JANUAET 23, 1914] 



SCIENCE 



141 



that a couple (for instance) is a vector and 

 the desired equations follow at once. If 

 economy of space means " economy of thought " 

 then the author has made his book very simple 

 indeed. To prove the equivalence of couples 

 it is only necessary to state : " Two couples 

 are equal when the vectors which represent 

 their torques are equal in magnitude and 

 have the same direction." This follows at once 

 from his definition of a vector. 



He states in his preface " that a subject like 

 mechanics should start with a few simple laws 

 and the entire structure of the science should 

 be based upon them. In the present work the 

 following law is made the basis of the entire 

 subject : 



" To every action there is an equal and oppo- 

 site reaction, or, the sum of all the actions to 

 which a hody or a part of a hody is subject at 

 ■any instant vanishes." He further states that 

 thus the " fundamental principle of mechanics 

 is put in the form of a single law, which is 

 equivalent to Newton's laws of motion and 

 -which has the advantages of the point of 

 view involved in D'Alembert's principle." 



Here is a unique departure for an elementary 

 book. Does he mean to say that this law, 

 ■whatever it may mean, is the only assumption 

 he will make and that Newton's laws of motion 

 as usually given will not be made use of? If 

 he does, he completely fails. On page 16, he 

 introduces the conception of " force " as an 

 " action," and without any hesitation applies 

 vector addition to a system of forces. What is 

 lie doing here, but assuming the " parallelo- 

 gram of forces " in its most general form. On 

 page 102 he assumes that a force is propor- 

 tional to the accleration produced. This as- 

 sumes Newton's second law of motion. In 

 fact he makes more assumptions than are 

 usually made in elementary text-books of 

 mechanics. 



What about the law itself? The first part 

 of the law is clear. " To every action there is 

 an equal and opposite reaction " is nothing 

 iut Newton's third law of 'motion. The word 

 " or " leads us to think that the second part 

 means the same thing as the first part. On 

 page 15 he states that " The fundamental law 



of mechanics is known as the law of action 

 and reaction." He then states Newton's third 

 law and gives the following illustration. " Let 

 us apply this law to the interaction between a 

 book and the hand in which you hold it. Tour 

 hand presses upward upon the book in order to 

 keep it from falling, while the book presses 

 downward upon your hand. The law states 

 that the action of your hand equals the re- 

 action of the book and is in the opposite direc- 

 tion. The book reacts upon your hand because 

 the earth attracts it. When your hand and the 

 earth are the only bodies which act upon the 

 book, the action of your hand equals and is 

 opposite to the action of the earth. In other 

 words, the sum of the two actions is nil. 

 Generalizing from this simple illustration, we 

 can put the law into the following form: 



" To every action there is an equal and oppo- 

 site reaction, or the sum of all the actions to 

 which a body or a part of a body is subject at 

 any instant vanishes." 



Now does he mean to say that the pressure 

 of the hand on the book and the force of 

 gravity acting on the book are equal because 

 they are action and reaction? If he does he 

 errs. They are not action and reaction and 

 are equal only in case equilibrium exists. He 

 has said nothing about equilibrium and if he 

 does not mean this then what does he mean? 



On page 100 he takes up the subject of 

 " motion of a particle." Here he says that 

 " we must extend the meaning of the term 

 reaction so as to include a form of reaction 

 which is known as kinetic reaction^ In his 

 illustration we see that by kinetic reaction he 

 means the so-called force of inertia. We also 

 see that he considers kinetic reaction as a real 

 force. To this, serious objections can be raised. 

 The meaning of the second part now becomes 

 clear. It is simply D'Alembert's Principle — 

 " The impressed forces together with the re- 

 versed effective forces form a system in 

 equilibrium." 



I fail to see, however, the advantage of 

 assuming D'Alembert's principle as a funda- 

 mental law of mechanics, especially since he 

 finds it necessary, in reality, to assume all of 

 Newton's laws besides. Moreover the law itseK 



