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Proceeding from the propositions to the axioms, he denied that 

 the human mind possesses the innate power of perceiving a general 

 truth : and asserted that, without a knowledge of all the cases to 

 which a statement may be made to apply, we are not safe in enun- 

 ciating it ; thus, adopting the definition of equality as implied in 

 Euclid's eighth axiom, the proposition *' if equals be added to equals 

 the sums are equal," is not true ; the sums may be equal or they 

 may be equivalent. And, as an instance of the ease with which we 

 may be led to admit the truth of a specially worded proposition, he 

 cited this one : — " perfect equality implies equality in size, in shape, 

 in weight, in colour, and in every respect in which we can compare 

 them, so that of two perfectly equal bodies, the one could not be 

 distinguished from the other ; perfect equality, then, must include 

 every inferior degree of resemblance." Such is an axiom to which 

 most people would assent as self-evident — yet it is not true. 



Even the axiom " things equal to the same thing are equal to 

 each other," is not to be admitted without examining the particular 

 kind of equality implied ; for though bodies similar to the same 

 body be similar to each other, and those equivalent to the same equi- 

 valent to each other, solids symmetric to the same solid are not 

 symmetric to each other. 



Passing from the axioms to the definitions, he pointed out the 

 necessity of establishing the possibility of the thing defined : thus 

 if we form such a nomenclature as this ; a solid with four trigonal 

 faces is called a tetrahedron, a solid with five trigonal faces is called 

 a pentahedron, one with six trigonal faces a hexahedron, and so on, 

 our definitions would be essentially vicious, since no such pentahe- 

 dron can exist ; and thus we see that our definitions of the tetrahe- 

 dron and hexahedron are only admissible after examination. 



Again it is necessary to take care that the definition of one ob- 

 ject be consistent with that of another. Thus, having defined a 

 straight line, we are not at liberty to use the straight line in defin- 

 ing a plane surface until we have made sure that this use is consis- 

 tent with what has already been predicated. Now the ordinary de- 

 finition of a plane surface is, that the straight line joining any two 

 points in it lies wholly on the surface. This definition, however, 

 implies a very abstruse property of straight hues ; namely, that if 

 two straight lines be drawn from one point, and if two points be as- 

 sumed in each of them, the two straight lines joining alternately the 



