August 14, 1896.] 



SCIENCE. 



203 



circle of uuit radius, the formulae are put in the 

 very convenient form 



ttin \ (In An'' A-in 2 \An Oin / ' 



where small letters denote the areas of in- 

 scribed, large letters the areas of circumscribed 

 regular polygons, the subscript in each case in- 

 dicating the number of sides. Several of the 

 proofs given deserve censure. For example, 

 that in which the author claims to prove that 

 the sum of the exterior angles of a polygon is 

 four right angles is worthless, and that which 

 establishes the area of a rectangle is very in- 

 complete. 



The Plane and Solid Geometry of Yan Velzer 

 and Shutts consists of eight chapters, five re- 

 lating to plane geometry and three to geometry 

 of three dimensions. The so-called suggestive 

 method is employed. Each theorem is illus- 

 trated by a figure, and a series of suggestions 

 follows, so arranged as to indicate the s.ucces- 

 sive steps of the demonstration. Frequently a 

 ' model ' is given showing how the suggestions 

 are to be combined so as to produce a complete 

 formal proof. Numerous oi-iginal exercises are 

 scattered through the text, the whole number 

 being almost four hundred. Each chapter 

 closes with a syllabus of the preceding proposi- 

 tions so as to facilitate the student's reference to 

 them. 



Although the Plane and Solid Geometry of Be- 

 man and Smith is more extensive than either of 

 the other works under consideration, its publish- 

 ers by giving it a very compact form have been 

 able to make it even less bulky than the others. 

 It consists of eight books, of which five are de- 

 voted to plane geometry and three to solid 

 geometry. An appendix to book III. discusses 

 the various methods of attacking geometrical 

 propositions, while an appendix to book V. 

 treats of the special topics, maxima and mini- 

 ma, concurrence and collinearity. The work 

 contains almost eight hundred original ex- 

 ercises, and at the end are appended a table of 

 formulae and numerical constants, a biographical 

 table, a table of etymologies and an index. A 

 slight but sufficient treatment is given of sym- 

 metry, the notion of positive and negative 

 magnitudes, the principle of continuity, duality, 



the method of loci and the parallelism between 

 certain propositions of geometry and algebra. 

 The work thereby gains interest and modern- 

 ness. Clearness and conciseness are given to 

 the proofs by breaking them up into successive 

 steps numbered, in order that they may be the 

 better referred to, somewhat after the manner 

 of the equations in many text-books of algebra. 

 From a scientific point of view, this work is one 

 of the best yet published. 



Thomas S. Fiske. 

 Columbia University. 



American Fonostenografy . — A modern system 

 of rapid shorthand * * * formulating and 

 applying an entirely original principle of legi- 

 bility and brevity — the fonostenografic root. 

 By William McDevitt, LL.M. Judd & 

 Detweiler, Printers, Washington, D. C. 

 The author of this work has sought to raise 

 shorthand to the rank of an art of expression 

 coordinate with speech and writing, and he 

 has wrought with masterly hand his materials 

 into a definite system based on practical scien- 

 tific principles. The requirements of a rapid 

 and legible system of graphic expression are 

 clearly set forth and the means for supplying 

 them are provided ; and the structural laws on 

 which the system is based are fundamental 

 principles in the genesis and development of 

 human expression, which the author seems to 

 have attentively studied. He grasps firmly a 

 cardinal principle of linguistic growth, i. e., 

 that meaning and sense inhere in context and 

 not in the isolated syllable or word, and teaches 

 that he who sets himself the task of mastering 

 the composition of phrases, clauses and sen- 

 tences acquires thereby the key to successful 

 fonostenografic practice. A fundamental dis- 

 tinction between the principles of contraction 

 employed in this system of shorthand and the 

 rules of abbreviation applied in older stenog- 

 raphies is that in the latter vowels are not spe- 

 cifically symbolized because they are vowels, 

 consonants because they are consonants, and 

 final initial syllables because they are such — for 

 in McDevitt' s system of vocalized fonostenog- 

 rafy, contraction is attained by the omission of 

 factors or elements which are not ' strong and 

 significant, ' without regard to the adventitious 



