Abstract
In order to provide a brief overview of some of the historical affiliations between geometry and arithmetic in the emergence of algebra, we discuss some hypotheses on the origins of Diophantus’ algebraic ideas, based on recent historical data. The first part deals with the concept of unknown and its links to two different currents of Babylonian mathematics (one arithmetical and the other geometric). The second part deals with the concepts of formula and variable. Our study suggests that the historical conceptual structure of our main modern elementary algebraic concepts, that of unknown and that of variable, are quite different. The historical discussion allows us to raise some questions concerning the role geometry and arithmetic could play in the teaching of basic concepts of algebra in junior high school.
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Notes
This diagram is implicitly referred to in the problem-solving procedure through the meaning of geometrical operations (see Høyrup, 1985, pp. 28–29).
The statement of this problem (based on Thureau-Dangin’s French translation, 1938a, p. 103–105) is as follows: “By bur, I perceived 4 kur. By second bur, I perceived 3 kur of grain. One grain exceeds the other by 8`20 <sila>. I added my fields: 30` <SAR> What are my fields?” (NB: bur is a measure of surface; kur and sila are measures of capacity. A complete commented solution of this problem can be found in Radford, 1995a).
The arithme is designated as the letter ς, probably, as Heath (1910/1964) suggested, because it is the last letter in the Greek word arithme (arithmos, αριθμος). It is important to note that the symbolism used to designate the arithme and its powers (square, cube, etc.) leads us to the first symbolic algebraic language ever known (see Radford, 1992).
There are other problems in Diophantus’ Arithmetica whose problem-solving procedures recall the false-position method: for example, Book “IV” problem 8 (Ver Eecke, 1959, pp. 119–120); Book “IV” problem 31 (Ver Eecke, 1959, pp. 155–157). These problems are discussed in Katz (1993, pp. 170–172).
For instance, if we refer to the first problem of tablet BM 13901, shown earlier in this chapter, we can see that the unknown is found by displacements of figures. The unknown is not really involved in calculations, which are performed or executed with known quantities (Radford, 1995a). Although sometimes the scribe takes a fraction of the unknown or the unknowns in order to cut a figure (for some examples, see Høyrup 1986, pp. 449–455), it does not constitute a truly extended or generalized calculation with or between unknowns. The limits of such a geometric calculus can be better circumscribed if it is compared to the algebraic calculus which Diophantus develops at the beginning of his Arithmetica (see Ver Eecke, 1959, pp. 3–9).
See Radford (1995b).
The next passage from Olympiodorus’ Scholia to Plato’s Gorgias explains the initial difference between logistic and arithmetic: “It must be understood that the following difference exists: arithmetic concerns itself with the kinds of numbers; logistic, on the other hand, with their material” (cited by Klein, 1968, p. 13).
The distinction between variables and unknowns is not stated explicitly by Diophantus. This distinction is explicitly made in the 18th century by Leonard Euler, who sees unknowns as objects belonging to “ordinary analysis” (i.e., algebra) while variables are seen as objects belonging to “new analysis” (i.e., infinitesimal analysis) (see Sierpinska, 1992, p. 37).
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Radford, L. (1996). The Roles of Geometry and Arithmetic in the Development of Algebra: Historical Remarks from a Didactic Perspective. In: Bernarz, N., Kieran, C., Lee, L. (eds) Approaches to Algebra. Mathematics Education Library, vol 18. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1732-3_3
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