OM THE ASSORITHM OP IMAGINARY QUANTITIES. Qll 



Ag r«n, to square our second formula (B) 



— 1-4- i-Z-S 



— I -f iV— 3 



the square of ( — i + t" */ — 3) 



^ — i </— 3 — I- = — V — i V'— 3 =: (— i 4- 4 \^— 3)* 



— i + i- a/— 3 ^ j^^ product of ( - i+ i a/— 3) (— 4 — i V*— 3) 



— T — ^ a/— 3\ 



+ i = 1 



— -r — i -/— 3 ? ^jjg 3^e of (— 4- — i V— 3) 



— I- — i-v^— 3S 



r+ i-/— 3 — |. = — f 4- 4v/—3 - (i — I'/— 3)* 

 And consequently our square (B) becomes 



» . - 3 a 



V — i — ^ v/_ 3 4- 2 V 1 + V — 4 4- i V— 3 = 



V--i — ta/— 3 4- 2 4- /- 4+ ^v'— 3 



Thus far likewise we have proceeded step by step in both 

 operations. 



And since our first formula is equal to — 4, and our se- 

 cond to — t -87938; the square of the former ought to be 

 equal to 4* = l6,and the latter to (1-87938)* = 3'532069; 

 that is we ought to find the following equalities obtain; 



YIZ. 



(A)...</~ 117 — 44 a/— 1 4- 10 4- V^— 117 4- 44-/— 1 = 16 



(B)... j/— ^— ^ a/— 3 4-2 4- y — i + I -•— 3 =: 3«S3206^ 

 Or by transposing 10 and 2 



(A).../— 117 — 44 a/— 1 +/— 117 4-44-/— 1=6 



(B).. V- t ~ i •-3 4- ^-4^ + 1 -/- 3 z: 1-532069 



P a N»ir 



