138 MEMOIR OF LEGENDRE. 



tion, but the abl)e felt it to be a dut}^ to indicate to men of science the passages 

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

 22 3^ears. Among these passages is the definition of accelerative forces, dis- 

 tinguished by a precision and clearness of expression which seem sometimes to 

 be among the happy privileges of youth. This definition is so natural, and 

 now so familiar to scientific minds, that, when recalled, it is with difliculty con- 

 ceived hew it could ever have presented anything of originality and novelty. 

 It is but just to say that it forms no exceptional featm-e 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 

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

 younger geometers; these therefore he sought 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 ; and scarcely had 

 the first Mmpses of genius given presage of what might be expected from the 

 disciple of the Abbe 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 

 l)ody of 3'ouths from which have sprung not a few of oiu* warlike celebrities, and 

 whose number would have been more considerable, had not circumstances 

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

 given b}' the young professor embraced the first elements of haUstics, the art, 

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

 Bezout, Borda, and other eminent men had pid)lished on these difficult problems ; 

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

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

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

 taining the range which corresponds to different initial velocities and to different 

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

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

 public meeting of June 6, 1782, and was pitl:)lished at Berlin under the title 

 oi Becherchcs sur la trajcctoirc dcs projectiles dans les nidieux 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 ai)proximations, and those but rough ones, for the trajectory 

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

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

 solution of the i:)roblem, supposing the resistance to be as any power whatever 

 of the velocity. Long after, Euler discussed the same qitestion 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 17G5, 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 wdiich 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 Cotirse 

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

 of his own to the trajectory of shells and bullets. 



* This memoir bore for its motto : TuUiintur in altum ut casu grnviore ruant. 



