134 EULOGY ON AMPERE. 



of cure under my own eyes might give me some confidence in this mode 

 of treatment. 



The first, according to date, of all the mathematical memoirs of Am- 

 pere, printed after his arrival in Paris, relates to a question of elemen- 

 tary geometry. This memoir, presented to the Academy of Lyons in 

 1801, appeared in the publication of the correspondence of the Polytech- 

 nic School in the month of July, 180G. A few words will suffice to de- 

 scribe the end Ampere proposed to himself 



There is in elementary geometry a proposition so evident that it may 

 properly be regarded as an axiom. It is this : 



If two lines situated in the same plane are parallel, or, in other words, 

 if, prolonged indefinitely, can never meet ; and if a third line, forming an 

 angle at any point with the first of the two parallels, be indefinitely ex- 

 tended from the point of intersection, it will cut the second. iSTo one can 

 feel a doubt about this theorem, although all the ettbrts of the most cele- 

 brated geometers, the Euclids, the Lagranges, the Legend res, &c., to add 

 to its natural evidence by way of demonstration, properly so-called, have 

 been fruitless. 



The geometry of solid bodies, had ofl\?red, up to tlie present time, a 

 projiosition whose truth was quite as evident, and that, nevertheless, 

 had never been demonstrated. I refer to the equality of volume of sym- 

 metrical polyhedrons. Two obliipie polyhedrons have the same base 

 situated on a horizontal plane; one is entirely above the plane, the 

 other entirely below. Their faces are similar and of the same' length ; 

 moveover, their inclinations correspond exactly to a common base. To 

 give the same idea in diifei'eut words — one of the two polyhedrons being 

 considered as an object, the other would be its image refieeteil on the 

 plane of the common base, if that plane were a mirror. 



The object of Ampere's treatise is to demonstrate the equality of these 

 two polyhedrons; and it can be affirmed that, on this point, in the sci- 

 ence of geometry there is nothing more to desire. 



In 1803 M. Ampere addressed to the institute a very finished Mork, 

 which, however, was not given to the public until much later, (1808,) 

 entitled " Treatise on the advanta/jes to be derived, in the theory of curves, 

 from due consideration of osculatcry paraholas.^^ We also find a treatise by 

 Ampere dated the 26th florial year 11 which was published in the first 

 volume of the collection of the foreign savants of the Academy of 

 Sciences. This is its title " Investigations on the application of the gen- 

 eral formulas of the valcidiis of variations to problems in mechanics^ 



The formulas of equilibrium, given by the immortal author of ana- 

 lytical mechanics, have a form analogous to that of the equations that 

 the calculus of variations furnishes for the determination of the max- 

 ima and minima of integral formulas. Ampere thought that this simili- 

 tude of form, previously noticed b\' Lagrange, would aflbrd him the 

 means of avoiding, in the solution of questions in statics, the trouble- 

 some integration by parts. The analogy is not as complete as might be 



