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 invading hosts."[1] The vectorists were denounced as a "clique" and ridiculed 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 ridicule 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 vectorial methods. The real point at issue in the controversy, the fundamental 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 average 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 difference 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."[2] This view is in perfect accord with the recent trend of mathematical teaching, European or American, which is to emphasize the meaning and interpretation of equations and formulae 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
