CONDORCET: a BIOGRAPnY. 185 



Tutej^ral calculus of Coudorcet appears to me well wortby of the praises 

 with which you have honored it." 



But settiagf these authorities aside, it is none the less established that 

 this work contains the first serious well considered discussions of the 

 conditionsof integration of the ordinary differential equations of all orders, 

 as well relatively to the integral of an immediately inferior order, as to 

 the deiiuitive integral. In it we also find the germs of several import- 

 ant works, since completed, on equations with finite differences. 



The volume of the Academy of Sciences for 1772 contains the memoir 

 in which the inventive spirit of Coudorcet is most brilliantly manifested. 

 The blind or systematic detractors of the mathematical ability of our 

 former secretary are again controverted by the following verdict of La- 

 grange upon this production : 



" The memoir is filled with great and fruitful ideas sufficient to have 

 supplied the material for several works. The last article has esi)ecially 

 impressed me by its elegance and utility. Recurrent series have so 

 frequently been treated before, that the subject-might have been con- 

 sidered exhausted. In this article, however, is a new application of 

 these series, more important, iu my opinion, than any which had been 

 made before. It opens to us, so to say, a new field for the perfection 

 of the Integral calculus." 



Without leaving the field of pure mathematics, I might find in the 

 academic collections of Paris, Berlin, Bologna, St. Petersburg, works bear- 

 ing ui)ou the most difficult questions of the science which would equally 

 attest the ability of our former secretary, but I must hasten to notice 

 some applications of £nalysis which did him no less honor. To do jus- 

 tice to the subject in any reasonable time I cannot proceed by order of 

 date. 



When we refl.ect upon the difficulties of all kinds astronomers have 

 to overcome iu order to determine \\ith precision the orbits of the 

 planets; when we consider, further, that it has been possible to har- 

 monize the positions taken by the planets at the apogee, at the perigee, 

 and all the intermediate points, only because they are constantly observ- 

 able, we can hardly dare to conceive the hope of ever tracing iu si)ace the 

 course of most of the comets, those vagrant stars which show themselves 

 for a few days, only to be lost for centuries. 



A very simple analytical calculus dissipates this impression. It shows 

 that, speaking theoretically, three observations are more than sufficient 

 to determine a comet's orbit, supposed to be parabolic, but the elements 

 of this orbit are found to be so entangled in the equations that it appears 

 difficult to free them without calculations of inconvenient length. 



The problem thus regarded was not really solved, even after Newton, 

 Fontaine, Euler, &c., had made it the subject of assiduous study. When 

 the Academy of Berlin proposed it as a prize subject, the astronomers, 

 instead of employing the methods of computation of these great geome- 

 ters, still pursued the graphic systems, in which parabolas of card-board 



