Janttabt 30, 1914] 



SCIENCE 



177 



of them difficult, and often containing im- 

 portant results, as is to be expected of a 

 Cambridge treatise) is included. 



Perhaps the most important and interesting 

 feature of the book is the long chapter dealing 

 with differential invariants, covariants, and 

 parameters. The algebraic method employed 

 is due to Forsjth himseK, and was first pub- 

 lished in a memoir in the Philosophical Trans- 

 actions, 1903. The calculations are arranged 

 very ingeniously, and the detailed results are 

 certainly useful. The geometric interpreta- 

 tions, however, are not always clear, and 

 sometimes they appear to be only partly geo- 

 metric. The practise of calling a result geo- 

 metric when it is in fact semi-geometric and 

 semi-algebraic is unfortunately rather preva- 

 lent. 



The author's terminology, in this connec- 

 tion, can not be recommended. He speaks of 

 the invariants of " a single curve," when he 

 really means a system of curves, simply in- 

 finite in number. His results have in fact no 

 meaning for a single curve. It would be ab- 

 surd to imply that the author's ideas are not 

 clear — it is merely a matter of careless termi- 

 nology. The distinction between a system of 

 curves and a curve is just as great as that 

 between a curve and a point. Of course a 

 system is made up of an infinitude of curves, 

 just as a curve is made up of an infinitude of 

 points, but that is no excuse for identifying 

 the configurations. 



The long list of invariants for "two 

 curves " refers actually to two simply infinite 

 systems of curves, a figure usually called a 

 net of curves. The author is not here dis- 

 cussing doitbly-infinite systems. It is best to 

 avoid the ambiguous term double system: it 

 refers sometimes to a double infinity (that is, 

 infinity times infinity) and sometimes to two 

 single infinities (that is, a net). 



After the discussion of " one curve " and 

 " two curves," the rather mysterious statement 

 is made that " we could not consider profitably 

 more than two independent curves." As a 

 matter of fact there are some very important 

 (and naturally very difficult) problems con- 

 nected especially with three systems, and, in 



the reviewer's opinion, the investigation will 

 have to be extended. 



Another more serious confusion of terms, 

 and even of ideas, is prevalent in geometric 

 literature. We refer to the distinction be- 

 tween a system of curves and a parametered 

 system of curves. The latter object arises fre- 

 quently in applications. For example, in a 

 topographic map we have to deal not merely 

 with the system of contour lines, but with the 

 particular numbers attached to these curves 

 indicating the heights above sea level. The 

 same system of curves with different numbers 

 would represent a different topography. In 

 most discussions in geometry we are con- 

 cerned merely with the system of curves; but 

 if the attached numbers or parameters are also 

 of significance, as they often are, the com- 

 pound object should be called not a system, 

 but a parametered system. 



Even in one dimension an analogous dis- 

 tinction is important. A curve consists of 

 a single infinity of points: if the points are 

 labeled with numbers, then we have a new 

 figure, a parametered curve. A straight line 

 with a logarithmic scale is certainly different 

 from a straight line with an ordinary uniform 

 scale. A correct and well developed termi- 

 nology is at hand in, for example, d'Ocagne's 

 Nomography. This branch of mathematics 

 was originated and developed almost entirely 

 by engineers (mainly the French school), 

 rather than pure mathematicians, but it is 

 certainly time for writers on geometry to take 

 advantage of their work. 



With this terminology, it is possible to state 

 very compactly a fundamental theorem in the 

 theory of functions of a complex variable, as 

 follows: Any (analytic) parametered curve 

 can be converted into any second parametered 

 curve by a unique direct (and a unique re- 

 verse) conformal transformation. This is 

 true of parametered curves but not of curves : 

 any curve can be converted into a second 

 curve by an infinitude of conformal trans- 

 formations. 



The author is to be commended for not 

 confining himself, as much as most writers do, 

 to questions of first and second order (of 



