9. Mathematics
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The mathematical collection, ca. 380 MSS, starts with the beginning of mathematics in the 27th c. BC, and ends with Einstein. 11 of the earliest examples from Sumer and Babylonia where mathematics was invented, are listed here.
The counting of tokens and goods for accounting purposes is earlier than both script and mathematics. 10 examples are included as an introduction.
9.1. Pre-literate counting and accounting
- MS 5087/08 Australia, 20000-3000 BC
- MS 5087/15 Australia, 20000-3000 BC
- MS 5067/1-8 Syria/Sumer/Highland Iran, ca. 8000-3500 BC
- MS 4522/1 Syria/Sumer/Highland Iran, 3500-3200 BC
- MS 4631Syria/Sumer/Highland Iran, ca. 3700-3200 BC
- MS 4632 Syria/Sumer/Highland Iran, ca. 3700-3200 BC
- MS 4638 Syria/Sumer/Highland Iran, ca. 3700-3200 BC
- MS 4523 Syria/Sumer/Highland Iran, ca. 3500-3200 BC
- MS 3007 Syria/Sumer/Highland Iran, ca. 3500 BC
- MS 4647 Syria/Sumer/Highland Iran, ca. 3500-3200 BC
- MS 3047 Sumer, 27th c. BC
- MS 1844 Sumer, ca. 2050 BC
- MS 3866 Babylonia, ca. 19th c. BC
- MS 2351 Babylonia, ca. 19th c. BC
- MS 2221 Babylonia, ca. 19th c. BC
- MS 3048 Babylonia, ca. 19th c. BC
- MS 2317 Babylonia, ca. 19th c. BC
- MS 5112 Babylonia, ca. 1900-1700 BC
- MS 3052 Babylonia, ca. 19th c. BC
- MS 3049 Babylonia, ca. 17th c. BC
- MS 2192 Babylonia, ca. 19th c. BC
9.1. Pre-literate counting and accounting
CYLCON (YURDA), A TALLY WITH MARKS POSSIBLY RECORDING THE NUMBER OF YOUNG MEN TO PASS THE INITIATION RITUALS TO MANHOOD OF THE "BORA"
MS on chalk-like stone, Werenia, Bourke, New South Wales, Australia, ca. 20000-3000 BC, 1 oval-conical triangular cylcon slanting rounded base, 19x10x6 cm, 3 series of regularly longitudinal lines of 12+9+14 evenly spaced parallel dashes, half of the surface worn away due to weathering.
Provenance: 1. Found in Werenia, Bourke, South West Wales, Australia (1969); 2. H. Gallasch Museum, Australia (1973-); 3. Sam Fogg Rare Books Ltd., London.
Commentary: Cylcons are earlier than churingas. There is no certain ways to date individual cylcons. The oldest cylcon/message stone found in a dateable archaeological context is about 20,000 years old. The simple line motifs of the oldest cylcons represent the earliest art of the Aborigines, from a very early period of occupation. In Australian nomenclature this is the colonizing period, or early Stone Age, ca. 50,000/40,000-3,000 BC. With the earliest rock-carvings and -paintings, the cylcons represent the oldest form of communication and art; and they represent the oldest religion still observed. Only 2 Aborigines have been able to communicate their name of the cylcons: Yurda, and Wommagnaragnara (Heart of the snake), respectively. Other uses as tallies are possible, such as counting of dead people, warriors, emus, measures of nardo seeds, or mapping purposes counting day-marches in various directions. Later the use could also change to other magic rituals, some involving the chipping off smaller flakes, and the practical use for pounding and crushing. Much more research is needed before the cylcons' real age and significance can be properly understood and appreciated. The term cylcon is derived from the title of R. Ethridge's publication: The Cylindro-conical and Stone Implements of Western New South Wales and their significance. Ethnological Series No. 2, Memoirs of the Geological Survey of New South Wales, 1916:1-41.
CYLCON (YURDA), A TALLY WITH MARKS POSSIBLY RECORDING THE NUMBER OF YOUNG MEN TO PASS THE INITIATION RITUALS TO MANHOOD OF THE "BORA"
MS on pink hard sandstone, New South Wales, Australia, ca. 20000-3000 BC, 1 cylindro- conical and cornute form cylcon flat base, 25x8x7 cm, 7 groups of 8-10 parallel lines in vertical rows converging at point, double lines of evenly spaced dashes below placed both transversal and longitudinal, 6 groups of 3+4+6 dashes, 6 groups of 6-9 parallel lines in vertical rows around base.
Provenance: 1. Found in New South Wales, Australia; 2. H. Gallasch Museum, Australia (1973-); 3. Sam Fogg Rare Books Ltd., London.
Commentary: Cylcons are earlier than churingas. There is no certain ways to date individual cylcons. The oldest cylcon/message stone found in a dateable archaeological context is about 20,000 years old. The simple line motifs of the oldest cylcons represent the earliest art of the Aborigines, from a very early period of occupation. In Australian nomenclature this is the colonizing period, or early Stone Age, ca. 50,000/40,000-3,000 BC. With the earliest rock-carvings and -paintings, the cylcons represent the oldest form of communication and art; and they represent the oldest religion still observed. Only 2 Aborigines have been able to communicate their name of the cylcons: Yurda, and Wommagnaragnara (Heart of the snake), respectively. Other uses as tallies are possible, such as counting of dead people, warriors, emus, measures of nardo seeds, or mapping purposes counting day-marches in various directions. Later the use could also change to other magic rituals, some involving the chipping off smaller flakes, and the practical use for pounding and crushing. Much more research is needed before the cylcons' real age and significance can be properly understood and appreciated. The term cylcon is derived from the title of R. Ethridge's publication: The Cylindro-conical and Stone Implements of Western New South Wales and their significance. Ethnological Series No. 2, Memoirs of the Geological Survey of New South Wales, 1916:1-41.
NEOLITHIC PLAIN COUNTING TOKENS POSSIBLY REPRESENTING 1 MEASURE OF GRAIN, 1 ANIMAL AND 1 MAN OR 1 DAY'S LABOUR, RESPECTIVELY
Counting tokens in clay, Syria/Sumer/Highland Iran, ca. 8000-3500 BC, 3 spheres: diam. 1,6, 1,7 and 1,9 cm , (D.S.-B 2:1); 3 discs: diam. 1,0x0,4 cm, 1,1x0,4 cm and 1,0x0,5 cm (D.S.-B 3:1); 2 tetrahedrons: sides 1,4 cm and 1,7 cm (D.S.-B 5:1).
Commentary: About 8000 BC the Palaeolithic notched tallies representing the simplest form of counting, in one-to-one correspondence, were superseded by Neolithic tokens of various geometric forms suited for concrete counting, including the type of commodity. This invention was used without any discontinuity for 5000 years, prior to the use of abstract numbers which lead to writing about 3300 BC, and then to mathematics ca. 2600 BC. When tokens were invented they were the first clay objects of the Near East, and they first exploited systematically most of the basic geometric forms, such as spheres, tetrahedrons, cones, cylinders, discs, quadrangles, triangles, etc. They were first kept in baskets, leather poaches, clay bowls, etc., and later within clay bullas, see MSS 4631, 4632 and 4638.
Exhibited: The Norwegian Intitute of Palaeography and Historical Philology (PHI), Oslo, 13.10.2003-06.2005.
COMPLEX COUNTING TOKEN REPRESENTING 1 JAR OF OIL
Counting token in stone, Syria/Sumer/Highland Iran, ca. 4000-3200 BC, 1 ovoid token, diam. 2,0x2,3 cm, circular line at the top and piercing at the bottom.
Context: For a drum shaped token with zigzag band, see MS 4522/2 (Schmandt-Besserat 3:72), and for disk type tokens, see MSS 4522/3-8.
Commentary: Same type as Schmandt-Besserat 6:14, but pierced at the bottom. The complex tokens were a natural development from the plain tokens (see MSS 5067/1-8) with new forms, added lines, dots and various designs to cover the more advanced accounting needs. They were first kept in baskets, leather poaches, bowls, etc., and then to some extent within bulla-envelopes (see MS 4631), but mainly attached to strings fastened to a bulla (see MS 4523). They lasted until ca. 3200 BC, when they were superseded by counting tablets and pictographic tablets. Some of the earliest tablets have actual tokens impressed into the clay to form numbers and pictographs, and many of the pictographs were illustrations of tokens. An account of 14 jars of oil would just be 14 tokens of the present type. On a pictographic tablet this representation would be substituted by the number 14 and the pictograph of a jar with lid looking similar to the token. This was the first break-through of the invention of writing. For such a pictographic tablet, see MS 4551. (All 8 tokens MSS 4522/1-8 are illustrated here, text: Counting tokens representing a Jar of oil and various textiles, Near East, ca. 4000-3200 BC.)
BULLA-ENVELOPE WITH 11 PLAIN AND COMPLEX TOKENS INSIDE, REPRESENTING AN ACCOUNT OR AGREEMENT, TENTATIVELY OF WAGES FOR 4 DAYS' WORK, 4 MEASURES OF METAL, 1 LARGE MEASURE OF BARLEY AND 2 SMALL MEASURES OF SOME OTHER COMMODITY
Bulla in clay, Syria/Sumer/Highland Iran, ca. 3700-3200 BC, 1 spherical bulla-envelope (complete), diam. ca. 6,5 cm, cylinder seal impressions of a row of men walking left; and of a predator attacking a deer, inside a complete set of plain and complex tokens: 4 tetrahedrons 0,9x1,0 cm (D.S.-B.5:1), 4 triangles with 2 incised lines 2,0x0,9 (D.S.-B.(:14), 1 sphere diam. 1,7 cm (D.S.-B.2:2), 1 cylinder with 1 grove 2,0x0,3 cm (D.S.-B.4:13), 1 bent paraboloid 1,3xdiam. 0,5 cm (D.S.-B.8:14).
Context: Total number of bulla-envelopes worldwide is ca. 165 intact and 70 fragmentary.
Commentary: While counting for stocktaking purposes started ca. 8000 BC using plain tokens of the type also represented here, more complex accounting and recording of agreements started about 3700 BC using 2 systems: a) a string of complex tokens with the ends locked into a massive rollsealed clay bulla (see MS 4523), and b) the present system with the tokens enclosed inside a hollow bulla-shaped rollsealed envelope, sometimes with marks on the outside representing the hidden contents. The bulla-envelope had to be broken to check the contents hence the very few surviving intact bulla- envelopes. This complicated system was superseded around 3500-3200 BC by counting tablets giving birth to the actual recording in writing, of various number systems (see MSS 3007 and 4647), and around 3300-3200 BC the beginning of pictographic writing (see MSS 2963 and 4551).
Exhibited: The Norwegian Intitute of Palaeography and Historical Philology (PHI), Oslo, 13.10.2003-06.2005.
BULLA-ENVELOPE WITH 17 PLAIN TOKENS INSIDE, REPRESENTING AN ACCOUNT OR WAGES OF TENTATIVELY 1 LARGE MEASURE OF BARLEY, 8 SMALL MEASURES OF BARLEY, 5 MEDIUM AND 3 SMALL MEASURES OF SOME OTHER COMMODITY
Bulla in clay, Syria/Sumer/Highland Iran, ca. 3700-3200 BC, 1 spherical bulla-envelope (complete), diam. ca. 7 cm, cylinder seal impressions of a row of men each carrying a sack on his head towards a large cauldron placed on a rounded stand; and of a line of tall ringstaffs and men; a 3rd impression of a large disk type token or the bottom of a large cone, diam. 2,2 cm, possibly representing the total sum of the complete set of plain tokens inside: 1 sphere diam. 1,5 cm (D.S.-B.2:2), 8 small spheres diam. 0,8 cm of which 1 still sticks to the inside of the bulla (D.S.-B.2:1), 5 cones diam.1,0x1,5 cm (D.S.-B.1:1), 3 small cylinders diam. 0,4xca.1,2 cm (D.S.-B.4:1).
Context: Only 25 more bulla-envelopes are known from Sumer, all excavated in Uruk. Total number of bulla-envelopes worldwide is ca. 165 intact and 70 fragmentary.
Commentary: 17 tokens is the largest number found inside a bulla-envelope. While counting for stocktaking purposes started ca. 8000 BC using plain tokens of the type here, more complex accounting and recording of agreements started about 3700 BC using 2 systems: a) a string of complex tokens with the ends locked into a massive rollsealed clay bulla (see MS 4523), and b) the present system with the tokens enclosed inside a hollow bulla-shaped rollsealed envelope, sometimes with marks on the outside representing the hidden contents. The bulla-envelope had to be broken to check the contents hence the very few surviving intact bulla- envelopes. This complicated system was superseded around 3500-3200 BC by counting tablets giving birth to the actual recording in writing, of various number systems (see MSS 3007 and 4647), and around 3300-3200 BC the beginning of pictographic writing (see MSS 2963 and 4551).
BULLA-ENVELOPE WITH 1 PLAIN TOKEN INSIDE, REPRESENTING AN ACCOUNT OR AGREEMENT OF TENTATIVELY 1 VERY LARGE MEASURE OF BARLEY
Bulla in clay, Syria/Sumer/Highland Iran, ca. 3700-3200 BC, 1 spherical bulla-envelope (complete), diam. 6,0-6,8 cm, cylinder seal impression of several men facing tall ringstaff; and another with animals; token inside: 1 large sphere diam. 2 cm (D.S.-B.2:2).
Context: Only 25 more bulla-envelopes are known from Sumer, all excavated in Uruk. Total number of bulla-envelopes worldwide is ca. 165 intact and 70 fragmentary.
Commentary: While counting for stocktaking purposes started ca. 8000 BC using plain tokens of the type here, more complex accounting and recording of agreements started about 3700 BC using 2 systems: a) a string of complex tokens with the ends locked into a massive rollsealed clay bulla (see MS 4523), and b) the present system with the tokens enclosed inside a hollow bulla-shaped rollsealed envelope, sometimes with marks on the outside representing the hidden contents. The bulla-envelope had to be broken to check the contents hence the very few surviving intact bulla- envelopes. This complicated system was superseded around 3500-3200 BC by counting tablets giving birth to the actual recording in writing, of the sexagesimal counting system (see MSS 3007 and 4647), and around 3300-3200 BC the beginning of pictographic writing (see MSS 2963 and 4551).
BULLA FOR HOLDING A STRING OF COMPLEX COUNTING TOKENS CONCERNING A TRANSACTION
Bulla in clay, Syria/Sumer/Highland Iran, ca. 3500-3200 BC, 1 oblong bulla, diam. 2,5x6,5 cm, rollsealed with a line of animals walking left or 2 men standing with arms raised, pierced for holding a string of counting tokens.
Context: For another bulla of the same type, see MS 5113.
Commentary: The bulla originally locked the ends of a string with a number of complex counting tokens attached to it, representing 1 transaction. The string with the tokens was hanging outside the bulla like a necklace. If the string had, say, 5 disk type tokens representing types of textiles, this number could not be tampered with without breaking the seal. The tokens could also be entirely enclosed in the centre of the bulla, see MSS 4631, 4632 and 4638. Tokens were used for accounting purposes in the Near East from the Neolithic period ca. 8000 BC until ca. 3200 BC, when they were superseded by counting tablets and pictographic tablets. Some of the earliest tablets have actual tokens impressed into the clay to form numbers and pictographs, and some of the pictographs were illustrations of tokens, see MS 4551.
NUMBERS 10 AND 5 +4 + 4 + 4 + 5 + 3
MS on clay, Syria/Sumer/Highland Iran, ca. 3500-3200 BC, 1 elliptical tablet, 6,7x4,4x1,9 cm, 2+1 compartments, 2 of which with 3 columns of single numbers as small circular depressions.
Commentary: Numerical or counting tablets with their more complex combination of decimal and sexagesimal numbers are a further step from the tallies with the simplest form of counting in one-to-one correspondence. They were used parallel with the bulla-envelopes with tokens. The commodity counted was not indicated in the beginning, but was gradually imbedded in the numbers system or with a seal or a pictograph of the commodity added, i. e. development into ideonumerographical tablets, the forerunners to pictographic tablets. There are only about 260 numerical tablets known. Most of them are found in Iran.
NUMBERS 3+4, POSSIBLY REPRESENTING 3 MEASURES OF BARLEY AND 4 MEASURES OF SOME OTHER COMMODITY, IN SEXAGESIMAL NOTATION
MS on clay, Syria/Sumer/Highland Iran, ca. 3500-3200 BC, 1 tablet, 4,4x5,0x2,3 cm, 2 lines with 3 small circular depressions and 4 short wedges.
Numerical or counting tablets with their more complex combination of decimal and sexagesimal numbers are a further step from the tallies with the simplest form of counting in one-to-one correspondence. They were used parallel with the bulla-envelopes with tokens. The commodity counted was not indicated in the beginning, but was gradually imbedded in the numbers system or with a seal or a pictograph of the commodity added, i. e. development into ideonumerographical tablets, the forerunners to pictographic tablets. There are only about 260 numerical tablets known. Most of them are found in Iran.
Exhibited: The Norwegian Intitute of Palaeography and Historical Philology (PHI), Oslo, 13.10.2003-06.2005.
| 1. | MULTIPLICATION TABLE FOR LENGTH MEASURES, WITH THE PRODUCTS EXPRESSED AS AREA MEASURES, THE LENGTH NUMBERS (5, 10, 20, ETC.) IN COLUMN 1 ARE MULTIPLIED WITH THE LENGTH NUMBERS (5X60, 10X60, 20X60, ETC.) IN COLUMN 2 TO GIVE THE COMPLICATED AREA NUMBERS IN COLUMN 3 |
| 2. | SUCCESSIVE MULTIPLICATION OF SEXAGESIMAL NUMBERS BY 2, FROM 11.5=675 (OR 3/16) IN LINE 2 TO 3.00.00 (=3X60X60=10800) IN LINE 6 |
MS in Old Sumerian on clay, Sumer, 27th c. BC, 1 tablet, 7,2x7,1x2,0 cm, 28 compartments in cuneiform script.
Commentary: The oldest known mathematical text. Only one nearly as old mathematical table text is known, a table of squares of length measures, with the products expressed as area measures, Berlin VAT 12593. There is a big difference between this kind of multiplication table with explicit lengths and areas and the 1000 years younger Old Babylonian multiplication tables with abstract sexagesimal numbers.
Published: Jöran Friberg in Nordisk Matematisk Tidskrift, 52:4, 2004, pp. 151-152. To be published by Jöran Friberg in the Manuscripts in The Schøyen Collection series.
SUM OF A GEOMETRIC PROGRESSION COMPUTED FROM THE BOTTOM UP. THE FIRST TERM IS 2, THE SECOND TERM IS 2X(1+1/6) = 2 1/3, WRITTEN AS 2;20. THE SUM IS GIVEN IN LINE 1, IN SEXAGESIMAL PLACE VALUE NOTATION; SCHOOL TEXT REPRESENTING AN INHERITANCE PROBLEM FOR 7 BROTHERS
MS in Neo Sumerian on clay, Babylonia, 20th c. BC, 1 round tablet, 11,0x3,5 cm, 9 lines in cuneiform script. Binding: tasut
Context: No other Old Babylonian mathematical text is written from the bottom up in this way.
Commentary: According to the subscript, the number in each line should be equal to the number in the line above it, minus a seventh of that number. Actually, the 7 numbers in lines 2-8 have been computed from the bottom up, beginning with 2 and then making the number in each line equal to the number in the line below it plus a sixth of that number. The sum of the 7 numbers is recorded in line 1. A numerical error in line 3 is propagated upwards, to lines 2 and 1. The recorded numbers look like very large integers, but are actually all a small integer plus a sexagesimal fraction. The youngest of the 7 brothers gets 2, the next gets 2x(1+1/6), the next 2.20x(1+1/6), etc., or read from the top each brother gets 1/7 less than the brother before him. The tablet certainly has been re-used, and there are traces of possible numerical notation from its previous use.
Published: To be published by Jöran Friberg in the Manuscripts in The Schøyen Collection series.
MULTIPLICATION TABLE FOR 1.12(=72), IN THE SUMERIAN SEXAGESIMAL SYSTEM
MS on clay, Babylonia, 19th c. BC, 1 tablet, 7,8x4,7x1,8 cm, single column, 15+8 lines in cuneiform script.
Commentary: The number 72 or 1 1/5 is the sexagesimal reciprocal of 50, which appears in the standard tables of reciprocals. Scholars have used the absence of any multiplication tables of 1 1/5 as evidence that they did not exist, and that Babylonians did not have multiplication tables for all sexagesimal numbers appearing in their standard table of reciprocals. The present unique tablet proves that making such assumptions is groundless.
Published: To be published by Jöran Friberg in the Manuscripts in The Schøyen Collection series.
EXTREMELY LARGE 15-PLACE SEXAGESIMAL NUMBER 13 22 50 54 59 09 29 58 26 43 17 31 51 06 40, EQUALLING THE 20TH POWER OF 20, WHICH IS 104,857,600,000,000,000,000,000
MS on clay, Babylonia, 19th c. BC, 1 tablet, 4,5x11,7x2,8 cm, single column, 2 lines in cuneiform script.
Commentary: The number is one of the largest numbers recorded on a cuneiform tablet.
Published: To be published by Jöran Friberg in A Remarkable Collection of Babylonian Mathematical Texts, Springer, New York, 2007.
MATHEMATICAL CALCULATIONS ON CARRYING BRICKS AND MUD, THE 4X4 TABLE LISTS CONSTANTS FOR CARRYING THE 3 MOST COMMON BRICK SIZES AND MUD, THE LOAD OF 6 BRICKS, 50 MINAS (25 KG), THAT ONE WORKER CAN CARRY, AND THE DAILY CARRYING DISTANCE, 45.60 LENGTH UNITS = CA. 10,8 KM
MS on clay, Babylonia, 19th c. BC, 1 tablet, 5,0x5,2x2,3 cm, 3 + 4 columns, 9+6 lines in cuneiform script.
Published: To be published by Jöran Friberg in the Manuscripts in The Schøyen Collection series.
TABLE WITH DATA FOR SOLVING CUBIC EQUATIONS, IN THE SUMERIAN SEXAGESIMAL SYSTEM
MS on clay, Babylonia, 19th c. BC, 1 tablet, 7,6x4,4x2,3 cm, 3 columns, 30 lines in cuneiform script.
Context: The only similar text known before is a Late Babylonian table text, where the numbers m at left take the values nxnx(n+1). Problems of the mentioned type are known from a large Old Babylonian clay tablet (BM 85200+VAT 6599).
Commentary: Every line of the table says, "m has the root n". The numbers n at right take the values 1 to 30. The numbers m at left take the corresponding values nx(n+1)x(n+2). In the 6th line, for instance, n = 6 and m = 6x7x8 = 336 = 5x60 + 36. The table was probably used to set up a series of problems leading to cubic equations guaranteed to have integers as solutions. The problems would have been of the form "An excavated room. Its length equals its width plus 1 cubit. Its height equals its length. Its volume plus its bottom area is ... (a given number)."
Published: To be published by Jöran Friberg in the Manuscripts in The Schøyen Collection series.
SEXAGESIMAL DIVISION SUM WITH A NON-REGULAR DIVISOR, 1 01 01 01 DIVIDED WITH 13 IS 4 41 37
MS in Old Babylonian on clay, Babylonia, 19th c. BC, 1 tablet, 2,9x2,9x1,4 cm, single column, 2 lines in cuneiform script.
Commentary: The meaning of the text is that the first number, 1 01 01 01 in sexagesimal place value notation, is exactly divisible by 13, and that the quotient is 4 41 37. A dressed up version is known from an early Old Babylonian tablet from Ur, where 1 01 01 01 sheep are divided between 13 shepherds.
Published: Transliteration by Jöran Friberg in Nordisk Matematisk Tidskrift, 52:4, 2004, p. 148. To be published by Jöran Friberg in the Manuscripts in The Schøyen Collection series.
| 1. | EQUATIONS FOR THE SIDES OF ONE, TWO, OR MORE SQUARES |
| 2. | EQUATIONS FOR THE SIDES OF A RECTANGLE |
MS in Old Babylonian on clay, Babylonia, probably later than 1700 BC, upper half of a tablet, 8,9x9,8x2,7 cm, 2+2 columns, 125 lines in a clear minute cuneiform script.
Commentary: A collection of 16, originally 23, mathematical problem texts. The problem texts were the higher mathematics of the time, and for the better students only. The tablet is probably post-Old Babylonian.
EIGHT MATHEMATICAL PROBLEMS WITH DRAWINGS OF SUBDIVIDED TRAPEZOIDS AND TRIANGLES; MADE AS PROBLEMS OF BRICKWALLS
MS in Old Babylonian on clay, Babylonia, ca. 19th c. BC, 1 tablet, 21,0x8,2x2,9 cm, 92 lines in cuneiform script, drawings to each problem.
Commentary: Problems about trapezoids or triangles divided into two or more smaller parts by transversals parallel to the base were popular in Old Babylonian mathematics. Such problems led to systems of linear or quadratic equations. One particular type of problems for divided trapezoids led to the equation square a + square b = 2 square c. Old Babylonian mathematicians could find solutions in integers to both this equation and the similar equation square a + square b = square c, at least 1200 years before Pythagoras.
Published: Jöran Friberg in Nordisk Matematisk Tidskrift, 52:4,
2004, pp. 156-157.
To be published by Jöran Friberg in the Manuscripts in
The Schøyen Collection series.
PROPERTIES OF CHORDS OF CIRCLES, HERE CALLED BOW STRINGS, AND DIAMETERS IN CIRCLES; PROBLEM OF A GATE IN THE CITY WALL, AND A SUMMARY OF 15 MORE PROBLEM TEXTS ON THE SAME TABLET
MS in Old Babylonian on clay, Babylonia, ca. 17th c. BC, upper left quarter of a tablet, 11,5x6,4x2,2 cm, single column, 43 lines in an expert cuneiform script, signed by the scribe, drawings of 2 circles with diameters and chords indicated.
Commentary: The complete tablet contained 16 different exercises on
5 subjects, 6 problems of the circle, 5 problems of quadrates, 1 problem for
the triangle, 3 problems for "brickforms" (parallel-trapezes), 1 problem of an
"inner diagonal", which is preserved here. This is a geometrical problem where
the three-dimensional Pythagorean rule came into play, long before Pythagoras
lived.
This is a high quality tablet possibly from a royal library.
Published: Jöran Friberg in Nordisk Matematisk Tidskrift, 52:4, 2004, pp. 154-156. To be published by Jöran Friberg in the Manuscripts in The Schøyen Collection series.
GIVEN 2 CONCENTRIC AND PARALLEL EQUILATERAL TRIANGLES WITH THE AREA BETWEEN THEM DIVIDED INTO 3 EQUALLY SHAPED TRAPEZOIDS; COMPUTE THE AREA BETWEEN THE 2 TRIANGLES AS THE SUM OF THE AREAS OF THE 3 TRAPEZOIDS; SCHOOL TEXT
MS in Old Babylonian on clay, Babylonia, 19th c. BC, 1 tablet, diam. 7,1x2,5 cm, 8+3 lines in cuneiform script, drawing of 2 concentric and parallel equilateral triangles with the sides given as 60 and 10.
Commentary: The sides of the trapezoids are correctly computed. The
text may have been an assignment to a student, but the answer to the problem
is not given. No parallel to this text has been published before.
This
text shows the difference between Babylonian and Greek gemetry; while the
classical Greek was abstract and reasoning, the Babylonian was concret and
numerical.
Published: Transliteration by Jöran Friberg in Nordisk Matematisk Tidskrift, 52:4, 2004, p. 150-151. To be published by Jöran Friberg in the Manuscripts in The Schøyen Collection series.