Euclid and His Modern Rivals (1879) es una comedia en cuatro actos –el trabajo más famoso de Lewis Carroll en geometría–, una defensa de la geometría de Euclides frente a teorías matemáticas modernas.
Se trata del análisis meticuloso de varios libros destinados a la enseñanza de la geometría elemental en las escuelas de aquella época. En estos manuales –y respecto al libro de Los Elementos de Euclides– se modifica a veces algún axioma o una definición, en otros se cambia el orden de los teoremas, en ocasiones se abordan las demostraciones de manera diferente, en algunos se modifica el tratamiento de la teoría de las paralelas...
Para realizar este análisis –que podría resultar tedioso de otra manera, como afirma el propio autor en la introducción– Carroll recurre a Minos y Radamantis –dos de los tres jueces de Hades–, árbitros estrictos que dialogan con dos fantasmas: el de Euclides –modesto, y aunque convencido de la calidad de su obra, no tiene inconveniente en que se analice– y el del profesor alemán Herr Niemand, portavoz de los 13 autores cuyos libros se examinan. Uno a uno, escena a escena, estos rivales modernos verán como sus textos se critican y se rechazan frente al manual de Euclides.
Lewis Carroll se sirve del humor y de los juegos de palabras para invalidar a los rivales de Euclides.
El texto finaliza con el discurso de despedida de Euclides, tras el cual los fantasmas desaparecen y Minos se va a dormir:
‘The cock doth craw, the day doth daw’, and all respectable ghosts ought to be going home. Let me carry with me the hope that I have convinced you of the importance, if not the necessity, of retaining my order and numbering, and my method of treating straight Lines, angles, right angles, and (most especially) Parallels. Leave me these untouched, and I shall look on with great contentment while other changes are made while my proofs are abridged and improved, while alternative proofs are appended to mine and while new Problems and Theorems are interpolated.
In all these matters my Manual is capable of almost unlimited improvement.
Como anécdota, el primer logotipo de Wikipedia[1] –conocido como Wiki logo Nupedia– se diseñó en 2001, superponiendo una frase de Lewis Carroll sobre un círculo, usando el efecto de ojo de pez para simular una esfera. La frase es una cita en inglés tomada del prefacio de Euclid and his Modern Rivals, que dice:
In one respect this book is an experiment, and may chance to prove a failure: I mean that I have not thought it necessary to maintain throughout the gravity of style which scientific writers usually affect, and which has somehow come to be regarded as an ‘inseparable accident’ of scientific teaching. I never could quite see the reasonableness of this immemorial law: subjects there are, no doubt, which are in their essence too serious to admit of any lightness of treatment – but I cannot recognise Geometry as one of them. Nevertheless it will, I trust, be found that I have permitted myself a glimpse of the comic side of things only at fitting seasons, when the tired reader might well crave a moment’s breathingspace, and not on any occasion where it could endanger the continuity of the line of argument.
Para concluir, incluyo la exhaustiva descripción –actos y escenas con detalle de lo que sucede en cada una de ellas– del argumento de la obra tal y como se incluye en el texto de Lewis Carroll.
ACTO I: Preliminaries to examination of Modern Rivals
ESCENA I [Minos y Radamantis] Consequences of allowing the use of various Manuals of Geometry : that we must accept
(1) ‘Circular’arguments (2) Illogical do.
Example from Cooley Example from Wilson
ESCENA II. [Minos y Euclides]
§ 1: A priori reasons for retaining Euclid's Manual We require, in a Manual, a selection rather than a complete repertory of Geometrical truths Discussion limited to subjectmatter of Euc. I, II One fixed logical sequence essential One system of numbering desirable A priori claims of Euclid's sequence and numeration to be retained New theorems might be interpolated without change of numeration
§ 2: Method of procedure in examining Modern Rivals Proposed changes which, even if proved to be essential, would not necessitate the abandonment of Euclid's Manual:
(1) Propositions to be omitted; (2) Propositions to be replaced by new proofs; (3) New propositions to be added.
Proposed changes which, if proved to be essential, would necessitate such abandonment:
(1) Separation of Problems and Theorems; (2) Different treatment of Parallels.
Other subjects of enquiry:
(3) Superposition; (4) Use of diagonals in Euc. II; (5) Treatment of Lines; (6) Treatment of Angles; (7) Euclid's propositions omitted; (8) Euclid's propositions newly treated; (9) New propositions ; (10) Style, &e.
List of authors to be examined, viz.: Legendre, Cooley, Cuthbertson, Henrici, Wilson, Pierce, Willock, Chauvenet, Loomis, Morell, Reynolds, Wright, Syllabus of Association for Improvement of Geometrical Teaching, Wilson's ‘Syllabus’Manual.
§ 3: The combination, or separation, of Problems and Theorems Reasons assigned for separation Reasons for combination:
(1) Problems are also Theorems; (2) Separation would necessitate a new numeration, (3) and hypothetical constructions.
§ 4: Syllabus of propositions relating to Pairs of Lines Three classes of Pairs of Lines:
(1) Having two common points; (2) Having a common point and a separate point; (3) Having no common point.
Four kinds of ‘properties’;
(1) common or separate points; (2) equality, or otherwise, of angles made with transversals; (3) equidistance, or otherwise, of points on the one from the other; (4) direction.
Conventions as to language Propositions divisible into two classes:
(1) Deducible from undisputed Axioms; (2) Deducible from disputable Axioms.
Three classes of Pairs of Lines:
(1) Coincidental; (2) Intersectional; (3) Separational.
Subjects and predicates of Propositions concerning these three classes: Coincidental Intersectional Separational TABLE I. Containing twenty Propositions, of which some are undisputed Axioms, and the rest real and valid Theorems, deducible from undisputed Axioms Subjects and predicates of other propositions concerning Separational Lines TABLE II. Containing eighteen Propositions, of which no one is an undisputed Axiom, but all are real and valid Theorems, which, though not deducible from undisputed Axioms, are such that, if any one de admitted as an Axiom, the rest can be proved TABLE III. Containing five Propositions, taken from Table II, which have been proposed as Axioms:
(1) Euclid's Axiom; (2) T. Simpson's Axiom; (3) Clavius' Axiom; (4) Playfair's Axiom; (5) R. Simpson’s Axiom.
It will be shown (in Appendix III) that any Theorem of Table II is sufficient logical basis for all the rest
§ 5: Playfair's Axiom Is Euclid's 12 Axiom axiomatic? Need of test for meeting of finite Lines Euclid's and Playfair's Axioms deducible, each from the other Reasons for preferring Euclid's Axiom:
(1) Playfair's does not show which way the Lines will meet; (2) Playfair's asserts more than Euclid's, the additional matter being superfluous
Objection to Euclid's Axiom (that it is the converse of I. 17) untenable
§ 6: Principle of Superposition Used by Moderns in Euc. I. 5 Used by Moderns in Euc. I. 24
§ 7:.Omission of Diagonals in Euc. II Proposal tested by comparing Euc. II. 4, with Mr.Wilson's version of it
ACTO II: [Minos y Niemand] Manuals which reject Euclid's treatment of Parallels
ESCENA I: Introductory
ESCENA II: Treatment of Parallels by methods involving infinite series LEGENDRE Treatment of Line Treatment of Angle Treatment of Parallels Test for meeting of finite Lines Manual unsuited for beginners
ESCENA III: Treatment of Parallels ly angles made with transversals COOLEY Style of Preface Treatment of Parallels Utter collapse of Manual
ESCENA IV: Treatment of Parallels by equidistances CUTHBERTSON Treatment of Line Attempted proof of Euclid's (tacitly assumed) Axiom, that two Lines cannot have a common segment Treatment of Angle Treatment of Parallels Assumption of R. Simpson’s Axiom Euclid's 12th Axiom replaced by a Definition, two Axioms, and five Theorems Test for meeting of two finite Lines Manual a modified Euclid
ESCENA V: Treatment of Parallels by revolving Lines HENRICH Treatment of Line Treatment of Angle Treatment of Parallels Attempted proof of Playfair’s Axiom discussed Attempted proof of Playfair’s Axiom rejected General survey of book:
Enormous amount of new matter Two ‘nonsequitur’ An absurdity proved à la Henrici Motion ‘per saltum’ denied A strange hypothesis A new kind of ‘open question’ Another ‘nonsequitur’ An awkard corner Theorems on Symmetry Summary of faults Euclid I, 18, 19, contrasted with Henrici A final titbit
Manual rejected
ESCENA VI: Treatment of Parallels by direction
§1: WILSON Introductory Treatment of Line Treatment of Angle Extension of limit of ‘angle’ to sum of four right angles ‘Straight’ angles Meaning of ‘direction’ ‘Opposite’ directions ‘Same’ and ‘different’ directions Axiom ‘different Lines may have the same direction’ discussed Property ‘same direction’, when asserted of different Lines, can neither be defined, nor constructed, nor tested ‘Separational directions’not identical with ‘identical directions ' Virtual assumption of ‘separational Lines are real’ (which Euclid proves in I. 27), as Axiom ‘different Lines may have the different direction’ discussed Axiom ‘different Lines may have the same direction’ rejected, and Axiom ‘different Lines may have the different direction’ granted with limitations Axiom ‘different which meet one another have different directions’ granted Axiom ‘Lines with different directions would meet’ discussed and rejected Diagram of ‘same’ and ‘different’ directions condemned ‘Different but with the same direction’ accepted as (ideal) definition by Pair of Lines ‘Parallel’ as used by Wilson, to be replaced by term ‘sepcodal’ Definition discussed Theorem ‘sepcodal’ related Lines do not meet accepted Theorem ‘Lines sepcodal related to a third, are so to each other’ discussed, and condemned as a ‘Petitio Principii’ Axiom ‘Angle may be transferred, preserving directions of sides’ discussed If angle be variable, it involves fallacy ‘Adicto secmdun Quid ad dictum Simpliciter’’ If it be constant., the resulting Theorem (virtually identical with the Axiom) involves fallacy ‘Petitio Principii’ If angle be constant, the Axiom involves two assumptions: viz. that
(1) there can be a Pair of different Lines that make equal angles with any transversal (2) Lines, which make equal angles with a certain transversal, do so with any transversal
Axiom rejected Ideas of ‘direction’ discussed Theory of ‘direction’ unsuited for teaching Test for meeting of finite Lines discussed:
it virtually involves Euclid's Axiom or if not, it causes hiatus in proofs
List of Euclid's propositions which are omitted General survey of book:
A false Corollary A plethora of negatives A superfluous datum Cumbrous proof of Euc. I. 24 An unintelligible Corollary A unique ‘Theorem of equality’ A bold assumption Two cases of ‘Petitio Principii’ A problem 3½ pages long > A fifth case of ‘Petitio Principii’ A sixth
Summingup, and rejection of Manual.
§2: PIERCE Treatment of Line Introduction of Infinitesimals Treatment of Parallels Angle viewed as ‘difference of direction’ Assumption of Axiom ‘different Lines may have same direction’ List of Euclid's Theorems which are omitted Manual not adapted for beginners
§3: WILLOCK Treatment of Parallels Virtual assumption of Axiom ‘different Lines may have the same direction’ ' Assumption of Axiom ‘separational Lines have the same direction’ General survey of book:
Difficulties introduced too soon Omission of ‘coincidental’ Lines ‘Principle of double conversion’ discussed, and condemned as illogical Mysterious passage about ‘incommensurables’
Manual rejected
ACTO III: Manuals which adopt Euclid's treatment of Parallels
ESCENA I
§1: Introductory
§ 2: CHAUVENET General survey
§ 3: LOOMIS General survey
§ 4: MORELL Treatment of Line Treatment of Angle Treatment of Parallels General survey:
‘Direct’, ‘reciprocal’, and ‘contrary’ Theorems Sentient points A false assertion A speaking radius Ratios and common measures Derivation of ‘homologous’ Mensuration of areas 146 A logical fiasco
Manual rejected
§ 5: REYNOLDS General survey List of Euclid's Theorems omitted
§ 6: WRIGHT Quotations from preface General survey
Specimen of verbose obscurity
ESCENA II
§ 1: Syllabus of the Association for the Improvement of Geometrical Teaching Introduction of Nostradamus, a member of the Association Treatment of Line Treatment of Angle Treatment of Parallels Test for meeting of finite Lines Rearrangement of Euclid's Theorems General survey:
A ’Theorem’ is a ‘statement of a Theorem’ Rule of Conversion Miscellaneous inaccuracies
Summingup
§ 2: Wilson's ‘Syllabus’Manual Introductory A Theorem is a ‘statement of a Theorem’ Rule of Conversion Every Theorem a ‘means of measuring’ ‘Straight angles’ Miscellaneous inaccuracies The Manual's one great merit No test for meeting of finite Lines Propositions discussed in detail:
An important omission An illogical conversion ‘Un enfant terrible’
Summary of results:
Of 73 propositions of Euclid, this Manual has 14 omitted; 43 done as in Euclid; 10 done by new but objectionable methods, viz.
1 illogical; 1 ‘hypothetical construction’; 2 needlessly using ‘superposition’; 2 algebraical; 4 omitting the diagonals of Euc. II.; 6 done by new and admissible methods.
No reason for abandoning Euclid's sequence and numeration Nor for regarding this Manual as anything but a revised Euclid Summingup
ACTO IV: [Minos y Euclides] Manual de Euclides
§ 1: Treatment of Pairs of Lines Modern treatment of Parallels Playfair's Axiom Test for meeting of finite Lines
§ 2: Euclid’s constructions ‘Arbitrary restrictions’ ‘Exclusion of hypothetical constructions’
§ 3: Euclid's demonstrations PAGE ‘Invariably syllogistic form’ ‘Too great length of demonstration’ ‘Too great brevity of demonstration’ ‘Constant reference to axioms’
§ 4: Euclid's style Artificiality, unsuggestiveness, and want of simplicity
§ 5: Euclid's treatment of Lines and Angles Treatment of' Line Treatment of Angle :
‘declination from’ accepted must be less than sum of two right angles ‘multiple angles’ in VI. 33 proof for Ax. 10 accepted
§ 6: Omissions, alterations, and additions, suggested ly Modern Rivals Omission of I. 7 suggested Reasons for retaining it:
needed to prove I. 8 not included in new I. 8 proves rigidity of Triangle . I. 7, 8 analogous to III. 23, 24 bears on practical science
Omission of II. 8 suggested
Reason for retaining it, its use in Geometrical Conic Sections
Alterations suggested:
New proofs for I. 5 ‘hypothetical construction‘ superposition treating sides as ‘obliques’ treating sides as radii of a Circle Inversion of order of I. 8, 24; rejected Inversion of order of I 18, 19, 20; do. Fuller proof of I. 24 ; accepted Algebraical proofs of II. ; rejected
Additions suggested:
New Axiom; accepted Two new Theorems ; do.
§7: The summingup Euclid's farewell speech
APPENDICES
I. Extract from Mr. Todhunter's essay on ‘Elementary Geometry’ included in ‘The Conflict of Studies, &c’
II. Extract from Mr. De Morgan's review of Mr. Wilson's Geometry, in the ‘Athenoeum’ for July 1 8, 1868
III. Proof that, if any one proposition of Table II be granted as an Axiom, the rest can be deduced from it
IV. List of propositions of Euc. I, II, with references to their occurrence in the manuals of his Modern Rivals:
§1. References to Legendre, Cuthbertson, Henrici, Wilson, Pierce and Willock §2. References to the other Modern Rivals
Nota:
[1] En 2000 Jimbo Wales creó Nupedia, un proyecto de enciclopedia libre basado en un ambicioso proceso de revisión por pares. Debido al lento avance del proyecto, en 2001 se creó un wiki –UseModWiki–vinculado a Nupedia cuya finalidad inicial era agilizar la creación de artículos de forma paralela, antes de que éstos pasaran al sistema de revisión por expertos. El éxito de aquel proyecto paralelo –Wikipedia– acabó eclipsando a Nupedia, que dejó de funcionar en 2003.
