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SCIENCE 



[N. S. Vol. XXXIX. No. ! 



nates as nul lines, a very good name in itself, 

 but already in use in an entirely different 

 sense in connection with the so-called nul- 

 system of statics and line geometry, and fur- 

 thermore unnecessary since the term minimal 

 lines is quite standardized in the literature.] 

 Again in connection with the problem of 

 geodesic representation, the usual discussion 

 of the real solutions of Beltrami and Dini is 

 followed by the imaginary solutions discovered 

 by Sophus Lie. If then imaginary figures are 

 allowed even when, as in this last instance, 

 they are, however interesting, really incon- 

 venient, why not, as Study advocates, intro- 

 duce them deliberately and systematically? 



The student usually gets the impression 

 that whatever is true in the real domain, will, 

 by some very nebulous principle of continuity, 

 also be true in the larger complex (real and 

 imaginary) domain. This is actually the case 

 in a remarkably large number of questions, but 

 certainly not in all. The exceptional character 

 of minimal curves has long been recognized, 

 but only recently the peculiar curves lying in 

 a minimal plane have been investigated by 

 Study and his students. 



Even in the domain of curves lying in 

 an ordinary (Euclidean) plane, the reviewer 

 has recently encountered a striking instance 

 of how imaginary figures may have essen- 

 tially difFerent properties from real figures. It 

 is a standard theorem in elementary calculus 

 that when one point approaches another on a 

 curve the arc and the chord becomes ultimately 

 equal, that is, the ratio of the arc to the chord 

 approaches unity as its limit. This property 

 of real analytic curves is true of most imagi- 

 nary curves, as can be verified by calculation, 

 but not of all. In the simplest class of ex- 

 ceptional imaginary curves, the limit is a cer- 

 tain irrational number, approximately .94. 

 Thus the arc, instead of becoming equal to 

 the chord, becomes less (of course in absolute 

 value, since both arc and chord are complex 

 quantities). Again, the statement is usually 

 made that the difference between the arc and 

 the chord is an infinitesimal of third order; 

 but in the present instance it is in fact of the 

 first order. The same class of curves shows 



that even when there is no cusp or singular 

 point the radius of curvature may vanish. It 

 is remarkable that whenever the limit men- 

 tioned is not unity, it is at most equal to .94. 

 For space curves the result is quite different, 

 since the limit may then be any number, real 

 or imaginary.^ 



The moral of all this is that if, originally, 

 imaginaries were introduced into geometry 

 because they made the statement of proposi- 

 tions, especially of algebraic geometry, easier, 

 and bore out the principle of continuity, we 

 have to pay for this nowadays by a syste- 

 matic treatment of the imaginary figures in 

 complete generality. We must look, with an 

 enlightened view, over the entire complex 

 domain, instead of restricting our attention, 

 from some more or less accidental motive, to 

 some cross section connected, more or less 

 closely, with the original real domain. Of 

 course there will always be justification for a 

 purely real geometry (as instanced say by 

 analysis situs, or the geometry of connection, 

 including the theory of knots) ; but differential 

 geometry has been guided mainly by the 

 theory of analytic functions (power series), 

 rather than the theory of functions of a real 

 variable, and the tendency toward a perfect 

 correspondence with the former theory seems 

 all-compelling. The present period is one of 

 transition, and that is always hard on both the 

 writer of text-books and his students. 



The author has certainly succeeded in get- 

 ting into one volume most of the more im- 

 portant standard topics. This is shown by 

 the chapter headings: Curves in space. Gen- 

 eral theory of surfaces, Organic curves of a 

 surface. Lines of curvature. Geodesies, General 

 curves on a surface and differential invari- 

 ants. Comparison of surfaces. Minimal sur- 

 faces. Surfaces with plane or spherical lines 

 of curvature and Weingarten surfaces. De- 

 formation of surfaces, Triply-orthogonal sys- 

 tems of surfaces. Congruences of curves. 



The treatment of each of these topics is 

 quite elaborate, in many instances the proofs 

 are more elegant than those usually given, 

 and an abundant selection of problems (many 



2 See Bull. Amer. Math. Soc, Vol. 20, p. 727. 



