Euclid and His Modern Rivals/Argument of Drama

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ARGUMENT OF DRAMA.


ACT I.


Preliminaries to examination of Modern Rivals.


Scene I.

[Minos and Rhadamanthus.]

PAGE

Consequences of allowing the use of various Manuals of Geometry: that we must accept
 
(1)
'Circular' arguments
....................................................................................................................................................................................................................................................
1
(2)
Illogical do
....................................................................................................................................................................................................................................................
2
 
Example from Cooley
....................................................................................................................................................................................................................................................
2
 
Example from Wilson
....................................................................................................................................................................................................................................................
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
Discussion limited to subject-matter of Euc. I, II.
....................................................................................................................................................................................................................................................
8
One fixed logical sequence essential
....................................................................................................................................................................................................................................................
8
One system of numbering desirable
....................................................................................................................................................................................................................................................
10
A priori claims of Euclid's sequence and numeration to be retained
....................................................................................................................................................................................................................................................
11
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:—
....................................................................................................................................................................................................................................................
13
(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:—
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
(1)
Separation of Problems and Theorems;
 
(2)
Different treatment of Parallels.
 
Other subjects of enquiry:—
....................................................................................................................................................................................................................................................
15
(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.:—
....................................................................................................................................................................................................................................................
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.
 

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


Reasons assigned for separation
....................................................................................................................................................................................................................................................
18
Reasons for combination:—
....................................................................................................................................................................................................................................................
19
(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:—
....................................................................................................................................................................................................................................................
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
Propositions divisible into two classes:—
....................................................................................................................................................................................................................................................
23
(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
....................................................................................................................................................................................................................................................
24
 
Intersectional
....................................................................................................................................................................................................................................................
26
 
Separational
....................................................................................................................................................................................................................................................
27
Table I. Containing twenty Propositions, of which some are undisputed Axioms, and the rest real and valid Theorems, deducible from undisputed Axioms
....................................................................................................................................................................................................................................................
28
Subjects and predicates of other propositions concerning Separational Lines
....................................................................................................................................................................................................................................................
33
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
Table III. Containing five Propositions, taken from Table II, which have been proposed as Axioms
....................................................................................................................................................................................................................................................
37
(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
....................................................................................................................................................................................................................................................
38

§ 5. Playfair's Axiom.


Is Euclid's 12th Axiom axiomatic?
....................................................................................................................................................................................................................................................
40
Need of test for meeting of finite Lines
....................................................................................................................................................................................................................................................
41
Considerations which make Euclid's Axiom more axiomatic
....................................................................................................................................................................................................................................................
42
Euclid's Axiom deducible from Playfair's
....................................................................................................................................................................................................................................................
45
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
Objection to Euclid's Axiom (that it is the converse of I. 17) untenable
....................................................................................................................................................................................................................................................
47

§ 6. Principle of Superposition.


Used by Moderns in Euc. I. 5
....................................................................................................................................................................................................................................................
48
Used by Moderns in Euc. I. 24
....................................................................................................................................................................................................................................................
49

§ 7. Omission of Diagonals in Euc. II.


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.

Introductory
....................................................................................................................................................................................................................................................
54

Scene II.


Treatment of Parallels by methods involving infinite series.


Legendre.

Treatment of Line
....................................................................................................................................................................................................................................................
55
Treatment of Angle
....................................................................................................................................................................................................................................................
57
Treatment of Parallels
....................................................................................................................................................................................................................................................
57
Test for meeting of finite Lines
....................................................................................................................................................................................................................................................
58
Manual unsuited for beginners
....................................................................................................................................................................................................................................................
59

Scene III.


Treatment of Parallels by angles made with transversals.


Cooley.

Style of Preface
....................................................................................................................................................................................................................................................
60
Treatment of Parallels
....................................................................................................................................................................................................................................................
62
Utter collapse of Manual
....................................................................................................................................................................................................................................................
63

Scene IV.


Treatment of Parallels by equidistances.


Cuthbertson.

Treatment of Line
....................................................................................................................................................................................................................................................
64
Attempted proof of Euclid's (tacitly assumed) Axiom, that two Lines cannot have a common segment
....................................................................................................................................................................................................................................................
65
Treatment of Angle
....................................................................................................................................................................................................................................................
66
Treatment of Parallels
....................................................................................................................................................................................................................................................
66
Assumption of R. Simpson's Axiom
....................................................................................................................................................................................................................................................
67
Euclid's 12th Axiom replaced by a Definition, two Axioms, and five Theorems
....................................................................................................................................................................................................................................................
68
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
 
Two 'non-sequiturs'
....................................................................................................................................................................................................................................................
85
 
An absurdity proved à la Henrici
....................................................................................................................................................................................................................................................
86
 
Motion 'per saltum' denied
....................................................................................................................................................................................................................................................
87
 
A strange hypothesis
....................................................................................................................................................................................................................................................
89
 
A new kind of 'open question'
....................................................................................................................................................................................................................................................
90
 
Another 'non-sequitur'
....................................................................................................................................................................................................................................................
90
 
An awkward corner
....................................................................................................................................................................................................................................................
91
 
Theorems on Symmetry
....................................................................................................................................................................................................................................................
91
 
Summary of faults
....................................................................................................................................................................................................................................................
92
 
Euclid I, 18, 19, contrasted with Henrici
....................................................................................................................................................................................................................................................
94
 
A final tit-bit
....................................................................................................................................................................................................................................................
96
Manual rejected
....................................................................................................................................................................................................................................................
96

Scene VI.


Treatment of Parallels by direction.


§ 1. Wilson.

Introductory
....................................................................................................................................................................................................................................................
97
Treatment of Line
....................................................................................................................................................................................................................................................
98
Treatment of Angle
....................................................................................................................................................................................................................................................
99
Extension of limit of 'angle' to sum of four right angles
....................................................................................................................................................................................................................................................
99
'Straight' angles
....................................................................................................................................................................................................................................................
101
Meaning of 'direction'
....................................................................................................................................................................................................................................................
103
'Opposite' directions
....................................................................................................................................................................................................................................................
105
'Same' and 'different' directions
....................................................................................................................................................................................................................................................
106
Axiom 'different Lines may have the same direction,' discussed
....................................................................................................................................................................................................................................................
108
Property 'same direction,' when asserted of different Lines, can neither be defined, nor constructed, nor tested
....................................................................................................................................................................................................................................................
109
'Separational directions' not identical with 'identical directions'
....................................................................................................................................................................................................................................................
110
Virtual assumption of 'separational Lines are real' (which Euclid proves in I. 27), as Axiom
....................................................................................................................................................................................................................................................
113
Axiom 'different Lines may have different directions' discussed
....................................................................................................................................................................................................................................................
114
Axiom 'different Lines may have the same direction,' rejected, and Axiom 'different Lines may have different directions' granted with limitations
....................................................................................................................................................................................................................................................
115
Axiom 'different Lines which meet one another have different directions' granted
....................................................................................................................................................................................................................................................
115
Axiom 'Lines with different directions would meet' discussed
....................................................................................................................................................................................................................................................
115
and rejected
....................................................................................................................................................................................................................................................
116
Diagram of 'same' and 'different' directions condemned
....................................................................................................................................................................................................................................................
117
'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
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
 
A plethora of negatives
....................................................................................................................................................................................................................................................
136
 
A superfluous datum
....................................................................................................................................................................................................................................................
137
 
Cumbrous proof of Euc. I. 24
....................................................................................................................................................................................................................................................
137
 
An unintelligible Corollary
....................................................................................................................................................................................................................................................
138
 
A unique 'Theorem of equality'
....................................................................................................................................................................................................................................................
139
 
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