202 THE POPULAR SCIENCE MONTHLY 



system than quaternions should be proposed. " The ideas which 

 flashed into the mind of Hamilton at the classic Brougham Bridge " 

 became the occasion of a joined battle between the perfervid clan- 

 loyalty of the Celt and the cool individualism of the Saxon ; on one side, 



" The broad Scots tongue that flatters, scolds defies, 

 The thick Scots wit that fells you like a mace," 



and on the other, the overconscientious, ethical arguments of a super- 

 sensitive spirit, obviously nettled at certain rough pleasantries which 

 were understood but not appreciated. In 1893 Heaviside, an English 

 vectorist, reports " confusion in the quaternionic citadel : alarms and 

 excursions and hurling of stones and pouring of water upon the invad- 

 ing hosts." 155 The vectorists were denounced as a " clique " and ridi- 

 culed especially for their lack of elegance, their alleged intellectual 

 dishonesty and the fact that their pupils were " spoon-fed " upon 

 mathematico-physical pap. But some of the notations held up to ridi- 

 cule turned out to be things like Poisson's theorem or the difficult 

 hydrodynamic problem "given the spin in a case of liquid motion to 

 find the motion," which Helmholtz solved with one of his strokes of 

 genius, and which Gibbs showed could be understood and interpreted 

 by the average student without genius by a simple application of vecto- 

 rial methods. The real point at issue in the controversy, the funda- 

 mental difference in the ideals of European and American education, 

 lies here. Both have their relative advantages and defects, but the 

 object of one has been to bring the best to the highest development, 

 while the other is concerned with increasing the efficiency of the aver- 

 age man. One has been exclusive, aiming at the survival of the fittest ; 

 the other is democratic and inclusive, and aims, in Huxley's words, to 

 make the greatest number fit to survive. The merits of the case are 

 well summed up in Gibbs's final statement : " The notions which we 

 use in vector analysis are those which he who reads between the lines 

 will meet on every page of the greatest masters of analysis, or of those 

 who have probed deepest the secrets of nature, the only diffrence being 

 that the vector analyst, having regard for the weakness of the human 

 intellect, does as the early painters who wrote beneath their pictures 

 ' This is a tree.' ' This is a horse.' " 156 This view is in perfect accord 

 with the recent trend of mathematical teaching, European or Amer- 

 ican, which is to emphasize the meaning and interpretation of equa- 

 tions and formulas rather than their demonstrations or manipulation; 

 in short, to substitute visualizing methods, the art of thinking straight 

 and seeing clear, for what is conventional and scholastic. A Harvard 

 professor is said to have told his students that the demonstration of a 

 theorem is no evidence that it is understood, but the intelligent use of 

 it is; and the object of such teaching as Gibbs's was to enable the 



™Ibid., 1892-3, XL VII., 534. 

 ™Ibid., 464. 



