Euclid and His Modern Rivals/Argument of Drama
ARGUMENT OF DRAMA.
ACT I.
Preliminaries to examination of Modern Rivals.
Scene I.
[Minos and Rhadamanthus.]
PAGE
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Consequences of allowing the use of various Manuals of Geometry: that we must accept
|
| (1) |
'Circular' arguments
....................................................................................................................................................................................................................................................
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1 |
| (2) |
Illogical do
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|
2 |
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Example from Cooley
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2„ |
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Example„ from„ Wilson
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|
4 |
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 |
6 |
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Discussion limited to subject-matter of Euc. I, II.
....................................................................................................................................................................................................................................................
|
8 |
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One fixed logical sequence essential
....................................................................................................................................................................................................................................................
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8„ |
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One system of numbering desirable
....................................................................................................................................................................................................................................................
|
10 |
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A priori claims of Euclid's sequence and numeration to be retained
....................................................................................................................................................................................................................................................
|
11 |
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New Theorems might be interpolated without change of numeration
....................................................................................................................................................................................................................................................
|
11„ |
§ 2. Method of procedure in examining Modern Rivals.
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Proposed changes which, even if proved to be essential, would not necessitate the abandonment of Euclid's Manual:—
....................................................................................................................................................................................................................................................
|
13 |
| (1) |
Propositions to be omitted;
|
| (2) |
Propositions„ to be replaced by new proofs;
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| (3) |
New Propositions to be added.
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Proposed changes which, if proved to be essential, would necessitate such abandonment:— |
15 |
| (1) |
Separation of Problems and Theorems;
|
| (2) |
Different treatment of Parallels.
|
|
Other subjects of enquiry:—
....................................................................................................................................................................................................................................................
|
15 |
| (3) |
Superposition;
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| (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.:—
....................................................................................................................................................................................................................................................
|
16 |
|
Legendre, Cooley, Cuthbertson, Henrici, Wilson, Pierce, Willock, Chauvenet, Loomis, Morell, Reynolds, Wright, Syllabus of Association for Improvement of Geometrical Teaching, Wilson's 'Syllabus'-Manual.
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§ 3. The combination, or separation of Problems and Theorems.
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Reasons assigned for separation
....................................................................................................................................................................................................................................................
|
18 |
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Reasons for combination:—
....................................................................................................................................................................................................................................................
|
19 |
| (1) |
Problems are also Theorems;
|
| (2) |
Separation would necessitate a new numeration,
|
| (3) |
and hypothetical constructions.
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§ 4. Syllabus of propositions relating to Pairs of Lines.
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Three classes of Pairs of Lines:—
....................................................................................................................................................................................................................................................
|
20 |
| (1) |
Having two common points;
|
| (2) |
Having a common point and a separate point;
|
| (3) |
Having„ no common point.
|
|
Four kinds of 'properties';
....................................................................................................................................................................................................................................................
|
21 |
| (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
....................................................................................................................................................................................................................................................
|
22 |
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Propositions divisible into two classes:—
....................................................................................................................................................................................................................................................
|
23 |
| (1) |
Deducible from undisputed Axioms;
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| (2) |
Deducible from„ disputable Axioms„
|
|
Three classes of Pairs of Lines:—
....................................................................................................................................................................................................................................................
|
23„ |
| (1) |
Coincidental;
|
| (2) |
Intersectional;
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| (3) |
Separational.
|
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Subjects and predicates of Propositions concerning these three classes:—
|
|
Coincidental
....................................................................................................................................................................................................................................................
|
24 |
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Intersectional
....................................................................................................................................................................................................................................................
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26 |
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Separational
....................................................................................................................................................................................................................................................
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27 |
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Table I. Containing twenty Propositions, of which some are undisputed Axioms, and the rest real and valid Theorems, deducible from undisputed Axioms
....................................................................................................................................................................................................................................................
|
28 |
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Subjects and predicates of other propositions concerning Separational Lines
....................................................................................................................................................................................................................................................
|
33 |
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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
....................................................................................................................................................................................................................................................
|
34 |
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Table III. Containing five Propositions, taken from Table II, which have been proposed as Axioms
....................................................................................................................................................................................................................................................
|
37 |
| (1) |
Euclid's Axiom;
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| (2) |
T. Simpson's Axiom;
|
| (3) |
Clavius' Axiom„
|
| (4) |
Playfair's Axiom„
|
| (5) |
R. Simpson's Axiom.
|
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It will be shown (in Appendix III) that any Theorem of Table II is sufficient logical basis for all the rest
....................................................................................................................................................................................................................................................
|
38 |
§ 5. Playfair's Axiom.
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Is Euclid's 12th Axiom axiomatic?
....................................................................................................................................................................................................................................................
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40 |
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Need of test for meeting of finite Lines
....................................................................................................................................................................................................................................................
|
41 |
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Considerations which make Euclid's Axiom more axiomatic
....................................................................................................................................................................................................................................................
|
42 |
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Euclid's Axiom deducible from Playfair's
....................................................................................................................................................................................................................................................
|
45 |
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Reasons for preferring Euclid's Axiom:—
|
| (1) |
Playfair's does not show which way the Lines will meet;
....................................................................................................................................................................................................................................................
|
46 |
| (2) |
Playfair's asserts more than Euclid's, the additional matter being superfluous.
....................................................................................................................................................................................................................................................
|
46„ |
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Objection to Euclid's Axiom (that it is the converse of I. 17) untenable
....................................................................................................................................................................................................................................................
|
47 |
§ 6. Principle of Superposition.
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Used by Moderns in Euc. I. 5
....................................................................................................................................................................................................................................................
|
48 |
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Used by Moderns„ in Euc.„ I. 24
....................................................................................................................................................................................................................................................
|
49 |
§ 7. Omission of Diagonals in Euc. II.
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Proposal tested by comparing Euc. II. 4, with Mr. Wilson's version of it
....................................................................................................................................................................................................................................................
|
50 |
ACT II.
[Minos and Niemand.]
Manuals which reject Euclid's treatment of Parallels.
Scene I.
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Introductory
....................................................................................................................................................................................................................................................
|
54 |
Scene II.
Treatment of Parallels by methods involving infinite series.
Legendre.
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Treatment of Line
....................................................................................................................................................................................................................................................
|
55 |
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Treatment of„ Angle
....................................................................................................................................................................................................................................................
|
57 |
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Treatment of„ Parallels
....................................................................................................................................................................................................................................................
|
57„ |
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Test for meeting of finite Lines
....................................................................................................................................................................................................................................................
|
58 |
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Manual unsuited for beginners
....................................................................................................................................................................................................................................................
|
59 |
Scene III.
Treatment of Parallels by angles made with transversals.
Cooley.
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Style of Preface
....................................................................................................................................................................................................................................................
|
60 |
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Treatment of Parallels
....................................................................................................................................................................................................................................................
|
62 |
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Utter collapse of Manual
....................................................................................................................................................................................................................................................
|
63 |
Scene IV.
Treatment of Parallels by equidistances.
Cuthbertson.
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Treatment of Line
....................................................................................................................................................................................................................................................
|
64 |
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Attempted proof of Euclid's (tacitly assumed) Axiom, that two Lines cannot have a common segment
....................................................................................................................................................................................................................................................
|
65 |
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Treatment of Angle
....................................................................................................................................................................................................................................................
|
66 |
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Treatment of„ Parallels
....................................................................................................................................................................................................................................................
|
66„ |
|
Assumption of R. Simpson's Axiom
....................................................................................................................................................................................................................................................
|
67 |
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Euclid's 12th Axiom replaced by a Definition, two Axioms, and five Theorems
....................................................................................................................................................................................................................................................
|
68 |
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Test for meeting of two finite Lines
....................................................................................................................................................................................................................................................
|
69 |
|
Manual a modified Euclid
....................................................................................................................................................................................................................................................
|
70 |
Scene V.
Treatment of Parallels by revolving lines.
Henrici.
|
Treatment of Line
....................................................................................................................................................................................................................................................
|
71 |
|
Treatment of„ Angle
....................................................................................................................................................................................................................................................
|
74 |
|
Treatment of„ Parallels
....................................................................................................................................................................................................................................................
|
76 |
|
Attempted proof of Playfair's Axiom discussed
....................................................................................................................................................................................................................................................
|
76„ |
|
Attempted„ proof of„ Playfair's Axiom„ rejected
....................................................................................................................................................................................................................................................
|
83 |
|
General survey of book:—
....................................................................................................................................................................................................................................................
|
83„ |
|
Enormous amount of new matter
....................................................................................................................................................................................................................................................
|
84 |
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Two 'non-sequiturs'
....................................................................................................................................................................................................................................................
|
85 |
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An absurdity proved à la Henrici
....................................................................................................................................................................................................................................................
|
86 |
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Motion 'per saltum' denied
....................................................................................................................................................................................................................................................
|
87 |
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A strange hypothesis
....................................................................................................................................................................................................................................................
|
89 |
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A new kind of 'open question'
....................................................................................................................................................................................................................................................
|
90 |
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Another 'non-sequitur'
....................................................................................................................................................................................................................................................
|
90„ |
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An awkward corner
....................................................................................................................................................................................................................................................
|
91 |
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Theorems on Symmetry
....................................................................................................................................................................................................................................................
|
91„ |
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Summary of faults
....................................................................................................................................................................................................................................................
|
92 |
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Euclid I, 18, 19, contrasted with Henrici
....................................................................................................................................................................................................................................................
|
94 |
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A final tit-bit
....................................................................................................................................................................................................................................................
|
96 |
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Manual rejected
....................................................................................................................................................................................................................................................
|
96„ |
Scene VI.
Treatment of Parallels by direction.
§ 1. Wilson.
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Introductory
....................................................................................................................................................................................................................................................
|
97 |
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Treatment of Line
....................................................................................................................................................................................................................................................
|
98 |
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Treatment of„ Angle
....................................................................................................................................................................................................................................................
|
99 |
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Extension of limit of 'angle' to sum of four right angles
....................................................................................................................................................................................................................................................
|
99 |
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'Straight' angles
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101 |
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Meaning of 'direction'
....................................................................................................................................................................................................................................................
|
103 |
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'Opposite' directions
....................................................................................................................................................................................................................................................
|
105 |
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'Same' and 'different' directions
....................................................................................................................................................................................................................................................
|
106 |
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Axiom 'different Lines may have the same direction,' discussed
....................................................................................................................................................................................................................................................
|
108 |
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Property 'same direction,' when asserted of different Lines, can neither be defined, nor constructed, nor tested
....................................................................................................................................................................................................................................................
|
109 |
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'Separational directions' not identical with 'identical directions'
....................................................................................................................................................................................................................................................
|
110 |
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Virtual assumption of 'separational Lines are real' (which Euclid proves in I. 27), as Axiom
....................................................................................................................................................................................................................................................
|
113 |
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Axiom 'different Lines may have different directions' discussed
....................................................................................................................................................................................................................................................
|
114 |
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Axiom 'different Lines may have the same direction,' rejected, and Axiom 'different Lines may have different directions' granted with limitations
....................................................................................................................................................................................................................................................
|
115 |
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Axiom 'different Lines which meet one another have different directions' granted
....................................................................................................................................................................................................................................................
|
115„ |
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Axiom 'Lines with different directions would meet' discussed
....................................................................................................................................................................................................................................................
|
115„ |
|
and rejected
....................................................................................................................................................................................................................................................
|
116 |
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Diagram of 'same' and 'different' directions condemned
....................................................................................................................................................................................................................................................
|
117 |
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'Different but with same direction' accepted as (ideal) definition of Pair of Lines
....................................................................................................................................................................................................................................................
|
118 |
|
'Parallel,' as used by Wilson, to be replaced by term 'sepcodal'
....................................................................................................................................................................................................................................................
|
118„ |
|
Definition discussed
....................................................................................................................................................................................................................................................
|
119 |
|
Theorem 'sepcodal Lines do not meet' accepted
....................................................................................................................................................................................................................................................
|
121 |
|
Theorem 'Lines, sepcodal to a thirds are so to each other,' discussed, and condemned as a 'Petitio Principii'
....................................................................................................................................................................................................................................................
|
121 |
|
Axiom 'Angle may be transferred, preserving directions of sides' discussed
....................................................................................................................................................................................................................................................
|
122 |
|
If angle be variable, it involves fallacy 'A dicto secundum Quid ad dictum Simpliciter'
....................................................................................................................................................................................................................................................
|
123 |
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If it be constant, the resulting Theorem (virtually identical with the Axiom) involves fallacy 'Petitio Principii'
....................................................................................................................................................................................................................................................
|
125 |
|
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
....................................................................................................................................................................................................................................................
|
127 |
| (2) |
Lines, which make equal angles with a certain transversal, do so with any transversal
....................................................................................................................................................................................................................................................
|
128 |
|
Axiom rejected
....................................................................................................................................................................................................................................................
|
129 |
|
Ideas of 'direction' discussed
....................................................................................................................................................................................................................................................
|
130 |
|
Theory of 'direction' unsuited for teaching
....................................................................................................................................................................................................................................................
|
131 |
|
Test for meeting of finite Lines discussed:—
....................................................................................................................................................................................................................................................
|
132 |
|
it virtually involves Euclid's Axiom
....................................................................................................................................................................................................................................................
|
133 |
|
or if not, it causes hiatus in proofs
....................................................................................................................................................................................................................................................
|
133„ |
|
List of Euclid's Propositions which are omitted
....................................................................................................................................................................................................................................................
|
134 |
|
General survey of book:—
....................................................................................................................................................................................................................................................
|
135 |
|
A false Corollary
....................................................................................................................................................................................................................................................
|
135„ |
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A plethora of negatives
....................................................................................................................................................................................................................................................
|
136 |
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A superfluous datum
....................................................................................................................................................................................................................................................
|
137 |
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Cumbrous proof of Euc. I. 24
....................................................................................................................................................................................................................................................
|
137„ |
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An unintelligible Corollary
....................................................................................................................................................................................................................................................
|
138 |
|
A unique 'Theorem of equality'
....................................................................................................................................................................................................................................................
|
139 |
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A bold assumption
....................................................................................................................................................................................................................................................
|
139„ |
|
Two cases of 'Petitio Principii'
....................................................................................................................................................................................................................................................
|
139„ |
|
A Problem 3½ pages long
....................................................................................................................................................................................................................................................
|
139„ |
|
A fifth case of 'Petitio Principii'
....................................................................................................................................................................................................................................................
|
140 |
|
A sixth
....................................................................................................................................................................................................................................................
|
141 |
|
Summing-up, and rejection of Manual
....................................................................................................................................................................................................................................................
|
141„ |
§ 2. Pierce.
|
Treatment of Line
....................................................................................................................................................................................................................................................
|
144 |
|
Introduction of Infinitesimals
....................................................................................................................................................................................................................................................
|
145 |
|
Treatment of Parallels
....................................................................................................................................................................................................................................................
|
145„ |
|
Angle viewed as 'difierence of direction'
....................................................................................................................................................................................................................................................
|
146 |
|
Assumption of Axiom 'different Lines may have the same direction'
....................................................................................................................................................................................................................................................
|
147 |
|
List of Euclid's Theorems which are omitted
....................................................................................................................................................................................................................................................
|
148 |
|
Manual not adapted for beginners
....................................................................................................................................................................................................................................................
|
148„ |
§ 3. Willock.
|
Treatment of Parallels
....................................................................................................................................................................................................................................................
|
150 |
|
Virtual assumption of Axiom 'different Lines may have the same direction'
....................................................................................................................................................................................................................................................
|
150„ |
|
Assumption of Axiom 'separational Lines have the same direction'
....................................................................................................................................................................................................................................................
|
152 |
|
General survey of book:—
....................................................................................................................................................................................................................................................
|
153 |
|
Difficulties introduced too soon
....................................................................................................................................................................................................................................................
|
153„ |
|
Omission of 'coincidental' Lines
....................................................................................................................................................................................................................................................
|
155 |
|
'Principle of double conversion' discussed, and condemned as illogical
....................................................................................................................................................................................................................................................
|
155„ |
|
Mysterious passage about 'incommensurables'
....................................................................................................................................................................................................................................................
|
157 |
|
Manual rejected
....................................................................................................................................................................................................................................................
|
157„ |
ACT III.
Manuals which adopt Euclid's treatment of Parallels.
Scene I.
§ 1.
|
Introductory
....................................................................................................................................................................................................................................................
|
158 |
§ 2. Chauvenet.
|
General survey
....................................................................................................................................................................................................................................................
|
160 |
§ 3. Loomis.
|
General survey
....................................................................................................................................................................................................................................................
|
162 |
§ 4. Morell.
|
Treatment of Line
....................................................................................................................................................................................................................................................
|
163 |
|
Treatment of„ Angle
....................................................................................................................................................................................................................................................
|
165 |
|
Treatment of„ Parallels
....................................................................................................................................................................................................................................................
|
165„ |
|
General survey:—
|
|
'Direct,' 'reciprocal,' and 'contrary' Theorems
....................................................................................................................................................................................................................................................
|
167 |
|
Sentient points
....................................................................................................................................................................................................................................................
|
165„ |
|
A false assertion
....................................................................................................................................................................................................................................................
|
169 |
|
A speaking radius
....................................................................................................................................................................................................................................................
|
170 |
|
Ratios and common measures
....................................................................................................................................................................................................................................................
|
165„ |
|
Derivation of 'homologous'
....................................................................................................................................................................................................................................................
|
171 |
|
Mensuration of areas
....................................................................................................................................................................................................................................................
|
172 |
|
A logical fiasco
....................................................................................................................................................................................................................................................
|
173 |
|
Manual rejected
....................................................................................................................................................................................................................................................
|
174 |
§ 5. Reynolds.
|
General survey
....................................................................................................................................................................................................................................................
|
175 |
|
List of Euclid's Theorems omitted
....................................................................................................................................................................................................................................................
|
176 |
§ 6. Wright.
|
Quotations from preface
....................................................................................................................................................................................................................................................
|
177 |
|
General survey:—
....................................................................................................................................................................................................................................................
|
178 |
|
Specimen of verbose obscurity
....................................................................................................................................................................................................................................................
|
179 |
Scene II.
§ 1. Syllabus of the Association for the Improvement of Geometrical Teaching.
|
Introduction of Nostradamus, a member of the Association
....................................................................................................................................................................................................................................................
|
182 |
|
Treatment of Line
....................................................................................................................................................................................................................................................
|
183 |
|
Treatment of„ Angle
....................................................................................................................................................................................................................................................
|
184 |
|
Treatment of„ Parallels
....................................................................................................................................................................................................................................................
|
187 |
|
Test for meeting of finite Lines
....................................................................................................................................................................................................................................................
|
187„ |
|
Re-arrangement of Euclid's Theorems
....................................................................................................................................................................................................................................................
|
188 |
|
General survey:—
....................................................................................................................................................................................................................................................
|
189 |
|
A 'Theorem' is a 'statement of a Theorem'
....................................................................................................................................................................................................................................................
|
189„ |
|
Rule of Conversion
....................................................................................................................................................................................................................................................
|
190 |
|
Miscellaneous inaccuracies
....................................................................................................................................................................................................................................................
|
191 |
|
Summing-up
....................................................................................................................................................................................................................................................
|
194 |
§2. Wilson's 'Syllabus'-Manual.
|
Introductory
....................................................................................................................................................................................................................................................
|
195 |
|
A Theorem is a 'statement of a Theorem'
....................................................................................................................................................................................................................................................
|
196 |
|
Rule of Conversion
....................................................................................................................................................................................................................................................
|
196„ |
|
Every Theorem a 'means of measuring'
....................................................................................................................................................................................................................................................
|
196„ |
|
'Straight angles'
....................................................................................................................................................................................................................................................
|
196„ |
|
Miscellaneous inaccuracies
....................................................................................................................................................................................................................................................
|
196„ |
|
The Manual's one great merit
....................................................................................................................................................................................................................................................
|
197 |
|
No test for meeting of finite Lines
....................................................................................................................................................................................................................................................
|
198 |
|
Propositions discussed in detail:—
|
|
An important omission
....................................................................................................................................................................................................................................................
|
199 |
|
An illogical conversion
....................................................................................................................................................................................................................................................
|
199„ |
|
'Un enfant terrible'
....................................................................................................................................................................................................................................................
|
201 |
|
Summary of results:—
....................................................................................................................................................................................................................................................
|
206 |
- 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
....................................................................................................................................................................................................................................................
|
207 |
|
Nor for regarding this Manual as anything but a revised Euclid
....................................................................................................................................................................................................................................................
|
207„ |
|
Summing-up
....................................................................................................................................................................................................................................................
|
207„ |
ACT IV.
[Minos and Euclid.]
Manual of Euclid.
§ 1. Treatment of Pairs of Lines.
|
Modern treatment of Parallels
....................................................................................................................................................................................................................................................
|
209 |
|
Playfair's Axiom
....................................................................................................................................................................................................................................................
|
210 |
|
Test for meeting of finite Lines
....................................................................................................................................................................................................................................................
|
210„ |
§ 2. Euclid's constructions.
|
'Arbitrary restrictions'
....................................................................................................................................................................................................................................................
|
212 |
|
'Exclusion of hypothetical constructions'
....................................................................................................................................................................................................................................................
|
213 |
§ 3. Euclid's demonstrations.
|
'Invariably syllogistic form'
....................................................................................................................................................................................................................................................
|
214 |
|
'Too great length of demonstration'
....................................................................................................................................................................................................................................................
|
214„ |
|
'Too great brevity of demonstration'
....................................................................................................................................................................................................................................................
|
215 |
|
'Constant reference to Axioms'
....................................................................................................................................................................................................................................................
|
215„ |
§ 4. Euclid's style.
|
Artificiality, unsuggestiveness, and want of simplicity
....................................................................................................................................................................................................................................................
|
217 |
§ 5. Euclid's treatment of Lines and Angles.
|
Treatment of Line
....................................................................................................................................................................................................................................................
|
217 |
|
Treatment of„ Angle:—
....................................................................................................................................................................................................................................................
|
|
'declination from' accepted
....................................................................................................................................................................................................................................................
|
218 |
|
must be less than sum of two right angles
....................................................................................................................................................................................................................................................
|
218„ |
|
'multiple angles' in VI. 33
....................................................................................................................................................................................................................................................
|
218„ |
|
proof for Ax. 10 accepted
....................................................................................................................................................................................................................................................
|
219 |
§ 6. Omissions, alterations, and additions, suggested by Modern Rivals.
|
Omission of I. 7 suggested
....................................................................................................................................................................................................................................................
|
220 |
|
Reasons for retaining it:—
|
|
needed to prove I. 8
....................................................................................................................................................................................................................................................
|
220 |
|
not included in new I. 8
....................................................................................................................................................................................................................................................
|
220„ |
|
proves rigidity of Triangle
....................................................................................................................................................................................................................................................
|
220„ |
|
I. 7, 8 analogous to III. 23, 24
....................................................................................................................................................................................................................................................
|
221 |
|
bears on practical science
....................................................................................................................................................................................................................................................
|
221„ |
|
Omission of II. 8 suggested
....................................................................................................................................................................................................................................................
|
221„ |
|
Reason for retaining it, its use in Geometrical Conic Sections
....................................................................................................................................................................................................................................................
|
221„ |
|
Alterations suggested:—
|
|
New proofs for I. 5:—
|
|
'hypothetical construction'
....................................................................................................................................................................................................................................................
|
222 |
|
superposition
....................................................................................................................................................................................................................................................
|
222„ |
|
treating sides as 'obliques'
....................................................................................................................................................................................................................................................
|
222„ |
|
treating sides„ as radii of a Circle
....................................................................................................................................................................................................................................................
|
223 |
|
Inversion of order of I. 8, 24 rejected
....................................................................................................................................................................................................................................................
|
223„ |
|
Inversion of order of„ I. 18,19,20; do.
....................................................................................................................................................................................................................................................
|
223„ |
|
Fuller proof of I. 24; accepted
....................................................................................................................................................................................................................................................
|
223„ |
|
Algebraical proofs of II.; rejected
....................................................................................................................................................................................................................................................
|
223„ |
|
Additions suggested:—
....................................................................................................................................................................................................................................................
|
|
New Axiom; accepted
....................................................................................................................................................................................................................................................
|
224 |
|
Two new Theorems; do.
....................................................................................................................................................................................................................................................
|
224„ |
§ 7. The summing-up.
|
Euclid's farewell speech
....................................................................................................................................................................................................................................................
|
225 |