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 Hitipanus . 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 124o . The methods which Leonardo made use of in solving those problems fill the Liber quadratorum, the Flos, and a Letter to Magisler 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) 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-I1;

Plutarch, Apophthegm . Lacon.; de malignitate Herodoti, 28-33; Pausanias i . 13, iii . 3, 4; Isocrates, Paneg . 92; Lycurgus, c . Leocr . 11o, III; Strabo i . 1o, 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 . II . 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 A'tertums, 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 . (M . N .