Page:Carroll - Euclid and His Modern Rivals.djvu/26

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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„