700 



SCIENCE. 



[N. S. Vol. V. No. 122. 



better if the real meaning of each expression 

 were clear from the beginning? 



The fourth book, by Professor Hathaway, 

 presents a much better introduction to the 

 method, and the student who masters it will find 

 that he has acquired some real knowledge, not 

 merely additional dexterity in formal manipu- 

 lations. The exposition, as a matter of logic 

 and of truth, is not all that can be desired, for 

 it is based partly on formal laws, partly on 

 mechanical truths. For example, the principle 

 that the addition of vectors is associative is 

 made to depend on an arbitrary definition of 

 the equality of vectors, but the same principle 

 for the product of quaternions is rested upon the 

 composition of rotations of a rigid body. 



The fifth book, by Mr. McAulay, has a dif- 

 ferent purpose from that of the others. It is 

 an essay, not an introduction or a treatise, and 

 the aim of the essay is to make good the fol- 

 lowing statements : First, that Quaternions are 

 in such a stage of development as already to 

 justify the practically complete banishment of 

 Cartesian geometry from physical questions of 

 a general nature ; and second, that Quaternions 

 will in physics produce many new results that 

 cannot be produced by the rival and older the- 

 ory. In the essay the author applies the quater- 

 nion analysis to the theories of elastic solids, 

 electricity and magnetism and hydrodynamics. 

 It is almost wholly a translation into quater- 

 nion notation of known results ; the author has, 

 however, endeavored to advance each of the 

 theories mentioned in at least one direction. 



It is evident that the utility of a method is 

 best proved not by any essay, but by its exten- 

 sive and fruitful use. How does it come about 

 that the method of quaternions is so far from 

 general and accepted use that it is still the sub- 

 ject of debate, misunderstanding and even ridi- 

 cule ? Not a few mathematicians agree with 

 the opinion expressed by a German mathema- 

 tician, that it is an aberration of the human in- 

 tellect. The answer to the above question I 

 believe to be as follows : 



In the books before us, and, indeed, in all 

 the works by members of the old school, it 'is 

 admitted, even proclaimed, that the Hamilto- 

 nian analysis is a rival of the Cartesian analy- 

 sis. Mr. McAulay talks of it as a new plant. 



independent of the old tree of analysis ; and in 

 their letterto Science proposing an international 

 association Dr. Molenbroek and Mr. Kimura 

 invited mathematicians to leave the old domain 

 of Cartesian analysis. Now, when one who 

 has been trained in the Cartesian analysis ap- 

 proaches the new method he finds that the no- 

 tation is strange and the conventions contra- 

 dictory of those to which he has been accus- 

 tomed ; consequently, he concludes, as David 

 did about Saul's armor, that it is better in 

 actual warfare to rely on a familiar weapon 

 than on one which may be superior but is un- 

 proved. 



What is the true relation of space-analysis to 

 the Cartesian analysis? The quaternionist 

 makes them rivals ; there is the blunder. Space- 

 analysis can be presented so as not to contra- 

 dict or rival the Cartesian analysis, but, on the 

 contrary, be consistent with and supplementary 

 to it. The relation of the former to the latter 

 is like that of algebra to arithmetic. Algebra 

 is universal arithmetic ; so space-analysis is 

 universal Cartesian analysis ; that is, it con- 

 siders the properties of vectors which are inde- 

 pendent of coordinates. Many theorems are 

 readily proved by algebra which it would be 

 difficult, if not impossible, to prove by arith- 

 metic ; similarly, many theorems can be read- 

 ily proved by space-analysis which it is 

 difficult, if not impossible, to prove by means 

 of coordinates. If we wish numerical results, 

 coordinates must be introduced, just as, if we 

 wish numerical results, numbers must be intro- 

 duced into the formula furnished by algebra. 



Some writers express the opinion that agree- 

 ment about notation is all that is required in 

 order to render space-analysis generally ac- 

 cepted. But it appears to me that the difficulty 

 is more deep-seated ; the fundamental princi- 

 ples need to be discussed, and no notation can 

 be adequate and lasting which is not built on 

 the simplest and truest principles. I may men- 

 tion briefly some points of principle which have 

 to be settled. 



It is unscientific to base the analysis partly 

 on formal laws, partly on physical principles. 

 By not distinguishing between simultaneous 

 and successive addition Hamilton failed to dis- 

 cover the true generalization for space of the 



