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Published by Good Press, 2020

goodpress@okpublishing.info

EAN 4064066066925

London

MACMILLAN AND CO.

Oxford

PREFACE TO SECOND EDITION.

PREFACE TO FIRST EDITION.

ARGUMENT OF DRAMA.

ACT I.

Preliminaries to examination of Modern Rivals.

ACT II.

Manuals which reject Euclid's treatment of Parallels.

ACT III.

Manuals which adopt Euclid's treatment of Parallels.

ACT IV.

Manual of Euclid.

APPENDICES.

II.

III.

IV.

ACT I.

Scene I.

ACT I.

Scene II.

§ 1. A priori reasons for retaining Euclid's Manual .

§ 2. Method of procedure in examining Modern Rivals .

§ 3. The combination, or separation, of Problems and Theorems .

§ 4. Syllabus of propositions relating to Pairs of Lines .

§ 5. Playfair's Axiom .

§ 6. The Principle of Superposition .

§ 7. The omission of diagonals in Euc. II. .

ACT II.

Manuals which reject Euclid's treatment of Parallels .

Scene I.

ACT II.

Scene II.

Treatment of Parallels by methods involving infinite series .

ACT II.

Scene III.

Treatment of Parallels by angles made with transversals .

ACT II.

Scene IV.

Treatment of Parallels by equidistances .

ACT II.

Scene V.

Treatment of Parallels by revolving Lines .

ACT II.

Scene VI.

Treatment of Parallels by direction .

ACT II.

Scene VI.

ACT III.

Scene I.

ACT III.

Scene I.

ACT III.

Scene I.

ACT III.

Scene I.

ACT III.

Scene I.

ACT III.

Scene I.

ACT III.

Scene II.

ACT III.

Scene II.

ACT IV.

§ 1. Treatment of Pairs of Lines .

§ 2. Euclid's Constructions.

§ 3. Euclid's Demonstrations .

§ 4. Euclid's Style .

§ 5. Euclid's treatment of Lines and Angles .

§ 6. Omissions, alterations, and additions, suggested by Modern Rivals .

§ 7. The summing-up .

ERRATUM.

'All for your delight
We are not here. That you should here repent you
The actors are at hand; and, by their show,
You shall know all, that you are like to know.'

SECOND EDITION

London

MACMILLAN AND CO.

Table of Contents

1885

Oxford

Table of Contents

PRINTED BY HORACE HART, PRINTER TO THE UNIVERSITY

Dedicated

to

the memory

of

Euclid

Table of Contents

Content(not individually listed)

Frontispiece

Preface

Argument of Drama

Appendices

ACT I.

Scene I.

Scene II.

ACT II.

Scene I.

Scene II.

Scene III.

Scene IV.

Scene V.

Scene VI. § 1.

Scene VI. § 2.

Scene VI. § 3.

ACT III.

Scene I. § 1.

Scene I. § 2.

Scene I. § 3.

Scene I. § 4.

Scene I. § 5.

Scene I. § 6.

Scene II. § 1.

Scene II. § 2.

ACT IV.

Appendix I.

Appendix II.

Appendix III.

EUCLID, BOOK I.

Arranged in Logical Sequence.

Euclid and His Modern Rivals Frontispiece.png

PREFACE TO SECOND EDITION.

Table of Contents

The only new features, worth mentioning, in the second edition, are the substitution of words for the symbols introduced in the first edition, and one additional review—of Mr. Henrici, to whom, if it should appear to him that I have at all exceeded the limits of fair criticism, I beg to tender my sincerest apologies.

C. L. D.

Ch. Ch. 1885.

PREFACE TO FIRST EDITION.

Table of Contents

'ridentem dicere verum
Quid vetat?'

The object of this little book is to furnish evidence, first, that it is essential, for the purpose of teaching or examining in elementary Geometry, to employ one textbook only; secondly, that there are strong a priori reasons for retaining, in all its main features, and specially in its sequence and numbering of Propositions and in its treatment of Parallels, the Manual of Euclid; and thirdly, that no sufficient reasons have yet been shown for abandoning it in favour of any one of the modern Manuals which have been offered as substitutes.

It is presented in a dramatic form, partly because it seemed a better way of exhibiting in alternation the arguments on the two sides of the question; partly that I might feel myself at liberty to treat it in a rather lighter style than would have suited an essay, and thus to make it a little less tedious and a little more acceptable to unscientific readers.

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 breathing-space, and not on any occasion where it could endanger the continuity of a line of argument.

Pitying friends have warned me of the fate upon which I am rushing: they have predicted that, in thus abandoning the dignity of a scientific writer, I shall alienate the sympathies of all true scientific readers, who will regard the book as a mere jeu d'esprit, and will not trouble themselves to look for any serious argument in it. But it must be borne in mind that, if there is a Scylla before me, there is also a Charybdis—and that, in my fear of being read as a jest, I may incur the darker destiny of not being read at all.

In furtherance of the great cause which I have at heart—the vindication of Euclid's masterpiece—I am content to run some risk; thinking it far better that the purchaser of this little book should read it, though it be with a smile, than that, with the deepest conviction of its seriousness of purpose, he should leave it unopened on the shelf.

To all the authors, who are here reviewed, I beg to tender my sincerest apologies, if I shall be found to have transgressed, in any instance, the limits of fair criticism, To Mr. Wilson especially such apology is due—partly because I have criticised his book at great length and with no sparing hand—partly because it may well be deemed an impertinence in one, whose line of study has been chiefly in the lower branches of Mathematics, to dare to pronounce any opinion at all on the work of a Senior Wrangler. Nor should I thus dare, if it entailed my following him up 'yonder mountain height' which he has scaled, but which I can only gaze at from a distance: it is only when he ceases 'to move so near the heavens,' and comes down into the lower regions of Elementary Geometry, which I have been teaching for nearly five-and-twenty years, that I feel sufficiently familiar with the matter in hand to venture to speak.

Let me take this opportunity of expressing my gratitude, first to Mr. Todbunter, for allowing me to quote ad libitum from the very interesting Essay on Elementary Geometry, which is included in his volume entitled 'The Conflict of Studies, and other Essays on subjects connected with Education,' and also to reproduce some of the beautiful diagrams from his edition of Euclid; secondly, to the Editor of the Athenæum, for giving me a similar permission with regard to a review of Mr. Wilson's Geometry, written by the late Professor De Morgan, which appeared in that journal, July 18, 1868.

C. L. D.

Ch. Ch. 1879.

ARGUMENT OF DRAMA.

ACT I.

Table of Contents

Preliminaries to examination of Modern Rivals.

Table of Contents

Scene I.

[Minos and Rhadamanthus.]

PAGE

Consequences of allowing the use of various Manuals of Geometry: that we must accept

(1)

'Circular' arguments

… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . …

(2)

Illogical do

Example from Cooley

2„

Example„ from„ Wilson

Scene II.

[Minos and Euclid.]

§ I. 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 subject-matter of Euc. I, II.

One fixed logical sequence essential

8„

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

11„

§ 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, &c.

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 others;

(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:—

23„

(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 be 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' mpsonAxiom„

(4)

Playfair's psonAxiom„

(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 12th Axiom axiomatic?

Need of test for meeting of finite Lines

Considerations which make Euclid's Axiom more axiomatic

Euclid's Axiom deducible from Playfair's

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.

46„

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

ACT II.

Table of Contents

[Minos and Niemand.]

Manuals which reject Euclid's treatment of Parallels.

Table of Contents

Scene I.

Introductory

Scene II.

Treatment of Parallels by methods involving infinite series.

Legendre.

Treatment of Line

Treatment of„ Angle

Treatment of„ Parallels

57„

Test for meeting of finite Lines

Manual unsuited for beginners

Scene III.

Treatment of Parallels by angles made with transversals.

Cooley.

Style of Preface

Treatment of Parallels

Utter collapse of Manual

Scene 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

66„

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

Scene V.

Treatment of Parallels by revolving lines.

Henrici.

Treatment of Line

Treatment of„ Angle

Treatment of„ Parallels

Attempted proof of Playfair's Axiom discussed

76„

Attempted„ proof of„ Playfair's Axiom„ rejected

General survey of book:—

83„

Enormous amount of new matter

Euclid and His Modern Rivals

Table of Contents

London

MACMILLAN AND CO.

Oxford

Dedicated to the memory ofEuclid

PREFACE TO SECOND EDITION.

PREFACE TO FIRST EDITION.

ARGUMENT OF DRAMA.

ACT I.

Preliminaries to examination of Modern Rivals.

ACT II.

Manuals which reject Euclid's treatment of Parallels.

Dedicated

to

the memory

of

Euclid