Jacob Klein - Greek Mathematical Thought and the Origin of Algebra

Jacob Klein

Translated by Eva Brann

Dover Publications, Inc.

New York

Copyright 1968 by The Massachusetts Institute of Technology.

All rights reserved.

This Dover edition, first published in 1992, is an unabridged and unaltered republication of the edition published by The M.I.T Press, Cambridge, Massachusetts, in 1968. The original German text was published as Die griechische Logistik und die Entstehung der Algebra in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien, Vol. 3, fasc. 1 (Berlin, 1934), ). The translation of Vietas Introduction to the Analytical Art which appears in the Appendix is by the Reverand J. Winfree Smith. This edition is published, by special arrangement with The M.I.T. Press, 55 Hayward Street, Cambridge, Massachusetts 02142.

Library of Congress Cataloging-in-Publication Data

Klein, Jacob, 18991978.

[Griechische Logistik und die Entstehung der Algebra. English]

Greek mathematical thought and the origin of algebra / by Jacob Klein; translated by Eva Brann.

p. cm.

Unabridged and unaltered republication of the edition published by the M.I.T. Press, Cambridge, Massachusetts, in 1968 T.p. verso.

Includes bibliographical references (p. ) and indexes.

ISBN 0-486-27289-3

1. Mathematics, Greek. 2. AlgebraHistory. I. Title.

QA22.K513 1992

Manufactured in the United States by Courier Corporation

27289306

www.doverpublications.com

T HIS STUDY was originally written and published in Germany during rather turbulent times. Were I writing it today, the vocabulary would be less scholarly, and the change from the ancient to the modern mode of thinking would be viewed in a larger perspective.

Some of the references could have been brought up to date and made more accessible to English-speaking readers, but this would have entailed a labor out of proportion to its usefulness. However, a few additions to the references as well as some minor changes in the text have been made by the translator and by myself.

JACOB KLEIN

St. Johns College

Annapolis, Maryland

November 1966

T HE GREEK word arithmos is rendered in the German text as Anzahl: a number of [things] to distinguish it from our modern Zahl: number. Since English approximations to Anzahl are either obsolescent (e.g., tale) or awkward (e.g., counting-number, numbered assemblage), Anzahl, like Zahl, has been rendered simply as number, although it is a chief object of this study to show that Greek arithmos and modern number do not mean the same thing, that they differ in their intentionality, for the former intends things, i.e., a number of them, while the latter intends a concept, i.e., that of quantity (cf. ). Intentionality and conceptualization are both used to translate the German word Begrifflichkeit.

The following Greek terms occur so frequently that they have been incorporated into the text in transliterated form:

asthesis sense perception

aisthetn object of sense

analoga proportion

aristos [dys] indeterminate, infinite [dyad]

peiron sb.: the limitless, infinite

apdeixis demonstration, [strict] proof

apora quandary

arch, pl. archa [governing] source, beginning

chorisms separation

dinoia [the faculty and activity of] thinking

dnamis power

eidetiks adjective from eidos

edos, pl. ede literally: looks; kind, form, species, idea; sometimes: figure

epistme Lat. scientia, knowledge, science

gnos, pl. gene genus, family, class; often: the higher eide

hen one

hle material

ida = eidos

koinn sb.: common thing

koinona ton eidn community of the eide

kat to plthos according to multitude

kath aut by itself

kathlou pragmatea general treatment or study

kat edos according to kind