228 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



of the science. The elementary functions of this class are x", a'', sin x; and 

 their converses x~", Xn, log x, sin^^x; and the complex are such as can be 

 formed from these by any of the operations which were used in forming these 

 functions themselves, certain restrictions being required when those operations 

 compose an infinite series. 



Now if we assume as our unit a number e, no otherwise assignable than as 

 being less than any number we use in the course of the inquiry, and as being 

 capable of indefinite diminution, the natural scale of numbers will become 



. . — 2 e, — l.e, O.c, l.c, 2 c, 3 f, . . . . x.e . . . 

 where each term exceeds the preceding by the indefinitely small portion s. 

 Such a scale is, in fact, employed by Laplace, under another form, at the com- 

 mencement of the Mec. Celeste, and, adding clearness to discussions concern- 

 ing infinitely small variations, will be used throughout this paper, wherein we 

 shall term it the natural scale to the unit e. 



But, substituting for x, either in the preceding elementary functions or their 

 converses, the successive numbers of this scale, or making the same substitu- 

 tions in any of the less complex functions that are obtained from these, our 

 knowledge of such functions assures us that the results would be one or other 

 of these three kinds, namely: 1. Real numbers, positive or negative, and in- 

 creasing and decreasing continuously; 2. Imaginary symbols of the form 

 V -f- w \/ — 1, where v and w continuously increase or decrease; and, 3. Ex- 

 pressions of the form ^ ^ , where w being exceedingly nearly equal to v, the 



continuous increase of w may cause the expression to become plus infinity; and 

 this increase continuing, and causing w to exceed v by a quantity infinitely 

 small, will make the value of the expression pass at once from plus i7ijinity to 

 minus infinity. 



Such, at least, then, are the degrees of variation that we must admit under 

 the notion of continuity, if we would regard the ordinary functions of algebra 

 as continuous; and we find that our elementary writers have, accordingly, in 

 speaking of continuous functions, limited themselves to these cases, framing a 

 rule that, in such functions, a number never passes from plus to minus rvithout 

 having previously passed through zero or infinity. 



The very simple expression y = tan — 5^ — will, however, show us that 



a — z 



such a rule is not exact, since, by making z respectively equal to a + e or a — e. 



