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 1197⁄144, 1697⁄144, and 697⁄144,—the squares of 35⁄12, 41⁄12 and 27⁄12; 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 = 10 22i 7ii 42iii 33iv 4v 40vi, 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, part ii., c. 40; E. Meyer, Geschichte des Altertums, iii., §§ 219, 220; G. Busolt, Griechische Geschichte, 2nd ed., ii. 666-688; J. B. Bury, “The Campaign of Artemisium and Thermopylae,” in British School Annual, ii. 83 seq.; J. A. R. Munro, “Some Observations on the Persian Wars, II.,” in Journal of Hellenic Studies, xxii. 294-332.