126 



NATURE 



[February i, 1894 



Euler or Lagrange. On the subject of mathematics the writer's 

 conclusion was that fruitful investigation seemed at an end, and 

 that there was little prospect of brilliant discoveries in the 

 future. To us, a century later, this judgment might seem to 

 illustrate the danger of prophesying, and lead us to look upon I 

 the author as one who must have been too prone to hasty con- [ 

 'ilusions. I am not sure that careful analysis would not show j 

 the P.uthor's view to be less rash than it may now appear. May i 

 we iiot say that in the special direction and along the special 

 lines which mathematical research was following a century ago 

 no very brilliant discoveries have been made? Can we really 

 say that Euler's field of work has been greatly widened since | 

 his time? Of the great problems which baffled the skill of the | 

 ancient geometers, including the quadrature of the circle, the ! 

 duplication of the cube, and the trisection of the angle, we have j 

 not solved one. Our only advance in treating them has been i 

 to show that they are insoluble. To the problem of three 

 bodies we have not added one of the integrals necessary to the i 

 complete solution. Our elementary integral calculus is two 

 centuries old. For the general equation of the fifth degree we ! 

 have only shown that no solution exists. We should, doubt- 

 less, solve many of the problems which the Bernoullis and their j 

 contemporaries amused themselves by putting to each other, 

 rather better than they did ; but, after all, could we get any 

 solution which was beyond their powers? I speak with some j 

 diffidence on such a point as this ; but it seems to me that pro- \ 

 gress has been made by going back to elementary principles, 

 and starting out to survey the whole field of mathematical in- 

 vestigation from a higher plane than that on which our pre- \ 

 decessors stood, rather than by continuing on their lines. : 



We may illustrate this passage to new modes of thought by 

 comparing Euclid's doctrine of ratio and proportion with our 

 own. No one questions the beauty or rigour of the process by 

 which Euclid developed this doctrine in his fifth book, and 

 applied it to the theory of numbers in his seventh book. But | 

 can we help pitying our forefathers who had to learn the 1 

 complex propositions and ponderous demonstrations of the 

 fifth book, all the processes and results of which we could 

 now write on a single sheet of paper ? As a mental discipline 

 the study was excellent ; but it seems hardly possible that 

 one could have remembered the propositions or the methods i 

 of demonstrating them if he had no other knowledge of 

 them than that derived from the work itself. When we care- 

 fully examine these propositions, we find that while Euclid 

 recognised the fact that one of two ratios might be greater than, 

 equal to, or less than another, yet he never regarded them as 

 mere quantities which could be treated as such. From his 

 standpoint a ratio was always a relation, and a relation cannot 

 exist without two terms. 



In pointing out this complexity of Euclid's doctrine, I 

 must not be taken to endorse the very loose way in which the 

 doctrine in question is usually treated in our modern text- 

 books. What we should aim at is to replace Euclid's methods 

 by those which pertain to modern mathematics. At the 

 present time we conceive that a relation between any two 

 concepts of the same kind may always be reduced to a single 

 term by substituting for it an operator whose function it is 

 to change one of these concepts into the other. In the case 

 of the relation between two lines, considered simply as one 

 dimensional quantities, which relation is called a ratio, we 

 regard the ratio as a numerical factor or multiple, which, oper- 

 ating on one line, changes it into the other. For example, 

 that relation which Euclid would have expressed by saying 

 that two lines were to each other as 5 to 2, or that twice one 

 line was equal to five times the other, we should now express 

 by saying that if we multiplied one of the lines by two and one 

 half, we should produce the other. This might seem to be 

 simple difference of words, but it is much more. It is a simpli- 

 fication of ideas ; a substitution of one conception for two. 

 Euclid needed two terms to express a relation ; we need but 

 one. 



But this is not the only simplification. A peculiarity of our 

 modern mathematics is that operators themselves are regarded 

 as independent objects of reasoning ; susceptible of becoming 

 operands, without specification of their particular qualities as 

 operators. 'J'hus, instead of considering the ratio which I have 

 just mentioned as an operation of multiplying a line by two and 

 one half, we finally reduce it to the simple quantity two and 

 one half, which we may conceive to remain inert until we bring 

 it into activity as a multiplier. It thus assumes a concrete form, 



capable of being carried about in thought, and operated upon as 

 if it were a single thing. 



This example may afford us a starting-point for a farther 

 illustration of the way in which we have broadened the con- 

 ceptions which lie at the basis of mathematical thought. Let us 

 reflect upon the relation between a straight line going out from a 

 certain point, and another line of equal length going out from 

 the same point at right angles to the first. Had this relation 

 been presented to Euclid as a subject for study, he would pro- 

 bably have replied that though much simpler than those he 

 was studying, he could see nothing fruitful in it, and would 

 have drawn no conclusions from it. But if we trace up the 

 thought we shall find a wide field before us, embracing the first 

 conception of groups, and with it an important part of our 

 modern mathematics. In accordance with the principle already 

 set forth, we replace the relation between these two lines by an 

 operator which will change the first into the second. We define 

 this operator by saying that its function is to turn a line through 

 a right angle in a fixed plane containing the line. This definition 

 permits of the operator in question being applied to any line in 

 the plane. Then let us apply it twice in succession to the same 

 line. The result will be a line pointing in the opposite direc- 

 tion from the original one. A third operation will bring it 

 again to a right angle on the opposite side from the second 

 position ; and a fourth will restore the line to its original 

 position, the result being to carry it through a complete circle. 

 If we now consider the operations which would have been 

 equivalent to these one, two, three, and four revolutions through 

 a right angle as four separate operators, we see that their results 

 will be either to leave the line in its original position, or to 

 move it into one of three definite positions. If we then repeat 

 one of these four operations as often as we please, or in any 

 order we please, we shall only bring the line to one of the four 

 positions in question. We thus have a group of the fourth 

 order, possessing the property that the repetition of any two 

 operations of the group is equivalent to some single operation 

 of it. 



I scarcely need call attention to the familiar homology between 

 these operations and successive multiplications by the imaginary 

 unit \'-i. This last concept, considered as a multiplier, has 

 the same properties as our rotating operator. Repeated twice, 

 it changes the sign or direction of the quantity on which 

 ; operates ; repeated four times, it restores it to its original 

 value. Let us extend this idea a little. Instead of 

 taking two lines at right angles to each other, let us con- 

 sider two which form an angle of 40°. As already re- 

 marked, this relation is homologous with an operator which 

 will turn a single line through that angle. If we continually 

 rep eat this operation, we .^hall bring the line into thirty-five 

 different positions, the thirty-sixth position being identical with 

 the original one. Thus we should have thirty-six positions in 

 all, expressed by that number of lines radiating from a single 

 centre, and making angles of 10 with each other. Now let 

 us imagine thirty-six operators whose function it is to turn a 

 line, no matter what, successively through an arc of 10", 20°, 

 30°, &c. up to 360°, the last being equivalent to an operator 

 which simply does nothing. These thirty-six operators will 

 form a group which we know to be strictly homologous with 

 multiplication by the thirty-six expressions 



e^'^, «-'</>, £"'</>, . 



''■(^ = e" — \, 



NO. 1266, VOL. 49] 



where <^ is the arc of 10" in circular measure. 



So far we have only considered operations formed by the 

 continual repetition of a single one ; in the language of the 

 subject, all our groups are constructed from powers of a single 

 operator. Now let us extend our process by substituting a 

 cube for our straight line. Through this cube we have an axis 

 parallel to four of its plane sides. By rotating the cube 

 through any multiples of 90° around this axis we effect an inter- 

 change of position between four of its sides. This process of 

 interchanging is homologous with rotation through 90°, being 

 in fact equivalent to it, and therefore it is also homologous 

 with multiplication by the imaginary unit. But there is also 

 another homology. Let us designate the four sides of the cube 

 parallel to the axis of rotation as A, B, C, D. Then our group 

 of rotations will be homologous with the powers of a cyclic 

 substitution between the four letters A, B, C, D. 



Let us next introduce a new operator, namely, rotation 

 around an axis at right angles to the first one, but always 



