6 



IMAGINARY QUANTITIES. 



= 1 _ W _1- + _^- +,&c. 



for Sin. x Cos. X 7 + Cos. x Sin. x>, (AftiTHMETic OF 

 SINES,) the product is equivalent to, 



Thete being substituted in the formula, we get 



expressions which are altogether free from the imagina- 

 TJ symbol. 



An example of the application of the same formulas 

 to the determination of the arc in terms of the tangent, 

 is given in the AhiTiniETic QF SINES, 30. They also 

 admit of various applications to the investigation of 

 rules in plane and spherical trigonometry. (See Le- 

 gendre, Element de Geometric, in App. to TRIG.) 



Because Cos. * + Sin. x y 7 1 = e x *~ ' there- 

 fore, Log. (Cos. x + Sin. x y 7 1 ) = x y 7 I. Let * 

 denote half the circumference of a circle, of which the 

 radius = 1, and suppose x = \ *, then Cos. x = 0, and 

 Sin. x = 1 ; hence we get Log. y 7 1 = it y 7 1, and 



_ 



This remarkable expression for the circumference of a 

 circle was first found by John Bernoulli. It is of no 

 use as a rectification of the circle ; but it shews, that if 

 the expression should occur in any investigation, we 

 may substitute, instead of it, the real quantity 2 r. 



The formula, Cos. * + Sin. x / I = e 



be- 



comes, in the case of x = ^ , yf 1 = e * * ^~~" ' 

 Now, let both sides of this equation be raised to a power 

 expressed by V- 1, then, observing that .] v y' 1 X 

 J 1 = \ *, we have, 



(y 7 1 ) *'~ 1 = e Z - 0.20Y879- 



This very singular imaginary expression was, we be- 

 lieve, first noticed by Euler. 



D'Alembert first demonstrated a remarkable proper- 

 ty of imaginary expressions, namely, that however 

 complicated they may be, they may always be reduced 

 to the form A -J- B y 7 1 ; so that every function of 

 an impossible quantity a + b y 7 1 has the form A + 

 B y 7 1, where, however, B may be equal 0. 



Let r :r y 7 (u* + 6 a ), and suppose x to be such an 



arc, that Cos. x = , then Sin. x = , where r, Cos. x, 



and Sin. x will be all real quantities ; by substitution, 

 the imaginary formula a + b y 7 1 will be transformed 

 into r (Cos. a -f Sin. a y 7 1 ). If, now, a' + b' y 7 1 

 be another imaginary expression, by a like substitution 

 it may be trantoriued into r' (Cos. x' + Sin. x 1 y 7 1), 

 where, as beioie, r", Cos. x and Sin x' are all real. Sup- 

 pose, now, an imaginary expression to be the product 

 of the formula- t, + b y 7 1, and a 1 + b' y 7 1, this, 

 by substitution, will be 



rr> (Cos. x + Sin. x y 7 -!) (Cos. x* + Sin. x" y 7 1). 



By actual multiplication and substitution of Cos. (x + x') 

 for Co. x C6. < Sin. * Sin. x' ; also Sin. (x + x') 



' { c s - (* + *') + Sin " 



+ *') </- 



which has manifestly the form A + B y 7 1. From 

 this it is evident, that the continual product of any 

 number of imaginary quantities, a -f- b y 7 I, a' -j- 

 b' y 7 1, &c. will be an imaginary quantity of the 

 same form. 



Consider now the fraction -rr-iT~7^T' This, ^7 



substituting the trigonometrical formulas, will be chang- 

 ed to 



r (Cos, x + Sin, x y 7 1 ) 

 r 1 (Cos. x' + Sin. x' y 7 1) 



Multiply the numerator and denominator by r' (Cos. x' 

 Sin. x' y 7 ) ), and substitute Cos. (x x') for Cos. x 

 Cos. x" + Sin. x Sin. x 1 , and Sin (x x') for Sin. x 

 Cos. x' Cos. x Sin. x', it then becomes, 



L j Cos. (x x') + Sin. (x x') y 7 1 1 



a quantity which has the form A 4- B y 7 1 as before. 

 The function (o + 6 y 7 1)" becomes, in like man- 

 ner, 



r (Cos. x + Sin. x y 7 1 ) * = 

 r" (Cos. n x + Sin. n x y 7 1 ) 



which is still a quantity of the same form. 



By employing the same substitution it is easy t* 

 prove, that the very general imaginary function 



is still of the form A + B y 7 1. 



In applying imaginary expressions to analytical in- 

 vestigations, it becomes an important question, what is 

 the nature of the evidence it affords for the truth of the 

 results ? It must be confessed that this part of their 

 theory is involved in some degree of obscurity, lohn 

 Bernoulli and Maclaurin allege, that when imaginary 

 expressions are put to denote real quantities, the ima- 

 ginary characters involved in the different terms of such 

 expressions do then compensate or destroy each other. 

 To this it has been objected, that an imaginary charac- 

 ter being no more than a mark of impossibility, such a 

 compensation is altogether unintelligible ; for to sup 

 pose that one impossibility can remove or destroy ano. 

 ther, would be to bring impossibility under the predica- 

 ment of quantity, and to make it the subject of arith- 

 metical computation. 



Professor Playfair of the university of Edinburgh, has 

 advanced a different theory in the London Pliiloidphi* 

 cal Transactions for 1778. He there observes, that im- 

 possible expressions occur only when circular arcs or hy- 

 perbolic areas are the subject of investigation, and that 

 corresponding to every imaginary formula, there is a real 

 formula perfectly analogous ; one of these is the analy- 

 tic expression of some property of the circle, and the 

 other the expression of a like property in the hyperbola. 

 Thus, supposing that in either curve the semiaxis =1, 

 let e denote any abcissa, reckoned from the centre, s the 

 ordinate, and x the double of the area contained by the 

 sum 'axis, a semidiameter drawn to the top ot the ordi. 

 nate, ami the intercepted hyperbolic arc; also let e de- 

 note the number of which Nap. log. = 1 : In the circle 



