110 Journal of the Mitchell Society [December 



in our geometries, it seems appropriate to call attention to it as an 

 original work of high scientific value that should prove not only help- 

 ful, but stimulating to teachers and students in geometry. 



In the Introduction, Dr. Henderson properly calls attention to 

 the lessons of the world war as to the prime necessity of Mathematics 

 relative to the scientific progress, welfare and preservation of a nation, 

 which naturally recalls the saying of Napoleon that, "The advance- 

 ment, the perfecting of mathematics, are bound up with the pros- 

 perity of the state." 



In discussing "The Aims and Results of Geometrical Study," the 

 author emphasizes the fact that the true purpose of instruction in 

 geometry is to develop the faculty of independent thinking in geome- 

 try, to acquire facility in working problems — "originals, "as they are 

 happily called — -in other words, to train the student as an investigator 

 — a research worker. There can be no question as to the correctness 

 of this point of view. Students so trained, take an increasing interest 

 in the study — they "make good — ■" whereas, those neglecting this 

 discipline, soon lose their mental self-reliance and intellectual courage 

 to tackle new problems and difficulties, become discouraged, resort to 

 mere memorizing, and thus, ultimately fail. 



The author next takes up "The Problem of Instruction," and ad- 

 vocates the natural method of instruction, even when it is longer than 

 the usual synthetic method. The usual text-books, following almost 

 exclusively the synthetic method, are models of "Compression, ele- 

 gance and rigor;" but they, too often, leave the student puzzled as to 

 what the author is driving at, until the end is reached; when, presto, 

 a conclusion is reached which comes as a distinct surprise to the reader, 

 who has not been prepared for the denouement l)y being given the 

 reasons for the successive steps of the demonstration. 



By the synthetical method, whether for the demonstration of 

 theorems or the solution of problems, we start from known theorems 

 or problems and endeavor to effect a solution; but, as it is often dif- 

 ficult to know from what known theorems or problems to start, a series 

 of fruitless essays may be made before hitting upon the proper solu- 

 tion. 



On the contrary, by the natural method — that of the "old an- 

 alysis" of Plato — we have the advantage of a starting point, the thing 

 to be proved; and from that, an endeavor is made, by logical processes, 

 to reach a known result. When this is attained, then if the successive 



