MEETINGS OF SOCIETIES. 229 



Papers. — (1) "Mill on demonstration and Necessary Truth," by 

 W. W. Carlile, M.A. (Abstract) In regard to the questions what 

 constitute necessary truths, English opinion was much divided. Hume 

 consistently put all the truths of pure mathematics in one class, truths 

 of matters-of-fact in another, and thus avoided the hair-splitting and 

 contradiction of the modern Humist school. Mill took all the axioms 

 out of the class of necessary truths and put them into the class of 

 truths of experience. Bain called " things that are equal to the same 

 thing are equal to one another " a truth of experience, but not, like 

 Mill " two straight lines cannot enclose a space." Mansell, a Kantian 

 philosopher, precisely reversed this. Bain affirmed that the axiom 

 "things that are equal to the same thing are equal to one another" was 

 a generalisation from experience. In support of this he said that 

 equality was properly defined as "immediate coincidence." If so, 

 "coincidence" could be used convertibly with "equality," but it was 

 plain that it could not. Equal lines are not lines that coincide, but 

 lines that would, if superimposed, coincide, a very different matter. 

 Coincidence was learnt by sense, equality only by thought. Probably 

 incomprehensible to the Damaraman or Bushman. The key to the 

 possibility of geometrical demonstration lay in this, in the power we 

 possessed of contemplating a line, for instance, as remaining the same 

 though in an altered position and environment. The want of a true 

 theory of identity in the opinion of Bosanquet was the great want of 

 philosophy of the English school. Spinoza asked what is the efficient 

 cause of a circle, answered it was the space described by a line 

 one point of which was fixed, the other moveable. We need only to 

 contemplate this line, the radius, as being the same in all its positions 

 to deduce all the properties of the circle. In the IV. proposition, 

 Euclid plainly postulated for the mathematical figures with which he 

 dealt the capacity of being lifted and moved about and put on top of 

 one another, or of themselves moved as others. This was the great 

 postulate of Euclid and should be prominently set forth as such, 

 instead of being merely taken for granted. If it were, it would be 

 seen, at a glance, that the construction in the V., the famous Pons 

 Asinorum was mere surplusage. If the big triangles formed by the 

 produced sides could be lifted up and put on top of one another, why 

 could not the isosceles triangle itself be lifted up, reversed, and put en 

 top of itself. If it were, we should have two triangles superimposed on 

 one another fulfilling all the requirements of the IV. proposition. It 

 was plain indeed, that a matter so simple as the equality of the angles at 

 the base did not really rest on anything so far fetched as the conventional 

 proof in Euclid. Mr. Mill seems always to contemplate the lines and 

 figures of geometry as if they were specimens picked up in our rambles 

 instead of being those which we supposed ourselves to have just con- 

 structed. To sum up, with regard to necessary truths. (1) They were 

 always concerned with abstractions. (2) The opposite of them was in 

 the strict sense inconceivable, not merely unbelievable. (3) This was so 

 because if we put their opposite in words, the last half of the proposition 

 " sublated " the first. (4) They were truths which could be seen to be 

 truths by merely thinking of them. (5) They were truths of sequence 

 only, not of fact. That brought them face to face with a difficulty that 

 might seem formidable, as of course geometrical propositions were used 



