Hidden lemmas in Euler's summation of the reciprocals of the squares
type: Item
instance of: scholarly article
title: Hidden lemmas in Euler's summation of the reciprocals of the squares
zbMATH Open document ID: 0879.01017
DOI: 10.1007/BF00376450
author: Mark McKinzie
author: Curtis Tuckey
published in: Archive for History of Exact Sciences
publication date: 1998-01-19T00:00:00Z
review text: Die Verff. zeichnen die verschiedenen Arbeiten von L. Euler nach, in welchen dieser die Summe der reziproken Quadrate zum Sechstel von \(\pi^2\) berechnete. ``We will conclude that if one acknowledges the arithmetic of infinite and infinitesimal numbers, then Euler's arguments are correct, and with the adjunction of a few elementary lemmas become a model of deductive reasoning.'' 1. Darstellung von Eulers Herleitung von 1734. 2. Die Herleitung von Euler in seinem Buch `Introductio in analysin infinitorum, 1748'. Euler verwendet die Produktzerlegung einer unendlichen Reihe. 3. Gaps in the Deduction. Es werden zwei Lücken in Eulers Ableitungen aufgewiesen, die von dessen unbedachtem Umgang mit unendlich kleinen Größen und der Produktzerlegung von unendlichen Reihen herrühren. 4. Avoiding the Gaps. Darlegung der Produktzerlegung unendlicher Reihen durch Euler. 5. The Hidden Lemmas. Formulierung der Hilfssätze über konvergente Reihen, die zur Rechtfertigung von Eulers Herleitung benötigt werden (nach Lakatos und Laugwitz). Sie betreffen einen von Euler verwendeten Begriff der Konvergenz. 6. Filling the Gaps. 7. Proofs of the Hidden Lemmas. 8. Understanding Euler's Arguments. ``If the goal is to understand Euler, then the arithmetic of infinitesimal numbers must be taken seriously'' \dots ``with the Hidden Lemmas, we have been able to follow Euler's arguments almost exactly, merely inserting a few words of explanation at the appropriate junctures''\dots ``Euler's arguments are beautiful, intelligible, and correct, as well as being brilliant contributions to the early foundations of analysis''.
Mathematics Subject Classification ID: 01A50
Mathematics Subject Classification ID: 11-01
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11-03
zbMATH DE Number: 1057318
zbMATH Keywords: convergent series
zbMATH Keywords: arithmetic of infinite
zbMATH Keywords: infinitesimal numbers
reviewed by: Johann Jakob Burckhardt
MaRDI profile type: Publication
Wikidata QID: Q124823141
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description: scientific article; zbMATH DE number 1057318