AND ON THE LOGIC OF RELATIONS. 357 



position, is of use to the learner when carried to the extent to which he and Mr John Mill 

 have carried it, that is, the exhibition of one proposition, to be repeated a few times for prac- 

 tice. But there is a far better logical exercise in Euclid. This great leader has, equally 

 with Aristotle, a style of his own, and one full of its own technicalities: but utterly divested 

 of any the smallest distinction between form and matter. This is most fortunate for that 

 student for whom a further guide is provided : the book before him is raw material on which 

 the exercise of thought about form and matter can be far more profitably carried on than it 

 could have been if Euclid had made the distinction. And this especially on two points. 



First, a geometrical proposition may either be a purely formal consequence of those which 

 precede, or it may require (as most do) a further infusion of geometrical matter. When 

 Euclid has proved that a won-central point inside a circle, or outside, is not a point to which 

 three equal lines can be drawn, he holds himself not to have completed the proof that a point 

 to which three equal lines can be drawn is the centre. But his demonstration is nothing 

 except his often repeated transition from one to the other of the contrapositive forms of a 

 universal affirmative proposition. It is not in his system to establish a purely logical inference 

 once for all : accordingly, ' not-X is always not-Y ' is converted into ' Y is always X ' by 

 one and the same train of thought whenever it is wanted. That the common end • of three 

 equal lines is the centre follows equally from the now-centre not being such common end, whether 

 or no the reasoner can say what a circle is, or a centre, or a common end. 



Again, this same want of admission of what logicians call contraposition gives rise to the 

 majority of the ex absurdo demonstrations : in fact the reductio ad absurdum is usually 

 nothing more than the mode of making the passage from the direct to the contrapositive form. 

 When (in I. 6) it is to be shown that equal angles give equal sides, what Euclid really 

 shows — that is, the geometrical matter of his proposition — is that unequal sides give unequal 

 angles. His unequal sides immediately produce unequal areas with a pair of sides equal, 

 each to each : whence, by I. 4 contrapositively taken, the included angles are unequal. 

 All that is ew absurdo serves only to show that ' unequal sides give unequal angles ' is 

 identical with ' equal angles give equal sides,' and to admit of the direct, instead of the 

 contrapositive, form of I. 4. 



From such instances, and many others, I derive my now long fixed opinion that geometry 

 is of little, though some, account for technical exercise in the syllogism ; of more for exercise 

 in the transformations of the proposition ; of most of all, and of very much, for exercise in the 

 separation of form and matter. 



It says but little for the truth of the views taken of logic that this science and geometry 

 lived so long in the same family — the old school of arts — without any attention being paid to 

 the bearing of the first upon the second. But it is to be remembered — to say nothing of 

 Euclid being kJjoios aTot^^eHOTi;? and above criticism — that the form of contraposition, though 

 known and duly registered, was, by reason of the neglect of contrary or privative terms, very 

 little used or thought about: and also that the distinction of form and matter was never 

 completely envisaged, though influential. That the logicians — and it must be remembered 

 that logicus meant student or graduate in arts, in all its intension — prone as they were to 

 syllogise, never threw the propositions of Euclid into technical form, must be taken as a point 



