246 G. H. KNIBBS. 



be more than symbolic results, 1 formally deduced, and their trans- 

 lation into geometry cannot transcend the conceptions of geometry 

 itself. 2 Therefore notwithstanding the advantages of the applica- 

 tion of algebra to geometry — i.e., the construction of an "analytic 

 geometry," algebraic in form — no fundamentally new conception, 

 not already implicitly contained in the original assignment of 

 meaning to the symbols, can really arise, inattention to which 

 matter may lead, and actually has led, to erroneous conclusions. 3 

 "We may say broadly, therefore, that geometrical conceptions are 

 essentially independent of all schemes of formulating them 

 symbolically, or translating them from symbols. 4 It is of course 



- l It is often forgotten that the significance of the algebraic expression 

 is not inherent, but is determined by pure convention. The reader 

 unfamiliar with the algebra of symbolic logic will realise the truth of this 

 from such expressions as a+a = a; a+0 = a; aa—a; a+ab=a; ai=ay x+x 

 = i; x+i = i; %xx = 0, which are some of the general formulae of that 

 algebra. See Universal Algebra — Whitehead, Bk. n., Cap. i., pp. 35 - 

 39, 1898. • 



2 For example, the algebraic expression y 2 = ax has per se no geometri- 

 cal meaning, but if y be intended to represent a line on a surface, parallel 

 to a y-axis thereon, and x a length on the a?-axis, the relation between the 

 two being expressed through the equation, then the expression represents 

 for plus values of x, a parabolic curve. But retaining the ordinary con- 

 ventions of abstract arithmetic, there is a difficulty as to whether the 

 expression has any meaning for minus values of x. The determination 

 that ior V - 1 shall be taken to represent a quantity perpendicular to the 

 surface is a purely arbitrary convention, or scheme, which will admit of 

 more or less consistent interpretations, and the algebra does not prove 

 that the imaginary parabola for minus values of x has its plane perpen- 

 dicular to the xy surface. Strictly the equation implies the existence 

 only of surface, viz. that in which x and y lines exist, and an xz surface 

 has no implied existence, and therefore there is no intelligible interpreta- 

 tion when x is negative. See § 33 hereinafter. 



3 Chrystal remarks : — "It seems to be forgotten by some writers that 

 the e in e 1 ^ is a mere nominis umbra — a contraction for the name of a 

 function, and not 2'71828. . . Oblivion of this fact has led to some strange 

 pieces of mathematical logic." Text Book of Algebra, Vol. n., p. 264. 



4 Tait says, "it was reserved for Hamilton to discover the use of V -1 

 as a geometric reality" — the italics are his; see Quaternions, chap, i., sec. 

 13 — " tied down to no particular direction of space." Is not the essence 

 of the matter more perfectly stated by Argand's title : — "Essai sur une 

 maniere de representer les quantites imaginaires dans les constructions 

 geometriques "? (1806). Wallis [1616-1703] proposed to construct 

 imaginary roots by going out of the line on. which if the roots were real 

 they would be constructed, see his Algebra 1685. H. Kuhn of Danzig in 

 1750-1 represented aV-lasa line of the length a perpendicular to the 

 line a. Eeference is made to this particular symbol because of the signal 

 part it has played, through the work of Gauss, Cauchy, Eiemann and 

 others, in the modern theory of functions. See § 33 hereinafter. 



