52 



SCIENCE. 



[N. S. Vol. VIII. No. 184. 



is as true now as ever : ' ' Sans les mathematiques 

 ou ne penetre point au fond de la pliilosophie ; 

 sans la pbilosophie on ne penetre point au fond 

 des mathematiques ; sans les deux, on ne penetre 

 au fond de rieu.'' 



The author bases the plan of the volume on 

 an opposition between pure mathematics and 

 applied mathematics. It is pure mathematics 

 which furnishes us with infallible formulas and 

 deductions ; it is applied mathematics which 

 completes the work by showing the necessary 

 existence of errors. The distinction made is 

 largely that between rational and empirical 

 laws in physics. It is not that the one is in- 

 dependent of experience and the other depen- 

 dent on it ; for both are based on experience 

 (p. 10). Why the formulas of the one are in- 

 fallible, while those of the other are not, is not 

 made very clear. 



The author is not satisfied with any of the 

 current definitions of mathematics, but prefers 

 to describe it by means of its object. The essen- 

 tial object of all science, according to M. Lais- 

 ant, is the study of the phenomena presented 

 by the external world ; and the mathematical 

 treatment of these phenomena consists of three 

 stages : forming the equations, solving the equa- 

 tions, interpreting the results. 



As regards the importance of mathematical 

 science, the author quotes the saying of Kant : 

 "A natural science is a science only so far as it 

 is mathematical," and that other saying at- 

 tributed to Napoleon the First : " The advance- 

 ment and progress of mathematics are bound 

 up with the prosperity of the State." Accord- 

 ing to M. Laisant it is the most marvelous in- 

 strument created by the genius of man to aid 

 in the discovery of the truth. 



At page 30 the author takes the ratio or rela- 

 tive view of algebraic quantity and is led to 

 look upon multiplication as the formation of a 

 number (the product) which has to the multi- 

 plicand the same ratio as that which the multi- 

 plier has to unity. While this may be sufiicient 

 for elementary arithmetic, it does not satisfy 

 the calculus of the mathematical physicist, for 

 with him a symbol stands for a magnitude of 

 some kind, not for its ratio to some supposed 

 unit. 



The author criticises in vigorous terms the 



procedure adopted by some of giving mathe- 

 matics a definitional or ideal basis. It is march- 

 ing in the direction opposite to progress, and 

 is a revival of the attitude of the sophist. 

 Algebra is not one, but several, because the 

 properties of the quantities which it treats are 

 different in their nature. Thus the rules of the 

 method of quaternions differ from those of 

 algebra not by arbitrary definition, but because 

 the geometry of space demands it. At another 

 place the author suggests that the method of 

 quaternions when more fully developed is cer- 

 tain to modify and advance the theory of func- 

 tions. 



The author says that the fundamental error 

 of the ancient geometers was that they did not 

 recognize the experimental element which is at 

 the base of geometry ; when it is recognized, the 

 non-Euclidean geometry becomes a logical pos- 

 sibility. 



What is said on the teaching of mathematics 

 refers to France, but much of it is valid every- 

 where. In the domain of primary mathematics 

 the pupil ought to be interested, encouraged to 

 make research, be given the feeling — the illu- 

 sion, if you will — that he is discovering for him- 

 self what he is being taught. This demands in 

 the first place a teacher of a high order, and in 

 the second place that the class be small and 

 homogeneous. When the class become a flock 

 and the teacher a shepherd's dog, no progress 

 is possible. The fundamental principles ought 

 to be singled out and exhibited in all their rela- 

 tions, and their meaning made clear by frequent 

 applications. Memory ought to play a very 

 subordinate part ; it is the understanding which 

 ought to be exercised. In the domain of the 

 more advanced instruction he does not favor 

 the study of a text-book, still less dictated les- 

 sons. He recommends notes taken by the pupil 

 of oral lessons. In the highest domain he 

 recommends the taking of notes, supplemented 

 by a text-book. 



For the first introduction to arithmetic 

 he recommends a rigorously experimental 

 method ; to have the child to make his own 

 notions in the presence of realities which he can 

 touch and see, to demonstrate nothing, and to 

 make the instruction have the appearance of 

 play rather than of work. In this way a firm 



