IMAGINARY QUANTITIES. 



tmtjfmmf greatest or leatt value* which this quantity can have, 

 producing the imaginary symbol y' 1 ' nto 



! T:''.U!.I-_ 



The x-o >i.' I i-laaa of imaginary expressions, or those 

 which ir.tlic.ite real quantities, involve the symbol 

 v/ 1 in such a manner, that, by suitable transforma- 

 tion*, it may at last be made to disappear. The two 

 following exprewioiis are of this kind, viz. 



,bJ\ ) ; 



-i )} 



By taking the square, and then again the square root 

 of each, toe former i* tran*formedto 



cad the latter to 



2 v/ (+*) -*], 



(co.. -T+ sin. *J I)- 4. (cos. *_ 



U equivalent to 8 coa. at x, or real quantity ; alee, that 

 imaginary uprise ten 



-J { (OJB.X + sin. V I ,- (cos-x sin. x*/ 



V' t 



>.t to 8 sin .x.wx.ther real quantity Theae 

 rxpreMion*. although the repreaentat.*., of reeJ 

 (amitie*, vi*.oo*.ix, andSsin * x. 

 dered l,y thematlvr*. are utterly without any 



Scat ion. It u impoeaible to tra 

 analytic formula? into the lingiiage of strict 



mill Ike symbol, ain. 



E*g^% ** ** **-**+**.. 



rial 



their aid, would have been altogether untractable. We Imgy 

 have given examples of their application to the theory . t ^|""" J ^ 

 of angular sections, and the investigation of that ele- 

 gant property of the circle called the Coteoian theorem, 

 in the ARITHMETIC OF SINES, $ 19 $ 25. 

 If in the formula 



(Cos. x.{. sin. TV/ I)" =: cos. w x+sin. mxn/ 1 

 we write for , it becomes 



(\ * X X 



Cos.jr-J-rin.xv' 1 l=r cos. \- sin. / 1. 



Suppose nown to be indefinitely great, then, x being sup- 

 posed a finite arc, the arc will be indefinitely small ; 



in this case it* cosine will be equal to the radius, and 

 it* sine equal to the arc itself, hence, n being indefi- 

 nitely great, 



which are both real quantities. 



The general expreaaion for the root* of a cubic equa- 

 tion ha* the form 



when it* root* are all real ; but. unlike the two former, 

 k cannot, by any mean*, be transformed into a real al- 

 gebraic expression, ronswting of a finite number of 

 terms ; tat its value may be found by an infinite se- 

 rie*. or a Uble of sine*. ( Autcaat. { 7 f 8SU) 



It ha* bem proved, in the ARITHMETIC or Sixts. f 

 19, that, m. being any whole number or fraction, 



(cot. x 4- sin. x y' !)- = cos. mi 4. sin. mt^/\. 



Thi* formula we* first given by De Moivre. (PkiL 

 Trait. 1707, and .Muttt .i^mtyll m, lib. 2. ) 

 calls k " a formula a* renmrkable for it* 

 and elegance a* it* generality and fertility 

 qurncev" ( Cafcai dti Food***, p. 1 16) ; and Laplace 

 coneintfi its invention a* of equal importance with the 

 binomial theorem, (J>;o*M * &** Aere^n.) A* 

 the *ign of the square root of a quantity may be either 

 -J- or I , we in*; pat in lw sutBMsU ^l inateed 

 of -f- v^ 1 ; it then become* 



(eoe,* ein.xV !)"=< MX sin. mt S 1. 



From this, and the former ripreeaiun. we find, by ad. 



[>reeMB 



/-!)- 



Cos. 



sin. .TV/ " = 



</ \ 



or putting cos. n -f- sin. x </ 1 v. 



and hence */V_t^ = Xv / 1. 



Now it ha* been demonetrated in the introduction to 

 the article FLUXION*. | 12, that is being supposed in. 



finitely great, (* i 1)= Nap. log. v, therefore 



Nap. log. *- = xv/_ 1 

 and hence, r denoting the basia of the system, (At- 



OEBR. J 333 and } 356) we have t = e '^'''that it, 



Thi* i* aaethir imaginary formula of great value, be. 

 eau*e k exhibit* under a finite form a relation between 

 an arc or angle, and its co-sine, tine, 6cc It was first 

 observed by EeJer, and is jiutly regarded a* one t t IR 

 moet important analvtic inventions of the last century. 

 Other invrvtigetions of thi* formula have been given in 

 ARITHMETIC of Sine*, f 89, and FLI/XIUN*, $ !>. Ob. 

 crving, ea before, that the square root of a quantity 

 may be caneidmd m negative a* well a* positive, we 

 have from the formula, 



Co*, x^sm. x i/- 1 = e~ * v ' ; 



and from the two expression*, by addition and ub. 

 traction, 



ry' 1 Jv /_] 



r= ' f *-! -*v^-l? 

 -V-l l r J 



These formula*, in their present state, are illusive ; for 

 the arc and sine, or co-sine, cannot, by mean* of them, 

 be found, the one from the other. However, by ex. 

 pn<liniz the exponential* into series, we have, ( ALOE. 

 ERA, { 357 ) 



1.2 



1.2.3 



