August 24, 1917] 



SCIENCE 



187 



point of view and that throws new light on the 

 philosophical questions which permeate the 

 various mathematical developments. Among 

 the chapters which might appeal especially to 

 such readers we may mention those bearing the 

 following headings : " The axiom of infinity," 

 " Mathematical productivity in the United 

 States," and " Concerning multiple interpre- 

 tations of postulate systems and the ' exist- 

 ence ' of hyperspace." 



In Chapter IX. Professor Keyser discusses 

 " Graduate mathematical instruction for grad- 

 uate students not intending to become mathe- 

 maticians," arguing that such courses need not 

 presuppose a first course in calculus, but could 

 be based upon the mathematical preparation 

 gained in a year of collegiate study. He would 

 begin such a course " with an exposition of the 

 nature and function of postulate systems and 

 of the great role such systems have always 

 played in the science, especially in the illus- 

 trious period of Greek mathematics and even 

 more consciously and elaborately in our own 

 time." 



The headings of the nine chapters which 

 have not been mentioned in what precedes are 

 as follows : " The human significance of mathe- 

 matics," " The humanization of the teaching 

 of mathematics," "The walls of the world; or 

 concerning the figure and the dimensions of 

 the universe of space," " Mathematical emanci- 

 pation ; dimensionality and hyperspace," " The 

 uxiiverse and beyond; the existence of the 

 hypercosmie," " The permanent basis of a lib- 

 eral education," " The source and function of 

 a imiversity," " Eesearch in American uni- 

 versities," and " Mathematics." 



Some of these titles are the subjects of ad- 

 dresses delivered by Professor Keyser before 

 large audiences, and many of those who recall 

 his stimulating language will doubtless wel- 

 come the opportunity to secure a collection 

 covering such a wide scope of interests which 

 are common to all, but which should appeal 

 especially to those devoted to the borderland 

 between philosophy and mathematics. One 

 finds here a mixture of the most modern theo- 

 ries and the emotional descriptions of past 

 generations, a charming flow of language il- 



luminating most recent advances and, above 

 all, an inspiring tableland of thought which is 

 easily accessible to all but which is closely re- 

 lated with fundamental questions of education. 



The mathematicians, as a class, are perhaps 

 too much inclined to put off the historic, philo- 

 sophic and didactic questions for later consid- 

 eration, following the example of the great 

 mathematical encyclopedias which are in 

 course of publication. As a result the major- 

 ity of them become so engrossed in the tech- 

 nical developments of their subjects as to find 

 little time for the postponed questions of the 

 most fundamental importance — a fate which 

 seemed to threaten the encyclopedias just men- 

 tioned. A work in which some of these funda- 

 mental questions are handled in an attractive 

 manner is therefore a valuable and timely ad- 

 dition to the mathematical literature. 



G. A. Miller 



Univebsitt of Illinois 



EQUATIONS AS STATEMENTS ABOUT 

 THINGS 



In the teaching of elementary physics and 

 mathematics, much trouble is often caused by 

 the fact that students who can readily solve 

 an equation given them are unable to formu- 

 late in mathematical terms the data occur- 

 ring in a practical problem. The purpose of 

 this paper is to report briefly the results of 

 several years' experience with a plan designed 

 to remove as much as possible of this trouble 

 by making the equations show more readily 

 their meanings as shorthand statements of the 

 facts. While there is probably nothing about 

 these ideas that has not been suggested before, 

 such suggestions, when applied at all to teach- 

 ing, seem to have been rather vague and in- 

 complete, or else applied only to one branch 

 of the subject. In this case the plan to be 

 outlined has been used in a general course of 

 physics and in a course in mechanics, with re- 

 sults much more satisfactory than those ob- 

 tained by the ordinary method. 



To illustrate the difference between the old 

 plan and the new, let us consider a single 

 equation, the falling body law 

 « = i fft-. 



