AS COMMONLY INVESTIGATED. 229 



or, which is the same, y = tan"^^ and — tan~^7, we observe that z, in 

 passing through the value a, changes y, at once, from + — to — — • 



The great abruptness of such changes might lead us to suppose their extent 

 of discontinuity not subject to any rule, but further inquiry will not warrant 

 us in this conclusion ; through all their changes, these functions have a con- 

 nexion that admits of measurement, and which, as we have before remarked, 

 is even made the standard of comparison for other changes; so that in the lan- 

 guage of algebra, variations of no greater extent than these, are reckoned as 

 within the limits of continuity, whilst changes more abrupt are regarded as 

 transcending those limits. 



But, although mathematicians may adopt this definition, it would evidently 

 be more correct to say that continuity is divided by such writers into orders : 

 geometrical continuity of the highest order may cease with the introduction 

 of imaginary signs, whilst analytical continuity admits such symbols of corre- 

 lation as indicative of a grouping of related problems. The expression, for 

 example, y = a + >/b^ — x^ ceases to "be geometrically continuous with the 

 value X = b; but this is far from being the case with the analytical function, 

 wherein that value indicates a mere transition to a second problem of the same 

 group. All functions y = Fx, of numerical logic, or which are expressed in 

 terms of those unbroken arrangements of " the calculus" that lead to the rules 

 of arithmetic — rules by which, as by a machine, the numerical values are com- 

 puted — will thus constitute series of continuous terms ; but it would seem that 

 analytical writers carry this notion further, and regard every arranged series, 

 whether reduced or not to the known arrangements of the calculus, as pos- 

 sessed of some continuity. 



The functions arithmetically direct, or that merely represent additions, to- 

 gether with all those which, in passing from one value to another, pass through 

 all the intermediate values, form the functions of the first degree; a continuity, 

 it may be observed, best understood by considering the branches which func- 

 tions admit. 



I have, in another place, proposed to denote functions as monoramic, or mul- 

 tiramic, according to their division into branches. And these divisions, when 

 applied to geometry or physics, would have again to be subdivided by their 

 quality of being real or apparent. The circle, or the common lemniscate, for 



VII. — 3 H 



