-Mi' 



MEMOIR OF LEGENNRE. 149 



Essay bad itself been preceded by a considerable wurk published in the Memoirs 

 of the Academy for 178o, and entitled licchcrchcs d'anali/se incUtcrmince ; which 

 relates principally to the study of the properties of numbers. In fine, we learn 

 from the mamiscript proceedings of the Academy, before cited, that, among the 

 memoirs which ^I. de Laplace, in the session of March 15, 1783, indicated as 

 having- been presented by M. Legendre, occur two memoirs on the resolution of 

 indeterminate equations of the second degree and on the properties of continual 

 fractions, and a memoir on the summation of these fractions. Now, from the 

 objects of which they treat, and indeed from the titles alone, these memoirs bear 

 a very natural relation to certain paragraphs of the great memoir of 1785. 

 They were probably the first rudiments of it. Hence we see that M. Legendre 

 had been occupied with the theory of numbers from his youth. He had labored 

 upon it for more than 50 years. Yet he concludes the advertisement of the 

 Theory of numbers, dated April 1, 1830, with the following words, which are 

 certainly modest enough : 



We shall not pretend that certain matters treated of in this work do not need to bo improved 

 or even rectified bv new researches. Nevertheless, the author has thought that it would be 

 better to leave them in this state of imperfection thau to suppress thein altogether ; they will 

 offer a subject of investigation to those who may be disposed in the future to occupy them- 

 selves with the advancement of the science. 



This part of the science has received in effect, since the publication of the 

 Theory of niwibers, important accessions ; but if we compare the contents of this 

 learned Avork with Avhat had been discovered during the 2,000 j^ears which 

 preceded 1785, Ave shall see that no sa\^ant has marked his passage in this 

 branch of mathematics by traces in any degree comparable to these efforts of 

 M. Legendre. It cannot surprise us that a science Avhich had advanced Avith but 

 slow and progressive steps under the hands of men as eminent as Euclid and 

 Uiophantes among the ancients, as Viete, Bachet, Fermat, Euler, and Lagrange 

 among the moderns, should not all at once have been carried to a point Avhich 

 comported Avith no further progress. It behooves us, on the contrary, candidly 

 to avow that M. Legendre, in speaking of new developments Avhich still awaited 

 it, gave proof of perspicuity almost as much as of modesty. 



The science of numbers is difficult, and it is above all difficult to convey an 

 idea of it to persons Avhose attention has never been occujiied with it. E\'-ery 

 one knoAvs that numbers are distinguished into tAvo great classes : even and odd 

 numbers, Avhich alternately succeed one another. The even numbers are divisible 

 by 2, Avhile the odd nvunbers are not, though they have often other divisors. 

 Whole numbers differ much from one another in the possibility of being divided 

 by other and smaller integers. It has been long ago remarked that the number 

 10, the basis of our decimal system, has but tAVO diAnsors, 2 and 5, the last of 

 Avhich is not subdivisil)le, while the number 8 has two divisors, 2 and 4, of which 

 the last is further subdivisible by 2, and the number 12 has three divisors, 2, 

 3, and 4, the last of Avhich is again subdivisilde by 2; Avhence it folloAvs thal^ 

 the number 8 and especially the number 12 have, as the basis of a system of 

 measiu'es susceptible of being successively subdivided, an incontestable supe- 

 riority over the number 10. This inferiority of the latter number is one of the 

 obstacles to the general adoption of the decimal system of weights and measures, 

 Avhich presents in other respects such great advantages. 



But the number 10 is more favored in this regard than the number 9, divisible 

 only by 3, of Avhicli it is the square. It is still more so than the numbers 3, 5, 

 7, il, 13, 17, Avhich have no divisors, or, to speak the language of science, have 

 no other divisors but themselves and unity. Number 7, Avhich enumerates the 

 seven days of the week, the seven Avonders of the Avorld, tho seven sages of 

 Grreece, passes for possessing a certain degree of excellence ; but number 13, as 

 well as 17, is looked upon as inauspicious, by reason, it may be, of this absence 

 of divisors Avhich renders both numbers refractory. All those numbers Avhich 

 have no other divisors but themselves and unity, are called prime numbers. 



