Fibonacci, a mathematician born in the Republic of Pisa who had studied in Béjaïa ( Bougie ), Algeria, promoted the Indian numeral system in Europe with his book Liber Abaci, written in 1202:

Special cases of the Chinese remainder theorem were also known to Brahmagupta ( 7th century ), and appear in Fibonacci's Liber Abaci ( 1202 ).

Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.

The Liber Abaci began the sequence with F < sub > 1 </ sub > = 1, omitting the initial 0, and the sequence is still written this way by some.

A page of Fibonacci's Liber Abaci from the National Central Library ( Florence ) | Biblioteca Nazionale di Firenze showing ( in box on right ) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.

* Leonardo Fibonacci writes Liber Abaci, about the modus Indorum, the numbering method of India ; it is the first major work in Europe toward moving away from the use of Roman numerals.

* Leonardo Fibonacci publishes Liber Abaci, introducing the Arabian zero to Europe.

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits ( chapter 13 of Liber Abaci, AD 1202 ) and later as losses ( in Flos ).

A page of the Liber Abaci from the National Central Library ( Florence ) | Biblioteca Nazionale di Firenze showing ( on right ) the numbers of the Fibonacci numbers | Fibonacci sequence.

Liber Abaci was among the first Western books to describe Arabic numerals.

The title of Liber Abaci means " The Book of Calculation ".

Although it has also been translated as " The Book of the Abacus ", writes that this is an error: the intent of the book is to describe methods of doing calculations without aid of an abacus, and as confirms, for centuries after its publication the algorismists ( followers of the style of calculation demonstrated in Liber Abaci ) remained in conflict with the abacists ( traditionalists who continued to use the abacus in conjunction with Roman numerals ).

In reading Liber Abaci, it is helpful to understand Fibonacci's notation for rational numbers, a notation that is intermediate in form between the Egyptian fractions commonly used until that time and the vulgar fractions still in use today.

In the Liber Abaci, Fibonacci says the following introducing the so-called " Modus Indorum " or the method of the Indians, today known as Arabic numerals.

ca: Liber Abaci

da: Liber Abaci

lt: Liber Abaci

pt: Liber Abaci

Napier's bones is an abacus created by John Napier for calculation of products and quotients of numbers that was based on Arab mathematics and lattice multiplication used by Matrakci Nasuh in the Umdet-ul Hisab and Fibonacci writing in the Liber Abaci.

By the 13th century, Hindu-Arabic numerals were accepted in European mathematical circles ( Fibonacci used them in his Liber Abaci ).

Some of the most widely circulating books, such as the Liber Abaci by Leonardo Fibonacci of Pisa, included applications of mathematics and arithmetic to business practice or were business manuals based on sophisticated numeracy and literacy.

This is a condensed form of the Italian censo di censo, used by Leonardo of Pisa in his famous book Liber Abaci of 1202.

Leonardo of Pisa ( Fibonacci ) devoted Chapter 13 of his book Liber Abaci ( AD 1202 ) to explaining and demonstrating the uses of double false position, terming the method regulis elchatayn after the al-khaṭāʾayn method that he had learned from Arab sources.

Practical numbers were used by Fibonacci in his Liber Abaci ( 1202 ) in connection with the problem of representing rational numbers as Egyptian fractions.

First recorded mention in Europe 976, first widely published in 1202 by Fibonacci with his Liber Abaci.

Until Arabic numerals were adopted and adapted from Indian numerals in the 8th and 9th century AD, and promoted in Europe by Fibonacci of Pisa with his 1202 book Liber Abaci, numerals were predominantly alphabetical.

It is sometimes erroneously stated that lattice multiplication was described by Muḥammad ibn Mūsā al-Khwārizmī ( Baghdad, c. 825 ) or by Fibonacci in his Liber Abaci ( Italy, 1202, 1228 ).

It is probably a translation from Latin into Scandinavian of some pages included in more ancient books such as Carmen de Algorismo by De Villa Dei of 1200, Liber Abaci by Fibonacci of 1202, and Algorismus Vulgaris by De Sacrobosco of 1230.

The greedy algorithm for Egyptian fractions, first described in 1202 by Fibonacci in his book Liber Abaci, finds an expansion in which each successive term is the largest unit fraction that is no larger than the remaining number to be represented.

He also drew on Josephus's Antiquities, and the works of Cassiodorus, and there was a copy of the Liber Pontificalis in Bede's monastery.

Hildegard also wrote nearly 400 letters to correspondents ranging from Popes to Emperors to abbots and abbesses ; two volumes of material on natural medicine and cures ; an invented language called the Lingua ignota ; various minor works, including a gospel commentary and two works of hagiography ; and three great volumes of visionary theology: Scivias, Liber vitae meritorum (" Book of Life's Merits " or " Book of the Rewards of Life "), and Liber divinorum operum (" Book of Divine Works ").

Liber vitae meritorum (" Book of Life's Merits " or " Book of the Rewards of Life ") and Liber divinorum operum (" Book of Divine Works ", also known as De operatione Dei, " On God's Activity ") followed.

The Liber Pontificalis also presents a list that makes Linus the second in the line of bishops of Rome, after Peter ; but at the same time it states that Peter ordained two bishops, Linus and Cletus, for the priestly service of the community, devoting himself instead to prayer and preaching, and that it was to Clement that he entrusted the Church as a whole, appointing him as his successor.

The Liber Pontificalis also says that he issued a decree that women should cover their heads in church, created the first fifteen bishops, and that he died a martyr and was buried on the Vatican Hill next to Peter.

The most important of his writings is the Liber censuum Romanae ecclesiae, which is the most valuable source for the medieval position of the Church in regard to property and also serves in part as a continuation of the Liber Pontificalis.

His father was called " Rufinus ", who was also said to be of Aquileia according to the Liber Pontificalis.

He also ordained Saint Willibrord as bishop of the Frisians, and the Liber Pontificalis states he also ordained Berhtwald as Archbishop of Canterbury.

His recounting of the period was remarkable for the rise of what 19th century papal historians saw as a " pornocracy ", or " rule of the harlots ", a reversal of the natural order as they saw it, according to Liber pontificalis and a later chronicler who was also biased against Sergius III.

The Liber Pontificalis attributes to Zosimus a decree on the wearing of the maniple by deacons and on the dedication of Easter candles in the country parishes ; also a decree forbidding clerics to visit taverns.

According to the notice in the Liber Pontificalis, Felix erected a basilica on the Via Aurelia ; the same source also adds that he was buried there.

Crowley retitled it Liber AL vel Legis in 1921, when he also gave the handwritten manuscript its own title, Liber XXXI.

Proserpina was subsumed by the cult of Libera, an ancient fertility goddess, wife of Liber and is also considered a life – death – rebirth deity.

* Italy: Emperor Frederick II promulgates the Constitutions of Melfi ( also known as Liber Augustalis ), a collection of laws for Sicily.

Possibly he did issue such an edict against the Gnostics and Montanists ; it is also possible that on his own responsibility the writer of the Liber Pontificalis attributed to this pope a similar decree current about the year 500.

Gregory's education was the standard Latin one of Late Antiquity, focusing on Vergil's Aeneid and Martianus Capella's Liber de Nuptiis Mercurii et Philologiae, but also other key texts such as Orosius ' Chronicles, which his Historia is a continuation of, and Sallust, all of which works he refers to in his own.

The Liber Pontificalis also relates that this pope organized the hierarchy and established the order of ecclesiastical precedence ().

His initiative in this direction was visible as early as the Assizes of Capua ( 1220, issued soon after his coronation in Rome ) but came to fruition in his promulgation of the Constitutions of Melfi ( 1231, also known as Liber Augustalis ), a collection of laws for his realm that was remarkable for its time and was a source of inspiration for a long time after.

It has also been suggested that the poem is dependent on Liber historiae Francorum ( 727 ), because it mentions the Attoarii, which in Beowulf become Hetware.