libro di

Antonio Billi," edited from MS. by G. de Fabriazy in _Archivio Storico Ital._ ser. v. vol. 7; "Breve vita di Leonardo da Vinci, scritto da un adnonimo del 1500" (known as the Anonimo Gaddiano), printed by G. Milanesi in _Archivio Storico Ital._ t. xvi. (1872), translated with notes by H. P. Horne in series published by the Unicorn Library (1903); Paolo Giovio, "Leonardi Vincii vita," in his _Elogia_, printed in Tiraboschi, _Storia della Lett. Ital._ t. vii. pt. 4, and in _Classici Italiani_, vol. 314; Vasari, in his celebrated _Lives of the Painters_ (1st ed., Florence, 1550; 2nd ed. ibid. 1568; ed. Milanesi, with notes and supplements, 1878-1885); Sabba da Castiglione, _Ricordi_ (Venice, 1565); G. P. Lomazzo, _Trattato dell' arte della pittura_, &c. (Milan, 1584-1585); _Id., Idea del tempio della pittura_ (Milan, 1591); Le Père Dan, _Le Trésor ... de Fontainebleau_ (1642); J. B. Venturi, _Essai sur les ouvrages physico-mathématiques de L. da V._ (Paris, 1797); C. Amoretti, _Memorie storiche sulla vita, &c. di L. da V._ (Milan, 1804), a work which laid the foundation of all future researches; Giuseppe Bossi, _Del Cenacolo di L. da V._ (Milan, 1810); C. Fumagalli, _Scuola di Leonardo da Vinci_ (1811); Gaye, _Carteggia d'artisti_ (1839-1841); G. Uzielli, _Ricerche intorno a L. da V._, series 1, 2 (Florence, 1872; Rome, 1884; series 1 revised, Turin, 1896), documentary researches of the first importance for the study; C. L. Calvi, _Notizie dei principali professori di belle arti_ (Milan, 1869); Arsène Houssaye, _Histoire de L. de V._ (Paris, 1869 and 1876, an agreeable literary biography of the pre-critical kind); Mrs Heaton, _Life of L. da V._ (London, 1872), a work also made obsolete by recent research; Hermann Grothe, _L. da V. als Ingenieur und Philosoph_ (Berlin, 1874); A. Marks, the _S. Anne of L. da V._ (London, 1882); J. P. Richter, _The Literary Works of L. da V._ (2 vols., London, 1883), this is the very important and valuable history of and selection from the texts mentioned above under MSS.; Ch. Ravaisson-Mollien, _Les Écrits de L. da V._ (Paris, 1881); Paul Müller Walde, _L. da V., Lebensskizze und Forschungen_ (Munich, 1889-1890); _Id._, "Beiträge zur Kenntniss des L. da V.," _in Jahrbuch der k. Preussischen Kunstsammlungen_ (1897-1899), the first immature and incomplete, the second of high value: the whole life of this writer has been devoted to the study of L. da V., but it is uncertain whether the vast mass of material collected by him will ever take shape or see the light; G. Gronau, _L. da V._ (London, 1902); Bernhard Berenson, _The Drawings of the Florentine Painters_ (London, 1903); Edmondo Solmi, _Studi sulla filosofia naturale di L. da V._ (Modena, 1898); _Id., Leonardo_ (Florence, 1st ed. 1900, 2nd ed. 1907; this last edition of Solmi's work is by far the most complete and satisfactory critical biography of the master which yet exists); A. Rosenberg, _L. da V._, in Knackfuss's series of art biographies (Leipzig, 1898); Gabriel Séailles, _L. da V. l'artiste et le savant_ (1st ed. 1892, 2nd ed. 1906), a lucid and careful general estimate of great value, especially in reference to Leonardo's relations to modern science; Edward McCurdy, _L. da V._, in Bell's "Great Masters" series (1904 and 1907), a very sound and trustworthy summary of the master's career as an artist; _Id., L. da V.'s Note-Books_ (1908), a selection from the passages of chief general interest in the master's MSS., very well chosen, arranged, and translated, with a useful history of the MSS. prefixed, _Le Vicende del Cenacolo di L. da V. nel secolo XIX._ (Milan, 1906), an official account of the later history and vicissitudes of the "Last Supper" previous to its final repair; Luca Beltrami, _Il Castello di Milano_ (1894); _Id., L. da V. et la Sala dell' Asse_ (1902); Id., "Il Cenacolo di Leonardo," in _Raccolta Vinciana_ (Milan, 1908), the official account of the successful work of repair carried out by Signor Cavenaghi in the preceding years; Woldemar von Seidlitz, _Leonardo da Vinci, der Wendepunkt der Renaissance_ (2 vols., 1909), a comprehensive and careful work by an accomplished and veteran critic, inclined to give perhaps an excessive share in the reputed works of Leonardo to a single pupil, Ambrogio Preda. It seems needless to give references to the voluminous discussion in newspapers and periodicals concerning the authenticity of a wax bust of Flora acquired in 1909 for the Berlin Museum and unfortunately ascribed to Leonardo da Vinci, its real author having been proved by external and internal evidence to be the Englishman Richard Cockle Lucas, and its date 1846. (S. C.)

LEONARDO OF PISA (LEONARDUS PISANUS or FIBONACCI), Italian mathematician of the 13th century. Of his personal history few particulars are known. His father was called Bonaccio, most probably a nickname with the ironical meaning of "a good, stupid fellow," while to Leonardo himself another nickname, Bigollone (dunce, blockhead), seems to have been given. The father was secretary in one of the numerous factories erected on the southern and eastern coasts of the Mediterranean by the warlike and enterprising merchants of Pisa. Leonardo was educated at Bugia, and afterwards toured the Mediterranean. In 1202 he was again in Italy and published his great work, _Liber abaci_, which probably procured him access to the learned and refined court of the emperor Frederick II. Leonardo certainly was in relation with some persons belonging to that circle when he published in 1220 another more extensive work, _De practica geometriae_, which he dedicated to the imperial astronomer Dominicus Hispanus. Some years afterwards (perhaps in 1228) Leonardo dedicated to the well-known astrologer Michael Scott the second edition of his _Liber abaci_, which was printed with Leonardo's other works by Prince Bald. Boncompagni (Rome, 1857-1862, 2 vols.). The other works consist of the _Practica geometriae_ and some most striking papers of the greatest scientific importance, amongst which the _Liber quadratorum_ may be specially signalized. It bears the notice that the author wrote it in 1225, and in the introduction Leonardo tells us the occasion of its being written. Dominicus had presented Leonardo to Frederick II. The presentation was accompanied by a kind of mathematical performance, in which Leonardo solved several hard problems proposed to him by John of Palermo, an imperial notary, whose name is met with in several documents dated between 1221 and 1240. The methods which Leonardo made use of in solving those problems fill the _Liber quadratorum_, the _Flos_, and a _Letter to Magister Theodore_. All these treatises seem to have been written nearly at the same period, and certainly before the publication of the second edition of the _Liber abaci_, in which the _Liber quadratorum_ is expressly mentioned. We know nothing of Leonardo's fate after he issued that second edition.

Leonardo's works are mainly developments of the results obtained by his predecessors; the influences of Greek, Arabian, and Indian mathematicians may be clearly discerned in his methods. In his _Practica geometriae_ plain traces of the use of the Roman _agrimensores_ are met with; in his _Liber abaci_ old Egyptian problems reveal their origin by the reappearance of the very numbers in which the problem is given, though one cannot guess through what channel they came to Leonardo's knowledge. Leonardo cannot be regarded as the inventor of that very great variety of truths for which he mentions no earlier source.

The _Liber abaci_, which fills 459 printed pages, contains the most perfect methods of calculating with whole numbers and with fractions, practice, extraction of the square and cube roots, proportion, chain rule, finding of proportional parts, averages, progressions, even compound interest, just as in the completest mercantile arithmetics of our days. They teach further the solution of problems leading to equations of the first and second degree, to determinate and indeterminate equations, not by single and double position only, but by real algebra, proved by means of geometric constructions, and including the use of letters as symbols for known numbers, the unknown quantity being called _res_ and its square _census_.

The second work of Leonardo, his _Practica geometriae_ (1220) requires readers already acquainted with Euclid's planimetry, who are able to follow rigorous demonstrations and feel the necessity for them. Among the contents of this book we simply mention a trigonometrical chapter, in which the words _sinus versus arcus_ occur, the approximate extraction of cube roots shown more at large than in the _Liber abaci_, and a very curious problem, which nobody would search for in a geometrical work, viz.--To find a square number remaining so after the addition of 5. This problem evidently suggested the first question, viz.--To find a square number which remains a square after the addition and subtraction of 5, put to our mathematician in presence of the emperor by John of Palermo, who, perhaps, was quite enough Leonardo's friend to set him such problems only as he had himself asked for. Leonardo gave as solution the numbers 11(97/144), 16(97/144), and 6(97/144),--the squares of 3(5/12), 4(1/12) and 2(7/12); and the method of finding them is given in the _Liber quadratorum_. We observe, however, that this kind of problem was not new. Arabian authors already had found three square numbers of equal difference, but the difference itself had not been assigned in proposing the question. Leonardo's method, therefore, when the difference was a fixed condition of the problem, was necessarily very different from the Arabian, and, in all probability, was his own discovery. The _Flos_ of Leonardo turns on the second question set by John of Palermo, which required the solution of the cubic equation x³ + 2x² + 10x = 20. Leonardo, making use of fractions of the sexagesimal scale, gives x = 1^0 22^i 7^ii 42^iii 33^iv 4^v 40^vi, after having demonstrated, by a discussion founded on the 10th book of Euclid, that a solution by square roots is impossible. It is much to be deplored that Leonardo does not give the least intimation how he found his approximative value, outrunning by this result more than three centuries. Genocchi believes Leonardo to have been in possession of a certain method called _regula aurea_ by H. Cardan in the 16th century, but this is a mere hypothesis without solid foundation. In the _Flos_ equations with negative values of the unknown quantity are also to be met with, and Leonardo perfectly understands the meaning of these negative solutions. In the _Letter to Magister Theodore_ indeterminate problems are chiefly worked, and Leonardo hints at his being able to solve by a general method any problem of this kind not exceeding the first degree.

As for the influence he exercised on posterity, it is enough to say that Luca Pacioli, about 1500, in his celebrated _Summa_, leans so exclusively to Leonardo's works (at that time known in manuscript only) that he frankly acknowledges his dependence on them, and states that wherever no other author is quoted all belongs to Leonardus Pisanus.

_Fibonacci's series_ is a sequence of numbers such that any term is the sum of the two preceding terms; also known as _Lamé's series_. (M. Ca.)

LEONCAVALLO, RUGGIERO (1858- ), Italian operatic composer, was born at Naples and educated for music at the conservatoire. After some years spent in teaching and in ineffectual attempts to obtain the production of more than one opera, his _Pagliacci_ was performed at Milan in 1892 with immediate success; and next year his _Medici_ was also produced there. But neither the latter nor _Chatterton_ (1896)--both early works--obtained any favour; and it was not till _La Bohème_ was performed in 1897 at Venice that his talent obtained public confirmation. Subsequent operas by Leoncavallo were _Zaza_ (1900), and _Der Roland_ (1904). In all these operas he was his own librettist.

LEONIDAS, king of Sparta, the seventeenth of the Agiad line. He succeeded, probably in 489 or 488 B.C., his half-brother Cleomenes, whose daughter Gorgo he married. In 480 he was sent with about 7000 men to hold the pass of Thermopylae against the army of Xerxes. The smallness of the force was, according to a current story, due to the fact that he was deliberately going to his doom, an oracle having foretold that Sparta could be saved only by the death of one of its kings: in reality it seems rather that the ephors supported the scheme half-heartedly, their policy being to concentrate the Greek forces at the Isthmus. Leonidas repulsed the frontal attacks of the Persians, but when the Malian Ephialtes led the Persian general Hydarnes by a mountain track to the rear of the Greeks he divided his army, himself remaining in the pass with 300 Spartiates, 700 Thespians and 400 Thebans. Perhaps he hoped to surround Hydarnes' force: if so, the movement failed, and the little Greek army, attacked from both sides, was cut down to a man save the Thebans, who are said to have surrendered. Leonidas fell in the thickest of the fight; his head was afterwards cut off by Xerxes' order and his body crucified. Our knowledge of the circumstances is too slight to enable us to judge of Leonidas's strategy, but his heroism and devotion secured him an almost unique place in the imagination not only of his own but also of succeeding times.

See Herodotus v. 39-41, vii. 202-225, 238, ix. 10; Diodorus xi. 4-11; Plutarch, _Apophthegm. Lacon.; de malignitate Herodoti_, 28-33; Pausanias i. 13, iii. 3, 4; Isocrates, _Paneg._ 92; Lycurgus, _c._ _Leocr._ 110, 111; Strabo i. 10, ix. 429; Aelian, _Var. hist._ iii. 25; Cicero, _Tusc. disput._ i. 42, 49; _de Finibus_, ii. 30; Cornelius Nepos, _Themistocles_, 3; Valerius Maximus iii. 2; Justin ii. 11. For modern criticism on the battle of Thermopylae see G. B. Grundy, _The Great Persian War_ (1901); G. Grote, _History of Greece_,