368 Mr DE morgan, ON A PROOF OF THE EXISTENCE OF 



taking D as small as he should please, before he began to shew that it might have been 

 taken yet smaller. 



Argand's proof is quite fundamental, and is the most direct of all. Its so called indirect 

 character is nothing but a case of the habit of the mathematicians not to admit the identity 

 of contrapositive forms without proof. To a logician the following forms, ' Every quantity 

 which is not is not the minimum,' and 'the minimum is 0' are identical, the existence of the 

 terms being known. If Cauchy's theorem were not to form part of a course, I should recom- 

 mend Argand's proof; and I should, in any case, insert Argand's as supplementary to the one 

 I have before given. 



Argand and Mourey were both in full possession of double algebra up to the interpreta- 

 tion of real exponents inclusive. The manner in which, at what is thereby proved to be the 

 due time, persons of all kinds, unconnected with each other and unknowing of each other's 

 existence, will take up a subject of speculation, of observation, or of experiment, is becoming 

 better and better known from day to day. Remembering that Mr Airy, more than five-and- 

 twenty years ago, casually told me that he had occupied himself with the interpretation of 

 ,y/ - 1 at a very early period of his studies, I lately begged of him to let me see any notes 

 which he might have made on the subject. The reply was the transmission of a manuscript, 

 drawn up in the form of a paper for a scientific society, dated January 21, 1820, and 

 therefore written in the first three months of the author's residence at Cambridge. It con- 

 tains, with many examples, a full interpretation of the roots of + 1 and of - l ; and commands 

 the full meaning of + and of -, and of x so far as relates to the formation of powers. 

 The idea on which it starts is, like that of Argand, the assumption of proportion, in the case 

 of lines, as involving equal differences of direction, as well as equal quotients of length. Of 

 Argand Mr Airy knew nothing ; of Buee as much as this, that he had been told a Frenchman 

 had treated the subject in the Philosophical Transactions. 



Mourey's* proof is as follows. It is much defaced in the original by peculiarities of 

 notation : the author had the idea that he was in possession of a new algebra, not the old 

 algebra under extension of interpretation. 



First, it is shewn that the equation which is expressed in my foregoing notation by 



must have a root or roots. 



The first side of the equation being altered as before, we have 

 rs,S2 ... $cos (0 + cTi +0-2 + ...) + sin {Q + ai + a^ + ...) ^ - l\.= m (cos^x + sin/xy^ - 1). 

 As before shewn we know that while Q changes from to Stt, no one of the angles a-,, o-^,... 

 loses value on the whole, while such as gain must increase by it, or by Sw, consequently 

 + o-j + 0-2 + ••• increases by 27r + (0 or some multiple of tt). At some value or values of 9, 

 then, we have cos (0 + o", + ...) = cos/u, sin {0 + a + ...) = sin ,«. Next, this value of B 

 being supposed to be determined, we have 



rs^s^... =ry/ {r" - 2ar cos (6 -a) + a^ . ^ {r^ - 2br cos (6 - (i) + b"].... 



' La vraie Theorie des quantiles nigatives et des quantites pretendues imaginaires. Dedii aux amis de i'evidence. Par 

 C. V. Mourey. Paris, Bachelier, 1828 ; pp. xii. + 144, 3 plates. 



