1921] Abstracts and Reviews 111 



results are reciprocal, or each can be obtained from the one that fol- 

 lows, the theorems or problem posited in the beginning is true; since it 

 can be proved by the synthetical method, by reversing the steps and 

 working from the first conclusion or known results.* 



The method of analysis is generally that of research and discovery; 

 and because of its importance to the investigator and because of the 

 slight attention accorded it in existing texts, Dr. Henderson has, pur- 

 posely, given a number of illustrations of its working in solutions of 

 theorems and problems. In fact, the treatment by the analytical 

 method of problems and theorems is one distinctive feature of the 

 paper, which should appeal particularly to the eager student who 

 aims at any research in geometry. 



"The Basic Problems of Construction" are treated by the author 

 in an interesting manner. Most of the constructions are simple; 

 some are complex, and are given as stimulating problems, since one 

 object of the paper is to give new and instructive points of view. 



This is especially noted in the section entitled "The Problem of 

 Research," where the various modes of approach to the solution of 

 geometrical problems are indicated, and numbers of illustrations are 

 given with a completeness and elegance of demonstration that causes 

 one to regard this section as the most interesting part of the essay. 



The scope of this section can be judged from the following sub- 

 headings: 



1. The Method of Analysis, 



2. The Method of Successive Substitutions, 



3. The Method of Reductis Ad Absurdum, 



4. The Method of Intersection of Loci, 



5. The Method of Construction of Loci by Points, 



6. The Method of Transformation; Construction of Auxihary Figures, 



7. The Method of Parallel Translation, 



8. The Method of Rotation Sjonmetry, 



9. The Algebraic Method, 

 10. The Method of Similarity. 



The final sections on "Procedure in Attacking Geometrical Prob- 

 lems" and a Biographical Note complete the subject. 



The paper is illustrated with 26 figures, and is written in the at- 

 tractive style one has come to associate with the author. It is plain- 

 ly intended for teachers and research workers, for the author is a firm 



* See Duhausel's "Des IMethodes dans les Science de Raisonnement," vols. 1 and 

 2; also, Cain's "Symbolic Algebra and Notes on Geometry" (D. Van Nostrand Co., New 

 York) for a full discussion of the analytic method, including the cases of "lost" solutions 

 and of those "strange" to the questions. 



