False position in Leonardo of Pisa's Liber Abbaci

Abstract

We examine the rhetorical methods of Leonardo of Pisa in his exposition of single false position in Liber Abbaci. For example, Leonardo makes extensive use of formulaic phrases in his solutions. Some of these formulas also seem to indicate whether a particular solution needs further justification. Although he prefers proofs in terms of the pseudo-Euclidean canon of al-Khwārizmī, sometimes such proof eludes Leonardo and he resorts instead to justification by experiment. We also look at the extent to which using symbolic representations might distort our view of Leonardo's thinking. SommarioSi esamina il metodo retorico di Leonardo da Pisa nella sua esposizione della singola falsa posizione nel Liber Abbaci. Per esempio, Leonardo fa un uso estensivo di frasi che sottintendono formule nelle sue soluzioni e alcune di queste formule sembrano dare una indicazione sulla necessitá o meno di una ulteriore giustificazione della particolare soluzione. Sebbene egli preferisca dimostrazioni in termini del canone pseudo-Euclideo di al-Khwārizmī, talvolta alcune dimostrazioni sono al di fuori della sue portata ed egli ricorre allora alla giustificazione per esperimento. Si discuterá inotre l'effetto che l'utilizzo della rappresentazione simbolica potrebbe avere su una nostra visione distorta del pensiero di Leonardo.

The Man of Numbers: Fibonacci's Arithmetic Revolution--A Book Review

“…But there are lots of interesting things to be found, not just isolated examples like the famous rabbit problem that gives rise to the Fibonacci numbers [11, pp. 404-405], but also extended treatments of topics that we might now see as early developments in solving systems of linear equations [12] or early acknowledgment of negative solutions [9]. Even when he uses methods that we now see as obsolete, such as simple false position, we recognize a fellow mathematician as he seeks proofs for techniques that seem to have been discovered experimentally [3].…”

“…These are explained in modern terms but he also lets Fibonacci speak for himself, with excerpts usually taken from Sigler's rather stilted translation [11]. 3 Chapter 6 deals with the episode at the court of Frederick II. In the absence of direct eye witness accounts, the description has to rely on averted vision: who was Frederick and what did he do?…”

Αλγοριθμικές Διαδικασίες Ως Κύρια Μαθηματική Γνώση

“…Στο δωδέκατο κεφάλαιο του «Liber Abaci», υπήρχαν διάφορες μέθοδοι επίλυσης μαθηματικών αινιγμάτων και προβλημάτων. Η πρώτη από αυτές τις μεθόδους ήταν η λαθεμένη παραδοχή (Hannah, 2007;Sigler, 2002). Ο Fibonacci καλούσε αυτή τη διαδικασία ως «μέθοδο δέντρων», επειδή, στο πρώτο παράδειγμα που χρησιμοποίησε για τη μελέτη και την αποσαφήνιση της μεθόδου, ζητούσε το μέγεθος ενός δέντρου: «Υπάρχει ένα δέντρο, του οποίου το 1/3 και το 1/4 βρίσκονται υπόγεια.…”

“…Πάντως, μερικά προβλήματα των Βαβυλωνίων, που επιλύονταν με τη μέθοδο της λαθεμένης παραδοχής και ήταν σαφώς πιο περίπλοκα από αυτά των Αιγυπτίων, περιείχαν συστήματα δυο γραμμικών εξισώσεων αλλά και μη γραμμικές μεταβλητές (Høyrup, 2013;Hannah, 2007). Ένα σχετικό βαβυλωνιακό πρόβλημα είναι και το εξής (Katz & Hunger Parshall, 2014;Høyrup, 2013;Katz, 2009): «Ένα χωράφι αποδίδει σπόρο 2/3 σιλά, ανά σαρ, ενώ ένα δεύτερο αποδίδει 1/2 σιλά, ανά σαρ (όπου σιλά ήταν μονάδα χωρητικότητας και σαρ μονάδα εμβαδού).…”