VOL. LXVIII.] VHILOSOPHICAL TRANSACTIONS. SQj 



This last coincides with the 41st of the curious and difficult propositions pub- 

 lished by Dr. Stewart, under the title of general theorems; it is given there 

 without a demonstration, but appears plainly to have been investigated, in a 

 manner altogether rigorous, by that profound geometer. It may therefore be 

 regarded as one of the instances, in which the conclusions of this imaginary 

 arithmetic are verified by the geometrical analysis. 



17. The two foregoing propositions being confined to the circle, and yet 

 having been investigated by the help of imaginary expressions, may, at first sight, 

 seem exceptions to the rule we have been endeavouring to establish. But it 

 needs only to be remarked, that they are particular cases of certain theorems be- 

 longing both to the circle and hyperbola, and that it was into the investigation 

 of those theorems, that the imaginary expressions were introduced. 



The conclusions therefore from the whole are these: that imaginary expressions 

 are never of use in investigation but when the subject is a property common to 

 the measures both of ratios and of angles; that they never lead to any conse- 

 quence which might not be drawn from the affinity between those measures; and 

 that they are indeed no more than a particular method of tracing that affinity. 

 The deductions into which they enter are thus reduced to an argument fi om ana- 

 logy, but the force of them is not diminished on that accovmt. The laws to 

 which this analogy is subject; the cases in which it is perfect, in which it suffers 

 certain alterations, and in which it is wholly interrupted, are capable, as may be 

 concluded from the specimens above, of being precisely ascertained. Supported 

 on so sure a foundation, the arithmetic of impossible quantities will always re- 

 main a useful instrument in the discovery of truth, and may be of service when 

 a more rigid analysis can hardly be applied. For this reason, many researches 

 concerning it, which in themselves might be deemed absurd, are yet not destitute 

 of utility. M. Bernoulli has found, for example, that if / be the radius of a 

 circle, the circumference = —^'~— r\ and the same maybe deduced from 

 art. 4. Considered as a quadrature of the circle, this imaginary theorem is 

 wholly insignificant, and would deservedly pass for an abuse of calculation; at 

 the same time we learn from it, that if in any equation the quantity -"^' '^~^. 



should occur, it may be made to disappear, by the substitution of a circular arch, 

 and a property, common to both the circle and hyperbola, may be obtained. 

 The same is to be observed of the rules which have been invented for the trans- 

 formation and reduction of impossible quantities:* they facilitate the operations 

 of this imaginary arithmetic, and thus lead to the knowledge of the most beau- 

 tiful and extensive analogy which the doctrine of quantity has yet exhibited. 



* The rules chiefly referred to are lliose for reducing the impossible roots of an equation to the 

 form A + B V— 1- — Orig. 



