OPERATIVE DIVISION 593 



NOTES 



I. The 4>° Fallacy. — It is curious that in these days when there is so much 

 analysis of the fundamental ideas of mathematics, the fallacy that $°, A , 2°, D°, E°, 

 ... all equal numerical unity should have been overlooked — Part I, p. 221. We 

 define that 4>°x = x, A°x — x, . . . , but then immediately assume that (p°x = 1 x x, 

 A°x = 1 x x. . . . Why not suppose that 4>°x = o + x or = x ] ? We have 

 evidently been misled by association of ideas, since, for numbers, juxtaposition 

 implies multiplication. But for operations, juxtaposition implies that the superior 

 element operates upon the inferior one, and this must still hold regarding <£°, as 

 well as regarding (f> h when h is infinitesimally small. Hence <f>° must be, not a 

 number, but an operator — and is in fact O ; and the recognition of this idea will 

 some day open a new chapter in mathematics. 



II. The Relations between Successive Roots and Critical Points (incompletely 

 studied in Section VII) are best indicated briefly as follows. From the algebraic 

 point of view a rational integral function is the algebraic product of a number of 

 linear or quadratic factors ; from an operative point of view it is an operative 

 product of operative factors — that is, 



fx = [p n + xo] [p a _ 1 + xo] . . . [p 2 + xo] |> + xo]p . 



The effect of operating with xo upon any function, that is, of multiplying it by 

 x, is to modify its current ordinates without disturbing either its roots or its critical 

 points, and also to bend the curve towards the origin and thus to add a new root, 

 zero. The effect of operating with p + xo is first to do this and then to shift the 

 entire curve away from the .r-axis to a distance p. And if fx is a rational integral 

 function it results from a succession of such operations. 



If pa, p\, . . . p r are all positive, the resulting curve commences, when x = o, at 

 the value p r , has only positive terms, increases for all positive values of x, and has 

 no positive root. Call this set of terms A. Next let p r + \ be negative; then a 

 positive root and a set of positive critical points are immediately introduced. If 

 fir + 'i, pr + s, ... all continue negative, no new positive root is introduced, but the 

 original one is, so to speak, pushed further and further from the origin along the 

 positive .r-axis and also new critical points, all referring to this single root, are 

 formed by each new term. Call this set of negative terms B. Next form another 

 set of positive terms C. Then a second root between the origin and the first root 

 may be introduced if the addition of the terms C does not push the curve too far 

 away from the axis; and also there will be in any case another set of critical points 

 between the C-terms and the Z?-terms. But if C is too large, not only will the 

 second root not be formed, but the first one will disappear. And the repetition 

 of these processes continues to produce the same results. Thus the number of 

 positive roots must either equal the number of changes of sign or fall short of it 

 by an even integer — which contains Descartes' Rule, of course. 



Again, arrange^ in an ascending series so that 



fx = A-B + C-D..., 



where A, B, C, . . . are successive sets of terms of the same sign and A now 

 contains the absolute term. Draw diagrams of the curves A, A — B, A — B + C, 

 etc., as in Figure 8, on the supposition that the variable possesses only positive 

 values. 



We observe first that when x is small enough A is nearly the same as the 

 absolute term ; that B, C, D, . . . which have no absolute term, are all nearly zero ; 

 and that A> B> C> D. . . . But when x is large enough A <B <C<D, . . . 



