Algebra in the scribal school—Schools in old builds on a source which is very close to Ab~ Bakr's…

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<p>0036-6978/93t040201-t8 $1.50 + 0.20 9 1993 Birkh~user Verlag, Basel R Algebra in the scribal school - Schools in old Babylonia algebra?* </p> <p>Jens Hcyrup </p> <p>Dedicated to Hubert L. L. Busard and Menso Folkerts in gratitude for so many important editions </p> <p>Zusammenfassung </p> <p>Eine Reihe von mittelalterlichen Schriften zur Landmessung (vom 9. islamischen Jahrhundert bis zu Fibonacci und Pacioli) enthiilt eine besonderr Art von ,,algebraischen" Aufgaben. Darin werden z.B. die Summe der Fl~iche und einer oder alle vier Seiten eines Quadrates beschrieben und nach der SeRe gefragt. Es zeigt sich erstens, dab dieser Aufgabentyp mindestens seit dem frfihesten 2. vorchristlichen Jahrtausend yon geometrischen Praktikem tradiert wurdr and zweitens, dab er die Entwicklung einer ,,Algebra" in der altbabylonischen Schreiberschule inspirierte. </p> <p>Der Aufsatz untersucht, in welcher Weise die Oberfiihrung der ,,sub-wissenschaftlichen" Prak- tikerlradition in einen systematischen Schulunterricht den mathematischen Inhalt mad den Denkstil ktinftiger Praktiker priigte. Im letzten Kapitel wird diskutiert, inwieweit es sinnvoll ist, irmerhalb dieser Schulalgebra von ,,Schulen" zu sprechen. </p> <p>I. From tradition to schooling </p> <p>In the second, geometrical part of his Summa de Arithmetica, Luca Pacioli [1] states that </p> <p>"Benche nela parle de arithmetica dicessimo de la regola dalghebra assai copiosamente: Niente dimeno e necessario alcuna cosa qui dime." </p> <p>The main interest of this passage lies in the assertion that it is necessary to say something about algebra when presenting practical geometry. As to the content, Luca follows the corresponding section of Leonardo Fibonacci's Pratica geome- trie (ed. [Boncompagni 1862: 56ff]) in a characteristic interest in problems dealing (e.g.) with a square area and the four sides, and has obviously copied </p> <p>Revised contribution to the symposium "Mathematische Schulen - Genese, Struktur, Wirkungsgeschichte", Oberwolfach, May 10-16, 1992 </p> <p>NTM N.S. 4(1993) 201 </p> <p>FORSCHUNG - RESEARCH J e n s H c y r u p </p> <p>directly (so much so that misprints in the diagrams of the 1523 edition are easily corrected with recourse to Leonardo's text). </p> <p>Leonardo's problems refer to genuine geometrical configurations, the linear extensions of which are given in a real unit (pertica). Often he asks, not for "the side" of a square but for "each of its sides". In many cases he has recourse to standard algebra, but in others he argues on geometrical diagrams (very often a problem is solved in both ways). The arguments from diagrams are tainted by familiarity with Elements II l, but basically most of them are elaborations of the same kind of naive-geometric considerations which constitute the fundamental method ofAb~ Bakr's Liber mensurationum (ed. [Busard 1968]), as I have argued elsewhere (e.g., [H0yrup 1986]) 2. A number of other affinities, some regarding mathematical content or style, some the terminology and the formulations, and some the organization of the subject-matter) also demonstrate that Leonardo builds on a source which is very close to Ab~ Bakr's work - maybe simply on another version than the one we know, and which is partially corrupt, maybe on a work equally used by Ab~ Bakr 3. </p> <p>Savasorda's Liber embadorum contains a similar though shorter "algebraic" section, also interested in square area and four sides and making use of similar diagrams (this time supported by explicit references to Elements 1I). </p> <p>Many aspects of Abfi Bakr's work, on its part, point back to the style and content of certain parts of Old Babylonian mathematical texts (cf. [Hcyrup 1986] and [1992a]), in particular to problems themselves pointing towards an origin in a surveyors' environment (see [Hcyrup 1990: 271-275, 309-314], and [I-Ioyrup 1990a: 79f]; to be explicated below), from which the Old Babylonian scribe school will probably have borrowed its interest in second-degree "algebraic" problems 4. </p> <p>Ab~ Bakr is separated from the Old Babylonian scribe school teachers by c. 2500 years, and Luca Pacioli from the earliest traces of "mensuration algebra" by almost 4000. Continuity and (at least concerning the central 2500 years) immutability of essential characteristics testify to the existence of a tradition. 5 It also suggests how we may distinguish a tradition from a school in the pre-modern context, in particular when we compare the characteristics of the surveying tradition with its scribe-school offspring. </p> <p>If Q signifies the area and s the side of the square, ss "its sides" (i.e., the sum of the four sides), s4 "its four sides" and Su "each of its sides" ([unum]quodque latus), the section on squares in Ab~ Bakr's work contains this sequence of problems: </p> <p>1. s = 10, Q? 2. s = 10, d? 3. s+Q = 110, s? 4. s4+Q = 140, su? 5. Q-s = 90, s? </p> <p>202 </p> <p>Algebra in the scribal school - Schools in old Babylonia algebra? F O R S C H U N G - R E S E A R C H </p> <p>6. Q-ss = 60, s,? 7. ss = ~r su? 8. ss=Q, su? 9. ss--Q = 3, Su? </p> <p>10. d = " ~ ~ ; s? 11. d = ~ ; Q? 12. ss+Q = 60, s? 6 13. Q-3s = 18, s? 14. Ss = 3/8.Q, Su? 7 15. Q/d = 71/2, Su? 16. d-s = 4 , s ? 17. d-s = 5, no question, refers to the previous case. 18. d = Su+4, no reference to N ~ 16. 19. Q/d = 71/14, s?, d? </p> <p>We observe that everything remains very close to the observable geometric configurations. With one exception, what occurs is either the side or the sides, and the single area. Similarly, Leonardo's text ordy brings one problem that in modem terms would be non-normalized (p. 60). He does not tell, however, that thrice the area added to the four sides yields 279, but that the sum of the square on the diagonal, the area and the sides equals this number. </p> <p>The picture offered by both authors in the sections on "quadrates one side of which is longer ''8 and on rhombi is analogous. Again, we encounter the length, the width, the two sides (i.e., the two different sides), the four sides, the diagonal, the diagonals, etc. </p> <p>Certain Old Babylonian algebra problems are strictly s imilar- thus BM 13901 N ~ 23, AO 8862 N ~ 1, and the single problem on Db2-146. The first is of the type s4+Qb = A, while the second tells the sum of the length and width of a rectangle and the sum of the area and the excess of length over width, and the third coincides with N ~ 27 of the Liber mensurationum (given area and diagonal of a rectangle; also in Leonardo, p. 64). </p> <p>But in spite of the strong similarities between Abfi Bakr's "mensuration algebra" and the Old Babylonian texts as concerns certain techniques (not least nor however solely the naive-geometric fundament) and the grammatical and rhetorical structure of the text, Old Babylonian algebra differs strongly from what we encounter in our Medieval treatises - so strongly that a direct descent from Old Babylonian scribe-school mathematics is highly improbable. This becomes visible when we ascend from the single problems to the larger structures in which they are organized. </p> <p>One such larger structure is the tablet BM 13901, which contained the problem s4+Q = A as its N ~ 23. Using the same symbols as before (and subscript numbers when several squares are involved), it contains the following problems (the length of the side is always asked for; "V]" between lines refers to the </p> <p>rcrM N.S. 40993) 203 </p> <p>F O R S C I ' I 1 J N G - R E S E A R C H Jens Hcyrup </p> <p>construction of a rectangular area; numbers are transcribed according to Thureau- Dangin's convention, ", " etc. indicating descending and ', " etc. ascending sexagesimal orders of magnitude): </p> <p>1. Q+s = 45' 2. Q-s = 14'30 3. (~/3)Q+(1/3)s = 20' 4. (2/3)Q+s = 4'46"41Y 5. Q+s+(I/3s = 55' 6. Q+(~C3)s = 35' 7. l lQ+7s = 6"15' 8. QI+Q2 = 21'40", sl+s2 = 50' (reconstructed) 9. QI+Q2 -- 21"40", s2 = sl+10' </p> <p>10. QI+Q2 = 21" 15', s2 = sl-(l/7)Sl 11. QI+Q2 = 28"15', s2 = sl+(1/7)sl 12. QI+Q2 = 21'40", sl-se = 10' 13. QI+Q2 = 28'20", s2 = (V4)sl 14. QI+Q2 = 25'25, s2 = (2/3)s1+5 15. QI+Q2+Q3+Q4 = 27'5, (s2,s3~4) = (7/3, 1/2, 1/3)sl 16. Q-(V3)s = 5" 17. QI+Q2+Q3 = 10' 12"45", s2 = (!~)Sl, s3 = (!/7)s2 18. Q~+Q2+Q3 = 23'20, s2 = sl+10, s3 = s2+10 19. Ql+Q2+(sl-s2)f-q(sl-s2) = 23'20, Sl+S2 = 50 20. [missing] 21. [missing] 22. [missing] 23. ss+Q = 41'40" 24. QI+Q2+Q3 = 29' 10, s2 = (~'3)s1+5, s3 = (1/2)s2+2"30' </p> <p>What most of all characterizes this list (and other Old Babylonian texts) in contrast to the one from the Liber mensurationum is a rather systematic variation of coefficients quite unfettered by the actual geometric configuration dealt with. Another difference, less conspicuous but none the less present, is the introduction of representation: Even though the entities appearing in the equations are measurable lines and surfaces, they can be used to represent entities of other kinds which are involved in structurally similar relationships; in precisely the same way, the x's and y's of modem elementary algebra, though conceptualized as pure numbers, may represent prices, weights, etc. N ~ 12, indeed, is solved in a way which lets the areas of the two squares be represented by the length and the width of a rectangle, whose area is determined as (s]Osz) 2. </p> <p>In principle, the difference between the two mathematical enterprises could be explained in two ways. Surveyors borrowing and continuing the algebraic tradition of the Old Babylonian scribe school might change its character, leaving </p> <p>204 </p> <p>Algebra in the scribal school - Schools in old Babylonia algebra? FORSCHUNG - RESEARCH </p> <p>out what had little appeal within theft professional environment. Alternatively, the scribe school might have been inspired by a pre-existing surveyors' "sub- scientific tradition ''9 and have developed a limited array of "algebraic riddles" dealing with real geometrical configurations into a mathematical discipline sui generis. </p> <p>The presence of isolated pieces of characteristic mensuration algebra in what appears to be the earliest Old Babylonian mathematical tablets (and hence the earliest algebraic texts at all), occasionally in a language which even inside this context seems to contain certain deliberate archaisms, speaks against a theory of gesunkenes Kulturgut. 1~ We are left with the conclusion that algebra originated as a set of surveyors' puzzles, and was only expanded, systematized and recast in the scribal school. 11 What we find in the Liber mensurationum will be a descendant of the original surveyors' tradition, most probably of course with significant admixture, first from the Babylonian and later from the Alexandrian school tradition. </p> <p>The organization of Old Babylonian algebraic texts can thus illustrate the process that takes place when a branch of sub-scientific mathematics is adopted into and transformed by an intellectually strict school environment. Further investigation may also reveal whether the process resulted in anything which can reasonably be regarded as a formation of a mathematical school - maybe even of different schools. </p> <p>II. The impact of schooling </p> <p>A first striking difference in character between the sub-scientific tradition and its scribe-school offspring is the contrast between the stability of the former and the relatively rapid change of the latter. The Old Babylonian era, in total, lasted from c. 2000 B.C. to 1600 B.C. Within this time span, at most one sixth of the distance that separates its termination from the epoch of Abfi Bakr, took place the transformation of the stock of surveyors' puzzles into the organized discipline reflected in BM 13901, and the further reorganization of this in the series texts, which is the theme of the following paragraphs. </p> <p>The contrast between the systematic variation of data in BM 13901 and the structure of the corresponding section of the Liber mensurationum 12 was already dealt with. Other texts, according to internal evidence of a somewhat later but still Old Babylonian date, bring this spirit of systematization to a culmination (and certainly a peak of boredom for the students, if they ever had to solve the problems in sequence!). </p> <p>These texts are the "series texts", thus called because the single tablets are elements of larger series. Series organization of tablets belonging together is known from other domains of Old Babylonian culture: Omina, incantations, lexical series. Like omina, incantations, and lexical series, the mathematical </p> <p>NTM N.S. 4(1993) 205 </p> <p>F O R S C H U N G - R E S E A R C H Jens H a y r u p </p> <p>series contain long lists of single cases, ordered according to some principle or principles. At times only the statements of problems are given, at times also the solutions. </p> <p>YBC 4714 - a texts which informs about the solutions - is told to be Tablet 4 of its series, and contains problems which are somewhat more intricate than those of BM 13901 but of the same character; we may imagine that the subject- matter of the latter tablet was dealt with in Tablet 3 of the series (only Tablet 4 is extant). The problems of Tablet 4 are ordered as follows (some of them are damaged and reconstructed from context; { L } stands for a set of linear equations involving the sides of the squares, always precisely as many equations as needed and often tediously complex; the "25 nindan" in N ~ 3 0 - 3 9 is presented as a "second width", the nindan being the basic unit of horizontal distance): </p> <p>. </p> <p>2. 3. 4. </p> <p>5. . </p> <p>7. 8. 9. </p> <p>10. 11. 12. 13. </p> <p>14. 15. 16. 17. 18. 19. 20. </p> <p>QI+Q2 = 21 '40, sl+s2 = 50 QI+Q2+Q3+Q4 = 1"30', sl+s2+s3+s4 = 2 '20 QI+Q2+Q3+Q4+Qs+Q6 = 1"52'55, Sl+S2+S3+S4+SS+S6 = 3 ' I5 QI+Q2+Q3 = 30 '50 , s2 = (1/7)si+15, s3 = (1/2)s2+5 QJ+Q2+Q3 = 1"8'5 (or Q~+Q2+Q3+sl+s2+s3 = 1'~)'46), {L } Ql+Q2+Q3+sl+s2+s3 = 27 '50 , { L } QI+Q2+Q3 = 1"17'30, {L} Q~+Q2+QJ+Q4 = 2"23 '20, I L } Ql+Q2+Q3+Q4+sl+s2+s3+s4 = [?], { L } QI+Qe+Q3+Q4 = 1"15'50, { L } </p> <p>~[+sl+s2+s3+s4] = [?], {L } QI+Qe+Q3+Q4 " QI+Q2+Q3+Q4 = i"36 ' 15, {L} QI+Q2+Q3 = 2"47'5. Sl = 1'20, {L} Q~+Q2+Q3 = 2 " 4 7 ' 5 QI+Q2+Q3 = 2"47'5. QI+Q2+Q3 = 2"47'5. QI+Q2+Q3 = 2 " 4 7 ' 5 QI+Q2+Q3 = 2~'47' 5. QI+Q2+Q3 = 2 " 4 7 ' 5 </p> <p>s2 = 45, {L} s3 = 40, {L} {L} {L} {L} {L} </p> <p>[too damaged for reconstruction] 21. QJ+Qe+Q3+Q4 = 52 '30 , si+~ = si+(I/7)Sl 2 2 . Ql+a 2 +a 3 +a 4 +s l+s2 +s3+s4 </p> <p>23. QI+Q2+Q3+Q4 = 52 '30 , Si+l 24. Ql+Q2+Q3+Q4+sl+s2+s3+s4 25. QI+Q2+Q3+Q4 = 52 '30 , Si+l 26. Ql+O2+a3+a4+sl+s2+s3+s4 27. QI+Q2+Q3+Q4 = 52 '30 , si+] </p> <p>= 54 '20 , Si+l = Si+(1/7)$1 = si+(Va)s4 = 54 '20 , Si+l = si+Q/4)s4 = si+(1/5)s3 = 54 '20 , Si+l = Si+(1/5)$3 = si+(1/2)183(V3)s2 </p> <p>28. Ql+Q2+Q3+Q4+sl+s2+s3+s4 = 54 '20 , Si+l = Si+(1/2)'(1/3)$2 29. QI+Q2 = 48 '45 , sl[Zs2 = 22 '30 30. Q1--Q2 = (25 nindan)l-qs2, {L} </p> <p>206 </p> <p>Algebra in the scribal school - Schools in old Babylonia algebra? FORSCHUNG - RESEARCH </p> <p>31. QI--Q2 = (25 nindan)Os2, { L } 32. Q1--Q2 = (25 nindan)I-ls2, {L} 33. Q~-Q2 = (25 nindan)E]s2, {L} 34. Q1-Q2 = (25 nindan)[]s2, {L} 35. Q J-...</p>