91

Plus ga change ...

The following quotation is reproduced verbatim, except for a few deletions that would too easily give the game away. The object of the game is to answer the following

questions: 1. Name the deleted mathematicians.

(Score 2 points each .) 2. When was the quotation published?

(Score max (0, 10 - [g - a I), where g is your guess and a the correct answer given below.)

3. Who wrote it, and where was it published? (Score 5 points for the author, 3 for title, 2 for publisher. In practice you either get 10 or 0.)

Recent Times

Never more zealously and successfully has mathematics been cultivated than in this century. Nor has progress, as in previous periods, been confined to one or two countries. While the French and Swiss, who alone during the preced- ing epoch carried the torch of progress, have continued to develop mathematics with great success, from other coun- tries whole armies of enthusiastic workers have wheeled into the front rank. Germany awoke from her lethargy by bringing forward G . . . . . . . . . . . J . . . . . . . . . . , D . . . . . . . . . . . and hosts of more recent men; Great Britain produced her D . . . . . . . . . . , B . . . . . . . . . . . H . . . . . . . . . . . besides champions who are still living; Russia entered the arena with her L . . . . . . . . . . ; Norway with A . . . . . . . . . . ; Italy with C . . . . . . . . . . ; Hungary with her two B . . . . . . . . . . s; the United States with

P . . . . . . . . . . . The productiveness of modern writers has been enorm-

ous. "It is difficult," says Professor C . . . . . . . . . . , " to give an idea of the vast extent of modern mathematics. This word 'extent ' is not the right one: I mean extent crowded with beautiful detail - not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful coun- try seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood, and flower." It is pleasant to the mathematician to think that in his, as in no other science, the achievements of every age remain possessions forever; new discoveries seldom disprove older tenets; seldom is anything lost or wasted.

If it is to be asked wherein the utility of some modern extensions of mathematics lies, it must be acknowledged that it is at present difficult to see how they are ever to become applicable to questions of common life or physi- cal science. But our inability to do this should not be urged as an argument against the pursuit o f such studies. In the first place, we know neither the day nor the hour when

these abstract developments will find application in the mechanic arts, in physical science, or in other branches of mathematics. For example, the whole subject of graphical statics, so useful to the practical engineer, was made to rest upon Von Staudt 's Geometrie der Lage; Hamilton's "prin- ciple of varying action" has its use in astronomy; complex quantities, general integrals, and general theorems in inte- gration offer advantages in the study of electricity and magnetism. "The utility of such researches," says S . . . . . . . . . . . "can in no case be discounted, or even imagined beforehand. Who, for instance, would have sup- posed that the calculus of forms or the theory of substitu- tions would have thrown much light upon ordinary equa- tions; or that Abelian functions and hyperelliptic trans- cendants would have told us anything about the properties of curves; or that the calculus of operations would have helped us in any way towards the figure o f the earth?" A second reason in favour of the pursuit of advanced mathematics, even when there is no promise of practical application, is this, that mathematics, like poetry and mu- sic, deserves cultivation for its own sake.

The great characteristic of modern mathematics is its generalizing tendency. Nowadays little weight is given to isolated theorems "except as affording hints of an unsus- pected new sphere of thought, like meteorites detached from such undiscovered planetary orb of speculation." In mathematics, as in all true sciences, no subject is con- sidered in itself alone, but always as related to, or an out- growth of, other things. The development of the notion of continuity plays a leading part in modern research. In geometry the principle of continuity, the idea of corre- spondence, and the theory of projection constitute the fundamental modern notions. Continuity asserts itself in a most striking way in relation to the circular points at infinity in a plane. In algebra the modern idea finds expres- sion in the theory of linear transformations and invariants, and in the recognition of the value of homogeneity and symmetry.

(Answers on page 92)

Two Problems

Problem 3.1. Let a, 3, 3' be complex numbers, each of unit norm and with a37 = 1. Find the set of all possible values for a+~3+ 7,

(A related problem is the following. For any n x n ma- trix define the normalized trace to be the usual trace divid- ed by n. Find the range of the normalized trace on all uni-

tary matrices.)