MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 523 



But also 



[A" 



and hence (cf. *30'3) we have 



Thence, by the same reasoning as that in (ix) for Case II., , it follows that Case II., ft, 

 cannot hold. 



(xi). Hence neither Case I. nor Case II. can hold ; and therefore the proposition 

 follows. 



*31'3. Proposition. -The point-ordering relation (R,, n ) satisfies all the axioms 

 satisfied by the essential relation of Concept III., except the axiom of persistence for 

 that concept. 



Proof. In order to prove this, we make a comparison, as in Concept III., with the 

 axioms of Concept I. 



For I Hp R of Concept I., cf. *25'12 and *2G'24 and *3ril. 



For II Hp R of Concept I. , cf. *3 1 '2 1 . 



For III Hp R of Concept I. , cf. *3 1 '23. 



For IV Up R of Concept I, cf. *31 "22. 



For V Hp R of Concept I. , cf. XIV Hp R (*22 7 1 ). 



For VI Hp R of Concept L, cf. *28'33 and *31'12. 



For VII Hp R of Concept I, cf. *28-01'll. 



For VIII Hp R of Concept L, cf. XVI HpR (*2273). 



For IX Hp R of Concept L, cf. *28'12. 



For X Hp R of Concept I. , cf. *28 -42. 



For XI Hp R of Concept I, cf. *XVII Hp R (*2274). 



For XII Hp R of Concept L, cf. *26'23 and *28'33. 



Note. -In order to complete this comparison, it must be noticed that it follows from 

 *31'12 that the punctual line, with its cogredient point excepted, is the line as 

 defined on the analogy of Concept I. Also, it follows, from *28'32 and *28'42 and 

 the propositions of *31, that the punctual plane, with its cogredient points excepted, 

 is the plane as defined on the analogy of Concept I. Then the transition to projective 

 geometry is made, not by constructing a fresh type of points (the projective points), 

 but simply by putting back the class ( < R( ) of cogredient points. Metrical geometry 

 is then constructed in the well-known way,t making the plane ( CO E( ) of cogredient 

 points to be the plane at infinity. 



The Extraneous Relation. For the purpose of enabling velocity and acceleration 

 to be measured, an extraneous relation is required, in all respects similar to those 

 required in Concepts III. and IV., and the description already given need not be 

 repeated. 



* Cf. VEBLEN, loc. tit., for a sketch of this method ; also CLEBSCH and LINDEMANN, loc. at. 



3x2 



