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 1197144, 1697144, and 697144,—the squares of 3512, 4112 and
2712; 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. (M. N. T.)
LEONTIASIS OSSEA, a rare disease characterized by an
overgrowth of the facial and cranial bones. The common form
is that in which one or other maxilla is affected, its size progressively
increasing both regularly and irregularly, and thus encroaching
on the cavities of the orbit, the mouth, the nose and
its accessory sinuses. Exophthalmos gradually develops, going
on later to a complete loss of sight due to compression of the optic
nerve by the overgrowth of bone. There may also be interference
with the nasal respiration and with the taking of food. In the
somewhat less common form of this rare disease the overgrowth
of bone affects all the cranial bones as well as those of the face,
the senses being lost one by one and death finally resulting
from cerebral pressure. There is no treatment other than
exposing the overgrown bone, and chipping away pieces, or
excising entirely where possible.
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, originally
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[1] 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
Römische 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.)
- ↑ 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.
