7Z^ 



NATURE 



[August 12, 1920 



Complex Elements in Geometry. 



The Theory of the Imaginary in Geometry, 

 together with the Trigonometry of the Imagin- 

 ary. By Prof. J. L. S. Hatton. Pp. vii + 215. 

 (Cambridg-e : At the University Press, 1920.) 

 Price 185, net. 



WHEN we interpret ^(.x, y) = o, ip^x, y) = o as 

 the point-equations of two loci, we are 

 bound to consider any vakies (jcj, yi) which satisfy 

 ' both equations as the co-ordinates of a point 

 common to both curves. The simplest case is 

 when <j>, rp are polynomials with ordinary integral 

 coefficients ; here the values (xi, ji) are determin- 

 ate, and can be calculated, either exactly or to 

 any desired degree of approximation. Abstractly, 

 {xi, yi) are a perfectly definite set of couples of 

 algebraic numbers. A couple {xi, yi) may be real, 

 and then corresponds to a real point ; but it may 

 be, and often is, complex. What is the most 

 appropriate and fruitful way, from a geometrical 

 point of view, of interpreting these complex solu- 

 tions of the given pair of equations? This is one 

 of the fundamental problems of analytical geo- 

 metry, and there are two ways in which it may 

 be attacked. Suppose that the coefficients of <f>, ^ 

 are real, complex intersections {xi, yi) fall into 

 conjugate pairs. The usual analytical formula 

 gives a real line as the join of two conjugate 

 points, and we may call this a common chord of 

 the two loci. The visible result of combining 

 <f) — o,ip = o may be said to be a certain number 

 of real intersections and a certain number of real 

 lines which, from an algebraical point of view, are 

 to be regarded as common chords. The most 

 familiar case is that of two circles and their 

 radical axis ; and here we have a geometrical 

 definition of the radical axis which applies whether 

 it meets the two circles or not. We can construct 

 a definition of a common chord of two conies by 

 analogy, whether it meets them in two real or 

 two conjugate complex points ; but the procedure 

 is artificial, and there is no obvious way of extend- 

 ing it to higher curves. 



The other way is to try to find, as the image or 

 representative of (x, y), when x, y are not both 

 real, some definite constructible geometrical entity 

 to which we can give the name of "point" with- 

 out violating the axioms of projective geometry 

 — e.g. it must still be true that any two points 

 determine a line, and so on. This, of course, 

 involves an appropriate definition of a complex 

 line. 



It is to von Staudt that we owe an absolutely 



perfect solution of this difficult problem. Its basic 



idea is this : Given a real conic, and a real line 



which does not cut it (in the ordinary sense), there 



NO. 2650, VOL. 105] 



is on the line an elliptic involution of pairs of 

 points conjugate to the conic. With this elliptic 

 involution we can associate either of two opposite 

 " senses " (or directions), and we can interpret the 

 involution, with either sense, as a complex point. 

 These complex points are distinct, and conjugate 

 in a sense analogous to the algebraic one. This 

 geometrical distinction of conjugate complex 

 points appears to have been the one thing with 

 which von Staudt had the greatest difficulty ; it 

 must be remembered that he was trying to find a 

 theory applicable to three dimensions as well as 

 to two, and that he wanted to define the line join- 

 ing any two points in space whether real or com- 

 plex, and this by purely projective considerations. 

 The "join" of two non-conjugate complex points 

 in space is von Staudt's "line of the second kind," 

 and the most difficult to realise of all his concepts. 

 What we may call a metrical, or Cartesian, 

 image of a complex point {a+hi, c+di) is a seg- 

 ment OP drawn from the real point {a, c) to the 

 real point (a+h,c + d). The conjugate point is 



represented by a segment OP' — — OP, and these 

 two conjugate points are on the real line PP'. 

 Poncelet, following that Will-o'-the-Wisp, the 

 "principle of continuity," very nearly hit vipon 

 this representation; for if we consider x^ + y" = a^, 

 x — h (f)>»a), we have as the intersections 

 (b, +i\/b"-a2), which, in this representation, are 

 the principal ordinates of the real hyperbola 



Prof. Hatton practically adopts this metrical 

 definition, but in doing so, as it seems to us, in- 

 troduces unnecessary vagueness, and occasionally 

 wabbles between the two points of view. He 

 begins by an " axiom " which von Staudt breaks 

 up into two definitions, and, so far as we can 

 see, ignores it in all his algebraical "verifications." 

 There is no such thing as an algebraical verifica- 

 tion in the true theory. The algebra is taken for 

 granted, and we have to show that our geo- 

 metrical definitions and postulates and axioms 

 agree with ordinary complex algebra. In the 

 Cartesian representation a point which we may 



call (OP), or more simply (OP), corresponds to von 

 Staudt's representation (O ooPP') , where O bisects 

 PP', and 00 is the point at infinity on the real 

 Une POP'. 



So long as we keep to von Staudt's projective 

 definition, the questions of such things as "dis- 

 tance," "angle," etc., do not arise. "Sense" 

 and "order" are essential, the latter especially 

 when we consider von Staudt's theory of "casts" 

 and cross-ratios. 



It is in connection with the Cartesian imagery 



