Translations in Convergence

Teaching with original sources is a key technique for bringing history into the mathematics classroom, one that often presents concepts within a historical context that makes the concepts more understandable for many students. Although Convergence is aimed chiefly at an English-speaking audience, obviously many mathematicians throughout history wrote about their ideas in other languages. Thus, our authors have often included translations in full or in part in their articles. These translations are collected here.

To guide students through the translation process, see "Investigating Euler's Polyhedral Formula Using Original Sources" by Lee Stemkoski. One major repository of translated original sources is the Euler Archive, also housed by the MAA and described in the Convergence article, "Teaching and Research with Original Sources from the Euler Archive," by Dominic Klyve, Lee Stemkoski, and Erik Tou.

Full Translations

Maximus Planudes' The Great Calculation According to the Indians translated by Peter G. Brown (ca 1300)
Introduces and explains how to use an Arabic form of Indian numerals.
The translation appears on pages 3–7 of the article.

Manuel Moschopoulos' "Magic Squares" translated by Peter G. Brown (ca 1315)
First Western contribution to the topic.
The translation appears on pages 7–12, with endnotes on page 13 and references on page 14.

Evangelista Torricelli's "Quadratura Parabolae per novam indivisibilium Geometriam pluribus modis absoluta" translated by Andrew Leahy and Kasandara Sullivan (1644)
Portion of Opera Geometrica dealing with finding the area of a segment of the parabola.
The translation appears on the last page of the article.

Jan Hudde's second letter to Frans van Schooten translated by Daniel J. Curtin (1658)
Discusses algebraic ways of calculating double roots for polynomials and rational functions.
A link to a pdf of the translation appears on page 1 of this article.

Alexis Clairaut’s “Quatre Problèmes sur de Nouvelles Courbes” translated by Taner Kiral, Jonathan Murdock, and Colin B. P. McKinney (1726, published 1734)
Examines the properties of four new families of algebraic curves.
A link to a pdf of the translation appears on page 1 of this article.

Leonhard Euler's letter to Gabriel Cramer translated by Robert E. Bradley and Lee Stemkoski (1744)
Euler resolves Cramer's Paradox.
The translation appears on the last content page of the article.

Leonhard Euler's "Recherches sur les racines imaginaires des equations" translated by Todd Doucet (1746, published 1751)
Provides text of E170, Investigations on the Imaginary Roots of Equations.
A link to a pdf of the translation appears on the last page of this article.

Leonhard Euler's "On divisors of numbers contained in the form mxx + nyy" translated by Paul Bialek and Dominic W. Klyve (1778, published 1812)
Provides text of E744, one of Euler's contributions to quadratic reciprocity.
A link to a pdf of the translation appears on page 1 of this article.

François-Joseph Servois's “Calendrier perpétuel" translated by Salvatore J. Petrilli, Jr. (1813)
A perpetual calendar.
A link to a pdf of the translation appears on page 9 of this article.

François-Joseph Servois's  "Reflections on the various systems of exposition of the principles of the differential calculus" translated by Robert E. Bradley and Salvatore J. Petrilli, Jr. (1814)
A philosophical essay on the foundations of differential calculus.
A link to a pdf of the translation appears on page 1 of this article.

François-Joseph Servois's "Essay on a new method of exposition of the principles of differential calculus” translated by Robert E. Bradley and Salvatore J. Petrilli, Jr. (1814)
Represents an extension of Lagrange's effort to place calculus on a foundation of algebraic analysis without recourse to infinitesimals.
A link to a pdf of the translation appears on page 1 of this article.

François-Joseph Servois's 1817 "Memoir on Quadratures" translated by Robert E. Bradley and Salvatore J. Petrilli, Jr. (1817)
Servois' response to a debate on numerical integration in Gergonne's Annales.
A link to a pdf of the translation appears on page 1 of this article.

Raffaele Rubini's "Application of the Theory of Determinants: Note" translated by Salvatore J. Petrilli, Jr., and Nicole Smolenski (1857)
Makes an effort to introduce Italian mathematicians to this topic.
A link to a pdf of the translation appears on page 1 of this article.

David Hilbert's radio address translated by James T. Smith (1930)
Argues that every mathematical problem is solvable.
A translated transcript appears on page 3 of this article, with side-by-side German and English text and the German audio recording on page 4.

Mark Kac's ''On a new way of solving equations of the third degree'' translated by David Derbes (1931)
A new derivation of Cardano's formula, discovered while still a high school student.
The translation and a link to a pdf of a transcription of the original Polish article on page 5.

Partial Translations

Eutocius of Ascalon's Commentary on Archimedes’ On The Sphere and the Cylinder translated by Colin McKinney (ca 500)
Includes passages on duplication of the cube.
Translated sections appear on pages 5–6, 8–20.

Wibold's Ludus regularis seu clericalis translated by Richard Pulskamp and Daniel Otero (10th century)
Rules for a dice game by a French archdeacon, as preserved in an 11th-century chronicle by Balderic.
Translated excerpts appear on pages 3–5, 7.

Moses ibn Tibbon's Hebrew translation of Abu Bakr al-Hassar's Kitāb al-Bayān wa-l-tadhkār translated by Jeremy I. Pfeffer (Book of Proof and Recall) (ca 1260)
Selections illustrating the treatment of arithmetical operations.
Manuscript excerpts and translated excerpts are interwoven throughout an analysis of the text.

Ibn al-Bannā's Lifting the Veil from the Face of the Operations of Arithmetic translated by Jeff Oaks (1301)
Historical analysis of selected theorems and problems from the treatise.
Translated excerpts are interwoven throughout an analysis of the text.

Levi ben Gershon's Maaseh Hoshev translated by Shai Simonson (1321)
The Art of Calculation, with theorems and problems on arithmetic, combinations, roots, and the like.
Translated excerpts are given toward the end of the linked pdf and in the "Resource 1" section of the on-screen article.

The Pamiers Mathematical Manuscript translated by Randy K. Schwartz (14th century)
Eleven representative story problems from this manual of commercial arithmetic, including an early acceptance of negative numbers.
Translated excerpts are interwoven with the translator's analysis on pages 4–11.

Frans van Schooten's Exercitationum mathematicarum libri quinque translated by Ed Sandifer (1657)
Excerpts from Book II, on ruler constructions.
Translated problems are linked from page 2 and listed directly on pages 3–4.

James Gregory's Geometriae Pars Universalis translated by Andrew Leahy (1668)
Selections containing a proof of the Pappus-Guldin Theorem.
Translated portions of the treatise on pages 1, 9, 11–12. Page 11 has a link to the translation in process of the full treatise.

Niels Henrik Abel's “Recherches sur les fonctions elliptiques” translated by Marcus Emmanuel Barnes (1827)
First part of this pioneering treatise on elliptic integrals.
A link to a pdf appears near the end of the article.

Georg Cantor's  "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen" translated by Nicholas Scoville (1872)
Tour of this paper and an 1870 one that are relevant to the origins of point-set topology.
Passages are interwoven with the translator's analysis on pages 4–8.

Giuseppe Peano’s "Sur le déterminant Wronskien" and "Sur les Wronskiens" translated by Susannah M. Engdahl and Adam E. Parker (1889)
Warnings that a zero Wronskian does not imply linear dependence, for use in a differential equations course.
Translations from relevant passages in the writings of Peano and others are interwoven with the translators' analysis on pages 2–5 and collected on page 8.

Émile Borel's “Sur quelques points de la théorie des fonctions” translated by Nicole R. Andre, Susannah M. Engdahl, and Adam E. Parker (1895)
Tour of the first statement and proof of the Heine-Borel Theorem, along with related proofs.
Relevant passages are translated and analyzed on pages 1–7.

A. A. Markov’s Calculus of Probabilities translated by Alan Levine (1900)
Five problems with solutions from Chapter IV of the first edition of Markov’s book, with suggestions for classroom use.
Translated portions of the book on pages 3–25. A link to a pdf of a translation of the entire book appears on page 1.