Aeeil 6, 1900.] 



SCIENCE. 



533 



"While the bulk of the mathematicians are 

 reveling in the new fields of thought which 

 are opening up on all sides, without any 

 thought in reference to any immediate prac- 

 tical application of their results, there is 

 fortunately a goodly number whose main 

 efforts are devoted towards making some of 

 these new results useful to the investigators 

 in some of the other sciences. As an in- 

 stance of fairly recent work of the latter 

 kind, we may mention the study of linkages 

 with a view towards describing a straight 

 line. Although the straight line is of such 

 fundamental importance both in pure and 

 applied mathematics, yet it seems it was 

 not until the latter half of the nineteenth 

 century that a mechanical device was dis- 

 covered by means of which such a line can 

 be described. 



In 1864 M. Peanceliier, an officer of engi- 

 neers in the French army, discovered the 

 well known device to describe a straight line 

 by means of an instrument composed of 

 seven links. " His discovery was not at 

 first estimated at its true value, fell almost 

 into' oblivion, and was rediscovered by a 

 Eussian named Lipkin, who got a substan- 

 tial reward from the Russian government 

 for his supposed originality. However M. 

 Peancellier's merit was finally recognized 

 and he has been awarded the great mechan- 

 ical prize of the Institute of France, the 

 Prix Montyon." * 



Although the straight line and the circle 

 occupy such a prominent place in elemen- 

 tary geometry and have been before the 

 eyes of the mathematicians for thousands 

 of years, yet less than half a century 

 has passed since the invention of a me- 

 chanical device by means of which the 

 straight line can be drawn. Such discov- 

 eries go far towards emphasizing the need 

 of investigations even in the most elemen- 

 tary subjects. Such investigations should, 

 however, be preceded by a thorough knowl- 



*A. B. Kempe, 'How to draw a straight line,' p. 12. 



edge of what has been done along the same 

 lines. 



If elementary mathematics is to continue 

 to furnish the best possible preparation for 

 the study of advanced mathematics, it is 

 evident that it has to adapt itself to the 

 rapid changes which are going on in the 

 different branches of mathematics. A need 

 is thus created for elementary text-books 

 which meet the new requirements, and we 

 are happy to be able to state that such books 

 are being produced in our midst. How 

 radical such changes may become cannot be 

 foretold. In his address before the New 

 York Mathematical Society, Simon New- 

 comb said, " The mathematics of the 

 twenty-first century may be very different 

 from our own ; perhaps the schoolboy will 

 begin algebra with the theory of substitu- 

 tion groups, as he might now but for in- 

 herited habits."* It is to be hoped that 

 our inherited habits will not furnish an in- 

 surmountable barrier to progress in this di- 

 rection. 



In modern times the continent of Europe 

 has always been the most progressive and 

 most of the new theories were first devel- 

 oped in these countries. The theory of in- 

 variants seems to be an exception to this 

 rule. The two great English mathemati- 

 cians, Cayley and Sylvester, developed this 

 theory with great vigor ; when their impor- 

 tant results became generally known on the 

 continent (largely through the work of 

 Clebsch), they aroused a great deal of in- 

 terest and they furnished the starting point 

 for many important investigations. 



One of the fundamental processes of 

 mathematics is transformation — the deduc- 

 ing of truths from given facts and relations. 

 The expressions which remain invariant 

 when given transformations are performed 

 are naturally objects of great interest and 

 of fundamental importance. Imbued with 



* Bulletin of the New York Mathematical Society, 

 1891, p. 95. 



