The Genesis of the Algebra Textbook: From Pacioli to Euler

Heeffer, Albrecht

doi:10.1484/j.alma.5.100794

Cited by 7 publications

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“…It was for others to develop Pacioli's first theorization of algebra and double entry into axiom‐based theory. The algebraists of the 16 th and 17 th centuries did so and developed his axiomatic approach to teaching algebra into axiom‐based theory (Heeffer, 2012, p. 39). The accounting masters did not, preferring to continue with an exemplar‐based approach that was augmented with rules in the 16 th century, firstly by Manzoni (1540) and then by those that followed, up to the present day.…”

Section: Figurementioning

confidence: 99%

The Life and Works of Luca Pacioli (1446/7–1517), Humanist Educator

Sangster

2021

Abacus

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Accounting has few heroes, but one that most acknowledge as worthy of that accolade is Luca Pacioli, the man who published the first printed exposition of double entry bookkeeping in 1494. This was the publication that led to the development of the accounting systems we use today. However, if we consider our literature on Pacioli, it is found to be not just confusing and contradictory, but undecided as to who he was and what he did, or did not do. This paper addresses this ambiguity and uncertainty by adopting a critical hermeneutic analysis of the literature in order to present a more authentic perception of the life and works of Luca Pacioli. Archival discoveries of the past 20 years are embraced with Pacioli's own writings, and those of others who have sought to identify the man and his ambition. The view it reveals shows Pacioli as far more than the peripatetic friar and journeyman teacher of mathematics described in our literature; and it fills a much larger gap than the contradictions and confusion implied in our knowledge of the man who many consider to be the ‘father of accounting’.

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Section: Figurementioning

confidence: 99%

The Life and Works of Luca Pacioli (1446/7–1517), Humanist Educator

Sangster

2021

Abacus

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“…For this reason I have not included algebra texts from the Seventeenth century and earlier. A history of early algebra is given by Stedall [21], and the classic history of mathematical notations is Cajori's [22], but see also Heeffer [23].…”

Section: Use Of Historic Textbooksmentioning

confidence: 99%

Textbook accounts of the rules of indices with rational exponents

Sangwin

2019

International Journal of Mathematical Education in Science and

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The rules of indices, e.g. a n b n = (ab) n , are a particularly important part of elementary algebra. This paper reports results from a textbook analysis which examined how the shift from integer to rational exponents in the rules of indices is discussed in school textbooks. The analysis also considered related issues, such as notation and the introduction of complex numbers. A selection of popular textbooks from the period 1800-2000 was examined and the nature of the justification given for the extension of meaning to rational indices considered. In both the definition and computational rules, when extending the domain of n in a n to rational numbers the (potential) contraction of the domain of a to positive numbers was often quietly ignored. A wide variety of approaches are used in choosing what is to be a definition, what is to follow, and how this is justified. The difference between computational rules for practical algebraic manipulation and a formal definition was often blurred.

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“…Through her archival studies, Ulivi has shed light on the teaching of abaco in the 14 th and 15 th centuries, and on Pacioli's life; and Ciocci (2009, 103-106) has reconstructed the process that transformed Pacioli from a practical mathematician in the 1470s into a theoretical mathematician by the 1490s. Yet more insights have been presented by other historians of mathematics, such as Fenny Smith (2008) (or Rankin 1992 and Albrecht Heeffer (2010Heeffer ( , 2012, for whom Pacioli's capacity to explain, to offer original solutions, and to organize his teaching in a manner that emphasized learning from single generalizable examples rather than repetitive rote, set him aside from his peers. Through their efforts, we now have a far clearer contextualization of the nature of Pacioli's writings and, thereby, his teaching.…”

Section: Pacioli's Motivationmentioning

confidence: 99%

“…After being awarded his degree in the early 1480s, Pacioli emulated Euclid's gathering of source material, spending over a decade seeking out abaco problems and solutions in the manuscripts of his predecessors, scouring libraries 23 for sources and ideas, particularly in Florence (Ciocci 2009;Heeffer 2012). In this, as he acknowledges in Summa Arithmetica, Pacioli was indebted to the works of Leonardo Pisano (Fibonacci) (c. 1170-c. 1240), widely recognized as the inspiration for abaco mathematics.…”

Section: Pacioli Euclid Algebra and Double Entrymentioning

confidence: 99%

Pacioli's Lens: God, Humanism, Euclid, and the Rhetoric of Double Entry

Sangster

2017

The Accounting Review

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This paper investigates why, in 1494, the Franciscan friar and teacher of mathematics, Luca Pacioli, published an instructional treatise describing the system of double entry bookkeeping. In doing so, it also explores the rhetoric and foundations of double entry through the lens of Pacioli's treatise. Recent findings on Pacioli's life and works, his writings, and the medieval accounting archives are combined to identify how he was inspired by his faith and his humanist beliefs to give all merchants access to the practical mathematics and the bookkeeping they required. The paper finds that Pacioli's teaching method was inspired by Euclid, his Franciscan education, and his humanist beliefs, and that Pacioli reveals a simplicity in the then-unrecognized axiomatic foundation of double entry that has been largely overlooked. The findings represent a paradigm shift in how we perceive Pacioli, his treatise, and double entry.

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The Genesis of the Algebra Textbook: From Pacioli to Euler

Cited by 7 publications

References 0 publications

The Life and Works of Luca Pacioli (1446/7–1517), Humanist Educator

The Life and Works of Luca Pacioli (1446/7–1517), Humanist Educator

Textbook accounts of the rules of indices with rational exponents

Pacioli's Lens: God, Humanism, Euclid, and the Rhetoric of Double Entry

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