358 PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



expressions ever so little complicated, is extremely obscure, if it be considered 

 that an imaginary character is no more than a mark of impossibility, such a com- 

 pensation becomes altogether unintelligible: for how can we conceive one impos- 

 sibility removing or destroying another? Is not this to bring impossibility under 

 the predicament of quantity, and to make it a subject of arithmetical computa- 

 tion? And are we not thus brought back to the very difficulty to be removed? 

 Their explanation cannot of consequence be admitted; but, on attempting an- 

 other, it behoves us to observe, that a more extensive application of this method, 

 than had been made in their time, has now greatly facilitated the inquiry. We 

 begin then with considering the manner in which the imaginary expressions, sup- 

 posed to denote real quantities, are derived; and the cases in which they prove 

 useful for the purposes of investigation. 



4. Let a be an arch of a circle of which the radius is unity, and let c be the 

 number which has unity for its hyperbolic logarithm, then the sine of the arch 

 a, or sm. a =■ ^ ; and cos. a = . 1 hese exponen- 

 tial and imaginary values of the sine and cosine are already well known to geo- 

 meters; and the investigation of them, according to the received aritlimetic of 

 impossible quantities, may be as follows. Let sin. a = z, then a = — jr. 



/v/(l — z^) 



To bring this fluxion under such a form that its fluent may be found by loga- 

 rithms, both numerator and denominator are to be multiplied by v^— 1 ; then a 

 = /— 1 X r_ -Yr» and (by form. 6. Harm. Men.) a = /— i x log. 



i±^'±zl\ Hence -^, or l X -^ = log. i±^^-l=lil, and because 1 is 



a 



the log. of c, c^~^ = ^ ~-^ ; therefore, if both parts of the fractional 



index of c be multiplied by v^ — 1, c— a-/— i = ~ ^j£. • Again, if the arch 



a be considered as negative, its sine becomes also negative, and therefore — a 

 = ^ _ 1 X log. Z±±^/Jf--L), or, -a^-l = - log. ^/Jf.-z2l, and 



av/-l =log. --+^-'-') ; whence also, c«^/-l = :zi±^^'zL!i. If from 



this equation the former be taken away, there remains ^^^ = cfV-i — 



' • V — 1 



c— aV— 1, whence dividing by 2\/ — 1 we have z =: sin. a = • ->-Z-' 



By adding together the equations a value of the cosine may be found in the same 

 imaginary terms which were assigned above. Now by means of these expressions 

 many theorems may be demonstrated; it may, for e^tample, be shown, that if a 



and b are any two arches of a circle, of which the radius is unity, then sin. a X 



, sin.« + i , sin. a — A t-< • c ja/ - ' — c-'' a^ ' , , 



cos. = 1 . tor sai. a = ^ , and cos. b = 



