# A sammelband of 5 mathematical texts, in Latin, illustrated manuscript on paper, [Italy, late 15th century].

A sammelband of 5 mathematical texts, in Latin, illustrated manuscript on paper, [Italy, late 15th century].

**An extremely rare testimony to the foundations of modern mathematics and algebra: a 15**

**e**

**– century illustrated sammelband of mathematical texts**

**,**

**including the most significant Fibonacci chapters**

**‘**

**it’s revolutionary**

*Liber Abaci*

**owned and bound by one of the most important figures in the history of the 19**

**e**

**mathematics of the last century, Prince Baldassare Boncompagni.**

217 x 158mm. v + 222 leaves + ii, complete, collation: 1-10^{8}11^{6}12-13^{8}14-15^{12}16^{ten}17^{12}18^{14}19-23^{12}, 19th century foliation in pen followed here, the first four texts ff.1-102 in two columns of 28-32 lines, line spacing: 173 x 56mm, the watermark a crossbow in a circle, closer to Bricquet 739, in use in Italy from 1468; rubrics and initials in red, capital letters touched in red, the Boethius illustrated with 125 diagrams in red and brown, the Grosseteste illustrated with 10 diagrams, the final text of Fibonacci ff.104-221 in a single column of approximately 26 lines in a Italian humanist cursive hand, ruled space: 141 x 94mm, the watermarks of a cross and a bull’s head similar to Bricquet 11806 and 1455a, both found in the Venetian region in the 1480s (folio opening with small repair margin and some soiling, occasional ink erosion of diagrams, some marginal stains and bumps). Mid-19th century marbled boards (scuffed and rubbed edges). In a red morocco shirt and case.*Origin*:* *(1) Pietro Girometti (1811-1859), gem engraver and Roman medalist. No. 25 in his catalog of

*Cartacei and Membranacei codes*

*dei secoli XIV*

*oh*

*th XV*

*oh*

*[*

*…*

*]*1856, sold with 33 other manuscripts to:

(2) Prince Baldassare Boncompagni-Ludovisi (1821-1894), one of the leading figures in 19th century mathematics, bibliophile and scholar responsible for the widespread dissemination and popularization of several key mathematical texts of the Middle Ages, including Fibonacci. His vast library consisted of some 650 manuscripts and more than 20,000 printed volumes: the manuscripts alone constituting one of the most important private collections of scientific and mathematical texts. The texts of this manuscript were assembled and bound by Boncompagni: remains of its label on the back and numbers 176 (see E. Narducci,

*Catalogo di Manoscritti Ora Posseduti D. Baldassarre Boncompagni*Rome, 1862, pp.74-75, no. 176) and 122 (E. Narducci,

*Catalogo di Manoscritti Ora Posseduti D. Baldassarre Boncompagni*, Rome, 1892, pp.77-78, no. 122). Sold by his heirs in

*Catalogo della Biblioteca Boncompagni. I Manoscritti. Facsimile, Edizioni del Secolo XV.*

*Abbachi Riviste*Rome, 1898, lot 99.

(3) Robert B Honeyman (1897-1987), metallurgical engineer and bibliophile. Purchased in 1932: its rating ‘Gen Sci 6 Ms23’. Its sale at Sotheby’s, May 2, 1979, lot 1109 to Nico Israel. Proposed in cat.22,

*Interesting books and manuscripts on various topics [*

*…*

*]*1980, n° 20, and purchased by:

(4) JG Bergart, and on loan to John Hay Library, Brown University (published in KP Harrington, ed.,

*Medieval Latin*1997, p.661 and p.31, where dated c.1390).

(5) Bonhams, June 22, 2011, lot 1009.

*Contents*: BOETHE, Anicius Manlius Severinus, *From Institutione Arithmetica*opening *‘*In dandis accipiendisc muneribus […]’, Book I ff.1-25, Book II ff.25-64; GROSSETESTE, Robert, *Compote*opening with the list of the twelve chapters at f.65, and the actual text ‘Computus e[st] science[en]you[i]a number[era]tionis and divisionis t[em]by[um]’ff.65-93; Comparison tables of Christian and Arabic years, conjunction and opposition tables with explanatory notes, overture ‘Tabula ad inueniendum annos arabum’, ff.93v-96; [DE PULCHRO RIVO, Johannes, attrib.], *Computus Manualis*opening ‘Inte[n]tio in hoc Capitulo e[st] artem […]’, ff.97-101; white ff.102-3; FIBONACCI, Leonardo, *Liber radicum* [chapters 14 and 15 from the *Liber Abaci*]opening ‘[Q]uidam numeri habent radices’, ff.104-221, the text largely corresponding to the Boncompagni edition of *Liber Abbaci by Leonardo Pisano*, 1857, pp.353-459, with the beginning of chapter 14 in the 1857 edition ‘Liceat mihi in hoc de radicum’ here at the end of the volume on f.220. The 1857 edition ends ‘[…] dragme pro quantitate rei’ (here at f.215v), the text continues here with references to Campanus of Novara [c.1220-1296] start: ‘Sum[m]a progression is […]’and ending'[…] divisus is igitur triang[u]I[u]sabc in tres partes equales ut proponit. Camp.’, ff.215v-220.

This sammelband brings together some of the essential mathematical texts of the High Middle Ages. Written by the 6th century Roman philosopher Boethius, *From Institutione Arithmetica* was the major pre-12th century Western European mathematics textbook, including a philosophical discussion of numbers, their relationships, and their meanings. One of the text’s most influential features was its division of the mathematical sciences into arithmetic, music, geometry, and astronomy, which it referred to together as the *quadrivium*. The *Compote *English statesman, scholastic philosopher, theologian, and Bishop of Lincoln Robert Grosseteste was innovative in incorporating Arab astronomical influences into computational theory. Grosseteste defines “computing” as a science of counting and dividing time, and his discussion of the solar year is a key contribution to the raging medieval debate over calendar reform. The text survives in 38 manuscripts (for the most recent critical edition, see A. Lohr and CPE Nothaft, *Robert Grosseteste**‘**s **‘**Compote**‘*, 2019). By far the rarest of the texts in this sammelband, however, is the Fibonacci excerpt *Liber Abaci.*

In 1852, Boncompagni listed 12 surviving manuscript copies – partial or fragmentary – of the *Liber Abaci* (B. Boncompagni, *Della Vita e delle Opera di Leonardo Pisano, matematico del secolo decimoterzo*, Rome, 1852, pp.25-69). A more recent 2017 survey by Italian mathematician Enrico Giusti lists eight complete or nearly complete manuscripts: one in Siena (Biblioteca Comunale L.IV.20, 13th century); one in Rome (Biblioteca Apostolica Vaticana, Pal.lat.1343, 13th/14th century); one in Milan, (Biblioteca Ambrosiana, ms. I.72 sup, 13th century); one in Naples (Biblioteca Nazionale, ms. VIII.C.18, 17th century); and five in Florence (Biblioteca Nazionale Centrale, Conv. Soppr. CI2616, 14th century; Magl. XI.21, 14th century; Fond. Princ. II.III.25, 16th century; Biblioteca Riccardiana, ms. 783, 15th century) . In addition to these are extracts, such as the present manuscript, containing the final and most significant chapters of *Liber Abaci*. Eight of these manuscripts survive in public institutions. Three are in Paris: Bibliothèque Mazarine, ms. 1256, 14th century; National Library, Paris, Lat. 7367, 15th century; and lat. 7225A, 16th century; three are in Florence: Biblioteca Riccardiana, ms. 2252, 14th century; Biblioteca Medicea Laurenziana, Florence, Gaddi 36, 14th century; and Biblioteca Nazionale Centrale, Magl. XI.38, 16th century; we are in Perugia, Biblioteca Augusta, ms. D68; and the other is in the Vatican, Biblioteca Apostolica Vaticana, Vat. Lat. 4606, 14th century. The current text of Fibonacci belongs to the Parisian trio of manuscripts, of which that of Mazarine is the main example: all three begin with “Quidam numeri habent radices”, end with “Liceat mihi in hoc de radicum” and contain the additional text with references to Campanus of Novara.

Leonardo Pisano, or Fibonacci, is rightly considered one of the greatest mathematicians of the Middle Ages. His most important work, the *Liber Abaci*, is the first comprehensive and systematic explanation of Hindu-Arabic numerals by a European writer. Little is known of Leonardo’s life, and most biographical details are taken from his introduction to his *Liber Abaci*. When he was young, his father was employed as a “publicus scriba” in Bugia, modern Algeria, and it was there that Leonardo apparently learned to use the abacus and familiarized himself with Hindu-Arabic numeration. He traveled widely through Egypt, Syria, Greece, Sicily and Constantinople, where he developed his skills as a mathematician, became involved in academic debates and fell into the circles of the court of Frederick II. , Holy Roman Emperor.

The *Liber Abaci *is divided into 15 chapters. The first section introduces the Hindu-Arabic numeral system, including methods of conversion between different systems of representation. This section also includes the first known description of trial division for testing whether a number is composite and, if so, factoring it. The second section deals with trading, currency conversions, and profit and interest calculations. The third and larger section deals with a number of mathematical problems, including the Chinese remainder theorem, perfect numbers and Mersenne primes as well as formulas for arithmetic series and for square pyramidal numbers. The final and most important section, which comprises chapters 14 and 15, and was in a small group of separate manuscripts (see above)—as in this one—extracted as a stand-alone text, deals with approximations, à la both numerical and geometric, of irrational numbers such as square roots, and introduced the algebraic method of Fibonacci, inspired by Euclid’s method *Elements* and the introduction of Arabic sources into European mathematics for the very first time.

Fibonacci’s impact on the history of Western mathematics is incalculable: even at the turn of the 16th century, Luca Pacioli would acknowledge his reliance on Fibonacci in his* Sum*. According to William Goetzman: “The five hundred years since Leonardo saw the development in Europe of practically all the tools of financial capitalism that we know today: shareholding in limited liability companies, long-term loans from governments and companies, liquidity and active international financial markets, life insurance, life annuities, mutual funds, derivative securities and depository banking. Many of these developments have their roots in contracts first analyzed mathematically by Fibonacci. (W. Goetzmann, ‘Fibonacci and the financial revolution’, *The origins of value: the financial innovations that created modern capital markets*2005, p.125).