RECENT ADVANCES IN SCIENCE 519 



where yjr is readily determinate by successive approximations. 

 We cannot recall a very practical problem in which the applica- 

 tion of this type of analysis to a new differential equation, and 

 its rigorous elucidation on more logical lines, is more urgently 

 needed, or more capable of leading to results of importance not 

 only in relation to applied mathematics. It is clear that 

 developments on these lines may be expected, for the dynamics 

 of a spinning shell is not, in its essence, different from the very 

 progressive dynamics, with all its associated problems for the 

 pure analyst, to which modern theory of aeroplanes has 

 introduced us. 



The appearance of Dr. Brodetsky's book entitled, A First 

 Course on Nomography (G. Bell & Sons, los.), marks an oppor- 

 tune moment at which to direct the attention of mathematical 

 readers to this subject. Hitherto it has hardly been possible 

 to study the matter except from the larger and very compre- 

 hensive work of D'Ocagne {Trait e de Nomographie, Gauthier 

 Villars, Paris, 1899), to whom so many useful nomograms are 

 due. The interest attaching to this subject is little known. 

 It is perhaps no exaggeration to say that many pure mathe- 

 maticians have never even met a nomogram, except, perhaps, 

 the slide-rule, which, in its essence, is a collection of simple 

 nomograms, though most of these are not quite of the more 

 usual types to which this name is applied. Nomograms are of 

 great utility in practice, and are in constant use in many 

 industrial problems in which the continued solution of 

 mathematical equations of certain types — and these by no 

 means always of the simplest — is required. They could be 

 made of equal service to the applied mathematician, and often 

 even to the pure mathematician, when numerical solutions of 

 a problem are of value. 



Descartes, in inventing Co-ordinate Geometry, laid the 

 foundations of the most powerful methods of mathematics. 

 Buache, in the eighteenth century, followed it up by the method 

 of contours, which made it possible for three quantities to be 

 dealt with at once instead of two only. Further developments 

 — stimulated by the extraordinary growth of railway systems — 

 of graphical methods were made in the nineteenth century by 

 Lalanne, Massau, and Lallemand. D'Ocagne, in 1884, hit on 

 the method of using collinear points, which is used in nomo- 

 graphy. 



The essence of nomography is this. By means of one 

 diagram we can read off the solutions of any equation of a 

 certain type. Take a simple case : x^ -\- ax -{■ b = o. Children 

 at school laboriously draw the graph of jy = x* -\- ax -\- b, notice 



