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The Khet

The Egyptians had a very well documented standard of measure called the khet which was 100 royal cubits in length and was the side of an 3ht or field called a st3t. In Greek and Roman times the Egyptian fields were generally farmed in clusters of three with one left fallow, one plowed and sowed in grain and another planted in hay for the plow animal.

The itrw as a multiple of the khet

The Egyptian value for the itrw or river journey was 21,000 royal cubits. This equated to one hour of travel along the river but also to 70 units of 3 khet which was defined as 100 Royal cubits. Eratosthenes calculations benefited from the fact that 700 stadia of 3 khet or ten itrw make a degree of the earths [[great circle\\.

the Rhind Papyrus

Gillings shows how a number of math problems in the rhind papyrus were calculated using the khet as a length, area and volume knowledgable mensurationists are aware that Mesopotamian measures would be sexigesimal and Egyptian measures septenary

Unit Fractions

The Egyptians calculated in unit fractions so to represent a number like Pi, rather than use 22/7 they might have represented it as 3 '8 '16 ... The khet seems to be in a relation to other Egyptian units such that it facilitates calculating the area of a circular field.

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_and_Egyptian.html

"Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation.

The Babylonians had an advanced number system, in some ways more advanced than our present system.

It was a positional system with base 60 rather than the base 10 of our present system.

Now 10 has only two proper divisors, 2 and 5. However 60 has 10 proper divisors so many more numbers have a finite form.

The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds.

This form of counting has survived for 4000 years.

To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds is just to write the base 60 fraction, 5 25/60 30/3600 or as a base 10 fraction 5 4/10 2/100 5/1000 which we write as 5.425 in decimal notation.

Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC. They give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives

    82 = 1 4 which stands for 82 = 1 4 = 1.60 + 4 = 64 

and so on up to 592 = 58 1 (= 58.60 +1 =3481).

One major disadvantage of the Babylonian system however was their lack of a zero.

This meant that numbers did not have a unique representation but required the context to make clear whether 1 meant 1, 61, 3601, etc.

The Babylonians used the formula

    ab = ((a + b)2 - a2 - b2)/2 

to make multiplication easier. Even better is the formula

    ab = (a + b)2/4 - (a - b)2/4 

which shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of two numbers that were looked up in the table.

Division is a harder process. The Babylonians did not have an algorithm for long division. Instead the based their method on the fact that

    a/b = a.(1/b) 

so what was necessary was a table of reciprocals. We still have their reciprocal tables going up to the reciprocals of numbers up to several billion. Of course the tables are in their number notation, but translating into our notation, but leaving the base as 60, the beginning of one of their tables would look like

     2      30
     3      20
     4      15
     5      12
     6      10
     8       7   30
     9       6   40
    10       6
    12       5
    15       4
    16       3   45
    18       3   20
    20       3
    24       2   30
    25       2   24
    27       2   13   20

Now the table had gaps in it since 1/7, 1/11, 1/13, etc. do not have terminating base 60 fractions.

This did not mean that the Babylonians could not compute 1/13, say. They would write

1/13 = 7/91 = 7.(1/91) =(approx) 7.(1/90)

and these values were given in the tables."

their cubit could be divided by fingers and hands and that five hands would be 25 fingers and six hands would be thirty fingers.

One possible issue is what is the value of the finger used.

In order to determine that I suggest we look at the value of the cubit in terms of the unit fractions used in several different cultures and see how many fingers, palms or hands that unit was divided into.

                                   19.275

value 20mm 18.75mm 18.487mm 18.487mm 19.05mm Unit Mesopotamia Egypt Greece Rome England fingers 15 16 16 16 16 palm 4 4 4 4 hand 3 foot 300mm 300mm 308.4mm 296mm 304.8mm 17 fingers 316mm 314.1428mm 18 f (remen) 360mm 19 fingers 365.24 mm cubit 3/5 2/3 2/3 2/3 2/3 yard great cubit 1/2 4/7 rc 1/6 orgia 1/5 passus 1/6 fathom

note that dividing by hands and palms results in different unit fractions

http://www.interpres.cz/metrolog/measures/stereometric.htm

4. "... the people of Mesopotamia conceived of units of volumes as cubes, whether they were as small as a shekel or as large as a cubic iku. When units had a size such that they could not be constructed as cubes with an edge measured in round numbers, they conceived of them as near-cubes, that is, as cubes increased or decreased in height. "

rulers may have been used with unit fractions as a sort of slide rule or if you prefer a table of values.

Collecting data on mensuration is not easy. Here is a site which gives evidence that there was a cubit of 28 fingers in use in ancient Babylon.

http://www.fordham.edu/halsall/ancient/greek-babylon.html

"The following is a description of the place: The city stands on a broad plain, and is an exact square, a hundred and twenty furlongs in length each way, so that the entire circuit is four hundred and eighty furlongs. While such is its size, in magnificence there is no other city that approaches to it. It is surrounded, in the first place, by a broad and deep moat, full of water, behind which rises a wall fifty royal cubits in width, and two hundred in height.

(The royal cubit is longer by three fingers' breadth than the common cubit.)"

We know the royal cubit of Egypt was longer by three fingers than the ordinary cubit of mesopotamia, but we also know the fingers that were used by these two cultures were different.

http://www.columbia.edu/~rcc20/ppedit.html

you can go to this site and compare the value of different unit fractions of an area called the aroura by the Greeks in Egypt.

Perhaps standards of measure were more widely influential than the kings that decreed them, but establishing that takes a lot of research.

First you need a reason that standards of measure would be widespread. I think that reason is trade. Rulers, rods, weights and measures needed to avoid any bronze age equivalent of the infamous hanging chards.

When you compare the standards of measures from different places and find they are even unit fractions of one another its reasonable to ask why.

http://users.aol.com/jackproot/met/antbible.html

Measures of length

These measures were generally derived from the human body. We may find :

    the digit (width of the forefinger - in Latin digitus = finger) 
    the inch (width of the thumb) 
    the foot 
    the cubit (theoretically the distance between the elbow and the middle finger) 
    the pace (or double step) 
    the fathom (finger-tip to finger-tip with arms outstreched)

Egypt

The basic unit seems to have been the royal cubit or "mh" estimated at 525 mm. Another unit was the double remen or the diagonal of a square having sides of 1 cubit. The remen (+/- 371 mm) was essentially used for land measure. The main subdivision was the digit or "zebo" with 28 digits in a cubit and about 40 in a double remen.

There was an "ordinary" cubit of 450 mm.

For those who still think in inches, 1 royal cubit = 20.62 " ; 1 remen = 14.6 " or about and 1 short cubit = 17.67 "

    1 digit or zebo (= 18.7 mm) 
    4 digits = 1 palm or shep 
    5 digits = 1 hand 
    12 digits = 3 palms = 1 small span 
    14 digits = 1 large span or 1/2 royal cubit 
    24 digits = 6 palms = 1 ordinary or small cubit (= 450 mm) 
    28 digits = 7 palms = 1 royal cubit or "meh" (= 524 mm) 
    100 royal cubits = 1 "khet" (= 52.4 m) 
    120 khet = 1 "ater" (later called a "skhoinos") (+/- 6288 m)

Mesopotamia

Also uses the cubit (some think it originated in Sumer). Its measure varies from 522 to 532 mm. They had a foot, equal to 2/3 cubit, and a digit equal to 1/30 cubit (therefore 20 digits to a foot.).

There is an exception in Assyria : the cubit is thought to have 640 mm and the foot was 1/2 cubit.

Let's give some examples :

    digit, "shusi" or "uban" (+/- 17.67 mm) 
    5 uban = 1 "qat" (= 3.18 m) 
    6 qat = 1 "ammat" or "kus" (cubit of 530 mm - 20.87 ")) 
    6 ammat = 1 "qanu" 
    60 qanu = 1 "sos" (= 191 m - 209 yards) 
    30 sos = 1 "parasang" (later unit ?) (= 5724 m - 3.6 miles) 
    2 parasang = 1 "kapsu"

According to findings in Khorsabad, we get another scale :

    the unit seems to have 275 mm (name unknown) 
    1/60 gives the "susi" (= 4.58 mm or about 1/4 digit) 
    6 units = 1"qanu" (= 1.65 m - fathom ?) 
    12 units make for a "sa" (= 3.3 m) 
    60 sa = 1 "us" (= 198 m - 217 yards) 
    and 30 us = 1 "kasbu" (= 5940 m) which is not very different from the previous parasang.

we may assume a "palm" equal to 1/7 Assyrian cubit (640 / 7 = 91.43 mm) The nameless unit would be 3 palms, and a palm contains 20 susi or 5 digits of 18.3 mm.

Encyclopaedia Brittanica.

In Persia we had :

    the cubit or "arasni" (520 to 543 mm) 
    1/2 cubit, "vitasti" or "charac" 
    2 cubits = 1 "guz" 
    360 cubits = 1 stadion or "asparsa" (187 to 195 m) 
    30 stadions = 1 "parathanka" (or parasang) (= 5610 to 5850 m) 
    there is also mention of a "mansion" equal to 80 000 Assyrian feet (= 25.6 km)

Greece

generally a foot of 309 mm (12.16 ") subdivided into 16 digits and equal to 2/3 of a (small) cubit - take or leave 4 %. There was also an older foot of 316 mm equal to 3/5 of a big cubit - 527 mm

    1 digit or "daktylos" - plural : "daktyloi" (= 19.3 mm) 
    2 digits = 1 "condylos" 
    4 digits = 1 "palaiste" 
    8 digits = 1 "dichas" 
    12 digits = 1 "spithame" 
    16 digits = 1 "pous" or foot - plural "podes" (= 309 mm) 
    20 digits = 1 "pygon" 
    24 digits = 1 "pechya" or small cubit 
    40 digits = 1 "bema" 
    72 digits = 4.5 feet = 1 "xylon" 
    6 feet = 1 "orgyia" (or fathom - 1.854 m) 
    10 feet = 1 "akaina" 
    100 feet = 1 "plethron" 
    600 feet or 6 plethra = 1 "stadion" (+/- 185.4 m) 
    2 stadia = 1 "diaulos" 
    6 diauloi = 1 "dolichos" 
    there was also a "stathmos", poorly defined - estimated by some 

authors as 25.8 km (16 miles) - is it another name for "mansion" ?

The Persian parasang was also adopted quite soon and seems to represent the distance walked in 1 hour.

The stadion - whatever its name - was quite widespread throughout antiquity. It is similar to the English furlong and close to 100 toises (fathoms) the optimal lenghtfor a plough furrow

Roman Empire

The foot was also widely used across Italy - estimated at 295 mm (11.6 ") give or take a few percents. It is found also in Etruria. The system absorbed several units from conquered territories.

    1 digit or "digitus" = 18.44 mm 
    1 inch or "uncia" = 24.58 mm (inch derives from uncia, 
    meaning 1/12 - same root as "ounce") 
    4 digits or 3 inches = 1 small palm or "minor palmus" 
    12 digits or 9 inches = 1 large palm or "major palmus" 
    16 digits or 12 inches = 1 foot or "pes" (= 295 mm) 
    24 digits or 18 inches = 1 cubit or "cubitus" 
    5 feet = 1 pace or "passus" - with its half = "gradus" and 
    its quarter = "palmites" 
    10 feet or 2 paces = 1 "decempeda" 
    120 feet or 24 paces = 1 "actus" 
    625 feet or 125 paces = 1 "stadium" (= 184.4 m) 
    1000 paces or 8 stadia = 1 "milliarium" or "mille passus" (+/- 1475 m)

Milliarium was actually the name of the military stones erected every 1000 paces along the Roman highways, to ease the localisation and the maintenance. The name is, of course, at the origin of "mile".

Rem : Let's come back to accuracy : when the Romans started to organize Northern Gaul and Germania, they used a "Drusian" or "Belgian" foot which was 2 digits longer, or 325 to 330 mm (12.9 ") - rather close to the feet of early medieval England.

Measures in the Bible

Essentially a composite of the neighbouring regions. Originally, the cubit was used - the same as the royal cubit in Egypt. Later, the smaller cubit took over.

    the digit or "esba" (18.75 mm) 
    4 digits = 1 palm or "tophah" 
    12 digits or 3 palms = 1 "zeret" 
    24 digits or 6 palms = 1 small cubit or "amma" (= 450 mm or 17.72 ") 
    7 palms = 1 old cubit 

measures borrowed from the Greeks or the Romans (fathom-like, stadium, mille, parasang, ... whatever the name used in the translation.)"

You can see there is a relation between the use of unit fractions for making calculations and the standards of measure of Egypt Mesopotamia, Greece, and Rome

the Mesopotamian cubit measuring 53 centimeters, is the Barley cubit given by Stecchini the Greek foot measuring 30.9 centimeters. is the short Greek foot (refer to Klein, p 61)

"The World of Measurements" H. Arthur Klein

the ancient sources, classic authors like Herodotus mention one unit in comparison to others

modern archaeologists have measured artifacts found in their digs.

A cube with a side of 1 Greek foot. 316 mm = 12.44" = 1925 cu in contained 7 BIG. Thats the foot that turns up most commonly in cubic measures. if we dispute that all we need do is run the numbers.

A cube formed on the side of a long Greek foot of 331 mm is one bushel

To derive the cubit of 530 mm from the short Greek foot once again we run the numbers. The same is true for the Mesopotamian wheat cubit of 600 mm

A field laid out with a side of 120 cubits of 530 mm is one acre

A cubit of 600 mm or 2 feet has a cube of 13181 cu in and contains 48 BIG

the Macedonian foot measured 279 mm (or 11")

the Ionian foot measured 3.483 mm

http://www.fordham.edu/halsall/ancient/greek-babylon.html

"The artaba is a Persian measure, and holds three choenixes more than the medimnus of the Athenians."

http://perseus.csad.ox.ac.uk/cgi-bin/ptext? doc=Perseus%3Aabo%3Atlg%2C0016%2C001&query=1%3A192%3A1

The Attic medimnus = about 12 gallons; it contained 48 choinikes.

Ancient measures of volume were the cube of units of length (You can refer to my previous posts for the discussion of why.)

12 American gallons = 231 cu in x 12 = 2772 cu in If measured in american gallons the choinikes was 2772/48 = 57.75 cu inches. 1 quart = 57.749 cu in

The artaba holds 3 choinixes more so it is 2945.25 cu inches or 51 quarts.

This cube has as its side a measure of one remen. 14.4", 364 mm

We know that the firkin and the metrete are related to the British Imperial Gallon (277.42 cu in) so that 9 BIG = 1 firkin = 2496.78 mm; its side is 13.57" = 344.58 mm

This is sometimes called the bath which was a Hebrew unit of measure

Expressed in unit fractions instead of unit factions its really an old story '...2 '4 '8 '160 '320...

Dd.in Smsw iqr wDA ib=k HAt-a mk pH.n=n Xnw Ssp xrpw Hw mnit HAt.t rdi.t Hr tA rdi Hknw dwA(.w) nTr s nb Hr Hpt sn.nw=f iswt=(t)n ii.t(i) ad.t(i) nn nhw n mSa.w=n

how Horus lost his Eye to Set at Sais....:)

http://newton.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_geometry.html http://www.math.buffalo.edu/mad/AMU/amu_chma_16.html#egypt vs greece

120 mesopotamian royal cubits of 530 mm

http://users.aol.com/jackproot/met/antbible.html 1 "ammat" or "kus" (cubit of 530 mm - 20.87 "))

"At the beginning of the nineteenth century it was determined that the Egyptian royal cubit is 525 mm. and hence it was concluded that Eratosthenes calculated by stadia of 300 Egyptian royal cubits. Newton too had tried quite successfully to ascertain the length of the Egyptian royal cubit from the dimensions of the Great Pyramid, in order to interpret Eratosthenes’ datum.

http://users.aol.com/jackproot/met/antbible.html 30 sos = 1 "parasang" http://www.geocities.com/Athens/8744/herhist.htm

"the nations whose territories are scanty measure them by the fathom; those whose bounds are less confined, by the furlong; those who have an ample territory, by the parasang; but if men have a country which is very vast, they measure it by the schoene.

Now the length of the parasang is thirty furlongs, but the schoene, which is an Egyptian measure, is sixty furlongs. Thus the coastline of Egypt would extend a length of three thousand six hundred furlongs. "

Now if thirty furlongs or stadions, equal 30 sos, equal 1 parasang and the stadion is 600 medium Greek feet of 318mm (Miletus) then the sos is 190.8m 60 qanu = 1 "sos" so 1 ganu = 3180 mm = 10 Greek feet of Miletus 6 ammat = 1 "qanu" so 1 ammat = 530 mm 30 uban or 6 qat = 1 "ammat" or "kus" (cubit of 530 mm - 20.87 ")) (thirty digits or six hands = 1 cubit) 5 uban = 1 "qat" (= 88.3 mm) 1 uban = 17.6 mm

But if thirty furlongs or stadions, equal 30 sos, equal 1 parasang and the stadion is 600 long Greek feet of 339mm (Egypt) then the sos is 203.4m 60 qanu = 1 "sos" so 1 ganu = 3390 mm = 10 Greek feet of Asia Minor 6 ammat = 1 "qanu" so 1 ammat = 565 mm 30 uban or 6 qat = 1 "ammat" or "kus" (cubit of 565 mm - 22.25 ") (thirty digits or six hands = 1 cubit) 5 uban = 1 "qat" (= 94 mm) 1 uban = 18.83 mm

On the other hand if thirty furlongs or stadions, equal 30 sos, equal 1 parasang and the stadion is 600 short Greek feet of 308.4mm (Athens) or 625 Roman feet then the stadion or stadium or furlong or sos is 185.04m 60 qanu = 1 "sos" so 1 ganu = 3084 mm = 10 Greek feet of Athens 6 ammat = 1 "qanu" so 1 ammat = 514 mm 30 uban or 6 qat = 1 "ammat" or "kus" (cubit of 514 mm - 20.23") (thirty digits or six hands = 1 cubit) 5 uban = 1 "qat" (= 85.67 mm) 1 uban = 17.13 mm

In the case of Herodotus we are told it is the Egyptian measure. Here is a different calculation of units with the cubit at .5 m

Smallest unit of length is the she (barleycorn), of about 1/360 meter. 6 she=1 shu-si (finger)=500/30 = 16.67mm 30 shu-si=1 kush (cubit - about 1/2 m.) 6 kush=1 gi / qanu (reed) 12 kush=1 nindan/ GAR (rod - 6 m.) 10 nindan=1 eshe (rope) 60 nindan=1 USH (360 m.) 30 USH=1 beru (10.8 km.)

120 great cubits of 600 mm (30 uban of 20 mm)

A Few Essential References

This is a short list of references for discussion of the Khet.

• GardinerTemplate:Section:Book reference after author
• Antonio Loprieno Template:Section:Book reference after author
• GillingsTemplate:Section:Book reference after author
• Somers Clarke and R. EnglebachTemplate:Section:Book reference after author

These are the original primary sources

• VitruviusTemplate:Section:Book reference after author
• Claudias PtolemyTemplate:Section:Book reference after author
• HerodotusTemplate:Section:Book reference after author

These are the mensurationists

• Lucas N. H. Bunt, Phillip S.Jones, Jack D. Bedient Template:Section:Book reference after author
• H Arthur KleinTemplate:Section:Book reference after author
• R. A. CordingleyTemplate:Section:Book reference after author