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 ferebatur electio, quid veritas haberet inquireret, nobisque omnia fideliter indicaret”). Passing on to the time of Gregory the Great, we find him sending two representatives to Gaul in 599, to suppress simony, and one to Spain in 603. Augustine of Canterbury is sometimes spoken of as legate, but it does not appear that in his case this title was used in any strictly technical sense, although the archbishop of Canterbury afterwards attained the permanent dignity of a legatus natus. Boniface, the apostle of Germany, was in like manner constituted, according to Hincmar (Ep. 30), a legate of the apostolic see by Popes Gregory II. and Gregory III. According to Hefele (Conc. iv. 239), Rodoald of Porto and Zecharias of Anagni, who were sent by Pope Nicolas to Constantinople in 860, were the first actually called legati a latere. The policy of Gregory VII. naturally led to a great development of the legatine as distinguished from the ordinary episcopal function. From the creation of the medieval papal monarchy until the close of the middle ages, the papal legate played a most important rôle in national as well as church history. The further definition of his powers proceeded throughout the 12th and 13th centuries. From the 16th century legates a latere give way almost entirely to (q.v.).

See P. Hinschius, Kirchenrecht, i. 498 ff.; G. Phillips, Kirchenrecht, vol. vi. 680 ff.

LEGATION (Lat. legatio, a sending or mission), a diplomatic mission of the second rank. The term is also applied to the building in which the minister resides and to the area round it covered by his diplomatic immunities. See.

LEGEND (through the French from the med. Lat. legenda, things to be read, from legere, to read), in its primary meaning the history or life-story of a saint, and so applied to portions of Scripture and selections from the lives of the saints as read at divine service. The statute of 3 and 4 Edward VI. dealing with the abolition of certain books and images (1549), cap. 10, sect. 1, says that “all bookes called processionalles, manuelles, legends shall be  abolished.” The “Golden Legend,” or Aurea Legenda, was the name given to a book containing lives of the saints and descriptions of festivals, written by Jacobus de Voragine, archbishop of Genoa, in the 13th century. From the original application of the word to stories of the saints containing wonders and miracles, the word came to be applied to a story handed down without any foundation in history, but popularly believed to be true. “Legend” is also used of a writing, inscription, or motto on coins or medals, and in connexion with coats of arms, shields, monuments, &c.

LEGENDRE, ADRIEN MARIE (1752–1833), French mathematician, was born at Paris (or, according to some accounts, at Toulouse) in 1752. He was brought up at Paris, where he completed his studies at the Collège Mazarin. His first published writings consist of articles forming part of the Traité de mécanique (1774) of the Abbé Marie, who was his professor; Legendre’s name, however, is not mentioned. Soon afterwards he was appointed professor of mathematics in the École Militaire at Paris, and he was afterwards professor in the École Normale. In 1782 he received the prize from the Berlin Academy for his “Dissertation sur la question de balistique,” a memoir relating to the paths of projectiles in resisting media. He also, about this time, wrote his “Recherches sur la figure des planètes,” published in the Mémoires of the French Academy, of which he was elected a member in succession to J. le Rond d’Alembert in 1783. He was also appointed a commissioner for connecting geodetically Paris and Greenwich, his colleagues being P. F. A. Méchain and C. F. Cassini de Thury; General William Roy conducted the operations on behalf of England. The French observations were published in 1792 (Exposé des opérations faites en France in 1787 pour la jonction des observatoires de Paris et de Greenwich). During the Revolution, he was one of the three members of the council established to introduce the decimal system, and he was also a member of the commission appointed to determine the length of the metre, for which purpose the calculations, &c., connected with the arc of the meridian from Barcelona to Dunkirk were revised. He was also associated with G. C. F. M. Prony (1755–1839) in the formation of the great French tables of logarithms of numbers, sines, and tangents, and natural sines, called the Tables du Cadastre, in which the quadrant was divided centesimally; these tables have never been published (see ). He was examiner in the École Polytechnique, but held few important state offices. He died at Paris on the 10th of January 1833, and the discourse at his grave was pronounced by S. D. Poisson. The last of the three supplements to his Traité des fonctions elliptiques was published in 1832, and Poisson in his funeral oration remarked: “M. Legendre a eu cela de commun avec la plupart des géomètres qui l’ont précédé, que ses travaux n’ont fini qu’avec sa vie. Le dernier volume de nos mémoires renferme encore un mémoire de lui, sur une question difficile de la théorie des nombres; et peu de temps avant la maladie qui l’a conduit au tombeau, il se procura les observations les plus récentes des comètes à courtes périodes, dont il allait se servir pour appliquer et perfectionner ses méthodes.”

It will be convenient, in giving an account of his writings, to consider them under the different subjects which are especially associated with his name.

Elliptic Functions.—This is the subject with which Legendre’s name will always be most closely connected, and his researches upon it extend over a period of more than forty years. His first published writings upon the subject consist of two papers in the Mémoires de l’Académie Française for 1786 upon elliptic arcs. In 1792 he presented to the Academy a memoir on elliptic transcendents. The contents of these memoirs are included in the first volume of his Exercices de calcul intégral (1811). The third volume (1816) contains the very elaborate and now well-known tables of the elliptic integrals which were calculated by Legendre himself, with an account of the mode of their construction. In 1827 appeared the Traité des fonctions elliptiques (2 vols., the first dated 1825, the second 1826), a great part of the first volume agrees very closely with the contents of the Exercices; the tables, &c., are given in the second volume. Three supplements, relating to the researches of N. H. Abel and C. G. J. Jacobi, were published in 1828–1832, and form a third volume. Legendre had pursued the subject which would now be called elliptic integrals alone from 1786 to 1827, the results of his labours having been almost entirely neglected by his contemporaries, but his work had scarcely appeared in 1827 when the discoveries which were independently made by the two young and as yet unknown mathematicians Abel and Jacobi placed the subject on a new basis, and revolutionized it completely. The readiness with which Legendre, who was then seventy-six years of age, welcomed these important researches, that quite overshadowed his own, and included them in successive supplements to his work, does the highest honour to him (see ).

Eulerian Integrals and Integral Calculus.—The Exercices de calcul intégral consist of three volumes, a great portion of the first and the whole of the third being devoted to elliptic functions. The remainder of the first volume relates to the Eulerian integrals and to quadratures. The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function. The latter portion of the second volume of the Traité des fonctions elliptiques (1826) is also devoted to the Eulerian integrals, the table being reproduced. Legendre’s researches connected with the “gamma function” are of importance, and are well known; the subject was also treated by K. F. Gauss in his memoir Disquisitiones generales circa series infinitas (1816), but in a very different manner. The results given in the second volume of the Exercices are of too miscellaneous a character to admit of being briefly described. In 1788 Legendre published a memoir on double integrals, and in 1809 one on definite integrals.

Theory of Numbers.—Legendre’s Théorie des nombres and Gauss’s Disquisitiones arithmeticae (1801) are still standard works upon this subject. The first edition of the former appeared in 1798 under the title Essai sur la théorie des nombres; there was a second edition in 1808; a first supplement was published in 1816, and a second in 1825. The third edition, under the title Théorie des nombres, appeared in 1830 in two volumes. The fourth edition appeared in 1900. To Legendre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de Fermat, and which was called by Gauss the “gem of arithmetic.” It was first given by Legendre in the Mémoires of the Academy for 1785, but the demonstration that accompanied it was incomplete. The symbol (a/p) which is known as Legendre’s symbol, and denotes the positive or negative unit which is the remainder when a$1⁄2$p(−1) is divided by a prime number p, does not appear in this memoir, but was first used in the Essai sur la théorie des nombres. Legendre’s formula x: (log x−1.08366) for the approximate number of forms inferior to a given number x was first given by him also in this work (2nd ed., p. 394) (see ). 