l38 MEMOIR OF LEGENDRE. 



tioii, l)ut the abbe felt it to be a duty to indicate to men of science tlie passages 

 wliicli had proceeded from the pen of the young Legendre, aged at that time 

 22 years. Among these passages is the definition of accelerative forces, dis- 

 tinguished bj' a precision and clearness of expression which seem sometimes to 

 be among the happ}^ privileges of youth. This definition is so natural, and 

 iiow so familiar to scientific minds, that, when lecalled, it is with difficulty con- 

 ceived how it could ever have presented anything of originalit}' and novelty. 

 It is but just to say that it forms no exceptional feature in the work of the Abbe 

 Marie, who, in many respects, was in advance of his age, and whose merit was 

 not limited to that of having divined the talents of Legendre. 



D'Alembert had said, with just foresight, that the fate of the new calculus 

 (differential and integral) would depend on the reception it met with from the 

 younger geometers; these therefore he souglit to allure to the method in ques- 

 tion, and which was as yet imperfectly comprehended, by the degree of esteem 

 and consideration which he accorded to such among them as evinced a capacity 

 for following it. He was not likely long to overlook the penetrating and pre- 

 cocious talent which disclosed itself in the young Legendre 5 and scarcely had 

 the first ^impses of genius given presage of what might be expected from the 

 disciple of the Al)be Marie, when he was named professor of mathematics at the 

 military school of Paris. Here, from 1775 to 1780, he continued to give les- 

 sons on the scientific grounds of the military art to that ardent and intelligent 

 body of 3'outlis from which have sprung not a few of om" warlike celebrities, and 

 whose number would have been more considerable, had not circumstances 

 forced a part of them into emigration. It may be inferred that the instruction 

 given by the young professor embraced the first elements of halistlcs, the art, 

 namely, of throwing projectiles, and that he studied the learned treatises which 

 Bezout, Borda, and other eminent men had published on these difficult problems ; 

 for when the Royal Academy of Sciences and Belles-lettres of Prussia proposed, 

 for the prize of 1782, the question of determining the curve described by balls and 

 shells, regard being had to the resistance of the air, and giving the rides for ascer- 

 taining the range tvhich corrcsjyonds to different initial velocities and to different 

 angles of ])rojeciion, M. Legendre was quite in readiness to enter into the compe- 

 tition. His memoir, prepared on this occasion, was crowned with success in the 

 pubfic meeting of June 6, 1782, and was published at Berlin under the title 

 oi Eecherches sur la trajectoire des projectdes dans Ics milieux resistants.* 



Newton, it is stated in this memoir, was the first who made researches respect- 

 ing trajectories in resisting mediums. He particularly considers that which takes 

 place on the hypothesis of a resistance proportional to the simple velocity; but 

 he gives merely approximations, and those but rough ones, for the trajectory 

 which results when the resistance is proportional to the square of the velocity. 

 The honor of the discovery is due to Jean Bernoulli, who published a general 

 solution of the problem, supposing the resistance to be as any power whatever 

 of the velocity. Long after, Euler discussed the same question in the Memoirs 

 of the Academy of Berlin for the year 1753. His object was to apply the 

 theory to balistics, and for that he proposes very ingenious means. In the 

 memoirs of the same Academy for the year 1765, and elsewhere, we find very 

 extended researches by Lambert with the same object. Borda, in the Memoirs 

 of the Academy of Sciences of Paris for the year 1769, has treated this ques- 

 tion with his usual elegance and ingenuity. Conformably with the idea of 

 Newton, he substitutes for the true trajectory that which would be described in 

 virtue of a density but slightly variable, and he obtains by this means an 

 approximation much superior to that of Newton. Lastly, Berout, in his Course 

 of Artillery, published in 1772, made a more particular application of methods 

 of his own to the trajectory of shells and bullets. 



* This memoir bore for its motto: ToUuntur in ultuin lit casu grnviure ruant. 



