128 



to which so many answers, more or less plausible, have been offered, 

 that nothing short of a complete exposition of a consistent scheme 

 of demonstration can be expected to carry conviction in the validity 

 of a fresh solution. The object of the present paper is accordingly 

 to complete the task undertaken in the foregoing publication by a 

 formal statement of the other definitions required in connexion with 

 those of straight and parallel lines and plane surface, and by a rigid 

 demonstration from these premises of the steps intervening between 

 those and the premises of the ordinary system ; and in additional 

 proof of the fundamental character of the proposed analysis, the de- 

 monstration is carried on through the geometry of the three first 

 books of Euclid by direct reasoning, without resort to the compara- 

 tively unsatisfactory method of ex absurdo proof, which, although 

 equally conclusive as to the necessity of the result, yet always leaves 

 a hankering in the mind for an answer why the case must be as the 

 demonstration shows that it cannot avoid being. 



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 adopted 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 necessary 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- 



