ON THJB AtfiOaiTHM OF IMAGINARY QUANTIT|[ES* QQQ 



XL 



A remarkable analytical Anomaly respectfully submitted to 

 the Consideration of Mathematicians. 



To Mr. NICHOLSON. 

 SIR, 



An your number for August last, I puWished a short On the algo* 

 paper on the defective algorithm of imaginary quantities, ^^}^^ of ima- 

 which has not at present been honoured by the remarks of ties, 

 any of your correspondents, though the importance of the 

 subject seems to demand the attention of every advocate for 

 the introduction of these expressions into mathematical 

 investigations : and as I am extremely desirous to have the 

 opinion of analysts on this subject, and particularly those, , 



who in their writings have maintained the legitimacy of re- 

 sults, in cases where imaginary quantities have been nearly 

 the only instruments employed in obtaining them ; I am 

 induced, in order to draw a reply from those quarters, to 

 consider the same under rather a different point of view. 

 For which purpose let us assume the two followiing ex- 

 pressions : 



f/—^ ± 11 V— 1 + l/— 2— 11 V—l — —4 



^~ i + 4 V'— 3 + */"— i — i -/— 3 = - 1'87938 



which equalities may be verified either by the develope- 

 ment of the above expressions into series, or by the solu- 

 tion of the equations of which they ar£ the roots, accordinjg 

 to Cardan's rule, viz. 



x^ — 15x = — 4 

 j; ' — 3 X = — 1 



Now let us square these formulce at full length, and by 

 precisely the same steps. Then we shall have the following 

 operations : 



Vol. XXX.-^Nov. UlU P (A) 



