230 THIRD REPORT — 1833. 



governed by necessary laws, except so far as those interpreta- 

 tions are dependent upon each other. Thus, if a be taken to 

 represent a line in magnitude, it is not necessary that (cos d 

 + -v/ — 1 '^in 6) a should represent a line equal in length to the one 

 represented by a, and also making. an angle 6 with the line re- 

 presented by a ; but if (cos 9 + V" — 1 sin 9) a, may, consistently 

 with the symbolical conditions, represent such a line, without 

 any restriction in the value of 9, then, if it does represent such 

 a line for one value of 9, it must represent such a line for every 

 value of 9 included in the formula. It is only in such a sense 

 that interpretations can be said in any case to have a necessary 

 and inevitable existence. 



It is this confusion of necessary and contingent truth which 

 has occasioned much of the difficulty which has attended the 

 theories of the interpretation of algebraical signs. It has been 

 supposed that a meaning could be transmitted through a suc- 

 cession of merely symbolical operations, and that there would 

 exist at the conclusion an equally necessary connexion between 

 the primitive definition and the ultimate interpretation, as be- 

 tween the final symbolical result and the laAvs which govern it. 

 So long as the definitions both of the meaning of the symbols 

 and of the operations to which they are required to be subject 

 are sufficient to deduce the results, those results will have a 

 necessary interpretation which will be dependent upon a joint 

 consideration of all those conditions; but whenever an operation 

 is required to be performed under circumstances which do not 

 allow it to be strictly defined or interpreted, the chain of con- 

 nexion is broken, and the interpretation of the result will be 

 no longer traceable through its successive steps. This must 

 take place whenever negative or other aflPected quantities are 

 introduced, and whenever operations are to be performed, 

 either with them, or upon them, even though such quantities 

 and signs should altogether disappear from the final result. 



This principle of interpretation being once established, we 

 must equally consider — I, V — \, cos 9 + ./ — l sin 9, as signs 

 of impossibility, in those cases in which no consistent meaning 

 can be assigned to the quantities which are affected by them, 

 and in those cases only : and it must be kept in mind that the 

 impossibility which may or may not be thus indicated, has re- 

 ference to the interpretation only, and not to the symbolical 

 result, considered as an equivalent form : for all symbolical 

 results must be considered as equally possible which the signs 

 and symbols of algebra, whether admitting of interpretation or 

 not, are competent to express. But there will be found to be 

 many species of impossibility which will present themselves in 



