## Libro de algebra en arithmetica y geometria.

Antwerp: heirs of Arnold Birckman, 1567.

First edition of this very rare and celebrated treatise on algebra, the superb de Thou copy bound in early seventeenth-century French citron morocco with his arms – scientific books in de Thou bindings very rarely appear on the market. “Considered the greatest of Portuguese mathematicians, Nuñez reveals in his discoveries, theories, and publications that he was a first-rate geographer, physicist, cosmologist, geometer and algebraist” (DSB). Nuñez’s work is distinguished from other algebra textbooks of the time by its greater level of abstraction and its use of letters rather than numbers. Indeed, it is the first modern work to use algebraic symbolism in something like the way it is used today. This marks Nuñez as one of the most important precursors of François Viète. “We notice first of all a generality in the demonstrations, an abstraction in the statements of the exercises, which is very exceptional for the time and which gives the *Libro de algebra* an already quite modern character. Thus, the 110 problems of algebra, which are the object of chapter 5 of the third part, are not drawn, as in the texts of other algebraists of the time, from commerce, industry, or other situations in real life; these are problems about numbers. When the author borrows from Pacioli or someone else a question in which players share their winnings, for example, he is careful to reduce the problem to one about numbers … This same quality of abstraction is also noticed in the 77 exercises on the application of algebra to geometry which form the subject of the last chapter of the third part” (translated from Bosmans (1907-8), pp. 157-158). The first part of the *Libro de Algebra* treats the solution of equations of the first and second degree. The second part is divided into three subsections: the first treats the addition, subtraction, multiplication, and division of polynomials; the second the same manipulations of radicals (square and higher roots); and the third the theory of proportion, treated algebraically rather than in the geometric language used by his predecessors. The third and longest part is devoted to the solution of polynomial equations (including one chapter on systems of linear equations). The work concludes with a chapter on the application of algebra to geometry. It was valued highly by most of the leading seventeenth century mathematicians, in England, France and Germany, as well as in Spain and Portugal, especially by John Wallis, Jacques Peletier, Elias Vinet, and Guillaume Gosselin. “Stevin was acquainted with it and credited Nuñez with leading him to apply Euclid’s algorithm to polynomials” (Malet, p. 193). Until the appearance of Christoph Clavius’s *Algebra* in 1608, the *Libro de algebra* was one of the most widely used algebra texts in the Jesuit colleges, and it was highly praised by Clavius himself. The manuscript of this work was prepared in Portuguese some thirty years before Nuñez published this Spanish translation (the preface is in Portuguese), but he added to the work substantially in the intervening years (for example, reference is made to Peletier’s edition of Euclid’s *Elements*, published in 1557). ABPC/RBH lists only three other copies since 1935: the Honeyman/Streeter copy (Christie’s NY 2007 & Sotheby’s 1978); Macclesfield (Sotheby’s 2005); and Hartung 2005 (a copy in a late 19^{th}-century binding with paper repairs). The Macclesfield copy, in a similar seventeenth-century morocco binding to the present copy, realized a little over $17,000. OCLC lists only Folger and Michigan in the US.

*Provenance*: The Peeters-Fontainas copy (the sale of his library, Sotheby’s London, May 23, 1978, lot 382), with the combined arms of J. A. de Thou and those of his second wife, Gasparde de La Chastre, on the covers.

Pedro Nuñez Salaciense (1502-78) was born in Alcácer do Sol, Portugal and studied at Salamanca, Spain, and in 1524 or 1525 at Lisbon. He held a professorship in Lisbon, which was moved to Coimbra in 1537. He taught Clavius, Nicolaus Coelho de Amaral, Manuel de Figueredo and Joao de Castro, one of the greatest Portuguese navigators. “Both as Royal Cosmographer under King John III (the Pious) of Portugal and as professor of mathematics at the University of Coimbra, Nunes gave instruction in the art of navigation to those associated with Portugal’s merchant and naval fleets. His *Libro *de *algebra *provided the mathematical underpinnings of that instruction — and much more — adopting Luca Pacioli’s abbreviated notational style and treating the solution not only of linear and quadratic equations but also that of a cubic equation of the type *x ^{3}* +

*cx*=

*d*following the spectacular mid-sixteenth-century work of the Italians Niccolò Tartaglia and Girolamo Cardano” (Katz & Parshall,

*Taming the Unknown*, p. 205).

“By 1533, Nunes had already translated most of the scientific works of Aristotle, Euclid and Ptolemy, and mathematical treatises by Arabic and Italian authors, and he had been appointed as the first Royal Cosmographer. It seems highly probable that his theories in geometry and algebra were by then well thought out, already sufficiently advanced for him to have a far better understanding of the workings of the globe, of cartography and of the differing shadows, as can be seen in his *Tratado da Sphera*, published in 1537, and his *Crepusculis*, published in 1542. Only four years later he was also well able to refute solutions to apparently insoluble problems for the Ancient Greeks, as propounded by France’s leading mathematician, Oronce Finé (*De erratis Orontii Finaei*, 1546). The major addition from his thirty years of research and teaching mathematics at the University of Coimbra was the collection of problems (187 in all), in Part 3 of his Spanish Algebra …

“When Nunes dedicated his *magnum opus* on algebra to Cardinal Henry, he dated it 1564, and located it not in Coimbra, but in Lisbon. It seems quite likely that at this stage the enlarged work was also written in Portuguese, like its introduction. His original teaching materials had been expanded tenfold with the necessary elucidation of what he could explain by word of mouth, and with the 187 problems in the final chapter. These problems must have been developed while he was using his algebra to teach his students at the University of Coimbra. Three years later, the book appeared in print, but in Spanish, and at Antwerp, by then a major commercial rival to Lisbon … Early in the dedication, Nunes refers to *nesta opulentíssima cidade de Lixboa, onde tanto negotio ha desde extremo oriente e occidente, e ilhas do mar Oceano, e onde el Rey nosso senhor tem quarenta contadores de sua fazenda* (‘this very rich city of Lisbon, to where so much commerce comes from the farthest Eastern and Western lands and islands of the Atlantic ocean, and where the King our Lord has 40 accountants for his treasury’) … However, his choice of Spanish and of Antwerp, hopefully to give his algebra greater publicity, proved to be disastrous. The busy port was overtly hostile to the Spanish, and no mecca for an abstract work on mathematics, and it seems that his text was only purchased by the professional mathematicians in Northern Europe” (Martin). This perhaps explains the great rarity of the book today.

As already noted, Nuñez refers to Italian authors such as Pacioli, Tartaglia, and Cardano, but the contemporary German algebraists, such as Christoph Rudolff (1499-1545) and Michael Stifel (1487-1567), are not mentioned and may have been unknown to Nuñez. However, his greatest debt is to Jordanus Nemorarius (c. 1225-60), who was perhaps the first to use letters to replace numbers in algebraic calculations. “Jordanus was ahead of his time.He hovered too high over his contemporaries to be understood.Two and a half centuries passed from the famous German to Viète.During this long period of time, only a Sicilian, Abbé Maurolico (1494-1575), seems to have appreciated the importance of the discovery.To his name, it will now be necessary to add that of Nuñez” (Bosmans (1908), p. 14). Maurolico’s contributions remained unpublished until the 20^{th} century, so Nunez’s *Libro de Algebra* can claim to be the first modern work to demonstrate the use of algebraic symbolism in something like the way it is used today, 24 years before Viète’s epoch-making *In artem analyticem isagoge*.

This is most clearly demonstrated in his proofs in the sections on radicals and proportion in the second part. “Let us admire, for example, the modern character of the following proof “ (Bosmans (1908), p. 14) [we use modern notation]. Suppose that

*a = b*^{1/c} and *b = g*^{1/f}.

We want to express *a* as a root of *g*. Nuñez argues as follows. From the two equations we have

*b = a *x* a *x … x* a* (*c* times)

and

*g = b* x *b* x … x *b* (*f* times)

Putting *c* x *f = k*, we then have

*g = a* x *a* x … x *a* (*k* times).

Hence,

*a = g*^{1/k}.

“Neither in Stifel, nor in Cardano, can one find a page written in this style” (*ibid*., p. 15).

Nuñez does not, of course, express his proof in such modern notation. He uses the terms *numero, cosa, censo*, *cubo, censo de censo, relato primo, censo de cubo* (or *cubo de censo*) … for our number (or constant), 1^{st}, 2^{nd}, 3^{rd}, 4^{th}, 5^{th}, 6^{th}, … powers. For example, the equation

12*x*^{3} + 65*x*^{2} – 7*x* = 10

was written by Nuñez,

12.cu.p.65.ce.m.7.co.yguales a.10.

He uses the notation R.V. (*Radix Vniversalis*) to denote a radical applied to the terms following it. For example,

R.V.cu.R.26.p.5

means

(√26 + 5)^{⅓.}.

Nuñez (like Maurolico) did not yet take the step of keeping track of the intermediate steps in his calculations: the sums, differences, products, quotients and roots are each replaced by a new letter as they appear during the calculation. This step was not taken before Viète.

In the third part of the *Libro de Algebra*, consisting of seven chapters devoted to the theory of equations, Nuñez gives an account of Tartaglia’s solution of cubic equations. In modern terms, Nuñez explains that to solve a cubic equation

*x*^{3} + *ax* = *b*,

determine two auxiliary unknowns *y* and *z* by means of the equations

*y – z = b*, *yz* = (*a*/3)^{3};

then the principal unknown will be

*x* = *y*^{1/3 }– *z*^{1/3}.

Nuñez states, however, that Tartaglia’s rule is impractical because of the sometimes unnecessary appearance of radicals. He demonstrates this by considering the equation

*x*^{3 }+ 3*x* = 36.

Clearly the root is 3 (the other two roots are not real). But Tartaglia’s rule gives

*x* = (√325 + 18)^{1/3} – (√325 – 18)^{1/3}.

Nuñez therefore proposes another method which, although to us it seems even less practical, proved fruitful in the hands of his successors. Essentially, Nuñez says that if we can guess a root of a cubic equation, we can use it to reduce the equation to a quadratic. Thus, in the equation

*x*^{3 }= *ax*^{2} + *bx + c*,

if *p* is known to be a root,

*p*^{3 }= *ap*^{2} + *bp + c*

and subtracting gives

*x*^{3} – *p*^{3} = *a*(*x*^{2} – *p*^{2}) + *b*(*x – p*);

dividing by *x – p* results in a quadratic equation for *x*. Nuñez gives several examples of types of cubic equations to which this method can be applied.

“This whole theory of lowering the degree of equations is very interesting.It strongly impressed Nuñez’s contemporaries who lavished praise upon it.These praises have even led some historians to believe that Nuñez was the inventor of the algorithm for finding the greatest common algebraic divisor by means of the successive division of one polynomial by another” (Bosmans (1907-8), pp. 164-5). Problem LIII in Stevin’s *L’Arithmetique* (1585), for example, asks for the greatest common divisor of two given polynomials. Stevin adds a Note that appears to credit Nuñez with the procedure. Bosmans interprets Stevin’s comment as meaning that Stevin derived the idea of the algorithm from Nuñez, though probably not the actual procedure.

Chapter 5 gives 110 algebra problems and their solutions. Bosmans (1908) (pp. 26-35) lists them all, writing that “the length of such an enumeration caused me to hesitate, but because of the rarity of the *Libro de Algebra* I overcame my hesitation.” He writes, “Considered as a whole, this chapter has no contemporary analogue. Nuñez on a certain point surpasses all his emulators, even the most illustrious, even Cardano and Stifel. From the first to the last without exception, the problems of chap.5 are abstract exercises on numbers.No more barrels of wine, no more alms of cloth, no more thieves' quarrels, no more mathematical recreations, which at times give the most learned algebras of the 16^{th} century the childlike character of primary school textbooks. Even Stifel’s *Arithmetica integra* (1544) and Cardano’s *De arte magna* (1545) are not completely exempt. Nuñez’s style seems to reflect that of Diophantus.Although the edition of Xylander [*Rerum arithmeticarum libri sex*, 1575] had not yet appeared, nothing prevented our author from knowing the Greek algebraist through his manuscripts.”

Chapter 7 is devoted to the application of algebra to geometry. The 77 exercises in this chapter fall into 6 groups (Bosmans (1908), p. 41): Squares (1-14); Non-square rectangles (15-31); triangles (32-64); rhombuses and rhomboides, i.e., lozenges and simple parallelograms (65-69); trapeziums (70-74); pentagons (75-77). The third group is the most interesting. Here Nuñez proposes to complete and perfect the second book of Regiomontanus’s *De triangulis* (1533). A singularly important role is played in Nuñez’s theory of triangles by the formula

*s* = √*p*(*p – a*)(*p – b*)(*p – c*)

for the area *s* of a triangle with sides *a*, *b*, *c* and semi-perimeter *p*. Today everyone knows this as ‘Heron’s formula’ (Heron, or Hero, of Alexandria, ca. 10-70 AD), but in 1567 Heron was very little known. Nuñez instead took the formula from Pacioli, but he writes that Pacioli’s proof is so obscure and difficult to understand that he will give a clear and rigorous proof himself. This occupies 21 pages of text.

At the start of his treatment of proportions in the second section, Nuñez makes an interesting digression on the subject of the ‘contingency’ or contact angle – the angle between two curves (or a curve and a straight line) at a point of tangency. Is the contact angle a magnitude? Is it the same as an angle between two straight lines? Can it be compared to such a rectilinear angle? This question generated a heated debate between Jacques Peletier (1517-1582), who maintained that ‘contact is not an angle’, and Clavius (1537-1612), who stated that the contact angle is an angle, but of a different nature to the recilinear angle. Nuñez agreed with Clavius that the two kinds of angles were of a different nature, since no (finite) multiple of the contact angle can exceed any rectilinear angle. The “arguments of Nuñez differ from those proposed by his contemporaries and present a real originality … The digression on the contingency angle provides an example of a very tight and incisive argumentation found in other passages of the *Libro de Algebra*. The Portuguese cosmographer condenses into a few lines demonstrations or arguments which sometimes occupy long passages in others (including Peletier and Clavius). On first reading, this brevity may suggest that Nuñez’s text is based on implicit information, or even that its author wrote it casually. On the contrary, it seems to me that it is the sign of a great mastery, of an in-depth knowledge of the literature available on the subject, and of a certainty still rare among his contemporaries that mathematical arguments are valid – that it is not necessary, in order to make them convincing, to embed them in the rhetoric used in other fields of knowledge. Without expressly affirming it, Nuñez adopts a point of view which, for most of his contemporaries appears at best as a paralogism, at worst (for Peletier) as a contradiction: the angle of contingency is a magnitude, but of another kind than the rectilinear. After him, Viète would adopt the same point of view, but it was Wallis and Newton who expressed it most clearly” (translated from Loget).

The edition was divided between the heirs of Birckman (as here) and the heirs of Johann Steelsius. In their description of a copy of the Steelsius version which they offered in 1931, Maggs stated that it was the second edition. However, Bosmans (1908) (p. 4) states: “Il ne faudrait pas croire pour cela à l’existence d’une deuxième édition. Seule le titre diffère.”

Jacques Auguste de Thou (1553-1617), was a French diplomat, lawyer, and historian. In spite of his immersion in political life, being a councillor of state to Henri III and Henri IV, and also president of the Paris *parlement*, in which position he attempted to promote the cause of religious toleration by helping to negotiate the Edict of Nantes with the Huguenots, his main achievement was to advance a scientific and impartial approach to history writing through his own *Historia sui temporis*. De Thou was also director of the Royal Library from 1593, and in the meantime also amassed a great library of his own, said to contain almost 13,000 volumes at the time of his death. It later became the property of Jean-Jacques Charron, marquis de Ménars (1643-1718) before being sold in 1789. Books acquired by de Thou when he was a batchelor bear his coat of arms on the bindings. The binding of Nunez’s *Libro de Algebra* shows his coat of arms intertwined with those of his first wife, Marie Barbançon; it incorporates a monogram which combines the initials ‘I A’ of his given names with the M of his wife’s first name. After Marie died in 1601, de Thou married Gasparde de La Chastre, and replaced Marie’s arms with Gasparde’s and Marie’s M with a G.

Jean Peeters-Fontainas (1891-1975), Belgian notary, scholar and bibliophile, compiled the *Bibliographie des impressions espagnoles des Pays-Bas 1520-1799* (Louvain-Anvers, 1933), revised and enlarged in 1965. This latter work was awarded the first ILAB Breslauer Prize for Bibliography. It listed 1417 Spanish-language books printed in the Low Countries before 1800; Peeters-Fontainas himself owned most of these books.

Frank Streeter 392 = Honeyman 2354; Macclesfield 1548; Peeters-Fontainas, *Bibliographie des impressions espagnols des Pays-Bas Méridionaux* 845 (this copy). Our description is based heavily on two articles by Bosmans: ‘L’Algèbre de Pedro Nuñez,’ *Annaes da Academia Polytechnica do Porto* 3 (1908), pp. 1-50; ‘Sur le ‘Libro de Algebra’ de Pedro Nuñez,’ *Bibliotheca Mathematica* 8 (1907-8), pp. 154-169. Loget, ‘La digression sur l’angle de contact dans le Libro de Algebra de Pedro Nuñez,’ *Quaderns d'història de l'enginyeria* 11 (2010), pp. 79-100. Malet, ‘Changing notions of proportionality in pre-modern mathematics,’ pp. 193-6 (http://asclepio.revistas.csic.es/index.php/asclepio/article/view/966/1587). Martin, ‘The teaching manual of Pedro Nuñes’ (uc.pt/fluc/eclassicos/publicacoes/ficheiros/humanitas43-44/15_Martyn.pdf).

8vo (167 x 104 mm), ff. [xvi], 339 (recte 341), woodcut printer’s device on title, woodcut diagrams in text, all edges gilt. Early seventeenth-century French citron morocco gilt, spine tooled in compartments with de Thou’s final monogram, covers with three-line roll border and in the centre the combined arms of de Thou and his second wife (slightly rubbed).

Item #5824

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Price:
$35,000.00
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