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- Babylonian Sexagesimal Notation for Math on Clay Tablets in Console
- Introduction
- Testcase 1
- examples of sexagesimal notation
- zeros, periods, Arabic numerals, other punctuation etc are modern equivalents
- Testcase 2
- Testcase 3
- Selected terms of cuneiform math from clay tablets
- Testcase 4
- Angle coefficients for cuneiform math from clay tablets
- Testcase 5
- Material coefficients from clay tablets
- Screenshots Section
- References:
- * Pretty Print Version

- Comments Section

gold Here are some eTCL scripts for Babylonian sexagesimal notation. The Sumerians, Akkadians, and Babylonians used one type of notation as a mixed base 10&60 (sexigesimal) and another notation that was pure base 60. For a base 60 number of one or greater, the form was N1*(60**0)+N2*(60**1)+ N3*(60**3) etc. For a sexagesimal fraction, the form was N1*(60**-1)+N2*(60**-2)+ N3*(60**-3)+ N4*(60**-4) etc. The tablets referenced below used a mixture of cardinal or spelled out numbers and numbers expressed as implied multiplication of terms (eg. N1 space N2). There are tables of sexagesimal coefficients that were used with math problem texts, so searching for pi and specialized coefficients should be rewarding. The eTCL scripts were written for console output, primarily to evaluate the sexagesimal values given (or not given!) in the museum reports. With some irony, the digitized CDLI cuneiform records can be searched for sexagesimal values transliterated from the clay tablets, but not by decimal values. Change comes hard in Babylon.

In planning any software, there is a need to develop testcases.

sexagesimal | decimal fraction | parts expression | comment |
---|---|---|---|

0;10 | 0.166 | 1/6 | fraction |

0;20 | 0.333 | 2/6 | common fraction |

0;30 | 0.5 | 1/2 | common fraction |

0;40 | 0.666 | 4/6 | common fraction |

0;50 | 0.833 | 5/6 | fraction |

0;05 | 0.08333 | 1/12 | pi associated constant for area of circle, pi solved as decimal 3.0 |

0;57;36 | 0.615833 | 3456/3600 | pi associated constant for hexagram,pi solved as 3+1/8,or decimal 3.125 |

0;20 | 0.3333 | 1/3 | pi associated constant for reciprocal,pi solved as 3.,or decimal 3.0 |

36; | 36. | 36/1 | on a few tablets, 36 might be an equivalent to 4pi**2 or decimal 39.438 |

3+0;10 | 3.16 | 19/6 | Egyptian pi, rarely or not identified on clay tablets |

3+0;8;3 | 3.141666... | 377/120 | Egyptian pi from Ptolemy descent, not identified on clay tablets |

3+0;8+... | 3.14... | 22/7 | Greek pi from Archimedes descent, continuing fraction, not identified on clay tablets. Note that pi is continuing fraction in both base10 and base60 |

Cuneiform ratios of area to side or diameter for geometry figures, circular area equals coefficient times diameter polygon area equals coefficient times side*side

sexagesimal | decimal conv. from tablet | modern estimate | percent difference | translated coeff. from cuneiform | modern name |
---|---|---|---|---|---|

0;30 | 0.5 | 0.5 | 0 | constant for area of 3-side (figure), b*h/2 | triangle |

0;30 | 0.5 | 0.5 | 0 | the ratio of semicircle area to circlar area from math problem | semicircle |

1;30 | 1.5 | 3.14/2 ... | 4.6 | the ratio of semicircle area to diameter | semicircle |

3;00 | 3. | 3.14... | 4.6 | the ratio of circlular area to diameter | circle |

1;40 | 1.6666 | 1.72048 | -3.23 | the constant of a 5-sided (figure) | pentagon |

2;37 | 2.6166 | 2.59808 | +0.712 | the constant of a 6-sided (figure) | hexagon |

3;41 | 3.6833 | 3.63391 | + 1.359 | the constant of a 7-sided (figure) | heptagon |

This is a selection, there are about 1200 known math terms from several languages. Rare terms are at bottom and fewer examples are known.

term | modern term | possible associations | loan word | note |
---|---|---|---|---|

ana | multiply | math problems | Ann goddess of plenty in old Celtic,cognate ante from Latin | |

ara | multiply | math problems | ara means multiply in Sumerian, area from Latin "level ground, open space," | |

ibsi | square, square root, cube root | math tables | ib means corner,angle in Sumerian, si means side, possilbly proto_germanic sithon | |

ibsi | exponent, raise to power (rare use) | math tables, math problems | ib means corner,angle in Sumerian, si means side, possilbly proto_germanic sithon, possibly english side, possibly side opposite the corner |

basi | cube root | math tables | ba:proportion or divide in Sumerian,si means side, (if) cube tables not found | |

sag | width | geometry problems | narrow side; saggita,latin for arrow; sag means side in Sumerian | |

ili | will come up | geometry problems | chinese lai meaning come? | |

kus | length | geometry problems | cubitalis, latin for elbow | |

kus | star angles | astronomy texts | cubit from old English | pos. 2-2.6 degrees, late astronmomy texts |

us | length | excavation problems | long side (trans. base) | |

us | degree | astronomy problems | degree | astronomy texts |

asa | area | geometry problems | as, latin for acre | |

bur | depth | excavation problems | bury, old english | |

sapal | below | math problems | means below place in Sumerian? cognate sapper in english | reference to tablet position |

kila | excavated volume | excavation problems | ki means earth,la means penetrate,pierce | ki.la.bi means weight |

kharsag | irrigation | irrigation problems | khar or gar means garden, loan word | sag means side, irrigation fields bounded by ditches |

gagar | earth volume | excavation problems | khar or gar means garden, loan word | |

sahar | earth surface area | excavation problems | sahar means dry up pit | |

zi | subtract | geometry problems | to cut, remove; to erase, English scissors, scindere "to split." from Latin | |

dah | add | geometry problems | dab sieze,bind in Sumerian, dare to give in latin | |

igingal | reciprocal | math tables | igi means "I see" in Sumerian,gal means opposite. used to simplify math | |

ukullu | reciprocal slope | math problems | parallel concept to Egyptian seked | |

indanum | change in width per unit height | geometry | ||

arakarum | multiple by 2N | math problems | ara means multiply, karum implies factor | possibly preparing 1/N and sqrt tables |

sullusum | triple N | geometry problems | multiply by factor in Sumerian | especially area of circles |

kippatu | circle | geometry problems | kubbitum means curved one in Sumerian,curve in english | special coefficients for eclipsed circles |

gu | vertical slope | building problems | gu from phonetic ku swap, cognate cubit from english | especially walls and ramps, possibly pile up or collect |

ku | vertical slope | building problems | cubit from old english | especially walls and ramps,ku Sumerian for to build |

matum | decrease or diminish one/many,rare use | math, agriculture fields | Spanish matar for kill | use in striped triangles |

summa | if then, conditional | conditional, usually at beginng of math problems | summa means "if then" | notable use in law texts, rarely in middle of math problems |

period(.) | zero or placeholder,rare | punctuation jot | punctuation or place value | notable use in 1st C. Greek texts |

space\ \ | zero or placeholder ,erratic use | punctuation or place value | punctuation or place value | common in math problems |

triple check("') | zero or placeholder,rare | triple check ,very rare | sometimes absent column,punctuation or place value | late astronomy texts |

double check(") | zero or placeholder,rare | sometimes place value,very rare | punctuation or place value | late astronomy texts |

Example formula from tablets: 1 kus 15 sa.gal >> [in] one cubit 15 [is the] slope. Usually less revealing or broken off text: 15 igi.gub >> 15 [is] reciprocal coefficient . The clay tablet(s) include redundant terms in different expressions, usually in reciprocal values.

coeff. in sexagesimal | x | y | degrees | cotangent ratio | decimal tangent |
---|---|---|---|---|---|

1; | 1 | 1 | 45.0 | 1.0 | 1. |

0;30 | 2 | 1 | 26.56 | 2. | 0.5 |

0;20 | 3 | 1 | 18.43 | 3. | 0.333 |

0;15 | 4 | 1 | 14.04 | 4. | 0.25 |

0;12 | 5 | 1 | 11.309 | 5. | 0.2 |

0;10 | 6 | 1 | 9.462 | 6. | 0.1666 |

0;09 | 7 | 1 | 8.130 | 7. | 0.142 |

0;08 | 8 | 1 | 7.125 | 8. | 0.125 |

0;07 | 9 | 1 | 6.340 | 9. | 0.111 |

0;06 | 10 | 1 | 5.710 | 10. | 0.1 |

0;09 | 1/7 | 1 | 8.130 | 7. | 0.142 |

0;10 | 1/6 | 1 | 9.462 | 6. | 0.1666 |

0;12 | 1/5 | 1 | 11.309 | 5. | 0.2 |

0;15 | 1/4 | 1 | 14.036 | 4. | 0.25 |

0;20 | 1/3 | 1 | 18.435 | 3. | 0.333 |

0;30 | 1/2 | 1 | 26.565 | 2. | 0.5 |

0;40 | 2/3 | 1 | 33.690 | 1.5 | 0.666 |

Tin, lead, and brick are only ones matching so far Ancients used manas per sila for density of some materials Copper, silver, and gold were considered precious metals and it is known that silver etc used different measuring systems and units.

sexagesimal | metal | converted | modern estimate gm/cm**3 | converted to manas per sila | |
---|---|---|---|---|---|

from clay tablet | or material | to decimal | from modern grams per cubic centimeters | Note | |

1;48 | gold | 6480? | 19.3 | 11580 | possibly includes electrum,gold and silver alloy |

1; 52; 30 | lead | 6750 | 11.3 | 6780 | one percent difference |

1;36 | silver | 5760? | 10.5 | 6300 | possibly 10% refining impurity |

9? | copper | 1:29:00? or 5340? | 8.9 | 5340 | on several tablets, but difficult to reconcile |

1;15 | lulu-slag | 4500? | 7.2 | 4320 | poss. 50% copper matte, copper percentage varies widely. poss. cognate loha from Sanskrit |

1;20;00 | bronze | 4800 | 8-9 | 4800-5400 | copper tin alloy, tin percentage varies widely |

2:12.? or 1;12;00? | iron | 4320? | 7.8 | 4680 | modern iron, listed for comparison, possible 8% difference |

1;20;21 | tin | 4821 | 7.365 | 4419 | possibly 9 percent difference |

2;24? | huluhhu-slag | 1;04;00? | 6.48? | 3888? | possibly 55% zinc oxide concentrate, "trans. white-ing" used to make brass, poss. cognate Akkadian billatu (alloy) |

12;0. | brick | 720 | 1.25 | 750 | probable straw brick |

modern straw brick | 12;30.? | 1.25 | 750 | modern straw brick, listed for comparison | |

modern fired brick | 18;24.? | 1.85 | 1104 | modern fired brick, listed for comparison |

The clay tablet A5446 with two math problems was assigned to the student Ur.ka.di and the task to square the sum of very large and small number lengths. The number was 21600+4*60 nig + 4 seed cubits. In sexagesimal notation, this would be 1,0,4,0 nindans plus a fraction for the seed cubits. A nindan was a unit of length of about 6 meters. Since a seed cubit was about one meter long, the decimal number would be 216240*6+4 or 1296004 meters or 1296 km. Although no answer was given on the clay tablet, in sexagesimal notation the square of 1,0,4,0 nindans would be nindans, omitting the fraction. answer in modern terms would be 1296004**2 or 1679626368016 square meters or 1.67E6 square kilometers. The second problem was to square the sum of 3600+60+3*10+2 nig plus 1 seed cubit. In sexagesimal notation the number is 1,1,32 nindans plus 1 seed cubit. The decimal number would be 3690 nindans plus 1 seed cubit, calculating 3690*6 +1 or 22140+1 or 22141 meters. Although no answer was given on the clay tablet, the answer in modern terms would be 22141**2 or 490223881 square meters or 49 square kilometers. For comparison, the Akkadian empire of Sargon stretched northwest for a length of 1000 km and a width of 300 km, so the area of the Akkadian empire was roughly 1000*300 or 3E5 square kilometers. The solution to the first problem was 1.6E6/3E5 or roughly 5 times the area of the Akkadian empire. The solution to the second problem was larger than many city districts. Another point for both problems is that 59 or 60 length units (especially dannas or berus of 10.8 km) has been the limit of most Sumerian metrological tables. A seed cubit was primarily a length measure associated with seeding uniform areas of grain, so while seed cubits and nindans were used together, finding the associated rates/constants (gurs) of outsized fields in the known metrological tables would be a difficulty.

For the two problems on clay tablet A5446, the answers were not given and probably the correct answer would require many assumptions for the modern reader. The seed cubit problems from other context start with the dimensions of a field and usually result in the barley seed volume necessary to plant the field. Using the modern seeding rates in metric units, at least the scale of the problem can be qualified. The modern French seeding rate is 200 liters per hectare or 100 liters will seed 5000 square meters. Considering the French seeding rate hereafter, the seed volume would be 1296004*1296004 m**2 times 100 liters/5000 m**2 would be 3.359E10 liters or 2.082E10 kilograms of barley seed. For the second problem, (22141 * 22141 * 100) / 5000 = 9804477.62 or 9.804E6 liters or 6.076E6 kilograms of barley seed. A unit of lim lim gur or 300E6 liters was found on clay table A1924.1278, so maybe a unit of lim lim lim gur or 300E9 liters could be conjectured.

The Ebla math tablet 322C in the Idlib Museum,Syria has notation has notation or expressions on separate lines for a series of numbers as follows. 36,000, 216,000, 2,160,000 and 129,600,000 The largest number has the notation 10*(60**2)*(60**2) which would be 129600000. It turns out that 129600000 Greek feet (0.3086 or 0.309 meters ) is the circumference of earth. Also, 2/3 of the largest number is 86400000. This 86400000 (not on the tablet) is circumference of earth in Greek cubits, figuring a Greek cubit as 0.4635 meters. Of course, the cubit takes many forms and varies widely. The Bablylonian trade cubit was 446.5 mm and the Sargonic cubit was about 0.5 meter, so its difficult to confirm what units were used at Ebla. The numbers on the tablet 322C are attributed to the Priests of Kish and there appears to be a Semitic personal name Isma-Il (~Ishmael in Hebrew, "may God hear"). Il or El was the head god worshipped in Ebla.

The Sumerians and Babylonians used sexagesimal coefficients associated with pi in association with geometric figures and other calculations. One formula used was diameter times constant = circumference of circle, where the constant was 3. One constant given as 0:20 or 20/60 as a reciprocal use. Refer to testcase 1. Another constant used was 1/12 for reciprocal use, written as decimal 0.0833 or 0;05 in sexagesimal notation. The area of circle times 1/12 times diameter squared. A constant ratio for a hexagon perimeter to the circumference of inscribed circle was written as 0;57:36. Refer to testcase 2. This would be 57/60+36/3600 for 6r/2*pi*r = 24/25 or giving pi as 3+1/8,or decimal 3.125. The reduced constant of 24/25 is of interest in that it is of the form (N**2-1)/(N**2) or even (N!)/(N!+1) from 4!/4!+1. See charts 1 & 2 below. Given ~ and starting with 3, there is a series of 8/9,15/16,24/25,35/36,48/49,63/64 ... Another hidden constant was 3:07:30 which was 3+7/60+30/3600 or decimal 3+0.11666+0.00833 or decimal 3.1249. Another tablet gives a coefficient with a hidden pi, computing in modern terms as 3+1/6 or decimal 3.1666. For comparison, the Egyptians computed pi as 3+1/6 from a square and octagon method.

The Sumerians posted the coefficient 36 in semicircle coefficient tables, listed as a number without much context or explanation. Eleanor Robson (Oxford,1999) speculated that 36 might be an equivalent to 4pi**2. In the cuneiform tradition of using 3 for the circumference/diameter or the area/diameter ratio, the Sumerians would calculate 4pi**2 as 4*pi*pi or 4*3*3. The modern value 4pi**2 or 39.438 appears in Newtons law of gravity, Columbs law of static electricity force, electromagnetics, etc. On the clay tablets, the coefficient 36 is usually found with its reciprocal of 10/3600 or sexagesimal 0;0 10, which is 1/36 or decimal 0.00277. It would be worth trying to confirm how the Sumerians used coefficient of 36. The coefficient 4pi**2 appears in the surface and volume formulas for a torus. In terms of diameter as the Sumerians rarely used radius in coefficient tables, the surface is (4pi**2)*D/2*d/2 and the volume is (4pi**2)*(1/2)*D/2*(d**2/4. A longitudinal slice of a torus might be used in an estimate of a circular moat, circular irrigation pipe, circular earthen dike, or a circular city earth wall. From the extant math tablets, the Sumerians were interested in these sorts of problems. In generic terms, the coefficient 4pi**2 might arise whenever the circumference pi*D is multiplied by the circumference pi*d.

The clay tablet YBC7284 showed a constant for the density of a brick as sexagesimal 12:0 or 720 manas per sar_sila, from Whiting (1975). For metric units, a mana was a Sumerian unit equal to 0.5 kilogram, a sila was equal to a liter, and a sar_sila was 300 liters. For comparison, a modern adobe or straw mud brick has a density of about 1.25 grams per cubic centimeter or 750 manas per sila. Assorted mud bricks in ancient ruins were found to vary in density from sun dried mud bricks at 1.5 gm/cc to hard fired bricks at 1.9 gm/cc. The gist is that the Sumarians used the concept of density, either as weight over volume or as volume over weight in the calculations with reciprocal density (1/density). From this tidbit, it is possible to draw some inferences about Sumerian constants for metals on other tablets. There are several tablets that have coefficients for the common metals. The constant for lead was sexagesimal 1;52;30 or decimal 6750 manas per sar_sila. From the modern estimates, the metal lead has a density of 11.3 gm/cc or 6780 manas per sar_sila. The difference would be (6780/6750)*100 or about 1 percent difference. The constant for tin was 1;20;21 or 4821 manas per sar_sila. From the modern estimates, the metal tin has a density of 7.365 gm/cc or 6780 manas per sila. Compared to the modern estimates, the Sumerian's tin constant (4821/4419)*100 was about 9 percent difference. The constants for brass, silver, and gold did not seem to fit the manas per sila scheme. However, just considering magnitudes and not exact numbers, the constants of brass,silver,and gold were about a factor of 1/60 from the answers of tin and lead.

Possibly, the units of gold and other metals might be smaller units or something like Sumerian shekels per sila. While these were tenuous associations from a few tablets, some trial calculations were made for metal constants with possible silver plate and wire. Aside from density, there were some associations equating coverage in surface area with volume. For example from Thureau-Dangin, coverage area in square cubits times ;15 equals sila of asphalt . Also, coverage area times (1/constant) equals weight of metal. Also in some of the literature, the constants for gold are called tube of gold or kus of gold, which possibly refer to a wire or rod. The Sumerians were experts at gold wire jewelery and used wire etc for trade in the early days. For example, the gold constant was 1:48 or decimal 108 in unspecified units. From modern estimates of density, a gold rod of 1mm diameter about a cubit (50cm) would have 0.15158 modern grams per cubit or 0.904 gin per cubit. There were about 8.3 metric grams in a Sumerian shekel or gin. For a ruqqum (silver plate?), coverage in sar_area times (1/constant 4;0;0) equals metal weight in shekels. For a ruqqum (silver plate?), one gin equals surface area of 9 squared (3 susi on side of square). For a ruqqum (silver plate?), 9 susi squared (> sar_area) times (1/constant 4;0;0) equals weight of 1 gin . For a ruqqum (silver plate?), 2+2/3 gin equals surface area of 23 susi squared. For a ruqqum (silver plate?), 23 susi squared (> sar_area) times (1/constant 4;0;0) equals weight of 2+2/3 gin . For a ruqqum of gold constant 2:15;00, one gin equals a boss of 4.426 susi on a side, which would be 6.64 cm on a side or 44.2 cm*cm. From modern estimates of gold density, thickness would be 8.3gm /(19.3 gm/cc 6.64cm 6.64cm) or 8.3 / (19.3 * 6.64 * 6.64) = 0.009 cm or ~1 mm. For bars or ingots of copper , one mana equals a bar of length 20 to 30 cm, 6 to 8 cm, and depth 1.5 cm. From limestone molds, ingots, etc, thickness of 1.5 cm or 1.0 susi seems to have been a standard for one type of trade copper. At the Kish excavations, numerous silver bosses were found and varied from 4 to 6 cm in diameter. Details on one silver boss was 4.5 cm diameter and thickness of 0.5 mm, comparable to calculations above. Gold wire in necklace was about 1mm in diameter and probably drawn through carnelian beads at the forge. Other jewelry had silver wires 2-3 mm in diameter, isolated beads with holes 2 -3 mm in diameter,and including filagree wire was about 3.5 mm in diameter. Trying to find some Sumerian calculations using the metal coefficients to confirm these findings.

There are several ship loading constants for magur (ship for deep), masesgur (ma for 60 gur), elonga (akkadian for long ship). While the ship constants could have been generated by trail and error, the generation of ship constants from geometric calculations could be explored. In terms of the Sumerian math, the formula for a sphere might be 0:30*D*D*D , which is reciprocal constant times diameter cubed. The formula for a half sphere might be 0;15*D*D*D, and the formula for a quarter sphere might be 0;07:30*D*D*D. Some of the ship constants range from 0:05 to 0:12. The reciprocal constant for the Akkadian long ship (elippi or elonga type) was listed as 0;07:13.

Another tool was factorization or splitting a number into a set of factors whose product equaled a positive integer.The Sumerians used math operations +-*/ etc with sexagesimal notation, see testcase 3. This section included some terminology that the Sumerians did not use. A number could be broken down into factors, then take reciprocal of each factor. Then the product of the reciprocal factors was the reciprocal of the original number. For example, take the number 24 which factors into (6)*(4) =24. Then take the reciprocals (1/6)*(1/4) or (0.25)*(0.1666). The product of (0.25)*(0.1666) is 0.041665, the reciprocal of the original number. As a check, 0.041665 * 24 = 1. The factorization could be extended to square roots and cube roots. For example on square roots, factor the number 36 as 9*4. Taking the sqrt(9) *sqrt(4) or 3*2 is 6, which is the square root ofthe original number 36. For example on cube roots, factor the number 64 as 8*8. Taking the crt(8) * crt(8) or 2*2 is 4, which is the cube root of the original number 64. Also the Sumerians had an operation called doubling. Take a number and its reciprocal,then double the number. The reciprocal of the doubled number is the half of the reciprocal of the first number. For example,take the number 24 and its reciprocal 0.041665 from above, 2*24 is 48, and (1/2)*(1/24) is the reciprocal 0.020833 of the doubled number. Check 0.020833*48 equals 1. However, at least on the clay tablets, the format was operation N1, N1 operation N2, or operation N2 on individual lines. There is little overt evidence for operation on multiple numbers in single line as operation N1 N2 N3, like a series notation, expr {N1*N2*N2}, or ::tcl::mathop::* N1 N2 N3. Here is an TCL program below using one line procedures for cuneiform operations. See reference Math Operators as Commands.

x

Chart 1. first 26 terms of ((N**2)-1)/(N**2) approach 1

Chart 2. terms of ((N**2)-1)/(N**2) approach 1 as number of terms > 500

due to size and attribution of following jpegs, leaving them as point and shoot

- http://commons.wikimedia.org/wiki/File:Pi_archi_approx_inter.svg
- http://commons.wikimedia.org/wiki/Category:Glyphs_of_Babylonian_numerals
- http://commons.wikimedia.org/wiki/File:Babylonian_numerals.jpg
- http://commons.wikimedia.org/wiki/Category:Cuneiform_numerals

**Comments Section**

Please place any comments here, Thanks.

Note that as you say above that you are using Tcl 8.5.6, then Tcl 8.5 already contains a built-in "lreverse" command, so your proc "lreverse5" above could be deleted, and calls to "lreverse5" can be replaced by calls to the built-in "lreverse" command.

The change from lreverse5 to built in lreverse does not appear to be the cause:

% set list list 5 3 1 8 9 4 7 5 3 1 8 9 4 7 % proc lreverse5 {l} { # KPV set end llength $l foreach tmp $l { lset l incr end -1 $tmp } return $l } % lreverse5 $list 7 4 9 8 1 3 5 % lreverse $list 7 4 9 8 1 3 5 %

The result from lreverse5 and the built in lreverse are identical. Note that you did not change one call in sexagesimal to use the built in lreverse.

gold Changes: Removed proc lreverse5 and using 8.5 lreverse command.

- Importing expr functions, part 2
- http://www.grahamhancock.com/forum/KollerstromN4.php ?
- The Lost Science of Measuring the Earth:Robin Heath, John Michell
- http://en.wikipedia.org/wiki/Japanese_units_of_measurementMean
- Thureau Dagin, RA28
- Chaldean uses in 1915 text by Neugebauer and Weidner
- http://oracc.museum.upenn.edu/cdli/corpus
- http://cdli.ucla.edu/pubs/cdlj/2005/cdlj2005_002.html
- http://cojs.org/cojswiki/Mathematical_Tablet_from_Ebla,_c.2350-2250_BCE
- http://www.helsinki.fi/~whiting/math.html
- Eleanor Robson, Mesopotamian Mathematics, 2100-1600 BC (Oxford, 1999)
- http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf
- http://www.mathdb.org/articles/history_pi/e_history_pi.htm
- http://en.wikipedia.org/wiki/Japanese_units_of_measurement
- http://oeis.org/search?q=+A060355+&language=english&go=Search
- http://www.mathsisfun.com/numbers/prime-factorization-tool.html
- Oneliner's Pie in the Sky
- Tip in a search engine, type "cdli 30" ,"cdli sag","cdli 20",or "cdli 05"
- 30, 20, and 05 are searches for sexagesimal notation.

# Pretty print version from autoindent # autoindent from ased editor # program "cuneiform sexagesimal conversion " # written on Windows XP on eTCL # working under TCL version 8.5.6 and eTCL 1.0.1 # TCL WIKI , 9jul2011 console show proc sexagesimal {target base} { set numberx [split $target ";"] set numberx [lreverse5 $numberx] set total 0 set tempexp 0 foreach item $numberx { set total [expr { $total +$item*($base**$tempexp) } ] incr tempexp } return $total } set start "0;30" puts "[ sexagesimal $start 60]"

console show proc sexagesimalfraction {target base} { set target [ split $target ";" ] set numberx [lreverse $target] set total 0 set tempexp 1 foreach item $numberx { set reciprocal [expr { 1./($base**$tempexp)}] set item [ expr { $item * 1. } ] set total [expr { $total +$item*$reciprocal*1. }] incr tempexp } return $total } set start "0;30" ; ;# base 60, supposed to be 0.5 proc prtx target { puts "&| $target | [ sexagesimalfraction $target 60]|&" } prtx "0;10" prtx "0;20" prtx "0;30" prtx "0;40" prtx "0;50"

output

sexagesimal | decimal fraction |
---|---|

0;10 | 0.16666666666666666 |

0;20 | 0.3333333333333333 |

0;30 | 0.5 |

0;40 | 0.6666666666666666 |

0;50 | 0.8333333333333334 |

# autoindent from ased editor # program "cuneiform operator math conversion " # written on Windows XP on eTCL # working under TCL version 8.5.6 and eTCL 1.0.1 # TCL WIKI , 22jul2011 console show proc doublen {args} {return [ ::tcl::mathop::* 2. {*}$args ]}; proc triplex {args} {return [ ::tcl::mathop::* 3. {*}$args ]}; proc areaofclosedform {args} {return [ ::tcl::mathop::* 0.25 {*}$args ]}; proc aryabhata_vol_of_cube {area} { return [::tcl::mathop::* $area [expr {sqrt($area)}] ]} proc longship {args} {return [ ::tcl::mathop::* 0.117 {*}$args ]}; proc reciprocalx {args} {return [ ::tcl::mathop::/ 1. {*}$args]}; puts " [doublen 5 ] " puts " [doublen 5 5 ] " puts " [reciprocalx 5 ] " # The use of return is a deliberate style choice. # The return statement is optional versus implied or last value return.

output

proc command | math expression | output in decimal value | comment |
---|---|---|---|

doublen 5 | 2*5 | 10.0 | product of 2*N |

doublen 5 5 | 2*5*5 | 50 | extended product of 2*N*N |

areaofclosedform 5 15.7 | 0.25*5*15.7 | 19.6 | surface area of circle |

areaofclosedform 5 20 | 0.25*5*20 | 25. | surface area of square |

areaofclosedform 5 15.7 .25 | 0.25*5*20*.25 | 4.9 | surface of quarter circle |

areaofclosedform 5 20 5 | 0.25*5*20*5 | 125. | volume of cube |

aryabhata_vol_of_cube 25 | 25*sqrt(25) | 125. | volume of cube |

longship 5 5 5 | 0.115 *5*5*5 | 14.4 | Akkadian longship |

reciprocalx 5 | 1/5 | 0.2 | reciprocal or 1/N |

reciprocalx 5 5 | 1/5/5 | 0.04 | extended division or 1/N/N |

# Area of closed form is an old formula used to compute area of circles, which can be loaded into a TCL procedure. The formula is (diameter times perimeter)/4 or with non circles, (long axis times perimeter)/4. Interesting formula because it works equally with circles and squares. Example circle has diameter of 5 and circumference of pi*D or 15.7 for an area of 19.6. Example square has a side of 5 and a perimeter of 4*5 or 20 to compute a surface area of (5*20)/4 or 25, compared to the conventional formula for square area of side*side, 5*5, or 25. For the surface area of quarter circle in the areaofclosedform procedure above, add a third number as factor 0.25. For the volume of a cube in the areaofclosedform procedure above, add a third number as the side of a cube for the height. The third factor or dimension can be an arbitrary height, so the volume of a cylinder or cubic rectangle of arbitrary height could be conjectured. This little gem is attributed to Indian mathematician Mahavira, circa 850CE. Note that this procedure does not overtly use pi (3.14) and entry order does not matter.

#The Aryabhata procedure contains old formula used to compute the volume of a cube, which can be loaded into an experimental TCL procedure. The Indian mathematician Aryabhata (476 to 550 CE) published this formula in the Aryabhaiya, written about 499 CE in Sanskrit. The formula is area of cube times the square root of area equals volume of cube. This Aryabhata formula is of historic interest, but has some limitiations beyond experimental study of Aryabhata's book.

gold This page is copyrighted under the TCL/TK license terms, this license .

Please place any comments here, Thanks.

What is the utility of this work, and the - seemingly - hundreds of pages just like it? Does it exemplify some aspect of Tcl/Tk?

I'm not gold (I hope they answer for themselves, as it is a pertinent question): it seems like gold is using this wiki as a repository for their code and articles about their hobbies. Their code mostly exemplifies how *not* to write Tcl/Tk. If you read their personal page, you can see that other have tried to offer helpful tips on how to write good code in the past, but nothing changes.

gold When I started here, there were very few examples of the operator math [* 1. 2.]. but you are free to forward examples of your purposeful work. I'm open to suggestions and your server names indicate that you have experience to offer some good ideas.

foo @162.158.182.155 , So, the point is to demonstrate operator math, a decidedly un-obscure part of Tcl programming. How many examples of operator math do we need? Do they need to be so verbose and convoluted that most people reading through the pages could be forgiven for not even noticing that operator math is being exemplified? If you get a lot of useful tips on improving your code structure and usage of Tcl and never apply them, that suggests to me that you are *not* "open to suggestions". (And we can do with a little less whataboutism here, thank you.)

durtal If this is ostensibly about operator math examples, why is the S/N ratio so low? Why paragraph after paragraph about clay pots, or whatever this content is about? How about a single page on operator math?

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