252 • G. H. KNIBBS. 



figures, two fundamental ideas of generation are naturally- 

 suggested; (a) the idea of summation or addition; (b) the idea 

 of fluxion or motion. The former idea was Cavalieri's, 1 the latter 

 Robervals' 2 and Newton's. 3 Since as defined, the point is of zero- 

 dimension, the continuous 4 generation of a'finite quantity by a 

 finite number of additions, is clearly not a conceptual possibility. 

 It is however by no means obviously inconceivable that such a 

 quantity can be continuously generated by an infinite number of 

 point-additions, since this was the essential feature of the Cavalieri- 

 Guldinus- 5 and Pascal- 6 controversy, which embraced also the same 

 question in respect to the possibility of lines and surfaces building 

 up respectively surfaces and solids. 7 



So long as points and infinitesimal lines are recognised as 

 different in kind, there can be no suggestion that any number, 

 finite or infinite, of points, point-elements, or groups, can consti- 

 tute a line or line element: 8 that is to say, there must be a generic 

 identity between the constitutive points and the line constituted. 



In considering the generation of geometrical figures as space, 

 whether by summational, fluxional, or other operations, it is to 

 be observed that the space itself is not the subject of generation. 

 Whatever its dimension, its existence is merely postulated as con- 

 ceivable, 9 and it is the geometrical figure therein that is alone the 



1 [1598 - 1647] Professor of geometry at Bologna. See his Georuetria 

 indivisibilibus continuorum nova quadam ratione promota. 1635. Also, 

 Exercitationes geometricae sex. 1647. 



2 [1602 - 1675] Mem. de l'Acad. roy. des Sciences, t. vi. 



3 [1642 - 1727] Philosophiae Naturalis Principia Mathematica, 1687. 



4 See Aristotle's Category No. 6 and Boscovich, [1711 - 1787] De con- 

 tinuitatis lege. 



5 [1577-1643] Author of 'Centrobaryca.' 6 [1623 - 1662] About 1658. 



7 Sensibly of course even & finite number of visible or sensuously per- 

 ceived points may appear to constitute a finite visible line : conceptually 

 the scale of separation is of no moment even though infinitesimal. 



8 The essence of the Guldinus 5 reply to Cavalieri, and to which the 

 rejoinder of Pascal is directed in his letter to de Carcavi, 1658. 



9 This may be put even more strongly. Thus Kant "Der Eaum ist 

 kein empirischer Begriff, der von ausseren Erfahrungen abgezogen 

 worden." Der transscendentalen Aesthetik. Absch. I § 2 (1). " Der 

 Eaum ist eine notwendige Vorstellung, apriori." ibid. (2). Quite recently 



Poincare writes, "Geometry is not an experimental science the 



geometrical ideas . . . preexist in us." Monist, Vol. ix., Oct. 1898 p. 41. 

 The matter has been fully discussed by Lotze. Metaphysik n., 1 and 2, 

 §99-137. 



