130 



SCIENCE. 



[N. S. Vol. I. No. 5. 



Sylvester afterward contracted Tclieby- 

 cliev's limits ; but the original paper remains 

 highly remarkable, especially as it depends 

 on very elementary considerations. 



In this respect it is in striking contrast 

 to the equally marvelous paper of the la- 

 mented Riemann, Ueber die Anzahl der 

 Primzahlen unter einer Gegegebenen Chvsse 

 presented to the Berlin Aeademia in 1859. 

 Tch6bychev had in 1848 presented a paper 

 with this very title to the St. Petersburg 

 Academie ; Stir la totalite des nombres pre- 

 miers hiferieurs a line limite doninee. (Giv- 

 en in Liouville's Joiu-nal, 1852, pp. 341- 

 365.) 



Eiemann speaks of the interest long be- 

 stowed on this subject by Gauss and Di- 

 richlet, but makes no mention of Tcheby- 

 chev. However, Sylvester speaks of ' his 

 usual success in overcoming diificiilties in- 

 superable to the rest of the world.' 



But though best known for his work in 

 the most abstract part of mathematics, in 

 reality Tchebychev was of an eminently 

 practical turn of mind. 



Thus it was his work, Theorie des meehan- 

 ismes eonniis sous le nom de parallelogrammes 

 (Memoirs des savants etrangers, Tom. 

 VII.), which led him to the elaborate dis- 

 sertation Sur les questions de minima qui se 

 rattachent a la representation approximate des 

 fondions, 91 quarto pages in Memoirs de 1' 

 Academic Imperiale des Sciences de Saint 

 Petersbourg, 1858. While the variable x 

 remains in the vicinity of one same value 

 we can represent with the greatest possible 

 approximation any function / (x) , of given 

 form, by the principles of the differential 

 calculus. But this is not the case if the va- 

 riable X is only required to remain within 

 limits more or less extended. The essen- 

 tially different methods demanded by this 

 case, which is just the one met in practice, 

 are developed in this memoir. 



The same line of thought led to his con- 

 nection with a subject which has since found 



a place even in elementary text-books, 

 namely rectilineal motion by linkage. 



He invented a three-bar linkage, whicli is 

 called Tchebychev's parallel motion, and 

 gives an extraordinarily close approxima- 

 tion to exact rectilineal motion ; so much 

 so that in a piece of apparatus exhibited by 

 him in the London Loan Collection of Scien- 

 tific Apparatus, a plane supported on a 

 combination of two of his parallel motion 

 linkages seemed to have a sti'ictlj' horizon- 

 tal movement, though its variation was 

 double that of the tracer in the simple par- 

 allel motion. 



Tchebychev long occupied himself with 

 attempting to solve the problem of produc- 

 ing exact rectilineal motion bj' linkage, un- 

 til he became convinced that it was impos- 

 sible and even strove long to find a proof of 

 that impossibility. What miist have been 

 his astonishment then, when a freshman 

 student of his own class, named Lipkin, 

 showed him the long sought conversion of 

 circular into straight motion. Tchebychev 

 brought Lipkin's name before the Russian 

 government, and secured for him a substan- 

 tial reward for his supposed original dis- 

 covery. 



And perhaps it was independent, but it 

 had been found several years pre^dously by 

 a French lieutenant of engineers, Peaucel- 

 lier, and first published by him in the form 

 of a question in the Annales de Mathema- 

 tiqne in 1864. When Tchebj'chev was on 

 a visit to London, Sylvester inquired after 

 the progress of his proof of the impossibility 

 of exact parallel motion, Avhen the Russian 

 announced its double discovery and made a 

 drawing of the cell and mounting. This 

 Sylvester happened to show to Manviel Gar- 

 cia, inventor of the laryngoscope, and the 

 next day received from him a model con- 

 structed of pieces of wood fastened \vith 

 nails as pivots, which, rough as it was, 

 woi-ked perfectly. Sylvester exhibited this 

 to the Philosopliical Club of the Royal So- 



