## A sammelband of five mathematical texts, in Latin, illustrated manuscript on paper. Comprising: Boethius, De Institutione Arithmetica; Grosseteste, Compotus; [Anon.] Tables for the comparison of Christian and Arabic years, tables of conjunction and opposition with explanatory notes; De Pulchro Rivo, Computus Manualis; Fibonacci, Liber Abbaci, chapters 14 & 15.

[Italy: late 15th century].

An extraordinary and highly important 15^{th}-century illustrated sammelband of mediaeval mathematical texts, including the two final and most advanced chapters of the *Liber Abbaci* of Leonardo of Pisa, also known as Fibonacci (c.1170-c.1250), “the first great mathematician of the Christian West” (DSB). This manuscript was owned and bound by one of the most important 19^{th}-century historians of mathematics, Prince Baldassare Boncompagni, who published the first modern critical edition of Fibonacci’s work (1857-62). *Liber Abbaci* was largely responsible for disseminating knowledge of the system of Hindu-Arabi numerals in Europe. “The first seven chapters dealt with the notation, explaining the principle of place value, by which the position of a figure determines whether it is a unit, 10, 100, and so forth, and demonstrating the use of the numerals in arithmetical operations. The techniques were then applied to such practical problems as profit margin, barter, money changing, conversion of weights and measures, partnerships, and interest. Most of the work was devoted to speculative mathematics—proportion, the Rule of False Position, extraction of roots, and the properties of numbers, concluding with some geometry and algebra” (Britannica). In the final two chapters, represented in the present manuscript, “Leonardo shows himself to be a master in the application of algebraic methods and an outstanding student of Euclid” (DSB). Fibonnaci is perhaps best known today for ‘Fibonnaci sequences,’ which appeared in chapter 13 of the *Liber Abbaci*. This sammelband contains several other essential mathematical texts of the early Middle Ages. Written by the 6th-century Roman philosopher Boethius (c.480-524), *De institutione Arithmetica* was the principal mathematical textbook of pre-12th century Western Europe, comprising a philosophical discussion of numbers, their relationships and meanings. One of the text’s most influential features was its division of the mathematical sciences into arithmetic, music, geometry, and astronomy, which it together designated as the *quadrivium*. The *Compotus *of the English statesman, scholastic philosopher, theologian and Bishop of Lincoln Robert Grosseteste (1175-1253) was innovative in that it incorporated Arabic astronomical influences into computistical theory. In it Grosseteste defines ‘computus’ as a science of counting and dividing time, and his discussion of the solar year is a key contribution to the raging medieval debate around calendar reform. The *Computus Manualis* of the thirteenth-century German master Johannes De Pulchro Rivo treats the method of carrying out these calendrical calculations on one’s fingers, rather than by consulting tables. All of these works are rare, but the Fibonnaci is the rarest, and this is probably the only copy in private hands of any substantial part of his *Liber Abbaci*.

*Provenance*: (1) Pietro Girometti (1811-1859), Roman gem-engraver and medallist. No 25 in his catalogue of

*Codici Cartacei e Membranacei*

*dei secoli XIVo e XVo […]*, 1856, sold with 33 other manuscripts to:

(2) Prince Baldassare Boncompagni-Ludovisi (1821-1894), one of the leading figures in 19th-century mathematics, a bibliophile and scholar responsible for the widespread propagation and popularisation of several key mathematical texts of the Middle Ages, including Fibonacci. His vast library consisted of some 650 manuscripts and over 20,000 printed volumes: the manuscripts alone constituting one of the largest private collections of scientific and mathematical texts. The texts in the present manuscript were assembled and bound by Boncompagni: remnants of his shelf label on spine 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. Facsimili, Edizioni del Secolo XV. Abbachi Riviste*, Rome, 1898, lot 99.

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

*Interesting Books and Manuscripts on Various Subjects […]*, 1980, no 20, and purchased by:

(4) J.G. Bergart, and loaned to the John Hay Library, Brown University (published in K.P. Harrington, ed.,

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

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

(6) Christie’s, 13 July 2022, lot 154.

“Little is known about Fibonacci’s life beyond the few facts given in his mathematical writings. During Fibonacci’s boyhood his father, Guglielmo, a Pisan merchant, was appointed consul over the community of Pisan merchants in the North African port of Bugia (now Bejaia, Algeria). Fibonacci was sent to study calculation with an Arab master. He later went to Egypt, Syria, Greece, Sicily, and Provence, where he studied different numerical systems and methods of calculation” (Britannica).

“*Liber abaci*, or the Book of Calculation, appeared first in 1202, and then again in a second version in 1228. Leonardo's stated intention was to introduce the Hindu number system and its operations to the Italian people. However, *Liber abaci* is much more than merely an introduction to the Hindu number system and the algorithms for working with it. Liber abaci is an encyclopedic work treating much of the known mathematics of the thirteenth century on arithmetic, algebra, and problem solving. It is, moreover, a theoretical as well as a practical work; the methods employed in *Liber abaci* Leonardo firmly establishes with Euclidean geometric proofs …

“One should here again make the point, that while derived from the word abacus, the word *abaci* refers in the thirteenth century paradoxically to calculation without the abacus. Thus, *Liber abaci* should not be translated as *The Book of the Abacus*. A *maestro d’abbaco* was a person who calculated directly with Hindu numerals without using the abacus, and *abaco* is the discipline of doing this. It was Leonardo's purpose to replace Roman numerals with the Hindu numerals not only among scientists, but in commerce and among the common people. He achieved his goal perhaps more than he ever dreamed. Italian merchants carried the new mathematics and its methods wherever they went in the Mediterranean world. The new mathematics also spread into Germany where it was propagated by the *cossists *(a corruption of the Italian *cosa*, or thing, the unknown of algebra) …

“In addition to teaching all of the necessary methods of arithmetic and algebra, Leonardo includes in *Liber abaci* a wealth of applications of mathematics to all kinds of situations in business and trade, conversion of units of money, weight, and content, methods of barter, business partnerships and allocation of profit, alloying of money, investment of money, simple and compound interest. The problems on trade give valuable insight into the medieval world. He also includes many problems purely to show the power and beauty of his mathematics; these problems are noteworthy for his choice of appealing vivid images and his ingenuity in solution” (Sigler, pp. 5-6).

The impact of Fibonacci on the history of Western mathematics is incalculable: even at the turn of the 16th century, Luca Pacioli would acknowledge his dependence on Fibonacci in his* Summa*. According to William Goetzman: ‘The five-hundred-year period following Leonardo saw the development in Europe of virtually all the tools of financial capitalism that we know today: share ownership of limited-liability corporations, long-term government and corporate loans, liquid and active international financial markets, life insurance, life annuities, mutual funds, derivative securities, and deposit banking. Many of these developments have their roots in contracts first mathematically analyzed by Fibonacci'. (W. Goetzmann, ‘Fibonacci and the Financial Revolution’, *The Origins of Value: The Financial Innovations that Created Modern Capital Markets*, 2005, p.125).

“Chapter 14, which is devoted to calculations with radicals, begins with a few formulas of general arithmetic. Called ‘keys’ (claves), they are taken from book II of Euclid’s *Elements*. Leonardo explicitly says that he is forgoing any demonstrations of his own since they are all proved there. The fifth and sixth propositions of book II are especially important; from them, he said, one could derive all the problems of the *Aliebra* and the *Almuchabala*. Square and cube roots are taught numerically according to the Indian-Arabic algorithm, which in fact corresponds to the modern one.

“Leonardo also knew the procedure of adding zeros to the radicands in order to obtain greater exactness; actually, this had already been done by Johannes Hispalensis (fl.1135-1153) and al-Nasawi (fl.ca. 1025). Next, examples are given that are illustrative of the ancient methods of approximation … The chapter then goes on systematically to carry out complete operations with Euclidean irrationals … The proof, which is never lacking, of the correctness of the calculations is presented geometrically …

“With respect to mathematical content Leonardo does not surpass his Arab predecessors. Nevertheless, the richness of the examples and of their methodical arrangement, as well as the exact proofs, are to be emphasized. At the end of chapter 15, which is divided into three sections, one sees particularly clearly what complete control Leonardo had over the geometrical as well as the algebraic methods for solving quadratic equations and with what skill he could use them in applied problems. The first section is concerned with proportions and their multifarious transformations … The second section first presents applications of the Pythagorean theorem, such as the ancient Babylonian problem of a pole leaning against a wall and the Indian problem of two towers of different heights. On the given line joining them (i.e., their bases) there is a spring which shall be equally distant from the tops of the towers. The same problem was solved in chapter 13 by the method of false position. Many different types of problems follow, such as the solution of an indeterminate equation x^{2} + y^{2} = 25, given that 3^{2} + 4^{2} = 25; or problems of the type *de viagiis*, in which the merchant makes the same profit on each of his journeys. Geometric and stereometric problems are also presented; thus, for example, the determination of the amount of water running out of a receptacle when various bodies, including a sphere (with π = 3 1/7), are sunk in.

“The third section contains algebraic quadratic problems (*questiones secundum modum algebre*). First, with reference made to ‘Maumeht,’ i.e., to al-Khwarizmi, the six normal forms [in modern notation] *ax*^{2} = *bx*, *ax*^{2} = *c*, *bx = c*, a*x*^{2 }+ *bx = c*, *ax*^{2} + *c = bx* (here Leonardo is acquainted with both solutions), and ax^{2} = *bx + c* are introduced; they are then exactly computed in numerous, sometimes complicated, examples … The numerical examples are taken largely from the algebra of al-Khwarizmi and al-Karaji, frequently even with the same numerical values. In this fourth section of the *Liber abbaci* there also appear further names for the powers of the unknowns.

“When several unknowns are involved, then (along with *radix* and *res* for *x*) a third unknown is introduced as *pars* (‘part,’ Arabic, *qasm*); and sometimes the sum of two unknowns is designated as *res*. For *x*^{2}, the names quadratus, *census*, and *avere *(‘wealth,’ Arabic, *mal*) are employed; for *x*^{3}, *cubus*, for *x*^{4}, *census de censu* and *censuum census*, and for *x*^{6}*cubus cubi*. The constant term is called *numerus*, *denarius*, or *dragma*” (DSB).

Boethius, Roman scholar, Christian philosopher, and statesman, was the “author of the celebrated *De consolatione philosophiae*, a largely Neoplatonic work in which the pursuit of wisdom and the love of God are described as the true sources of human happiness. He belonged to the ancient Roman family of the Anicii, which had been Christian for about a century and of which Emperor Olybrius had been a member. Boethius’s father had been consul in 487 but died soon afterward, and Boethius was raised by Quintus Aurelius Memmius Symmachus, whose daughter Rusticiana he married. He became consul in 510 under the Ostrogothic king Theodoric. Although little of Boethius’s education is known, he was evidently well trained in Greek” (Britannica).

“The mathematical works by Boethius reproduced Greek works. Although it is not as clear as it has been thought, partly on the basis of what Boethius himself says, exactly which Greek works were reproduced, it is clear that the neo-Pythagorean theory of number as the very divine essence of the world is the view around which the four sciences of the quadrivium are developed. Number, qua multitude considered in itself, is the subject matter of arithmetic; qua multitude applied to something else (relations between numbers?), the subject matter of music; qua magnitude without movement, of astronomy; The *Arithmetic* develops here and there what was too concise in Nicomachus and abbreviates what was too diffuse. Further, it passes on to the Latin reader many of the basic terms and concepts of arithmetical theory: prime and composite numbers, proportionality, *numeri figurati* (linear, triangular, etc.; pyramidal and other solid numbers), and ten different kinds of *Medietates* (arithmetical, geometrical, harmonic, counterharmonic, etc.). His interest in proportions is perhaps connected with the story according to which, while in prison, he thought out a game based on number relations. Here it is noticeable, however, that his understanding of arithmetic, and possibly of Greek, was limited: the more advanced propositions and proofs in Nicomachus, such as the proposition that cubic numbers can be expressed as the successive sums of odd numbers and the proposition expressing the relation between triangular numbers and the polygonal numbers of polygons with *n* sides, are missing from the *Arithmetic*. He does not, however, miss such elementary things as the multiplication table up to ten” (DSB).

“Grosseteste was educated at the University of Oxford and then held a position with William de Vere, the bishop of Hereford. Grosseteste was chancellor of Oxford from about 1215 to 1221 and was given thereafter a number of ecclesiastical preferments and sinecures from which he resigned in 1232. From 1229 or 1230 to 1235 he was first lecturer in theology to the Franciscans, on whom his influence was profound. The works of this, his pre-episcopal career, include a commentary on Aristotle’s *Posterior Analytics* and *Physics,* many independent treatises on scientific subjects, and several scriptural commentaries” (Britannica).

“More original were Grosseteste's four separate treatises on the calendar: *Canon in kalendarium* and *Compotus*; correcting these, *Compotus correctorius*, probably between 1215 and 1219; and *Compotus minor*, with further corrections, in 1244. He showed that with the system long in use, according to which nineteen solar years were considered equal to 235 lunar months, in every 304 years the moon would be one day, six minutes, and forty seconds older than the calendar indicated. He pointed out in the *Compotus correctorius* (cap. 10) that by his time the moon was never full when the calendar said it should be and that this was especially obvious during an eclipse. The error in the reckoning of Easter came from the inaccuracy both of the year of 365.25 days and of the nineteen-year lunar cycle.

“Grosseteste's plan for reforming the calendar was threefold. First, he said that an accurate measure must be made of the length of the solar year. He knew of three estimates of this: that of Hipparchus and Ptolemy, accepted by the Latin computists; that of al-Battani; and that of Thabit ibn Qurra. He discussed in detail the systems of adjustments that would have to be made in each case to make the solstice and equinox occur in the calendar at the times they were observed. Al-Battani’s estimate, he said in the Compotus correctorius (cap. 1), ‘agrees best with what we find by observation on the advance of the solstice in our time.’ The next stage of the reform was to calculate the relationship between this and the mean lunar month. For the new-moon tables of the *Kalendarium*, Grosseteste had used a multiple nineteen-year cycle of seventy-six years. In the *Compotus correctorius* he calculated the error this involved and proposed the novel idea of using a much more accurate cycle of thirty Arab lunar years, each of twelve equal months, the whole occupying 10,631 days. This was the shortest time in which the cycle of whole lunations came back to the start. Grosseteste gave a method of combining this Arab cycle with the Christian solar calendar and of calculating true lunations. The third stage of the reform was to use these results for an accurate reckoning of Easter. In the *Compotus correctorius* (cap. 10), he said that even without an accurate measure of the length of the solar year, the spring equinox, on which the date of Easter depended, could be discovered ‘by observation with instruments or from verified astronomical tables’” (DSB).

Little is known of John of Pulchro Rivo (John from the beautiful Stream), also known as John of Brunswick and John the German. “Born in Brunswick at an unknown date, John appears to have spent some time as a student or master at the University of Paris. where in 1289 he started to write his most popular work, a brief *Compotus manualis* that survives in at least 28 copies. By 1297 he was back in Goslar in his native Saxony, where he compiled a *Compotus novus* as well as a detailed commentary on the same work completed on 1 April 1298. In the parlance of his day. John of Brunswick was a *compotista*, someone who practice the art of *compotus*, the thirteenth-century spelling of *computus*. On the surface *computus *was there simply to teach aspiring members of the clergy how to calculate the date of Easter and other mobile feast days of the liturgical year. In practice, however, it often made them grapple with the problems of time reckoning on a much broader scale, as *computus *texts (or *computi*) were capable of drawing knowledge from a diverse array of disciplines: astronomy and mathematics always occupied front and centre, but the flanks were habitually lined by history, etymology and theology, sometimes even by astrology, cosmography, poetry or medicine” (Nothaft, *Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe* (2018), pp. 1-2).

As noted earlier, the Fibonacci is the rarest of the manuscripts in this sammelband. In 1852 Boncompagni listed 12 surviving manuscript copies – partial or fragmentary – of the *Liber Abaci* (B. Boncompagni, *Della Vita e delle Opere di Leonardo Pisano, matematico del secolo decimoterzo*, Rome, 1852, pp.25-69). A more recent 2017 survey by the Italian mathematician Enrico Giusti lists eight complete, or near-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. C.I.2616, 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; Bibliothèque Nationale, Paris, Lat. 7367, 15th century; and Lat. 7225A, 16th century; three are in Florence: Biblioteca Riccardiana, ms. 2252, 14th century; Biblioteca Medicea Laurenziana, Firenze, Gaddi 36, 14th century; and Biblioteca Nazionale Centrale, Magl. XI.38, 16th century; one is in Perugia, Biblioteca Augusta, ms. D 68; and one is at the Vatican, Biblioteca Apostolica Vaticana, Vat. Lat. 4606, 14th century. The present Fibonacci text belongs to the Parisian trio of manuscripts, of which the one at the Mazarine is the primary exemplar: all three begin ‘Quidam numeri habent radices’, end with ‘Liceat mihi in hoc de radicum’ and contain the extra text with references to Campanus of Novara.

Five works in one volume (217 x 158mm), ff. v, 222, ii, complete. BOETHIUS, Anicius Manlius Severinus, De Institutione Arithmetica, opening ‘In dandis accipiendisque muneribus […]’, Book I ff.1-25, Book II ff.25-64; GROSSETESTE, Robert, Compotus, opening with a list of the twelve chapters on f.65, and the text proper ‘Computus e[st] sci[en]t[i]a num[era]tionis et divisionis t[em]por[um]’ ff.65-93; Tables for the comparison of Christian and Arabic years, tables of conjunction and opposition with explanatory notes, opening ‘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; blank 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 with the Boncompagni edition of Il Liber Abbaci di 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 on f.215v), the text here continues with references to Campanus of Novara [c.1220-1296] beginning: ‘Sum[m]a progressionis […]’ and ending ‘[…] divisus est igitur triang[u]l[u]s a b c in tres partes equales ut proponit. Camp.’, ff.215v-220. The first four texts ff.1-102 in two columns of 28-32 lines, ruled space 173 x 56mm, the watermark a crossbow in a circle, closest to Bricquet 739, in use in Italy from 1468; rubrics and initials in red, capitals touched in red, the Boethius illustrated with 125 diagrams in red and brown, the Grosseteste illustrated with 10 diagrams. The final text by Fibonacci ff.104-221 in a single column of c.26 lines in an 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; 19th-century foliation in pen, probably at the time of binding, giving collation 1-108, 116, 12-138, 14-1512, 1610, 1712, 1814, 19-2312 (opening leaf with small marginal repair and some soiling, occasional ink erosion to diagrams, some marginal staining and thumbing). Mid-19th century marbled boards (edges scuffed and rubbed). In a chemise and red quarter-morocco case.

Item #6038

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Price:
$650,000.00
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