HISTORIA MATHEMATICA 23 (1996), 418±436 ARTICLE NO. 0040 Gauss and the De®nition of the Plane Concept in Euclidean Elementary Geometry KONSTANTINA ZORMBALA View metadata, citation and similar papers at core.ac.uk brought to you by CORE Skyrow 49ÐKipseli, 113 63 Athens, Greece provided by Elsevier - Publisher Connector On July 28, 1797, Gauss wrote in his diary: ``Plani possibilitatem demostravi'' [``I have proved the possibility of the plane'']. On January 27, 1829, in a letter to Bessel, he was skeptical about a de®nition of the plane in Euclidean geometry according to which the plane is a surface having the property that the connecting line of two of its points lies on it. He voiced the same criticism of this de®nition some years later in a letter to Wolfgang Bolyai dated March 6, 1832. In his Nachlaû, a note was found with the title ``Zur BegruÈ ndung des Planum'' [``On the Founding of the Plane'']. All his letters and notes on this topic became known only after Gauss's Nachlaû had been published, and they have not yet been discussed in their totality. Here, I present and thoroughly discuss all the letters and notes in which Gauss struggled with the concept of the plane, especially with his own de®nition of the latter within Euclidean elementary geometry. 1996 Academic Press, Inc. Am 28. Juli 1797 trug Gauû in seinem Tagebuch ein: ``Plani possibilitatem demonstravi'' [ich habe die MoÈ glichkeit der Ebene bewiesen]. Am 27. Januar 1829 aÈuûerte er in einem Brief an Bessel Bedenken gegenuÈ ber einer De®nition der Ebene in der euklidischen Geometrie, nach der die Ebene eine FlaÈche ist, die die Eigenschaft hat, daû die Verbindungsgerade je zwei ihrer Punkten auf ihrer liegt. Dieselbe Kritik an diese De®nition uÈ bte er einige Jahre spaÈter, am 6. MaÈrz 1832, in einem Brief an Wolfgang Bolyai. In seinem Nachlaû ®ndet sich eine Notiz unter dem Titel ``Zur BegruÈ ndung des Planum.'' Alle Briefe und Notizen zu diesem Thema wurden erst nach der VeroÈ ffentlichung des Gausschen Nachlasses bekannt, und wurden bis heute in ihren Gesamtheit nicht diskutiert. Der vorliegende Aufsatz moÈ chte alle diese Briefe und Notizen, in denen sich Gauû mit dem Begriff der Ebene bzw. mit seiner De®nition derselben in der euklidischen Elementargeometrie auseinandersetzte, darstellen und einge- hend diskutieren. 1996 Academic Press, Inc. Sti§ 28 IoyliÂoy 1797 she«iÂgn«oGauû sto he«%olo cio toy: ''Plani possibilitatem demostravi`` (af«Â d«ija thn dynatoÂthta toy «fif«Â doy). Sti§ 27 Ianoya%iÂoy 1829 s««Âna c%aÂeea toystoBessel «Â kw%az««fiwyla j«i§ af«Ânanti s««Ânao%iseo toy «fif«Â doy sthn «ykl«iÂdiac«ge«t%iÂa, sy ewgnae«tonofoiÂoto«fiÂf«do«iÂnaieia«fiwaÂn«ia, foy «Â x«ithn idio thta, h «yq«iÂa, foysynd«Â«idyoofoiadhÂfot«she«iÂaayth §th§ «fiwa n«ia§ na k«iÂtai «z' oloklh %oys«ayth thn «fiwa n«ia. Thn iÂdia k%itikh af«Â nanti s' ayto ton o%iseo a skhs« e«%ika x%o niaa%coÂt«%a,sti§ 6 Ma%tiÂoy 1832, s««Âna c%aÂeea toystoWolfgang Bolyai. Sto a%x«iÂo toy Gauû b%«Â qhk« eia she«iÂgsh e« `` ton tiÂtlo ''Zur BegruÈ ndung des Planum (Gia thn q«e«liÂgsh toy «fif«Â doy). `Ola ta c%a eeata kai oishe«igÂs«i§ s' ayto to q«Â ea «cinan cngsta e«ta thn dheosi«yoh toy a%x«iÂoytoyGauû kai d«n «Â xoyn syzhthq«i sto syno lo toy§ e«Â x%i ti§ e«Â %«§ ea§.Hfa%oyÂsa «%casiÂa qafa%oysia s«ikaiqasyzhth s«itac%aÂeeata kaiti§ she«iÂgs«i§, sti§ ofoi«§ o Gauû «Â %x«tai antie«Â tgfo§ e« thn «Â nnoia toy «fif«Â doy, kaisyck«k%ie«Ânae«tono%iseo toysthn «ykl«iÂdia stoix«ig dh c«ge«t%iÂa. 1996 Academic Press, Inc. MSC 1991 subject classi®cations: 01A55 51-03. KEY WORDS: Gauss, Euclid, plane, Euclidean elementary geometry, Pasch, axiomatics. 418 0315-0860/96 $18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. HM 23 GAUSS AND THE DEFINITION OF THE PLANE CONCEPT 419 INTRODUCTION . ist darunter ein Gegenstand, der nicht minder unvollkommen, obschon gewiû nicht minder ein¯uûreich auf alles UÈ brige sein duÈ rfte, als irgend ein anderer: ja, der selbst noch unvollkom- mener, obgleich eben so wichtig ist, als die Parallelen-Theorie. Dieser Gegenstand ist die Theorie der Ebene. August Crelle 1834 to the Berlin Academy The ideas developed by Gauss (1777±1855) penetrated almost all ®elds of contem- porary mathematics and provided impulses toward their further development. I brie¯y review some of Gauss's contributions. In 1796, he solved an age-old geometric problem, the construction of the regular 17-gon with straightedge and compass. In 1799, he provided a ®rst proof of the fundamental theorem of algebra, a task at which d'Alembert (1717±1783), Lagrange (1736±1813), and Euler (1707±1783) had tried and failed. Further studies on higher mathematics and the publication of his Disquisitiones arithmeticae (1801) followed. In 1801, he developed a new method for determining the orbit of a newly discovered planet. In 1821, Gauss tackled problems of differential geometry and published the results of this work in 1828 in his Disquisitiones generales circa super®cies curvas [General Studies Concerning Curved Surfaces]. At this time, he was already convinced that Euclid's parallel postulate could not be proved and that another, non-Euclidean geometry was mathematically possible. Discovering non-Euclidean geometry was a breakthrough which in¯uenced the development of not only mathematics but also science as a whole. Historians have located Gauss's contribution to foundational questions of geometry mainly in non-Euclidean geometry. The editor of his Nachlaû, Paul StaÈckel, thus states: ``If the studies are mentioned which Gauss undertook with regard to the foundations of geometry, the discovery of non-Euclidean geometry is the ®rst thing which comes to mind. Gauss, however, did not con®ne himself to the parallel axiom; rather, he dealt with a number of questions which today would be assigned to axiomatics as well'' [24, 63]. Even as a student at the University of GoÈ ttingen (1792±1798), Gauss tackled the geometric concepts which are fundamental for Euclidean geometry. How to explain the concept of the straight line was the topic Gauss and his fellow student and later close friend, Wolfgang Bolyai (1775±1865), discussed in 1796 on the occasion of their ®rst walk. As Johann Bolyai later said, Gauss responded to his father's proposi- tions: ``Yes the straight line is indeed treated perversely; it is indeed the line which turns on itself'' [24, 64]. As one of his students, H. B. LuÈ bsen reported [20, 11], while lecturing on astronomy at GoÈ ttingen, Gauss himself had used this very de®nition in introducing the straight line. The Euclidean de®nition of the straight line was considered to be inappropriate for the treatment of geometry, and other de®nitions had been proposed. In his Elements (about 300 B.C.), Euclid de®ned the straight line as the line ``which lies evenly with the points on itself'' [3, 153]. First, from the perspective of the deductive method according to which Euclid structured his own geometry, this de®nition was not precise enough: it referred to a property of ``evenness'' which was not explained beforehand. Second, Euclid nowhere used this de®nition in his Elements, for in- 420 KONSTANTINA ZORMBALA HM 23 stance, in order to prove the properties of the straight line. This challenges its role as a de®nition. Since antiquity, there have been attempts to clarify the imprecise Euclidean account of the straight line and to replace it by a more suitable one. Gauss's alternative to the Euclidean view is similar to a presentation proposed by Heron of Alexandria (about A.D. 62), Proclus (410±485), Leibniz (1646±1716), and Girolamo Saccheri (1667±1733) [3, 168]. Gauss turned his critical eye not only to this disputed geometric concept but also to other geometrical concepts which, though much used, had never been explained properly. In 1832, he spoke of that ``part of a plane which is situated between three straight lines ab, cd, ef,'' noting, ``[s]uch terms like `between' must ®rst be reduced to clear concepts, which is easily done, but which I ®nd achieved nowhere'' [5, 222]. The issue raised here is the unsatisfactory description of the concept ``between'' for coplanar straight lines. In other texts, Gauss noted a similar situation regarding the concept of plane itself. In 1829, he wrote: ``It is curious that, besides the well- known gap in Euclid's geometry which people hitherto have vainly attempted to close, there is yet another de®cit in the latter, which to my knowledge nobody has as yet criticized, and which is by no means easy (although possible) to remedy. It is the de®nition of the plane . .'' [6, 226]. Gauss left behind an important number of letters and unpublished notes. Some of them contain critical remarks concerning a certain de®nition of the plane in elementary Euclidean geometry. In this paper, I discuss these letters and notes. The reconstruction is based on the following sources: July 28, 1797 Note in his diary November 5, 1825 Letter to Gerling January 27, 1829 Letter to Bessel 1829 Note ``Euclid's Theorem I,7''1 March 6, 1832 Letter to W. Bolyai 1832 Note ``BegruÈ ndung des Planum'' [Foundation of the Plane] Most of these letters and notes are contained in volume 8 of Gauss's Werke, edited by Paul StaÈckel in 1900.