In order to investigate how a 19th century mathematical textbook (in contrast to a contemporary one) would be experienced by a novice reader, we embarked on the following project: In the summer of 2013, a student with no previous training in college-level mathematics (the first author) set out to learn projective geometry from Pasch\u27s 1882 textbook Lectures on Modern Geometry. Afterwards, he studied the same material from Coxeter\u27s 1994 popular undergraduate textbook Projective Geometry. We report here some of his experiences and impressions contextualizing them along the way

Dirk Schlimm

Douglas Roland Campbell

Journal of Humanistic Mathematics

Campbell, Douglas Roland

Schlimm, Dirk

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Journal of Humanistic MathematicsVolume 4 | Issue 2 July 2014A Subjective Comparison Between a Historical anda Contemporary Textbook on GeometryDouglas Roland CampbellMcGill University, doug.roland.campbell@gmail.comDirk SchlimmMcGill University, dirk.schlimm@mcgill.caFollow this and additional works at: http://scholarship.claremont.edu/jhm©2014 by the authors. This work is licensed under a Creative Commons License.JHM is an open access bi-annual journal sponsored by the Claremont Center for the MathematicalSciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/Recommended CitationCampbell, D. R. and Schlimm, D. "A Subjective Comparison Between a Historical and a Contemporary Textbook on Geometry,"Journal of Humanistic Mathematics, Volume 4 Issue 2 ( July 2014), pages 58-66. DOI: 10.5642/jhummath.201402.06 . Available at:http://scholarship.claremont.edu/jhm/vol4/iss2/6A Subjective Comparison Between a Historical and aContemporary Textbook on Geometry1Douglas Roland CampbellDepartment of Philosophy, McGill University, Montreal, CANADAdouglas.campbell2@mail.mcgill.caDirk SchlimmDepartment of Philosophy, McGill University, Montreal, CANADAdirk.schlimm@mcgill.caSynopsisIn order to investigate how a 19th century mathematical textbook (in contrast to acontemporary one) would be experienced by a novice reader, we embarked on thefollowing project: In the summer of 2013, a student with no previous training incollege-level mathematics (the first author) set out to learn projective geometryfrom Pasch’s 1882 textbook Lectures on Modern Geometry. Afterwards, hestudied the same material from Coxeter’s 1994 popular undergraduate textbookProjective Geometry. We report here some of his experiences and impressionscontextualizing them along the way.1. IntroductionIt is generally accepted that mathematics underwent a radical transforma-tion around the turn of the 20th century: it became more rigorous, abstract,structural, or simply modern. For a general account of this development, see[5]; for the developments in algebra and set theory, see [1] and [4]. In some1This research was supported by an Arts Research Internship Award (ARIA) from theFaculty of Arts, McGill University, and by a grant from Fonds Que´be´cois de la Recherchesur la Socie´te´ et la Culture (FQRSC).Journal of Humanistic Mathematics Vol 4, No 2, July 2014Douglas Roland Campbell and Dirk Schlimm 59disciplines, the turning point can be identified fairly clearly. For example,in geometry it happened in 1899. Gray writes “Modernism in geometry ar-rives with Hilbert, and the difference between his Grundlagen der Geometrie(Foundations of Geometry) and the earlier work of Pasch” [5, page 19].The rise of modern mathematics is frequently described in terms of achange in the content of mathematics, its objects and our knowledge aboutthem. However, this development also accompanied a change in the waymathematics is presented, for instance, in textbooks. Long and verboseexplanations that meander towards specific results were replaced by crispexpositions following more or less a definition-theorem-proof structure. Atthe same time, the use of formal symbolisms, such as set-theoretic notation,increased. As a result, we take it that the reader will have no difficultiesin determining whether a mathematical textbook was published in 1850 or1950 by just flipping through and quickly glancing at the pages.The previous observation poses certain difficulties for a contemporaryreader who is interested in learning about the historical development of math-ematics directly from the sources. This is particularly relevant for the use ofhistorical sources in mathematics education; see [3, 13]. If one already hassome mathematical background, then the modern form of presentation is thefamiliar one that one has grown up with, and the historical exposition will al-most inevitably feel tedious and cumbersome. We deliberately describe thisreaction as a feeling, because it is spontaneous, natural, and unconscious.However, such learned attitudes can unwittingly distort one’s understandingof unfamiliar representations. Discussions of computations with Roman nu-merals can serve to illustrate this point. To support his claim that additionwith Roman numerals is extremely cumbersome, Hankel in 1874 invited hisreaders to “just imagine what it would take” to perform such a calculation[6, page 51]; and almost a century later in 1969, Menninger echoes this sen-timent: “The fairly simple multiplication [. . . ] looks clearly impossible tothe uninitiated reader” [8, page 294]. Had they relied less on their precon-ceptions, they would have found that both addition and multiplication areindeed not more complicated with Roman numerals than with our decimalplace-value system [12]. The years of rote learning, instruction, and prac-tice that it took us to become proficient in using our system of numeralshave been internalized and, in contrast, it has become very difficult to evenimagine how computations could be done differently, even for historians ofmathematics. The situation of being faced with a 19th-century mathematics60 A Subjective Comparison Between Geometry Textbookstextbook is analogous. Only with conscious effort can we overcome our bi-ases, but even then we will retain the modern perspective in the back of ourminds.The above considerations led us to the following question: How wouldsomeone experience learning geometry in Moritz Pasch’s classroom in thelate 19th century and following his un-modern lecture course on projectivegeometry? We reasoned that the situation would be somewhat similar to thatof a contemporary student, with no formal training in college-level mathe-matics, who would set out to learn geometry from Pasch’s 1882 textbookLectures on Modern Geometry [9]. And so, this became an undergraduatesummer project in 2013: For a student without prior knowledge of higher-level mathematics to work through the first twelve chapters of Pasch’s bookand, afterwards, to study the same material from a contemporary presen-tation, namely Coxeter’s popular undergraduate text Projective Geometryfrom 1994 [2], while paying careful attention to one’s initial reactions to thetwo presentations.2. Reading NotesIn the following we describe some aspects in which Pasch’s and Cox-eter’s presentations differ and how these differences affected the learning andunderstanding of the material from the point of view of a mathematicallyuntrained reader. Notice, however, that these comments are not the result ofextended deliberations on these issues; instead they reflect the first impres-sions of approaching the texts.2.1. Presupposed background knowledgeBoth Pasch and Coxeter claim explicitly that no mathematical back-ground is required for understanding their texts. Coxeter does not merelysay that he does not presuppose a prior acquaintance with geometry (andmathematics, generally), but he insists that it is actually best that the readerlack this familiarity. He suggests that “the best possible advice to the readeris to set aside all his previously acquired knowledge (such as trigonometryand analytic geometry) and use only the axioms [presented in Projective Ge-ometry] and their consequences” [2, page 15]. He explains that attemptsto understand the content of Projective Geometry with methods previouslyinculcated in the reader will lead to either cumbersome calculations or toDouglas Roland Campbell and Dirk Schlimm 61downright failures. Pasch formulates similar expectations for his reader. Hewrites that the lectures do “not presuppose any kind of knowledge that iscustomarily gained first in geometry” [9, page 7]. Rather, Pasch asks onlyfor vigilance in dutifully neglecting “in thought those things which one isfamiliar with and to return to a point of view from which one has distancedoneself considerably” [9, page 7]. Pasch’s attitude adopted in these lecturesin consistent with comments he made later in his essay On the educationalvalue of mathematics from 1894 [10], where he observes that the “languagein which we find mathematics presented nowadays lacks the clarity by whichthe ancients distinguish themselves and remain our role models” [10, page12]. Accordingly, one finds in Pasch’s work an attempt to re-introduce theclarity demonstrated first by the ancients, i.e., Euclid. Thus, both Pasch andCoxeter present the material in an axiomatic fashion and seem to think thatprevious exposure to related concepts may make learning the new conceptsintroduced in their works more difficult.2It seems that Pasch lived up to his promise more than Coxeter did. Letus illustrate this claim by an example. Coxeter tends to leave gaps in his ex-planations that, presumably, could be filled by one who has a superior (prior)grasp of the material. In Subsection 7.6, called A self-conjugate quadrilateral,he introduces Hesse’s theorem in one sentence, proves it in four sentences,and then concludes the section. The notion of a self-conjugate quadrilateralis not explained explicitly, but has to be inferred from previous discussions.In the mind of someone without much mathematical background, it may havebeen preferable to identify more clearly what is happening, as well as, per-haps, some implications of the theorem, such that one could discern preciselywhy it is in the text.In general, Pasch proceeds more slowly and introduces new concepts moregradually than Coxeter does. Also Pasch’s treatment is significantly morethorough, which makes it much easier for an inexperienced reader to under-stand. Indeed, what might appear as long-winded from a modern perspectiveactually appears to be beneficial for the learner. Seventeen pages into Pro-jective Geometry, Coxeter says that he has laid out a foundation that suffices“for the erection of the whole system of projective geometry” [2, page 17].For the inexperienced reader this appears to be too fast. Even if Pasch takes2It is interesting to note that this goes against common educational advice, accordingto which it is better to connect new material to previous knowledge.62 A Subjective Comparison Between Geometry Textbookslonger and goes more slowly (and thus one dedicates more time to it), thelatter also ends up being more successful, at least in the eyes of one (novice)reader. Again, Pasch has, in his work on mathematical education, made itclear that he wanted to avoid some of the problems that beset Coxeter’s pre-sentation. He affirms that, sometimes, there are gaps in one’s presentation—perhaps just for the sake of concision and elegance—, and that “the lessknowledgeable reader is misled into overlooking the gaps or into filling themout incorrectly” [10, page 16]. He makes an effort to avoid this situation inhis lectures on projective geometry, much to the student’s benefit.2.2. Organizational structureThe division and order of chapters, too, make Pasch’s work more accessi-ble on a first read than Coxeter’s. For instance, Pasch’s first chapter is calledOf the straight line and the content of the chapter reflects the title, whichmade it easier for the student to follow along. This effect was produced in twoways. Firstly, it was possible to anticipate the material in the chapter justafter having read the title. Secondly, every time a new concept or theoremwas encountered, it could immediately be related to the straight line, whichgenerated a strong (and helpful) sense of coherence and unity. In contrast,Coxeter’s work is organized less according to basic geometric objects, butmore according to higher-level concepts and principles. For example, Cox-eter presents perspectivity at the end of the first chapter, whereas Pasch onlypresents it in the tenth chapter; whereas Pasch dedicates a chapter to thenotion, Coxeter gives it a few pages as a subsection. Another example: Thethird chapter of Coxeter’s book is called The Principle of Duality. However,the content of the chapter is not merely an explanation of this principle, butit also features expositions of such notions as the Desargues configuration,a discussion of the invariance of the harmonic relation, and introductions toharmonic nets and trilinear polarity. At the time of the first reading, however,it was not obvious to the (inexperienced) reader what any of these conceptshave to do with the principle of duality, or what relation obtains betweenthem in a way that justifies their arrangement together. As such, Coxeter’swork ostensibly requires some interpolation and abstraction to figure out howthese ideas are united by the principle of duality. This necessary extra effortmade it uncomfortably hard for the student to become acquainted with thematerial, which was not the case with Pasch’s presentation.Douglas Roland Campbell and Dirk Schlimm 632.3. NotationThere is a marked difference between the texts of Coxeter and Pasch in theextent to which they use notation. Pasch was sceptical about the usefulnessof too formal a presentation [11, page 104]. He knew it would be difficultfor the first-time reader to go through the book, when the use of formalismswould be perceived as a needless obstacle. Thus, in [9] he avoids the useof notation entirely (except for the use of variables to label and refer to thediagrams). Coxeter, on the other hand, makes liberal use of it. For instance,he writes some theorems entirely formally; see for instance his Theorem 3.33:“If ABCF=∧ ABCF and H(AB,CF ), then H(A′B′, C ′F ′).”What is striking is that some of these theorems are written as early as thethird chapter, before the reader can even adjust to the content, let alone thestyle.This heavy reliance on notation in Coxeter’s book felt like an impedimentto developing an understanding of the material for the inexperienced reader.In addition, it probably contributed to the impression that Coxeter movesmore quickly through the material than one would have liked. While weacknowledge that, once the mathematical understanding is developed, beingable to freely and comfortably use the notation is an advantage, a gentlerand more gradual introduction of notation than Coxeter’s would have beenbeneficial. Perhaps, this could also be augmented with some explicit remarkson the role of the notation.2.4. Historical commentsAnother way in which Coxeter’s presentation differs from Pasch’s is thatthe former includes short remarks on the history of mathematics, whereasthe latter does not. Pasch only once mentions something that can be charac-terized as historical, whereas Coxeter’s work is crowded with such remarks.The decision to include such details may stem from the thought that thebook would be easier to read if there were informal breaks for the reader andthat tracing the genealogy of mathematical concepts enriches the reader’s un-derstanding. In practice, however, none of these advantages were conferred.The book felt bloated, the presentation suffered in terms of momentum, andnothing important seemed to be added to the discussion.64 A Subjective Comparison Between Geometry TextbooksFor example, Coxeter often writes such pithy comments as “in Figure2.3A, the collineation is a homology (so named by Poncelet) [. . . and] inFigure 6.2A, it is an elation (so named by the Norwegian geometer SophusLie, 1842–1899)” [2, page 53]. These sorts of comments do not obviously addto one’s understanding of the mathematics; they disrupt the rhythm instead.As Coxeter broaches Pappus’ theorem, he writes that Pappus himself lived in4th century Alexandria and “used a laborious development of Euclid’s meth-ods” [2, page 38]. This detail, and others of its kind, do not make it easier tolearn the material, Pappus’ theorem in this instance. On the contrary, theymight even impede the learning process by unnecessarily bloating the text.In contrast, the absence of such comments in Pasch’s text makes this 19thcentury presentation easier to follow for the uninitiated reader, since thereare no similar interruptions of the momentum. Pasch makes one historicalcomment, observing that “the empirical origin of the geometric axioms hasbeen investigated in detail by Helmholtz” [9, §1], but this is made in a foot-note and Helmholtz was still alive at the time of publication. In any case, itis easier to follow the material when the presentation of that material is notinterrupted by lessons in history.3. DiscussionWe have reported above the subjective experiences of one reader of Pasch’sand Coxeter’s textbooks on projective geometry. Now, what if anything canwe conclude from these observations?First of all, it appears that the modern mathematical presentation style3is indeed something that one has to learn and get used to, and that themore abstract and formal presentations are not necessarily more intuitiveor easier to follow for the uninitiated learner. Secondly, somebody who islearning mathematics encounters a text from a very different perspectivethan somebody who is already familiar with the material and the style ofpresentation. It appears that historical textbooks reflect more closely the waythe material was actually presented in class, whereas contemporary textbooksserve more to complement the lectures, rather than to reproduce them (this isalso supported by the fact that Coxeter has exercises following each section).3For a discussion of style in mathematics, see [7].Douglas Roland Campbell and Dirk Schlimm 65These observations should caution us against approaching and assessinga historical text in the same way as we read a modern presentation, and theyteach us that sometimes even an old text can reveal something new whenread with fresh eyes.References[1] L. Corry, Modern Algebra and the Rise of Mathematical Structures, Vol-ume 17 of Science Networks, Historical Studies, Birkhau¨ser, Basel, 1996.[2] H. S. M. Coxeter, Projective Geometry, 2nd edition, Springer, New York,1994.[3] J. Fauvel and J. van Maanen, History in Mathematics Education,Kluwer, New York, 2002.[4] J. Ferreiro´s, Labyrinth of Thought: A History of Set Theory and its Rolein Modern Mathematics, 2nd revised edition, Birkhau¨ser, Basel, 2007.[5] J. Gray, Plato’s Ghost: The Modernist Transformation of Mathematics.Princeton University Press, Princeton, N.J., 2008.[6] H. Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter,B. G. Teubner, Leipzig, 1874.[7] P. Mancosu, “Mathematical style,” in Zalta, E. N., editor, TheStanford Encyclopedia of Philosophy. Spring 2010 edition. Avail-able at http://plato.stanford.edu/archives/spr2010/entries/mathematical-style/, accessed on July 4, 2014.[8] K. Menninger, Number Words and Number Symbols, MIT Press, Cam-bridge, MA, 1969.[9] M. Pasch, Vorlesungen u¨ber neuere Geometrie, B. G. Teubner, Leipzig,1882. Reprinted in 1912 and in a second edition in 1926. English trans-lation by Dirk Schlimm (manuscript).[10] M. Pasch, “Ueber den Bildungswerth der Mathematik,” Rector’s Ad-dress, Grossh. Hof- und Universita¨ts-Druckerei Curt von Mu¨nchow,Giessen, July 2, 1894. Reprinted in Mitteilungen des mathematischenSeminars Giessen, Number 146 (1980), pages 20–39.66 A Subjective Comparison Between Geometry Textbooks[11] D. Schlimm, “Pasch’s philosophy of mathematics,” Review of SymbolicLogic, Volume 3 Issue 1 (2010), pages 93–118.[12] D. Schlimm and H. Neth, “Modeling ancient and modern arithmeticpractices: Addition and multiplication with Arabic and Roman numer-als,” in V. Sloutsky, B. Love, and K. McRae, editors, Proceedings ofthe 30th Annual Meeting of the Cognitive Science Society, Austin, TX.Cognitive Science Society, 2008, pages 2007–2012.[13] F. Swetz, J. Fauvel, O. Bekken, B. Johansson, and V. Katz, eds., Learnfrom the Masters, The Mathematical Association of America, Washing-ton, DC., 1995.

A Subjective Comparison Between a Historical and a Contemporary Textbook on Geometry

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