232 THIRD REPORT— 1833. 



Signs of transition are those signs which indicate a change 

 in the nature or form of a function, when considered in the 

 whole course of its passage through its different states of ex- 

 istence. Such signs, if they may be so designated, are gene- 

 rally sero and infinity. 



Zero and infinity are negative terms, and if applied to desig- 



he denies the correctness of the reasoning by which it is inferred that the 

 second term of the first, and the even terras of the second members of this 

 equation are equal to oue anotlier (when x is less than 1), because they are 

 the only terms whicli are homogeneous to each other, in as much as we thus 

 ascribe real properties to ideal quantities ; and he endeavours to make this 

 equality depend upon an assumed arbitrary relation between x and y, though 

 it is obvious that if y = cos &, we shall find x = cos n 6, and that, therefore, 

 this relation is determinate, and not arbitrary. A little further examination 

 of this conclusion would show that it did not depend upon any assumed 

 homogeneity of the parts of the members of this equation to each other, but 

 upon the double sign of the radical quantity which is involved upon both 

 sides. 



In arithmetical algebra, where no signs of affection are employed or recog- 

 nised, both negative and imaginary quantities become the limits of operations ; 

 and when this science is modified by the introduction of the independent 

 signs + and — and the rule for their incorporation, the occurrence of the 

 square roots of negative quantities, by presenting an apparent violation of the 

 rule of the signs, becomes a new limit to the application of this new form of 

 the science. The same algebraists who have acquiesced in the propriety of 

 making the first transition in consequence of the facility of assigning a meaning 

 to negative quantities, at the same time that they retained the definitions and 

 principles of the first science, were startled and embarrassed when they came 

 to the second ; for it was very clear that no attempt could be made to recon- 

 cile the existence and use of such quantities, consistently with the main- 

 tenance of that demonstrative character in our reasonings which exists in 

 geometry and arithmetic, where the mind readily comprehends the nature of 

 the quantities employed, and of the operations performed upon them. The 

 proper conclusion in such a case would be that the operations performed, as 

 well as the quantities employed, were symbolical, and that the results, though 

 they might be suggested by the primitive definitions, were not dependent 

 upon them. If no real conclusions had been obtained by the aid of such 

 merely symbolical quantities, they would probably have continued to be re- 

 garded as algebraical monsters, whose reduction under the laws of a regular 

 system was not merely unnecessary, but altogether impracticable. But it was 

 soon found that many useful theories were dependent upon them ; that any 

 attempt to guard against their introduction in the course of the progress of 

 our operations with symbols would not merely produce the most embarrassing 

 limitations, when such limitations were discoverable, but that they would 

 present themselves in the expression of real quantities, and would furnish at 

 the same time the only means by which such quantities could be expressed. 

 A memorable example of their occurrence under such circumstances presents 

 itself in what has been called the irreducible case of cubic equations. 



In the Philosophical Transactions for 1778 there is a paper by Mr. Playfair 

 on the arithmetic of impossible quantities, in which the definable nature of 

 algebraical operations is asserted in the most express terms, and in which 

 the truth of conclusions deduced b}^ the aid of imaginary symbols is made to 

 depend upon the analogy which exists between certain geometrical properties 



