Cambridge Philosophical Society. 443 



In the execution of the foregoing plan, the whole of the problems 

 ,of Euclid are omitted as irrelevant to the demonstration of the other 

 propositions. The grounds on which they were iido})ted in the 

 system of Euclid appear to be these. It frequently happens that it 

 is necessary in the course of demonstration to make some new con- 

 struction not included in the figure which forms the original subject 

 of the proposition, and it was evidently thought that the geometer 

 would not in strictness be entitled to take such a step until he had 

 demonstrated the means of executing it with exactitude. The stu- 

 dent was accordingly in the postulates put in possession of a ruler 

 and a pair of compasses ; and wherever any additional construction 

 was required in the proof of a proposition, a problem was premised, 

 showing the means by which the construction might be made by the 

 aid of those implements. 



But it should be recollected that the figure by which the demon- 

 stration is commonly accompanied is not the actual subject of the 

 reasoning, but a mere illustration to aid the imagination and the 

 memory, the exactitude of which is matter of comparative indiffer- 

 ence. Moreover, the principle on which the problems are introduced 

 is not consistently carried out to its legitimate conclusion even in 

 Euclid. There is no difference in the reasoning between the figure 

 which forms the original subject of the proposition, and the addi- 

 tional construction which is made in the course of demonstration ; 

 and therefore if it were necessary for the validity of the conclusion 

 to demonstrate the means of executing the latter figure, it would be 

 equally necessarj' in the case of the former. The student would not 

 be entitled to move a step in the demonstration of the equality of 

 two triangles having two sides and the included angle equal, until 

 he had been taught how to construct two such triangles, and con- 

 sequently how to describe an angle equal to a given angle. The 

 demonstration in Euclid begins with perfect legitimacy. " Let ABC, 

 DEF be two triangles in such and such conditions," without the 

 necessity of indicating the means by which those conditions may be 

 mechanically executed, or indeed of their possibility of actual exist- 

 ence ; and it may with equal legitimacy proceed to exemplify in 

 like manner any further construction which may be found necessary 

 in the course of demonstration. 



The question of motion has commonly been considered so essen- 

 tially distinct from that of position, that all reference to the former 

 subject has rigorously been excluded from the field of geometrical 

 inquiry. But tl:-' position of every point must ultimately be deter- 

 mined by motion from points antecedently known, and to the inci- 

 dents of motion we should accordingly look for the original source 

 of the relations of position. Now motion (in as far as it hifluences 

 position) admits of variation in two ways ; viz. in the direction of 

 the motion at each indivisible instant of time, and in the length of 

 the track accomplished in a finite period ; whence it has been^aid 

 by Sir John Herschel that space (which is primarily known as the 

 receptacle of motion) is reducible in ultimate analysis to distance 

 and direction. 



The relations of extent are simply those of equal, greater, and less. 



