THE MATHEMATICAL SCIENCES 119 



rectangular blocks carefully cut in such a way as to be 

 able to interchange them in their superposition. 

 Further, the obscure definition of a straight line given 

 in the Elements (definition IV) takes on a new light if 

 considered in connection with the art of the mason. 

 The latter in order to verify the facing of a chiselled 

 surface applies to it a stone rule coated with red oil. 

 If the facing is perfect, the imprint made by the rule 

 appears without any break; if not, there are gaps. 

 Hence, the definition of the straight line "as a line 

 lying equally between its points." However, it seems 

 that Greek geometry, as it progressed, was able to free 

 itself from the shackles laid on it by the age-long use of 

 the rule and compass, and to conquer new and vaster 

 realms by adopting figures constructed by other 

 means. 



If it has not accomplished this, it is doubtless because 

 of the contempt in which tools fashioned and handled 

 by slaves were held ; x but it is probably also because 

 the geometrical tracings obtained by these instruments 

 raised problems insoluble by logic, for the following 

 reasons : The instruments by which figures can be 

 described mechanically may be divided into two groups : 

 the first comprises the instruments whose arrangement 

 remains exactly the same whilst the figure is described ; 

 for example, the legs of a pair of compasses keep the 

 same length and the same opening, while one of them 

 traces the circle. In the same way a triangle which 

 generates a cone remains identical in area and length 



1 As M. E. Meyerson reminds us, Plato, speaking of the 

 geometrical demonstrations into which mechanics enter, 

 declares that this is to degrade geometry by making it pass, 

 like a fugitive slave, from the study of things incorporal and 

 intelligible to that of objects perceptible by the senses, and 

 by using, besides reasoning, objects laboriously and slavishly 

 fashioned by manual labour. Bulletin de la Societe frangaise 

 de -philosophic, Feb. -Mar., 191 4, p. 101. 



