650 Transactions, — Miscellaneous. 



all, are not the same." If we do, we plainly deny our own 

 assumption that they are the same. The key to the possibility 

 of geometrical demonstration lies in this : in the power that we 

 possess of contemplating one attribute, such as length, as 

 remaining "the same," though in a varied environment. The 

 want of a true theory of identity is indicated, I think, rightly 

 by Mr. Bosanquet (" Mind," li., 3) as being at the root of most 

 of the mistakes of the English school ; and this is a case in 

 point. From an identical proposition in the sense of one 

 which affirmed that the length of any given line in any given 

 position is the same now as it was five minutes ago no geo- 

 metrical deduction, at any rate, could be drawn ; but with an 

 identical proposition, in the sense of one which affirms that the 

 length of a line can be the same though its position is altered, 

 the case is wholly different. We need no other concession 

 than this to deduce all the properties of the circle. If we in- 

 quire, with Spinoza,''' what is the efficient cause of a circle, we 

 might answer, with him, " It is the space described by a line 

 of which one point is fixed and the other movable." This line 

 is the radius ; and when we conceive of it as the same line, but 

 in varied positions, even Professor Huxley would hardly deny 

 that it would be an identical proposition, with a proof resting 

 on the law of contradiction, to affirm that this line in all its 

 positions is equal to itself. This identity in varied positions 

 would be a not inapt definition of equality. It is a character 

 fyom which innumerable new truths — of sequence, at any 

 rate — can be drawn ; yet it is plain that we arrive at 

 it by the ordinary process of abstraction. When we ab- 

 stract our attention from the irrelevant circumstances of 

 its position and direction, and contemplate its length 

 alone, we can then, of course, contemplate the length as 

 remaining identical whatever the position and direction 

 may be. 



In the Fourth Proposition, which is proved by the method 

 of superposition, Euclid very plainly postulates for the mathe- 

 matical figures with which he deals this characteristic of being 

 capable of being lifted and moved about and put on top of one 

 another — in other words, of being capable of being regarded as 

 identical in spite of difference of position. This postulate is 

 one which, mstead of being taken for granted, ought, I think, 

 to be specifically stated at the beginning of every treatise on 

 Euclid, and clearly kept in view in all geometrical demonstra- 

 tions. If it were, it would be plain that the construction and 

 the proof resting on it in the Fifth Proposition — -the celebrated 

 pons asinuritm — are mere surplusage. We have in any case 

 to postulate the possibility of taking up the large triangles 



* Letter to Schirnhausen. 



