LEONTINI (mod. Lentini)  , an ancient town in the south-east of Sicily, 22 M . N.N.W. of Syracuse direct, founded by Chalcidians from Naxos in 729 B.C . It is almost the only Greek settlement not on the coast, from which it is 6 m. distant . The site, origin-ally held by the Sicels, was seized by the Greeks owing to its command of the fertile plain on the north . It was reduced to subjection in 498 B.C. by Hippocrates of Gela, and in 476 Hieron of Syracuse established here the inhabitants of Catana and Naxos . Later on Leontini regained its independence, but in its efforts to retain it, the intervention of Athens was more than once invoked . It was mainly the eloquence of Gorgias (q.v.) of Leontini which led to the abortive Athenian expedition of 427 . In 422 Syracuse supported the oligarchs against the people and received them as citizens, Leontini itself being forsaken . This led to renewed Athenian intervention, at first mainly diplomatic; but the exiles of Leontini joined the envoys of Segesta, in persuading Athens to undertake the great expedition of 415 . After its failure, Leontini became subject to Syracuse once more (see Strabo vi . 272) . Its independence was guaranteed by the treaty of 405 between Dionysius and the Carthaginians, but it very soon lost it again .

It was finally stormed by M .

Claudius Marcellus in 214 B.C . In Roman times it seems to have been of small importance . It was destroyed by the Saracens A.D . 848, and almost totally ruined by the earthquake of 1698 . The ancient city is described by Polybius (vii . 6) as lying in a bottom between two hills, and facing north . On the western side of this bottom ran a river with a row of houses on its western bank under the hill . At each end was a gate, the northern leading to the plain, the southern, at the upper end, to Syracuse . There was an acropolis on each side of the valley, which lies between precipitous hills with flat tops, over which buildings had extended . The eastern hill' still has considerable remains of a strongly fortified medieval castle, in which some writers are inclined(though wrongly) to recognize portions of Greek masonry . See G .

M .

Columba, in Archeologia di Leontinoi (Palermo, 1891), reprinted from Archivio Storico Siciliano, xi.; P . Orsi in Romische Mitteilungen (1900), 61 seq . Excavations were made in 1899 in one of the ravines in a Sicel necropolis of the third period; explorations in the various Greek cemeteries resulted in the discovery of some fine bronzes, notably a fine bronze lebes, now in the Berlin museum . (T . As.) i As a fact there are two flat valleys, up both of which the modern Lentini extends; and hence there is difficulty in fitting Polybius's account to the site . 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 1114'4, 16,9474 and 61414, the squares of 3A, 4,11 and 2,72; and the method of finding them is given in the Liber quadratorurn . 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 x'+2x2+IOx=2o . Leonardo, making use of fractions of the sexagesimal scale, gives X=10 22' 7" 42'" 33" 4° 40°", after having demonstrated, by a discussion founded on the loth 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 Lame's series . (M .