CORRESPONDENCE, 1856-66. 329 



For myself, it is my tasfce, if you please, that I will have a 1865, 

 formal algebra, in which, every form, every law of transformation 

 is universal. I admit that we have not yet arrived at it, bnt I 

 have entire faith in the future. I was in the spirit on the day of 

 Wellington's funeral, when, wife, children, and servants being 

 away to see the remains of the glorious old man carried upon 

 what was so justly called a cross between a locomotive and a 

 fire-engine, I was sole master of my house and myself. So I 

 sat down to eviscerate the following difficulty, and I believe I did 

 it. If the forms of algebra be universal, then %x=x gives 



=- or 2=1. I should not have been ashamed of myself if 

 x x 



resolved on a formal algebra, I had invented a generalisation 

 of = to meet this case. But I had no occasion for any such 

 thing. I found that, by only taking permission to lay down as a 

 canon what mathematicians never scruple to do when they want 

 it, I was master of the field. 



I have a paper now at Cambridge which explains my views, 

 so you see I have taken twelve years to think about it. In brief 

 as follows. Admitting in theory as full and free a use of infinites, 

 finites, and infinitesimals, as is made in practice, I say that 



1. The sign = is that of undistinguisJiability, say indistinction. 

 A=B means that A and B are not distinct. Equality is but a 

 case, though the most common one. 



2. Distinction implies the use of a standard or metre. 

 Quantities infinitely great with respect to the standard, or 

 infinitely small, or unmeasurable by it, are undistinguishable. And 

 this is the origin of Leibnitz's equation dx=dx + dx~~*, the metre 

 having finite ratio to dx . 



3. Whenever we divide or multiply both sides of an equation 

 by anything above or below the standard, the new equation takes 

 a different standard. When we multiply by an infinite we must 

 take a standard which was infinite, &c. Now 2a? and x can only 

 be undistinguishable when x is infinitely small or infinitely 

 great. Let x be infinitely small. In passing from 2#=& to 2=1 

 we change our standard into an infinitely small quantity, and 

 2=1 is true, that is 2 and 1 cannot be distinguished. The same 

 when x is infinitely great. 



The difficulties which will suggest themselves are many and 

 obvious ; but I think that they are superable, and also that, 

 looking at actual algebra, they are not so great as the difficulties 

 which actually occur. For all these things I refer to the paper 



