp's new inch standard" ... ha-haa, yes, but I'm afraid I'd be rather too late to be given that honour. Though perhaps I might have a chance at having my name attached to a certain decimal foot i.e. ten of these inches; a quarter of a metre; closer to the average length of an adult foot than 36 or even 35 barleycorns.

In fact being a quarter of a metre, this would fit nicely into a hexidecimal metric system. There's a third hexidecimal foot for you ... though, of course, this doesn't fit into your scheme of dividing the Earth's circumference but it does give a nice interger conversion to the current metric system.

No, you suggest dividing the Earth's circumference up by powers of two: a laudable plan but let me remind you of something. When they devised the metric system they had a system of decimal divisions of the arc to go along with it. The latter has been discarded and poeple still use nautical miles as a rival to the kilometre.

There's another possible base unit for you: 1024th of a nautical mile. But you really would have to settle on one hexidecimal foot. Having one scientific one and a different technical one would be a nightmare. This is one advantage of the metric system: it's unambiguous.

However if we're going to have a hexidecimal system, why not base 32 instead? 1024 is already used extensively in computing as a mulitplier. 32 is the square root of 1024. Of course, if my new foot causes aeroplanes to crash will your hexidecimal foot do any better?

I was ignoring such difficulties but ain't it about time the we all metricate anyway ... ;-> ? The problems you mention about the 25 mm inch are problems of metrication. May nations and industries have metricised successfully. The problems are, admittedly, worsened because the new unit has the same name as the old one.

Anyway this is all well and good but what are the chances that any of these inovations of ours will ever be adopted on a large scale? I don't share your optimism on this. As I see it, the (decimal) metric system will continue to gain ground and eventually completely replace English units. You say "in technical fields, the English inch is always omnipresent." This may be true in North America & perhaps the British Isles but it's not the case universally.

Jimp 4 December 2005

Talk:Ancient Roman weights and measures/Archive 2#The new Roman foot

## Related topos

• Forget the nautical mile ! This is a wayward measure. It is great time to abolish it for ever ! Dishonour for us that we always use this measure. It corresponds to nothing ! It's a scholar example for the bad contemporary standards. The theoretical value of this unit is 40,000,000 / 360 = 1851.851 m. Then, metre rounded to 1852, one adds exactly one 0.008 %. You can calculate yourself at which latitude this definition is correct. Fine ! The former GB Admiralty Mile and the Old U.S. nautical mile definitions were quite better. Both can be considered, rightly, to be "nautical submarine miles", upon condition that you find a U-boat going to depths of about 24,000 ft.
But there is a second reason: Why do you want that somebody who builts a new hexadecimal system, first divide the hexadecimal degree (= 11° 15') into 675 parts?? This would signify, that the "twice sixteen hexadecimal hours a day" and its corresponding meridians would be, each one, shared first into 0x2A3 or 675 parts, before continue with the hexadecimal divisions !?? This proposal did not seem me to be consistent, sorry. (Cf. The hexadecimal Earthgrid.)
You wrote at this topo too: "How about we make the Roman foot 296.32 mm exactly then ten Roman stadions will be exactly one international nautical mile?"  Yes, that's correct, since one Roman stadium is 10,000 Roman digits, the Roman stadion with a digit of 296.32 mm would give 1852 m exactly one. But, any "international nautical mile" of 1852 m (1/21,600 Earth circumference superior equal to 1855.325 m) is only 99.82 %, just "out of range". In other words: A hypothetical foot of 18.553 25 x 16 equal 296.852 mm. This is not a Roman foot, anymore! Furthermore, all theories trying to proof that the old system is deduced from the Earth circumference are rightly accused to be pseudo-scientific.
• You wrote above: "When they devised the metric system they had a system of decimal divisions of the arc to go along with it. The latter has been discarded and poeple still use nautical miles as a rival to the kilometre." The 100 grade system of arc has been discarded because the decimal time is not consistent. The number ten has no integer with division by four. So the decimal clock-face even not marked all the four cardinal points with a main point. This lack of the decimal time format was so obvious that any reasonable man couldn't defend this one. Because the decimal system did not conquere the time format, the respective meridian system had also to stay in sexagesimal system.
That's quite normal.
• In the case of your place in history as creator of the "25 cm measure shared into ten pseudo-inches". (Pseudo-inches, because everybody knows that "inch" comes from latin "uncia". This word signify literally "divided into twelve parts". Thus, beside, also the pound of 16 ounces is the same linguistic nonsense.) Also here you are too late! Some German länder allied to Napoleon did so before. Others defined a "new Fuß" shared in ten "new Zoll" of respectively 30 and 3 cm. All in perspective to switch to the metric system. But people are not stupid. Did you ever see a thumb with a breadth of 3 cm? Perhaps 25 cm (size 37.5) is "closer to the average length of an adult [woman] foot than 36 or even 35 barleycorns", but you forget something. Generally we put all on our shoes! An average length of foot with shoes, very good matches, since ever, with about 30 cm.
• You wrote: "Why not base 32 instead? 1024 is already used extensively in computing as a mulitplier. 32 is the square root of 1024." and just before: "But you really would have to settle on one hexidecimal foot. Having one scientific one and a different technical one would be a nightmare. This is one advantage of the metric system: it's unambiguous."

Concerning your first quotation, I understand you. I'll come back on in a moment. No, I don't share your expressed opinion of your second quotation.
There are not only two variants of the same digital foot, but there are also two terms.
Both, the "scientific digital foot" and the "technical digital foot" can each one – orally – designed by the popular term "foot".
Either you have the intelligence to exprime you without possible misunderstanding or – in the most of cases – the difference of 0.0054 % is without any significance.

However, scientists will never use terms like "digit", "foot", "aune", "mile" or "league"; moreover different in each foreign language. This would be a real regression in comparison with one of the most progressive acquirement of the decimal SI, will say: the logical and practical prefixes of the same base unit like the milli-, the mega-, the microunit etc.
This must be preserved, if not, none proposed system can successfully pretend to replace the former decimal system.
It's true 1024 is very important. Because this is 16 power 2.5, the number 1024 seems not to match with the hexadecimal system of units. It's a false prejudice.

Because the ingenious system of my friend Michael Florencetime integrates the factor 1024 in an easy and practical way.
I have neither the time nor the place to explain all exhaustively (even if you don't speak the French language, surely this document can help you.)
Nevertheless in short :
The hexadecimal million is obviously 1024 x 1024, the mega. Like 1024 power 4, the tera, is the hexadecimal billion.
(Of course compared to the decimal billion of the good old, consistent long scale which also has been said dead in France over decades by all the encyclopaedia of the 19th and in the beginning of the 20th century. This scale will certainly come back world-over, even in U.S.A. and in Brazil.)
Because the hexadecimal million is 16 power 5, Michael groups great numbers always into five digits, what's practical, five digits are still easy to distinguish. For the five so-called "ranks" of each magnitude (kilo-, mega-, giga- etc.) he uses the five vowels not employed with his good and unambiguous omni-literal digits (much better than the IBM standard).
The five vowels are arranged by linguistic criterions signifying respectively "i" = 1x, "e" = 16x, "a" = 256x, "o" = 4096x and finally "u" = 65536 times the magnitude of the unit.
Examples: 1 x 1024 x 1024 equal one mega-i-unit, 256 x 1024 x 1024 equal one mega-a-unit. In the same way: 256 / (1024 x 1024) is the micro-a-unit.
The magnitude between milli and kilo – in the opposite of the decimal system – has its own prefix: "hexa". Thus one hexadecimal is the hexa-i-unit.
No problem also with the odd powers of 1024 which form an easy to understand deduced subsystem. So: 256 x 1024 equal one kilo-a-unit.
So, let's come back to the topo:
In their works and professional use, scientists never will employ the term "foot", but always the term "milli-a-metre", which refers, without any ambiguity, invariably to the scientific definition, never to the technical definition. In practice, if possible, they even prefer the basic unit: the hexa-i-metre (= 19.109 257 m / 16) exactly one.
This basic unit, in its technical variant measures 1.194 393 6 current SI decimal metres exactly one and is called "aune", the appropriated popular name of this four-feet-ell.
You see, not a soupçon of ambiguity! All is clear, limpid and well-defined. Ready for a worldwide use. This will come without fail and probably quicker than you now imagine.
At last I made you the following table.

## The new digital foot

The hexadecimal foot is here compared to the Roman foot, Austrian foot, i.e. the Greek "pous metrios", to the English and French foot measures :

The English foot, after 1066, is defined 16/15 French foot. The Roman foot and the English foot entertain the ratio 36 to 35 exactly one.
An Austrian foot is 16/15 Roman foot. The new digital foot is always  48/49 English foot, but contests the current English foot definition.
 Definitively : 1575 French feet equal 1620 Austrian feet equal 1680 English feet equal 1715 digital feet equal 1728 Roman feet.
Depending on the attributed decimal meter value for each one of these five foot measures, the comparative distances differ slightly.
Comparison distances
in millimetres, like all the other values of this table.
1575 x 9 / 0.27706
511 622.031
1620 x 1896.5 / 6
= 512 055
1680 x 304.8
= 512 064
1715 x 19109.257 / 64
= 512 068.371 171 875
1728 x 296.352
= 512 096.256
Ratio Foot French
definition
(1799)
Austrian
definition
(1872)
Anglo-saxon
definition
(1959)
Scientific
definition
(1990)
Idealistic
definition
(2003)
1575 French   324.839 385   325.114 286 =  325.120 000 =  325.122 775 347 222 222 222 222 =  325.140 480
1620 Austrian   315.816 069 =  316.083 333 =  316.088 888 =  316.091 587 143 132 716 049 382 =  316.108 800
1680 English   304.536 923   304.794 643 =  304.800 000 =  304.802 601 888 020 833 333 333 =  304.819 200
1715 Digital   298.321 884   298.574 344   298.579 592 =  298.582 140 625 000 000 000 000 =  298.598 400
1728 Roman   296.077 564 =  296.328 125 =  296.333 333 =  296.335 862 946 686 921 296 296 =  296.352 000
Percentages : 99.912 836 % 99.997 389 % 99.999 146 % 100.000 000 000 % 100.005 445 %
 The new hexadecimal foot has a modern, scientific definition. From its definition, scientific values for the four other feet are deduced. The new worldwide hexadecimal foot has been conceived in A.D. 1989. Since 1990, the value of this foot is 19.109 257 m / 64 exactly one. This value of the digital foot is obtained by the adequate definition: Math.round ( r0 * Math.PI * Math.pow(10,6) / Math.pow(16,5) ) / 64. The radius r0 is the World Geodetic System 1984 value of the mayor radius of our planet Earth, scilicet: 6 378 137 metres ± 1 m. Thus, the 64-digital-feet chain is very flimsy over-defined since ± 1 m values almost ± 3 µm in this chain definition. But a definition is a definition ! The idealistic definition of 2003 is only a licence, outside of scientific and extremely high-level accuracy fields. A idealistic digital foot of 1728 x 0.1728 = 298.5984 mm gives easy values for all the other historical foot measures. The anglo-saxon compromise foot of 1959, even very close to the scientific value, has no rational well-founded definition. The percentages give the slight errors of the former partial (national) definitions. The difference for the idealistic definition is assumed for always obtain simple prime factors.

For the article, I'm confident, we'll find solutions. In particular the Nippur cubit reference seems me to be inevitable. It's the attested historical truth.

I'm aware, for reading and studying my long reply, I demanded much of your precious time. Never the less, I hope you account it as profitable for yourself.

For the rest, I really think that we'll go hexadecimal. We have to understand our now day main tool, the computers. Binary divisions are omni-present in ancient systems.

At the last, the modest but aright devise of the hexadecimal system of units: " Two thumbs and eight other fingers. Twice eight equal sixteen. "
is quite more sympatric than the bloated and hubristic devise of the decimal SI: " For all people and forevermore. " Such pretensions not prove true.
Retrospectively, the decimal metre would have been a parenthesis of not much more than two centuries in our about five millennia old history of metrology.

Have a good day Jimp, Paul Martin 17:38, 11 December 2005 (UTC)

PS. I forgot one reply: Aeroplanes will not crash by the presence of the hexadecimal units. Because the English foot is uncial and the digital foot always digital. No conflit.

Talk:Ancient Roman weights and measures/Archive 2#Reply form Jimp

I'll make this brief for I've got to go. I'm still not convinced that having two new feet would be of any use. It would be better to settle on one. Remember that most scientists would not be so concerned about their foot's being a close approximation to the circumference of the Earth divided by some power of two. This kind of relationship would be useful to airlines, pilots, sailors, and the like. Having two feet would just be cumbersome: nothing but a stumbling block.

"This will come without fail and probably quicker than you now imagine." if it comes at all it'll be quicker than I imagine. The beauty of ten is that we count this way. Had the ancients come up with your fancy style of counting to sixteen on their fingers then things may be different. Note also that you can count to 1023 (210-1) on your fingers if you use binary. I don't see it happening.

One last point. Although we seem to disagree on every thing there is one place where we can agree. A billion is a million million. Long live the so-called long scale. I hope you're right that this one-billion-is-one-thousand-million nonsense is quickly & forever forgotten.

Jimp 12Dec05

No, I'll agree that we shouldn't simply forget the false billion (of 109). You're right: we don't want to "misunderstand historical texts of our now present times."

Also, no, I won't suggest we count on our fingers. However, fingers are still quite useful for non-verbal signalling of numbers.

It is an interesting theory that the prehistoric Europeans used octal. I guess we'll never know for sure though.

Whilst on the topic of octal numbers ... one millilitre is one cubic centimetre, one litre is one cubic decilitre and one kilolitre is one cubic metre. This is the beauty of the thousand. It doesn't quite work with hexadecimal.

One thousand is ten to the three. Sixteen is two to the four. The number of dimensions that the universe we live in has is three (well, the String Theorists may have something to say about this but it ain't important here). There's an advantage of base eight over base sixteen. Eight is two to the three.

Perhaps you'll want to throw areas in there as well, though. This'll suggest sixty-four. Sixty-four is eight to the two whilst still being four to the three. Of course, on the other hand, octal fits in considerably worse with the 1024 scheme and base 64 worse still.

What kind of units of area and volume will we be using in this new age of hexadecimal? A cubic hexadecimal foot will be kind of large. One sixteenth of this won't correspond to the cube of any rational number of hexadecimal digits. An eighth of it will, of course, but are we not deviating from the spirit of hexadecimal? Area will, be no problem here, of course, for not only is sixteen the square of four but it is the square of the square of two.

"No, not only pilots and sailors, we all, hence, will know that the Earth circumference at the equator is – by definition – ... two thousand hexadecimal leagues." Yes, and this will be useful to all of us as and no less useful to the rest of us as it will be to the scientist. Glad you see the sense in having only one foot.

Anyhow, leaving aside powers of two for a moment. One of the drawbacks of using base ten is that you end up with a recurring decimal when you take the inverse of any prime number other to five or two. Hexadecimal, of course, does worse here. There is no solution to this: all bases are limited somehow. However, two and five, it may be argued are not optimal.

Not too long ago I had a bottle of beer. Guess how much beer was in the bottle: ... 333 millilitres. One might suggest that dividing things by three is more important than dividing things by five. In this way perhaps base six is better than either base ten or sixteen. If we're going for base six, then why not base twelve?

Dozenal counting is not all that alien to English speakers. The dozenal hundred and even the dozenal thousand have names in English viz. a gross and a great gross. I'm not to sure about how things are in other languages though except that a dozen is ﾄﾞｰｽ (doosu) in Japanese.

However, if we're going this far, why not get the good old base sixty system up and running again? I think the Babylonians were on the right track. Sixty divides evenly by two, three, four and five. Six sixties are three hundred and sixty the number of degrees in an arc and almost the number of days in a year. The division of the hour and the degree of arc is already in base sixty. We can also keep our nautical miles and knots. If 1852 metres is too wayward, then we've only got to redefine it.

P.S. Might it not be worthwhile splitting this Talk page up? We are discussing two quite different things. Perhaps the second one could find a new home at Talk:Ancient Roman units of measurement/Hexadecimal foot.

Jimp 15Dec05

P.S. Only one quick answer to the volumes. This, I'll try you to show by means of mass:
The central unit of mass, called hexadecimal pound, is deduced by the mass of water in the palm (= ¼ foot) cube. Like the kg makes exception in SI, also this pound – in the opposite to the six other basic units – is not called "hexa-i-gramme", but "hexa-a-gramme" 415.922 g.
Thousand (×4096) hexadecimal pounds (BQQQ hag) is the hexadecimal tonne (cf. four-feet cube) 1.7 t. One thousandth of the hexadecimal pound (Q.qqb hag), a mass called "ace" is 0.102 g. Half an ace (Q.qqqt hag) is the hexadecimal grain of about 50.771 683 5 mg.
An analogic high precision balance weighting for example two pounds maximum has exactly fifteen weights, also called "stones". Each stone has its popular name. (But attention: one stone, as a specific weight unit, equals 16 pounds.) The two last ones of this complete set of weights, the two grain-weights have, of course, an identical mass.
But you are right, the mass deduced from the foot cube equals 64 pounds. That's the digital talent 26.619 kg. Two aces equal – by the way – the hexadecimal carat.

Other point:
One of the advantages of the hexadecimal system is that "the inverse of any prime number other to [...] two" gives often simple recurrent values.
Just like all we know that 0.3333... means 1/3, we'll rapidly learn by heart the most important hexadecimal recurrent values.

 16/ 2 = T 16/ 3 = Z.z 16/ 6 = P.c 16/ 4 = F 16/ 5 = V.v 16/ 10 = B.j 16/ 8 = P 16/ 15 = B.b 16/ 12 = B.z

Sure 16/ 7, 16/11, like 16/13 and 16/14 are more complicated. 16/ 9 is with B.kdb not too complex for retaining it.

Other point:
You wrote: "However, fingers are still quite useful for non-verbal signalling of numbers." Yes, up to sixteen, this makes sens. If you are sure that your vis-à-vis knows that now you give a hexadecimal number, you can signalling him the number four either by the four fingers of one hand excepting the thumb or only by the thumb itself. Thus, the thumb and the index finger means five hexadecimal etc. It's easy to understand how to count up to sixteen hexadecimal on both hands. For all numbers above it seems me superfluous to toggle out any fancy and useless system; let alone to learn it.

Later more, Paul Martin 11:09, 15 December 2005 (UTC)

PS2. All other bases – exept the base sixteen – are not modern, not at the "height of our times". Especially the duodecimal has been often proposed in the past, always unsuccessfully. No any chance, that's certain.
For the antiquated sexagesimal the same applies. Furthermore, do you dispose of 60 practicable digits? In the improbable affirmative case: Who would learn it by heart?
Even the numerous proposers of the duodecimal, since several centuries, never have been able to present a convincing proposal for the two lacking digits.
Even if: Let's not talk about the persistent ambiguity between par example: 10010 and 10012.
Excepting the problems of other binary bases rightly discerned by you, in the cases of hypothetical bases 32, 64 and – why not – 1024 the same representation problem is posed.

An more exhaustive answer for the other topos however will come. Yes, we'll see how to split this page.
I'm glad to have find an interesting, competent and constructive discussion partner like you are.