202 



SCIENCE. 



[N. S. Vol. IV. No. 85. 



Madison, Wis., Tracy, Gibbs & Co. 1894. 

 8°. Pp. viii+395. 

 Plane and Solid Geometry. By Woosteb Wood- 

 ruff Beman, Professor of Mathematics in 

 the University of Michigan, and David Eu- 

 gene Smith, Professor of Mathematics in the 

 Michigan State Normal School. Boston, 

 Ginn & Co. 1895. 8°. Pp. ix+320. 

 These three text-books of geometry all show 

 points of interest and excellence, and bear testi- 

 mony scientifically and pedagogically to a spirit 

 of progress. It is natural to turn at once to the 

 first few pages of each book, and examine the 

 manner in which the diflficult problem of giving 

 the beginner a good start is treated. It is not 

 easy to communicate to the student the funda- 

 mental principles upon which the geometric 

 structure is to be based. A rigorous scientific 

 analysis or discussion of these principles is en- 

 tirely beyond his comprehension. It. is gener- 

 ally necessary for him to commit to memory 

 the first dozen pages, to exercise his powers of 

 reason by tracing the subsequent development, 

 and, having become somewhat familiar with the 

 methods of logical argument, to return to the 

 beginning of the text-book for a more complete 

 appreciation of the fundamental ideas. 



In order that the student may have some ac- 

 quaintance with geometric notions before tak- 

 ing up the study of demonstrative geometry, it 

 is desirable that such study should be preceded 

 by simple mechanical drawing and a course in 

 inventional geometry so-called. But the train- 

 ing that he will gain only from the study of 

 demonstrative geometry is necessary before he 

 can appreciate to any extent the character and 

 content of the definitions, axioms and postulates 

 of geometry. It is, of course, very satisfactory 

 to have the first few pages of the text-book con- 

 structed so that they meet all the needs of the 

 mature student who turns back to them for a 

 rigorous scientific discussion ; but it is of prime 

 importance that these pages should convey to 

 the beginner intelligibly and helpfully the in- 

 formation which is neccessary for his first steps 

 in the demonstration of geometrical truth. 

 That the definition of a straight line should be 

 based upon the notion of direction, as it is done 

 in the text-books of Wentworth and Wells in 

 Davies's Legendre, in Byerly's edition of 



Chauvenet and in other popular works, or that 

 it should be based on the notion of distance, as 

 is done in the original edition Chauvenet, are 

 points which may seem api^ropriate for criti- 

 cism to the mature student who seeks to sub- 

 ject every definition to a scientific analysis ; but 

 the beginner will accept at once certain notions 

 of direction and distance, of straight lines and 

 curved lines, and it will be fatal to his progress 

 to stop him at the threshold of the subject for a 

 complete discussion of these ideas. 



In the introductory portions of the three 

 books before us intuitive methods are used 

 most largely by Edwards; scientific accuracy of 

 treatment is maintained most fully by Beman 

 and Smith. For example, Edwards uses direc- 

 tion in defining a straight line, and assumes that 

 a straight line is the shortest distance between 

 two points. Van Velzer and Shutts avoid the 

 use of direction as a fundamental notion, but 

 adopt as an axiom that the straight line is the 

 shortest distance between two points. Beman 

 and Smith, however, wishing to make no un- 

 necessary assumptions, demonstrate at length, 

 like Euclid, that one side of a triangle is less 

 than the sum of the two other sides. The last 

 mentioned work will probably be most satisfac- 

 tory to the advanced student, but it is quite 

 likely that the beginner may prefer one of the 

 others. 



The Elements of Geometry by Edwards con- 

 sists of fourteen chapters, of which the first 

 eight relate to elementary plane geometry, the 

 following five to solid geometry, and the last 

 one to the conic sections. The propositions are 

 not numbered in the traditional manner, but 

 the work is divided into articles which are con- 

 secutively numbered. 



In many of the propositions the demonstra- 

 tion is preceded by a discussion, or ' analysis,' 

 in which are obtained the materials for the con- 

 struction of the formal proof. In many others 

 the demonstration is of an informal character, 

 a mere outline being given, which, however, 

 the student will have no difficulty in comple- 

 ting. Unusual and ingenious methods of proof 

 are frequent. The surfaces of the cylinder and 

 cone are measured by unwrapping them upon a 

 plane. In applying the method of successive 

 approximation to calculating the area of the 



