VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 357 



to suppose that b- is greater than \d-, is to suppose that the rectangle ac x cb 

 is greater than the square of half the line ab, which is impossible. The same 

 holds wherever expressions of this kind occur. Thus, when it is asserted that 



unity has the 3 cube roots 1, , , no more is meant than 



that when the general equation x^ — ax"- -\- bx — r = is, by a change in the 

 data, reduced to the particular state x^ — 1 ^ O, a: is then equal to unity only, 

 and admits not of any other value, as it does in more general forms of the equa- 

 tion. The natural office of imaginary expressions is, therefore, to point out 

 when the conditions, from which a general formula is derived, become incon- 

 sistent with each other; and they correspond in the algebraic calculus to that 

 part of the geometrical analysis, which is usually stiled the determination of 

 problems. 



3. This however is not the only use to which imaginary expressions have been 

 applied. When combined according to certain rules, they have been put to de- 

 note real quantities, and though they are in fact no more than marks of impos- 

 sibility, they have been made the subjects of arithmetical operations ; their ratios, 

 their products, and their sums, have been computed, and, what may seem 

 strange, just conclusions have in that way been deduced. Yet, the name of rea- 

 soning cannot be given to a process into which no idea is introduced. Accord- 

 ingly geometry, which has its modes of reasoning that correspond to every other 

 part of the algebraic calculus, has nothing similar to the method we are now 

 considering; for the arithmetic of mere characters can have no place in a science 

 which is immediately conversant with ideas. 



But though geometry rejects this method of investigation, it admits, on many 

 occasions, the conclusions derived from it, and has confirmed them by the most 

 rigorous demonstration. Here then is a paradox which remains to be explained. 

 If the operations of this imaginary arithmetic are unintelligible, why are they 

 not also useless? Is investigation an art so mechanical, that it may be conducted 

 by certain manual operations? or is truth so easily discovered, that intelligence 

 is not necessary to give success to our researches? These are difficulties which 

 it is of some importance to resolve, and on which much attention has not hitherto 

 been bestowed. Two celebrated mathematicians, Bernoulli and Maclaurin, have 

 indeed touched on this subject; but being more intent on applying their calculus, 

 than on explaining the grounds of it, they have only suggested a solution of the 

 difficulty, and one too by no means satisfactory. They alledge,* that when 

 imaginary expressions are put to denote real quantities, the imaginary characters 

 involved in the different terms of such expressions do then compensate or destroy 

 each other. But beside that the manner in which this compensation is made, in 



* Op. J. Bern. torn. i. N° 70. Maclaur. Flux. art. 699— 763.— Orig. 



