404 THE SCIENTIFIC MONTHLY 



tion, but this difficulty was overcome by Laguerre, Cayley and their 

 successors. 



Differential geometry is in its essence a study of the properties of 

 geometrical figures by investigating the properties of small elements of 

 those figures. Such properties as curvature of a single curve or sur- 

 face, and those which depend on classes, such as the envelope of a 

 systems of curves, the surfaces which cut systems of surfaces at right 

 angles are the natural subjects of investigation under this head. In 

 1828 Gauss published a memoir which immensely extended the range 

 of this subject, by introducing new ideas which have been applied to 

 such topics as the deformation of surfaces under given conditions and 

 more particularly to those properties which remained unchanged by 

 deformation. The subject is closely allied to many problems in 

 physics. 



Many futile attempts to deduce the axioms of parallels from the 

 other axioms laid down by Euclid led Lobatchewski and Bolyai to see 

 what would happen if a geometry free from this axiom were con- 

 structed. The results showed that it could be made quite consistent 

 in all its theorems, that some of the Euclidean theorems would still 

 hold while others would be changed or generalised. Their chief succes- 

 sor was Riemann who showed that all previous geometries were special 

 cases of a more general system. In our own time the subject has been 

 developed in the direction of finding the properties which are possessed 

 by the different geometries and also by investigation of the different 

 sets of axioms which can be used as a basis for constructing different 

 classes of geometries. In the applications to physics the most import- 

 ant has been perhaps the recognition that our own space is not neces- 

 sarily Euclidean and that we can only find out its nature by examining 

 properties which we are able to observe and measure. 



The new developments have not prevented further research on the 

 older lines. Geometry is still much used as a convenient language for 

 the development and expression of analytical results. As seen below, 

 the plane is the home of the complex variable, but in the theory of 

 functions of this variable, it has become many-storied with ladders 

 reaching from one story to another. Most of our physical problems, 

 however, demand the use of three dimensional space and here the 

 complex variable is not sufficient because with our ordinary algebraic 

 rules, there are only two different kinds of numbers, the real and the 

 imaginary, which are used to deal with two different directions in a 

 plane. Hence the theory of vectors which deals with straight lines 

 in any number of dimensions and particularly three has been evolved 

 and is coming more and more into use on account of its brevity and 

 compactness. It requires, however, a new notation to be learnt and a 

 certain degree of familiarity with the operations which are possible. 



