Euclid and his Modern Rivals

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Euclid and his Modern Rivals (English for Euclid and his modern rivals ) is a book by Charles Lutwidge Dodgson, better known as Lewis Carroll , first published by Macmillan Publishers in 1879 . In the form of a drama , the work defends Euclid's elements against modern geometry textbooks, especially in the treatment of parallels.

content

Act 1, scene 1

Minos corrects student exercises. In a monologue he gets excited about a student who, following a textbook by Legendre, deduces Euclid's Theorem 19 from Theorem 20, but in the previous exercise has proven Euclid's Theorem 20 from Theorem 19 and thus uses a circular argument . Enter Rhadamanthus . He too has problems with tasks that he has to correct.

Act 1, scene 2

While Minos is sleeping, the ghost of Euclid enters and asks Minos what is most important in a textbook on geometry. Minos is surprised, but replies that clear definitions and logical conclusions are more important than a complete representation. Euclid asks Minos for comments on the points on which a textbook could represent geometry better than its elements and names a series of textbooks that Minos should compare with his own work. So that he doesn’t have to read the books himself, Euclid’s Geist offers to fetch the ghost of a German professor who has read all these books and defends them on behalf of their authors, called “Herr Nobody”.

The main points that Minos should pay attention to are:

  • The order of the sentences should be maintained if possible.
  • Construction tasks and sentences should not be separated.
  • Definition and treatment of straight lines and angles
  • Treatment of the parallel axiom

Before Minos examines the individual works, he and Euclid discuss these and a few other points in general. Euclid also introduces his elements , first the propositions that can be proved without the postulate of parallels, then those in which this axiom is applied. He adds a number of equivalent formulations of the axiom suggested by other authors. Euclid proves part of the propositions presented, the rest of the proofs can be found in an appendix.

Act 2, scene 1

Mr. Nobody appears and immediately begins to discuss the works of Euclid's rivals.

Act 2, scene 2

Mr. Nobody begins with Legendres Éléments de Géométrie . He presents the definitions and evidence that are very different from Euclid's approach. Minos concludes that although the work is elegant for advanced readers, it is unsuitable for beginners.

Act 2, scene 3

The next work is Cooley's The Elements of Geometry, simplified and explained . Minos notes that a much stronger definition of parallel is used in the evidence than was defined and rejects the work.

Act 2, scene 4

It follows Cuthbertson's Euclidian Geometry . This work attempts to fill in some of the gaps in Euclid by providing evidence for some explicit or implicit axioms. But Minos himself finds gaps or axioms in all of these proofs that are no more plausible than the statements that they prove. He concludes that the work is not very different from Euclid's Elements and has a clear style but is not superior to the Elements .

Act 2, scene 5

The next work (only from the second edition) is Elementary Geometry: Congruent Figures by Olaus Henrici . Mr. Nobody has much trouble finding the definitions Minos wants to hear. Minos finds a number of contradictions in the various definitions of curve, straight line and angle. Over Henrici's attempt to justify the parallel postulate, Minos and Herr Nobody get into a discussion about non-Euclidean geometries. Finally, Minos cites a number of passages in which Henrici argues imprecisely. Finally, he elaborately explains that he considers Henrici's book to be absolutely unsuitable.

Act 2, scene 6

This scene is divided into three sections where Minos and Mr Nobody discuss the following books that use similar approaches:

As a result, Minos again rejects all books as unsuitable textbooks.

Act 3, scene 1

In this scene, too, Minos and Mr Nobody discuss several works that all have Euclid's elements as models:

Minos thinks the first two works are good, if not superior to the elements , but rejects the other three because of different problems.

Act 3, scene 2

Then discuss Minos and Mr. Nobody the curriculum of the Association for the Improvement of Geometrical Teaching and Wilson's textbook that follows this curriculum. Mr Nobody claims to be a member of this association because nobody was admitted the day before. In this role he takes on the name Nostradamus , Nero (with a plan to light and warm Rome), Guy Fawkes (to improve the position of Members of Parliament), Marie-Madeleine de Brinvilliers (with digestive alternatives) and F. Gustrell (who felled Shakespeare's mulberry tree to improve literary taste). Finally Newton's dog Diamond appears with a half-burned manuscript.

In the discussion with Mr. Nobody (or Nostradamus), Minos initially rejects the curriculum, then also Wilson's Elementary Geometry, following the Syllabus prepared by the Geometrical Association .

Lord Nobody and the other spirits go away.

Act 4

Euclid reappears, with him Archimedes , Pythagoras , Aristotle , Plato and others. Minos reports that he has not found a textbook better than Euclid's. However, he discusses with him some weaknesses of the elements , such as the fact that the use of a collapsing compass is cumbersome, whereas Euclid replies that - after the equivalence has been shown - an ordinary compass can also be used. Euclid also has good replies to further objections. Euclid urges Minos to keep his work as a textbook and disappears with the other spirits.

Work history

Euclid and his Modern Rivals was first published in March 1879. Essays by Augustus De Morgan and Isaac Todhunter are reprinted in two appendices . In 1885 a supplement was published that dealt with Henrici's book and contained eight reviews of the first edition. In the same year a second revised edition appeared, which also took over the new scene for Henrici's book from the supplement. In addition to these editions during Dodgson's lifetime, there are a number of modern new editions, such as the one from 1973 with a foreword by HSM Coxeter .

background

In Britain, Euclid's Elements were the standard textbook in schools and universities in Dodgson's day. It served not only as a textbook for geometry, but also for logical reasoning and - if it was used in the original - as a Greek exercise text. The first reform proposals came up in the 1860s, for example from Thomas Hewitt Key or Thomas Archer Hirst . Among the mathematicians who defended the elements as a textbook were, besides Dodgson, above all De Morgan and Todhunter. Dodgson bases his criticism on 25 years of experience teaching geometry.

The Association for the Improvement of Geometric Teaching, the forerunner of today's British Mathematical Association , was founded in 1871 to bundle the measures to improve geometry teaching . The British Association also set up a similar committee. However, the first curriculum to compete with Euclid from 1877 met with widespread rejection among universities. Despite revisions, in 1890 only two British universities ( London and Edinburgh ) allowed any textbooks on geometry, while all other universities continued to insist that the axioms and order of the theorems should largely follow Euclid's elements . It was not until the beginning of the 20th century that the elements were slowly abandoned as a binding textbook.

Dodgson himself published a number of books on the elements that follow his own suggestions, that is, are only slightly modified editions of Euclid. With Curiosa Mathematica, Part I: A New Theory of Parallels , he also presented his own approach in 1888 to deal with parallels, which is very different from Euclid. Instead of the usual axiom, it assumes that the area of ​​an equilateral triangle inscribed in a circle (a hexagon in the first editions) is larger than the area of ​​each of the three segments of the circle .

reception

Contemporary reviews were largely in agreement on two points of criticism: On the one hand, Dodgson used a symbolic language that was difficult to understand in the first edition (he did not use it for the second edition). On the other hand, the criticism concerns the one-sidedness of the work. Only arguments for the elements are given, Mr. Nobody acts as a front man . It was also criticized that Dodgson criticized long-revised editions, although the criticisms raised had already been resolved in the current editions. Some of the objections are just linguistic quibbles. Even so, most reviewers recommended the book as worth reading because it presented interesting content in a humorous way.

Stuart Dodgson Collingwood reports that Euclid and his Modern Rivals itself was also used as a textbook.

Today's reviewers see the book as a valuable source on the history of mathematics didactics on the one hand, and on the other hand it provides an interesting alternative view of the author, who is best known for books like Alice in Wonderland .

Nupedia logo

In addition to the content, the work was also received artistically: The first logo from Wikipedia, which Bjørn Smestad had designed for the previous project Nupedia and was used until the end of 2001, shows an excerpt from the foreword to Euclid and his Modern Rivals in a fisheye projection .

expenditure

  • Euclid and his Modern Rivals. Macmillan, 1879.
  • Supplement to Euclid and his Modern Rivals. Macmillan, 1885. ( Full text in Google Book Search - USA )
  • Euclid and his Modern Rivals. Macmillan, 1885. ( online )
  • Euclid and his Modern Rivals. Dover Phoenix Editions, 2004. ISBN 0-486-49566-3 .

Individual evidence

  1. ↑ a b Amirouche Moktefi: Review: Euclid and his modern rivals, by Charles L. Dodgson. In: The Mathematical Gazette. Vol. 89, No. 516 (November 2005), pp. 556-558. ( JSTOR 3621985 )
  2. ^ WH Brock: Geometry and the Universities: Euclid and his Modern Rivals 1860-1901. In: Journal of the History of Education Society. 1975, Vol. 4, No. 2. pp. 21-35. ( doi: 10.1080 / 0046760750040203 )
  3. ↑ Reviews in English Mechanic and World of Science of May 2, 1879, Saturday Review of May 10, 1879, Journal of Science of July 1879, Nature of July 10, 1879, The Examiner of October 25, 1879, reprinted in Supplement to Euclid and his Modern Rivals. Macmillan, 1885.
  4. ^ Review in Nature . July 1879, Volume 20, Number 506, pp. 240–241 ( online, PDF )
  5. ^ Stuart Dodgson Collingwood: The Life and Letters of Lewis Carroll. 1899. Chapter V. ( The Life and Letters of Lewis Carroll in Project Gutenberg ( currently not available to users from Germany ) )
  6. ↑ Logo history on meta.wikimedia.org, accessed on December 22, 2014.