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.

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.

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8000 BC the Palaeolithic notched tallies representing the simplest form of counting, in one-to-one correspondence, were superseded by Neolithic tokens of various geometric forms suited for concrete counting, including the type of commodity.

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).

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.

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.

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 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 4551.

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.

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.

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.

Exhibited: The Norwegian Intitute of Palaeography and Historical Philology (PHI), Oslo, 13.10.2003-06.2005.

MSin Old Sumerian on clay, Shuruppak, Sumer, 27th c. BC, 1 tablet, 7,2x7,1x2,0 cm, 28 compartments in cuneiform script.

Context: The tablet probably comes from Early Dynastic IIIa Shuruppak like the previously published TSS 926 (1937). TSS 188 (1937), and VAT 12593 (1923 & 1993), also metro-mathematical school texts dealing with areas of quadrilaterals.

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.

Mentioned: Jöran Friberg: A remarkable Collection of Babylonian Mathematical Texts. Springer 2007. Sources and Studies in the History of Mathematics and Physical Sciences. Manuscripts in the Schøyen Collection, vol. 6, Cuneiform Texts I. pp. 73-76, 87.

MSon 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. This tablet can be interpreted as a continuation of MS 2205, since the 2 texts together exhibit the squares of squares of the 3rd, 4th and 5th power of 20.

Published: Jöran Friberg: A remarkable Collection of Babylonian Mathematical Texts. Springer 2007. Sources and Studies in the History of Mathematics and Physical Sciences. Manuscripts in the Schøyen Collection, vol. 6, Cuneiform Texts I. pp. 43-44.

MS on clay, Babylonia, 19th c. BC, 1 tablet, 7,6x4,4x2,3 cm, 3 columns, 30 lines in cuneiform script.

Binding: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)."

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: Jöran Friberg: Matematiska kilskriftstexter i den norska Schøyensamlingen; in: Nordisk Matematisk Tidskrift, 52:4, 2004, p. 148. Jöran Friberg: A remarkable Collection of Babylonian Mathematical Texts. Springer 2007. Sources and Studies in the History of Mathematics and Physical Sciences. Manuscripts in the Schøyen Collection, vol. 6, Cuneiform Texts I. p. 155.

Commentary: A Collection of 13+3 partly preserved of originally 23 metric algebra problem texts.

Colophon: Total: 23 hand tablets - 22. The problem texts were the higher mathematics of the time, and for the better students only.

Published: Jöran Friberg: A remarkable Collection of Babylonian Mathematical Texts. Springer 2007. Sources and Studies in the History of Mathematics and Physical Sciences. Manuscripts in the Schøyen Collection, vol. 6, Cuneiform Texts I. pp. 308-341.

MS in Old Babylonian on clay, Uruk, Babylonia, 1763-1739 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.

Colophon: 5 Mud walls, 1 cross-over, 1 excavation, 1 equalside (quadrangle). Together 8 hand tablets (assignments). This kind of subscript with a detailed summary of the topics in the text is only known on the present tablet and MS 3049.

Published: Jöran Friberg: Matematiska kilskriftstexter i den norska Schøyensamlingen; in: Nordisk Matematisk Tidskrift, 52:4, 2004, pp. 156-157. Jöran Friberg: A remarkable Collection of Babylonian Mathematical Texts. Springer 2007. Sources and Studies in the History of Mathematics and Physical Sciences. Manuscripts in the Schøyen Collection, vol. 6, Cuneiform Texts I. pp. 254-278.

MS in Old Babylonian on clay, Uruk, 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.

Colophon: Its name: 6 arches (circles), 5 squares, 1 peghead (triangle), 3 brick moulds, 1 inner cross-over (diagonal) of a gate. This kind of subscript with a detailed summary of the topics in the text is only known on the present tablet and on MS 3042.

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 geometry; while the classical Greek was abstract and reasoning, the Babylonian was concrete and numerical.

Published: Jöran Friberg: Matematiska kilskriftstexter i den norska Schøyensamlingen; in: Nordisk Matematisk Tidskrift, 52:4, 2004, p. 150-151. Jöran Friberg: A remarkable Collection of Babylonian Mathematical Texts. Springer 2007. Sources and Studies in the History of Mathematics and Physical Sciences. Manuscripts in the Schøyen Collection, vol. 6, Cuneiform Texts I. pp. 202-05.

MS in Neo Sumerian on clay, Babylonia, 20th c. BC, 1 round tablet, 11,0x3,5 cm, 9 lines in cuneiform script.

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.

MS on clay, Babylonia, 19th c. BC, 1 tablet, 5,0x5,2x2,3 cm, 3 + 4 columns, 9+6 lines in cuneiform script.

Published: Jöran Friberg: A remarkable Collection of Babylonian Mathematical Texts. Springer 2007. Sources and Studies in the History of Mathematics and Physical Sciences. Manuscripts in the Schøyen Collection, vol. 6, Cuneiform Texts I. pp. 93-95, 169-171.

MS in Old Babylonian on clay, Babylonia, 2000-1700 BC, 1 tablet, 6,7x7,3x2,3 cm, 4 columns, 5 lines in cuneiform script.

Context: This tablet is a parallel to MS 2830.

Commentary: In a market economy before the invention of money, it was more convenient to operate with market rates. Money was not invented until 7th c. BC in Lydia in Western Asia Minor. The text can also be expressed as: To compute a number R such that 1 shekel of silver is the total price of R units of each of 4 different commodities with the individual market rates 1, 2, 3 and 4 units per shekel of silver.

Published: Jöran Friberg: A remarkable Collection of Babylonian Mathematical Texts. Springer 2007. Sources and Studies in the History of Mathematics and Physical Sciences. Manuscripts in the Schøyen Collection, vol. 6, Cuneiform Texts I. pp. 159-161.