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{ 345 ) 



XXH- — ^w Attempt to Elucidate and Apply the Principles of Goniometry, as pub- 

 lished by Mr Warren in his Treatise on the Square Roots of Negative Quan- 

 tities. By the Right Reverend Bishop Terrot. 



(Read 18th January 1847.) 



1 . The symbol ^/'—i. is called an impossible or imaginary quantity, because, 

 in analogy with the received laws of algebraic symbolism, it must mean such a 

 quantity as, being multiplied into itself, gives for a product — ] . Assuming, then, 

 that every quantity must be either plus or minus, it follows that the square of 

 every real quantity must be plus ; and hence V — I, which gives its square minus, 

 is called an imaginary or impossible quantity. 



If, however, we consider the most simple application of algebra to geometry, 

 we shall perceive that the assumption that every line must be considered and sjm- 

 bohzed as either + or — , is inconsistent with fact. In algebraic geometry, +a 

 or + 1 X a symbolizes a line whose numerical length is a, drawn in some given 

 direction ; while —a or — 1 x a, symbolizes a line of the same length, drawn from the 

 same extremity in the same straight line, but in a directly opposite direction. To 

 say, then, that aU lines must be either + or — , is as much as to say that all 

 lines drawn from the common extremity must be drawn in this one assumed line ; 

 and that it is impossible any line should be drawn making an angle with it. 

 But it is evident that an infinite number of such inclined lines may be drawn, 

 and none of them can have + 1 or - 1 as a factor, in accordance with the defini- 

 nition just given of those symbols. 



The assumption, therefore, upon which pig. i. 



'/^^ is considered and spoken of as an im- 

 possible quantity, is unfounded. All lines 

 drawn from C (Fig. 1.) are as real and pos- 

 sible as CA, which we symbolize by + 1 x a, 

 or CB, which we symbolize by — 1 x c. None 

 of them, however, except CA and CB, can be 

 symbolized, as to length and position, by a or 

 c multiplied into either a positive or a nega- 

 tive quantity ; since that would be equivalent 



to saying that they are coincident with CA g 



orCB. 



VOL. XVI. PART III. 



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