A ROOT IN EVERY ALGEBRAIC EQUATION. 26*7 



Q = always intersect orthogonally ; and do not, therefore, contain so much as all pairs of 

 straight lines. There are other conditions of intefsection and of sequence on which I do not 

 here enter. 



I now proceed to give the proof of the fundamental proposition which Argand gave 

 (1815) in the fifth volume of Gergonne''s Annales, p. 204. I repeat this proof here, first to 

 separate it entirely from the interpretation by double algebra which it was Argand's principal 

 object to illustrate, and which he did illustrate with great effect : secondly, to remark that, in 

 a much more simple form, it is the proof which Cauchy afterwards hit upon, and published, 

 first (1820) in the Journ. de I'Ecole Polytech. vol. xi. p. 411, and afterwards (1821) in the 

 Cours d' Analyse, p. 331, a work, to which, as a student, I was much indebted. Argand's 

 proof rests upon the easily proved proposition, that r^ signifying r- cos ^ + ri,\nQ y/ - 1, &c. 

 and p, q, &c. being ascending positive exponents, the length or modulus of aj^" + ft^r/ + c^Vg' + . .. 

 may, by taking r small enough, be made as nearly equal as we please to that of ay^^ and the 

 angle of the first as nearly equal as we please to that of the second. This, under the inter- 

 pretations of the complete, or double algebra, is instantly perceptible, and the pure algebraical 

 proof is very easy. This being premised, let us take aj"^ + b^r^~^ + ... + k^r^ + l^, which call 

 ?7(co8 Y + sin Y \/ — 1). If it be impossible to take r^ so that U = 0, it follows that values 

 of r and 9 exist which give for U a value which cannot be lessened. Let m^ be this value of r^, 

 and for r^ write m^j. + h^, which, D^ being the value of least modulus just mentioned, changes 

 the expression into the form 



where, p, q, &c. are ascending positive exponents. Take h so small that the effect produced 

 on A^.h by the succeeding terms shall be useless in the following considerations. The first 

 two terms give 



Z> cos A + Z> sin A v^ - 1 + \ AhP cos {prj + a) + Ah^ sin {pri + a) . ^ - 1 ^ . 



Here rt is at our pleasure. Assume jo>j + a = A + tt, the preceding then becomes 



{D - Ah) (cos A + sin A . \/ - 1), 



which, A and h being positive, as they may be, the angles furnishing negative signs when 

 wanted, has a modulus less than that which cannot be lessened ; which is a contradiction. 

 No less acute a person than Servois did not see that this contradiction deduced from the 

 assumption of one of two necessary alternatives, is final in favour of the other. He pleaded 

 to the contradiction that it was not shewn to be large enough ; and in so doing he has added 

 one to the many cases which prove that a severer study of pure logic would be useful to the 

 mathematicians. He contends that Argand was bound not merely to shew a less than the 

 least, but to shew that this less than the least might be made as near as we please to zero. 

 Argand''s argument was precisely that of Euclid in the proof that pyramids of equal bases and 

 altitudes have equal solidities ; the difference is nothing because, whatever else may be named 

 for the difference, it can be shewn to be too large. The minimum modulus must be nothing, 

 because, whatever else may be taken for the least modulus, it can be shewn to be too large. 

 Servois forgot that the opponent who undertook to convince him had allowed him to begin by 



34—2 



