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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 See also:

Italy and published his See also:

GREAT

great See also:

WORK

work, See also:

LIBER

Liber abaci, which probably procured him See also:

ACCESS (Lat. accessus)

access to the learned and refined See also:

court of the See also:

EMPEROR (Fr. empereur, from the Lat. imperator)

emperor See also:

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 See also:

Michael See also:

Scott the second edition of his Liber abaci, which was printed with Leonardo's other See also:

WORKS

works by See also:

Prince Bald. Boncompagni (See also:

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 See also:

NOTICE

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 See also:

KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind of mathematical performance, in which Leonardo solved several hard problems proposed to him by See also:

John of See also:

PALERMO (Greek, Havopuos; Latin, Panhormus, Panormus)

Palermo, an imperial See also:

NOTARY, or NOTARY PUBLIC

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 See also:

LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

Letter to Magisler See also:

Theodore. All these See also:

TREATISES

treatises seem to have been written nearly at the same See also:

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 See also:

FATE

fate after he issued that second edition. Leonardo's works are mainly developments of the results obtained by his predecessors; the influences of See also:

Greek, Arabian, and See also:

INDIAN

Indian mathematicians may be clearly discerned in his methods. In his Practica geometriae See also:

PLAIN (O. Fr. plain, from Lat. plenum)

plain traces of the use of the See also:

ROMAN

Roman agrimensores are met with; in his Liber abaci old See also:

EGYPTIAN

Egyptian problems reveal their origin by the reappearance of the very See also:

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 See also:

CUBE (Gr. K46os, a cube)

cube roots, proportion, See also:

CHAIN (through the O. Fr. citable, chcene, &c., from Lat. catena)

chain See also:

RULE

rule, finding of proportional parts, averages, progressions, even See also:

compound See also:

INTEREST

interest, just as in the completest See also:

MERCANTILE (or COMMERCIAL) AGENCIES

mercantile arithmetics of our days. They See also:

TEACH [THATCH OR THACH], EDWARD (d. 1718)

teach further the See also:

SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of problems leading to equations of the first and second degree, to determinate and indeterminate equations, not by single and See also:

double position only, but by real See also:

ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

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 See also:

CENSUS (from Lat. censere, to estimate or assess; connected by some with centum, i.e. a count by hundreds)

census. The second work of Leonardo, his Practica geometriae (1220) See also:

STRATEGY

strategy, but his heroism and devotion secured him an almost unique See also:

PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place in the See also:

IMAGINATION

imagination not only of his own but also of succeeding times. See See also:

HERODOTUS (c. 484-425 B.C.)

Herodotus v. 39-41, vii.

202-225, 238, ix. 10; Diodorus xi. 4-I1; See also:

Plutarch, See also:

APOPHTHEGM (from the Gr. It rogOey ya)

Apophthegm. Lacon.; de malignitate Herodoti, 28-33; See also:

PAUSANIAS

Pausanias i. 13, iii. 3, 4; Isocrates, Paneg. 92; See also:

Lycurgus, c. Leocr. 11o, III; See also:

STRABO (born c. 63 n.c.)

Strabo i. 1o, ix. 429; See also:

Aelian, See also:

Var hist. iii. 25; See also:

CICERO

Cicero, Tusc. disput. i.

42, 49; de Finibus, ii. 30; See also:

Cornelius See also:

Nepos, See also:

THEMISTOCLES (c. 514-449 B.C.)

Themistocles, 3; See also:

VALERIUS, PUBLIUS

Valerius See also:

Maximus iii. 2; See also:

JUSTIN (JUNIANUS JUSTINUS)

Justin ii. II. For See also:

MODERN

modern See also:

criticism on the See also:

BATTLE

battle of See also:

THERMOPYLAE (Gr. OepµSs, hot, and 1rf)X,

Thermopylae see G. B. See also:

Grundy, The Great See also:

Persian See also:

War (1901); G. See also:

GROTE, GEORGE (1794–1871)

Grote, History of See also:

GREECE

Greece, See also:

PART

part ii., c: 40; E. See also:

Meyer, Geschichte See also:

des A'tertums, iii., §§ 219, 220; G. Busolt, Griechische Geschichte, 2nd ed., ii. 666-688; J. B.

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