478 DR. A. N. WHITEHEAD ON 



Axioms of Concept I. 



The axioms, it must be remembered, are merely an enumeration of various 

 propositions concerning the properties of the fundamental relations, which will occur 

 as hypotheses in the propositions of the fourth stage. In this instance we are merely 

 considering the axioms of geometry, and these concern the essential relation (R) only. 

 The axioms will be named systematically thus, I Up R, II Hp R, III Hp R, and so on. 

 Their enumeration will take the form of defining these names as abbreviations 

 standing for the various statements, which will be used subsequently as hypotheses. 



I Hp R is the statement that there is at least one set of entities, a, b, c, such that 

 R ; (fec) is true. The definition in symbols is 



. = . a !R ; (;;;) Df 



II Hp R is the statement that R ; (fcc) implies R ; (c&). The definition in symbols is 



II Up R . = : (a, b, c) : R(abc) . 3 . R : (c&) Df 



III Hp R is the statement that R : (6c) is inconsistent with R ; (&ra). The definition 



in symbols is 



IIIHpR . = : (a, 6, c) : R'(afcc) . 3 . R'(&ca) Df 



IVHpR is the statement that R ; (6c) implies that is distinct from c. The 

 definition in symbols is 



IV Hp K . = : (a, b, c) : R ; (a6c) . 3 . a y* c Df 



V Hp R is the statement that, if and b are distinct points, the segmental 

 prolongation of ab beyond b possesses at least one member. The definition in 

 symbols is 



VH^R. = :(a, &):,& eR'(;;;). a* 6.3.3! {R'(o&;)} Df 



VI Hp R is the statement that, if c and d are distinct points, possessed by the line 

 defined by the points a and b, then a is possessed by the line defined by c and d. 

 The definition in symbols is 



VIHpR. = : (, b, c, d): c, d tftab . c * d . 3 . aeR'ceT Df 



VII Hp R is the statement that there exist at least three points forming a triangle. 

 The definition in symbols is 



VII Hp R . = . (g a, b, c) . A K -(abc) Df 



VIII Hp R is the statement that, if a, b, c be three points forming a triangle, and 

 R'(&cd) and R ; (cea) hold, then there exists a point possessed both by the segment ab, 

 and by the line defined by d and e. The definition in symbols is 



VIII HpR . = :<, 6, c, d, e): A H : (afec) . ft(bcd) . R : (cea) . = . a!{R ; denR ! (a;fe)} Df 



