940 



SCIENCE 



[N. S. Vol. XXXIX. No. 1017 



pharmacology and Dr. William Darrach has 

 been appointed assistant professor of surgery. 



Dr. Ross A. Gortner, since 1909 resident 

 investigator in biological chemistry at the 

 station for experimental evolution of the Car- 

 negie Institution of Washington, has been ap- 

 pointed associate professor of soil chemistry in 

 the University of Minnesota. 



Dr. Karl F. Meyer, vs^hose special field is 

 the tropical diseases, has been promoted to be 

 professor of bacteriology and protozoology in 

 the University of California. 



Dr. J. Howard Agnew, formerly first 

 assistant in the department of medicine, Uni- 

 versity of Michigan, has accepted the full time 

 professorship in medicine in the University of 

 Alabama, School of Medicine, at Mobile. 



At Dartmouth College, Drs. E. J. Eowe and 

 E. S. Allen have resigned as instructors in 

 mathematics, the latter to accept an instructor- 

 ship at Brown University. Dr. E. D. Beetle, 

 of Princeton University, and Dr. L. 0. 

 Mathewson, of the University of Illinois, have 

 been appointed instructors in mathematics. 



D. K. PiCKEN, professor' of mathematics in 

 Victoria College, University of New Zealand, 

 has been appointed master of Ormond College, 

 Melbourne University. 



DISCUSSION AND COBBESPONDENCE 

 dadourian's analytical mechanics 



In the issue of Science of April 3, Dr. 

 Dadourian replies to my criticism of his 

 " Analytical Mechanics." His reply was read 

 with interest. It was hoped that he would 

 clear up several points in this reply that 

 seemed to the reviewer as unsatisfactory. I 

 do not wish to get into a controversy, but it 

 seems to me that his standpoint is untenable. 

 He says in his reply: 



It is a fact that I have applied vector addition 

 to forces without hesitation, but I have shown as 

 little hesitation in treating velocities, accelerations, 

 torques, linear momenta and angular momenta as 

 vectors. Why did not Professor Eettger accuse me 

 of having assumed the ' ' parallelograms ' ' of these 

 magnitudes? Is the "parallelogram of forces" 

 more of a dynamical law than the parallelogram of 

 torques, for instance? The parallelogram law ap- 



plies to any vector and is not at all^a characteris- 

 tic of forces, therefore, it is not a dynamiSal law. 

 It does not even deserve being called a "law" 

 when applied to a special type of vectors. In its 

 most general form the "parallelogram law" is the 

 principle of the independence of mutually per- 

 pendicular directions in space, a purely geometrical 

 principle. . . . After devoting an entire chapter to 

 vector addition and after defining force as a 

 vector, to introduce the ' ' parallelogram of forces ' ' 

 as a new law, as Professor Rettger would have it, 

 could^ serve only to show that the man who did it 

 could not have a clear conception of the meanings 

 of the terms he was using. 



Let us assume that a body, originally in 

 the position 0, moves first through a distance, 

 a, in a given direction and then through a 

 distance, h, in another direction. Assume the 

 body finally to be in the position C The re- 

 sultant displacement then is 00 = c. The 

 body would be in the same position, C, if it 

 had moved first through the distance, h, and 

 then through the distance, a, that is, its final 

 position, or its final displacement is independ- 

 ent of the order in which the two displace- 

 ments take place. They may take place, there- 

 fore, simultaneously, and the final or resultant 

 displacement is still equal to c. If then we 

 recognize that the two displacements have no 

 mutual eilect on each other, or, what amounts 

 to the same thing, that the displacements are 

 independent of each other, then the resultant 

 displacement may be represented by the diag- 

 onal of a parallelogram of which the two dis- 

 placements are adjacent sides. As soon as this 

 " Principle of Independence " is once recog- 

 nized, then the " parallelogram law " can be 

 proved to hold also for velocities, accelera- 

 tions and other conceptions of kinematics. The 

 parallelogram law as applied to these quan- 

 tities is then equivalent to the " principle of 

 the independence of motions " and as such is 

 a purely " geometric principle." These quan- 

 tities, displacements, velocities and accelera- 

 tions are therefore vectors in accordance with 

 the definitions of a vector, and the principles of 

 vector analysis may be applied advantageously. 



Vector analysis may be called an algebra 

 that rests on certain (arbitrary) assumptions, 

 and the " parallelogram of vectors " is one of 



