Abstract
The subject matter of this chapter should be placed normally among the issues treated in the previous chapter, dealing with Greek Algebra (and Logistic). Its separate treatment is due to the fact that the history of fractions has recently become a topic of independent study among historians of ancient mathematics, though a comprehensive book on the topic does not yet exist, as Jim Ritter points out in his introduction of the collective work Histoire des fractions, fraction d’histoire.1 This bibliographic lacuna is particularly notable for the period from the end of the Middle Ages until the utilization and popularization of techniques involving decimal fractions in Western Europe at the end of the sixteenth century. With respect to Antiquity the gap is less significant, though the disagreement among historians as to the appropriate criteria of what constitutes a fraction leaves room for simplistic views, which, in particular, occurred frequently in the works of the older historiography.
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Notes
P. Benoit, K. Chemla, J. Ritter (eds.), Histoire des fractions, fraction d’histoire,Basel/Boston/Berlin, 1992, p. x.
W.R. Knorr, “Techniques of fractions in ancient Egypt and Greece”, Historia Mathematica, 9 (1982), pp. 133–171.
D.H. Fowler, “Logistic and fractions in early Greek mathematics: a new interpretation”, in P. Benoit, K. Chemla, J. Ritter (eds.), Histoire des fractions, fraction d’histoire, pp. 133–147; “Ratio and proportion in Early Greek Mathematics”, in A. Bowen (ed.), Science and Philosophy in Classical Greece, New York, 1991; “Hibeh papyrus I 27: an early example of Greek arithmetical notation” Historia Mathematica, 10 (1983), pp. 344–359 (joint paper with E.G. Turner).
D.H. Fowler, The Mathematics of Plato’s Academy. A New Reconstruction, second edition, Oxford, 1999.
D.H. Fowler, The Mathematics of Plato’s Academy. A New Reconstruction, 1st ed., 1990, p. 265, note 91.
D.H. Fowler, “Logistic and fractions in early Greek mathematics: a new interpretation”, p. 134.
D.H. Fowler, The Mathematics ofPlato’s Academy, p. 195.
D.H. Fowler, The Mathematics ofPlato’s Academy, p. 226.
D.H. Fowler, The Mathematics ofPlato’s Academy, p. 264.
B. Vitrac, “Logistique et fractions dans le monde hellénistique”, in Histoire des fractions, fraction d’histoire, p. 162, note 38 (the emphasis is mine).
D.H. Fowler, The Mathematics ofPlato’s Academy, pp. 264–5.
R.D. Carmichael, Analyse indéterminée, traduit de l’Anglais par A. Sallin, Paris, 1929, p. 3.
See J. Christianidis, “Les interprétations de la méthode de Diophante”, Neusis, 3 (1995), pp. 109–132 (in Greek); of the same author, “Une interprétation byzantine de Diophante”, Historia Mathematica, 25 (1998), pp. 22–28.
J. Christianidis, “On the History of Indeterminate problems of the first degree in Greek Mathematics”, in K. Gavroglu, J. Christianidis, E. Nicolaidis, (eds.), Trends in the Historiography of Science, Dordrecht/Boston/London, 1994, pp. 237–247.
According to Jacob Klein “by a fraction Diophantus means nothing but a number of fractional parts” (Greek Mathematical Thought and the Origin of Algebra, translated by E. Brann, New York, 1992, p. 137).
I would like to thank Fabio Acerbi for bringing to my attention that W.R. Knorr has arrived at the same conclusion in his paper “What Euclid Meant: On the Use of Evidence in Studying Ancient Mathematics”. See A. Bowen (ed.), Science and Philosophy in Classical Greece, pp. 119–163.
We use the French translation of Arithmetica by Paul ver Eecke because Heath’s English translation is not literal. The only modification to the French translation is that I have substituted “en partie de” instead of “fractionnés par”, the expression by which Ver Eecke renders the Greek “en moriô ”.
Algebraic transcription: XY= 3 · (X+ Y), YZ = 4 · (Y+ Z), XZ = 5 · (X+ Z).
P. ver Eecke has “premier”.
It refers to the lemma that precedes immediately this proposition: “To find two numbers indeterminately such that their product has to their sum a given ratio”.
Algebraic transcription: Let Y= x; therefore X = 3x in part of x — 3, and Z = 4x in part of x — 4.
Algebraic transcription: XZ = (3x in part of x — 3) X (4x in part of x — 4) = 12x2 in part of x2+12~7x.
Algebraic transcription: X+ Z= (3x in part of x — 3) + (4x part of x — 4) = 7x2 — 24x in part of x2 + 12 — 7x.
Algebraic transcription: (3x in part of x — 3) + (4x in part of x — 4) = 3x · (x — 4) + 4x · (x — 3) in part of (x~4)·(x-3)=7x2–24xin part of x2+12–7x.
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Christianidis, J. (2004). Introduction. In: Christianidis, J. (eds) Classics in the History of Greek Mathematics. Boston Studies in the Philosophy of Science, vol 240. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2640-9_18
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