## 2.1 The author Jordanus de Nemore

Jordanus de Nemore, or Jordanus Nemorarius as he is called in some manuscripts, was the author of several treatises completed before 1260. Nothing specific is known about his personal life. The historical period and the range of his scholarly activities are only circumscribed by the inclusion of his works in the *Biblionomia*, a catalog of the library of Richard de Fournival, the chancellor of the Amiens Cathedral,^{1} compiled between 1246 and 1260.^{2}

Codex 43 of this catalog lists several works attributed to Jordanus:

**1***Philotegni* or *De triangulis*,

**2***De ratione ponderum*,

**3***De ponderum proportione*, and

**4***De quadratura circuli*.

Codex 45 lists:

**1***Practica* or *Algorismus*,

**2***Practica de minutiis*, and

**3***Experimenta super algebra*.

Codex 47 refers to:

**1***Arithmetica*.

Codex 48 lists:

**1***De numeris datis*,

**2***Quedam experimenta super progressione numerorum*, and

**3***Liber de proportionibus*.

Codex 59 refers to:

**1***Suppletiones plane spere*.

Many more later manuscripts are attributed to Jordanus, but this may have been due to the close association of his name with the subjects he established in the Latin tradition, in particular also the *science of weights*. It is therefore often difficult to assess which works he actually authored, which works he adopted himself from an earlier tradition and just commented upon, which works were partly authored by him but then extended by later commentaries, and which works were simply ascribed to him because of his role as an authority in a particular field.

Jordanus thus emerges as one of the most important mathematicians of the Middle Ages. His extant mathematical writings deal with geometry, algebra, and arithmetics. His work clearly draws both on ancient and on Arabic sources, as is the case for his contributions to mechanics. He thus represents for the mathematical sciences, in a sense, a parallel figure to his contemporary Albertus Magnus,^{3} who established Aristotelianism as a frame of reference for theological and philosophical discourse, benefitting from the Arabic-Latin translation movement of the preceding century.^{4} More specifically, Jordanus flourished in a period in which Latin Europe was about to establish its own institutional and intellectual structures for absorbing the rich knowledge inherited from the Arabic world. By bringing subjects such as the science of weights, known to him through the Arabic tradition, into a more rigorous, Euclidean form, he elevated them to the scientific standards of the emerging scholastics, a transformation that did not take place without leaving traces on the contents with which it was concerned.^{5} Here we claim, in particular, that the concept of *positional heaviness* which Jordanus introduced in order to distinguish between a weight and its positional effect was exactly such a trace of the framework of emerging scholastics (see page

The starting point of Jordanus' work on mechanics was probably the *Liber karastonis*, ascribed to Thābit Ibn Qurra^{6} in a translation that may go back to Gerard of Cremona,^{7} as well as the *Liber de canonio*,^{8} probably a Latin translation of a text going back to a Greek source. Both texts deal with the balancing of the steelyard and take into account the fact that a balance has a material beam which itself possesses weight. The *Liber karastonis* provides a proof of the law of the lever from an Aristotelian foundation and the *Liber de canonio* focuses on the material beam. Taken together, they constituted the challenge of constructing a sophisticated theory of the balance on an Aristotelian foundation in a Euclidean form, in other words, just the kind of challenge that Jordanus also addressed in his other works.

Three major groups of manuscripts on the *science of weights* attributed to Jordanus can be distinguished.^{9} The *Elementa super demonstrationem ponderum* contain seven postulates and nine propositions together with extensive proofs. They also contain a reference to one of Jordanus' mathematical works. The *Elementa* are often found in medieval manuscripts together with the *Liber de canonio*. They may thus be considered as providing a theoretical foundation, anchored in an Aristotelian framework, for the treatment of the material beam in the *Liber de canonio*.

The *Liber de ponderibus* begins with precisely the same postulates and propositions, albeit their wording is partly different.^{10} Furthermore the postulates are preceded by a prologue, explicitly introducing the term *gravitas secundum situm*, i.e. *positional heaviness*; at the end four additional propositions are appended, which stem from the *Liber de canonio*. This treatise, however, does not contain the extended proofs of the *Elementa* but instead, in the various forms in which it is extant, two types of explanatory commentaries to the propositions, one short in a scholastic style, one longer involving mathematical arguments as well.

Finally, the *Liber de ratione ponderis* is a much longer treatise divided into four parts containing ten, twelve, six and seventeen propositions respectively. The text begins with seven postulates that are similar to those of the other two treatises. The first postulate adds reference to the “virtus,” i.e. “force” of tending downward and resisting motion. The last postulate explains the horizontal equilibrium position of the beam in terms of angles with the vertical. Also, most of the propositions of the first part are similar to those of the other treatises, with a reference to upward motion omitted in the first proposition, a reference to unequal weights added in the second proposition, with proposition 3 of the *De ratione ponderis* taking the place of proposition 4 of the *Elementa* and vice versa, proposition 5 remaining identical, with proposition 6 of the *De ratione ponderis* taking the place of proposition 8 of the *Elementa*, and with proposition 7 of the *De ratione ponderis* taking the place of proposition 9 of the *Elementa*. Propositions 6 and 7 of the *Elementa*, dealing with the bent lever in a way that is problematic from a modern perspective, however, are replaced by the “correct” proposition 8 of the *De ratione ponderis*. Propositions 9 and 10 of the *De ratione ponderis* deal with the descent of a weight along a rectilinear path and with the inclined plane. Part 2 of the *De ratione ponderis* treats the material beam, Part 3 further cases of the bent lever, while Part 4 adresses various subjects of motion.

As mentioned above, Jordanus' work became known in the sixteenth century through printed editions of his *Liber de ponderibus* by Petrus Apianus^{11} and of the *De ratione ponderis* by Tartaglia.^{12} The latter comprises the two kinds of commentaries mentioned above, the longer one obviously based on knowledge of other manuscripts by Jordanus as well.

The present volume deals mainly with Apianus' edition of the *Liber de ponderibus*, that is, the edition annotated by the marginalia of Guidobaldo. In manuscripts of the *Liber de ponderibus*, but not in the Apianus edition, the text concludes with the formula:

Explicit tractatus de ponderibus magistri Jordanis.

Here ends the treatise on weights of Master Jordanus.^{13}

The authorship of Jordanus de Nemore is nevertheless controversial, even for the postulates and the theorems. Some manuscripts ascribe the postulates and the first nine theorems not to Jordanus but to Euclid. Indeed, the final sentence of the second comment to the ninth theorem of the Apianus edition reads:

Hic explicit secundum aliquos liber Euclidis de ponderibus.

Here ends, according to some, Euclid's book on weights.^{14}

Since in medieval manuscript traditions, propositions, proofs, and ascriptions of authorship led a life of their own, rather independently from each other, there is little one can conclude with certainty from these circumstances.^{15}

The postulates and the first nine theorems in fact form a rather closely knit deductive system centered on the notion of *positional heaviness* that could have hardly arisen without some explicit technical proofs in the first place.^{16} For this reason, we will treat the core theory of the various treatises, the *Elementa*, the *Liber de ponderibus*, and the *De ratione ponderis* as the work of Jordanus. We have to leave open, in particular, whether the prologue of the *Liber de ponderibus* with its apparent or real echoes of the Aristotelian *Mechanical Problems*, or the substantial improvements found in the *De ratione ponderis* are the accomplishment of Jordanus himself or of a later commentator. For most of our arguments it is sufficient to associate them with the paradigm he created.

## 2.2 The author Giovanni Battista Benedetti

Giovanni Battista Benedetti was born in Venice on August 14, 1530 and died on January 20, 1590 in Turin.^{17} He belonged to a patrician family and was educated in philosophy, music, and mathematics by his father, who, according to Gaurico, was a Spaniard interested in philosophy and the natural sciences.^{18} At the age of 23 Benedetti published his first scientific treatise, the *Resolutio omnium Euclidis problematum*,^{19} offering the solution to geometrical problems using a compass with a fixed opening. The work reacted to a challenge that emerged from a controversy between Niccolò Tartaglia^{20} and Ludovico Ferrari^{21} in the years 1546–1548. The letter of dedication, addressed to Gabriel de Guzman, a Spanish priest, contains some autobiographical remarks by Benedetti. According to these remarks he did not receive any formal education, nor did he have a master. However, he acknowledged that Tartaglia had introduced him to the first four books of Euclid's *Elements*, probably between 1546 and 1548.^{22} Tartaglia may also have familiarized the young Benedetti with the problems of mechanics as he had treated them in his own book, *Quesiti, et inventioni diverse* of 1546.^{23} Benedetti later also became acquainted with the edition of a work by Jordanus that Tartaglia had prepared and that contained an analysis of the bent lever by the principle that Benedetti was using in his own work to determine the *positional heaviness* of a body.^{24}

According to an epigraph preserved in Turin, Benedetti had a daughter who died in childbirth in 1554 at the age of 26. In the same year he published another work, the *Demonstratio proportionum motuum localium*.^{25} Here he developed a theory of the motion of fall, first proposed in the dedicatory letter of the *Resolutio* of 1553.^{26} According to this theory, bodies of the same material fall through a given medium with the same speed and not with speeds in proportion to their weights, as Aristotle had claimed. Benedetti thus tried to overcome the fallacies of the Aristotelian theory of fall by employing the Archimedean concept of buoyancy, assuming that the motion of fall depends on the specific rather than the absolute weight. The use of Archimedean notions to correct Aristotle's physics was probably stimulated by Tartaglia's Italian translation of the first book of Archimedes' treatise on bodies in water in 1543.^{27} Benedetti's challenge to Aristotle apparently raised considerable discussion. In his *Demonstratio* he discussed Aristotle's views at length and responded to his critics. In the second edition of the *Demonstratio*, also published in Venice in 1554,^{28} Benedetti argued that the resistance incurred by a falling body in a medium depends not on its volume, but on its surface area. This is also the view that Benedetti presented in *Diversarum speculationum mathematicarum et physicarum liber*, published in Turin in 1585 and issued again under slightly different titles in Venice in 1586 and in 1599.^{29} He explained the acceleration of the motion of fall in terms of an increasing *impetus* of the falling body. Such examples show how he dealt with new challenging problems, which were difficult and sometimes impossible to treat using the mainstream theories of his time, by bringing forth and promoting new ideas.
In spite of Benedetti's efforts to secure priority for his ideas by repeated publication, they were plagiarized by Jean Taisnier^{30} in 1562 and spread without recognition of his authorship.^{31}

In 1558 Benedetti joined the court of Ottavio Farnese, the Duke of Parma,^{32} as “lettore di filosofia e matematica.”^{33} There, he performed astronomical observations and built sundials whose construction he later described in his own book on the subject in 1574.^{34} In two letters to Cipriano da Rore, choirmaster at the Court of Parma, Benedetti explained the musical consonance and dissonance of two tones by the ratio of oscillations of waves of air generated by the strings of musical instruments. He claimed that the frequency of two strings of equal tension must have an inverse ratio to the lengths of the strings, and thus proposed to mathematically describe the degree of consonance or dissonance of two tones. These letters were only published much later in Benedetti's comprehensive *Diversarum speculationum mathematicarum et physicarum liber*. In January 1567 Benedetti left Parma with a letter of recommendation from the Duke.

In the same year Benedetti was invited by the Duke of Savoy, Emanuele Filiberto,^{35} to the Court in Turin.^{36} The Duke, after the invasion and devastation of his territory by French and Spanish troops, was engaged in a renewal of the civic and military infrastructure that included political and economic reforms, but also an increased support for education and the sciences.^{37} Benedetti became involved in this renewal as an advisor to the Duke, as a court mathematician, and as an engineer-scientist. In Turin he constructed mathematical instruments such as sundials, calculated horoscopes, built a fountain, and executed other public tasks.^{38} Concurrently, he possibly taught at the new University of Turin and educated the son of the Duke, the later Carlo Emanuele I, in mathematics. In recognition of his services to the court he was made a nobleman in 1570.

In his book *De gnomonum umbrarumque solarium usu liber* of 1574^{39} he dealt at length with the construction of sundials with faces of varying inclinations and also with cylindrical and conical surfaces. His treatise *De temporum emendatione opinion* of 1578 aimed at correcting and reforming the calendar. In 1578 the Duke initiated a public disputation at the University of Turin, at which Benedetti argued with Antonio Berga on whether there was more water or more land covering the surface of the earth. The views which Benedetti brought forth against Berga in this debate were published in Turin in 1579 under the title *Consideratione di Gio. Battista Benedetti, filosofo del sereniss. S. Duca di Savoia, d'intorno al discorso della grandezza della terra e dell'acqua del eccellent. sig. Antonio Berga*.^{40} In 1580, after the death of Emanuele Filiberto, Benedetti was confirmed in his position by Carlo Emanuele I. There is evidence that, by 1585, he was married. In 1581 he wrote a lengthy letter in which he reacted to a treatise questioning the reliability of astrology and ephemerides, later published in *Diversarum speculationum mathematicarum et physicarum liber*.^{41} Benedetti was, as this book shows, an admirer of Copernicus and developed cosmological views of his own, which were remarkably close to the views of his correspondent Francesco Patrizi (the fluidity of space and the infinity of the universe outside the sphere of the fixed stars) and to Giordano Bruno (Copernicanism and plurality of worlds) who visited Turin and Chambéry around 1578.^{42}

In astrological accounts, Benedetti predicted his own death for the year 1592, as one reads in the conclusion of the *Diversarum speculationum mathematicarum et physicarum liber*. As he lay on his deathbed in January 1590, he tried to account for his premature death with a calculational error of four minutes that he must have made in his horoscope.

## 2.3 The critic Guidobaldo del Monte

Guidobaldo del Monte was born on January 11, 1545 in Pesaro, in the territories of the Duke of Urbino and died on January 6, 1607 in nearby Montebaroccio (today Mombaroccio).^{43} He studied mathematics at the University of Padua in 1564. After serving in the army for some time and participating in the campaign of the Holy Roman Emperor Maximilian II^{44} against the Turks, he joined the circle of Federico Commandino^{45} in Urbino. Commandino was a key figure of a scientific humanism that aimed at restoring the ancient mathematical sciences by editing and translating works of Euclid, Archimedes, Pappus,^{46} and others. In later life Guidobaldo pursued his studies, writing several books and constructing and producing scientific instruments at the family castle in Montebaroccio.

In his own work Guidobaldo built on the restoration of ancient science inaugurated by Commandino and, in 1577, published the comprehensive and influential *Mechanicorum liber*.^{47}
The book focuses on Heron's^{48} five simple machines – the lever, the pulley, the axle in a wheel, the wedge, and the screw – complemented by the balance as a sixth one. Following Heron and Pappus, Guidobaldo claimed that every mechanism can be reduced to one of these machines and that their properties can in turn be derived from those of the balance and the lever. He saw himself as pursuing an approach that could be traced directly to Archimedes. In fact, the latter's concept of center of gravity plays a key role in his treatise. But Guidobaldo also followed the Aristotelian tradition by attaching great importance to the concept of the center of the world, deriving mechanical properties from the mutual relation of the three centers: the point of suspension or support of a body (its fulcrum), its center of gravity, and the center of the world. In 1581 Guidobaldo's book was published in Italian,^{49} translated and introduced by Filippo Pigafetta.^{50} In 1588 Guidobaldo published a commentary on Archimedes' book on the equilibrium of planes,^{51} followed in 1600 by a major treatise on perspective.^{52}

The Urbino school of engineer-scientists to which Guidobaldo belonged was characterized by a strict focus on classical antiquity as the only legitimate model for science as well as by an *esprit de corps* that found its most prominent expression in Bernardino Baldi's^{53} posthumously published *Cronica de' Matematici*.^{54} Baldi was Guidobaldo's friend and a fellow-disciple of Commandino. His *In mechanica Aristotelis problemata exercitationes*, also published posthumously,^{55} built on Guidobaldo's mechanics and constituted another attempt to demonstrate the harmony between Archimedean and Aristotelian approaches to mechanics and thus of the integrity of the ancient tradition. The members of the Urbino school were unanimous in their rejection of what they considered medieval contaminations of the ancient tradition by writers such as Jordanus and his early modern followers Tartaglia and Cardano.^{56} Accordingly, the judgement on authors such as Benedetti who was considered a proponent of this tradition was harsh. This is evident from Guidobaldo's marginal comments presented in this volume, but also from the short biography of Benedetti in Baldi's *Cronica de' matematici*:

Gio: Battista Benedetti Venetiano attese alle Matematiche, nelle quali servì i Duchi di Savoja. Scrisse un libro di Gnomonica, il quale toccò molte cose appartenenti alle dimostrationi della detta disciplina, se non che viene ripreso da più esquisiti di non haver'osservato quel metodo, e quella purità nell'insegnare, che ricercano le Matematiche, ed è stato osservato da gl'ottimi Greci, e da gl'Imitatori loro. Scrisse anco alcune altre cose leggiere, e di non molto momento.

The Venetian G.B. Benedetti occupied himself with mathematics, a field in which he served the Dukes of Savoy. He wrote a book on gnomonics which deals with many themes belonging to the proofs of this discipline. It is, however, reproached by more distinguished scholars for not having followed that method and that purity in teaching which mathematics requires and which has been observed by the great Greeks and those who followed them. He furthermore wrote some other light things of little import.^{57}

In 1589 Guidobaldo became Visitor General of the fortresses and cities of the Grand Duke of Tuscany. A year earlier he had come in contact with the young Galileo. They frequently exchanged letters about mechanical subjects and probably met for the first time when Guidobaldo visited Tuscany in the late Spring of 1589,^{58} and again in 1592 in Montebaroccio. Guidobaldo became Galileo's mentor and patron, securing him university positions first in Pisa (1589) and later in Padua (1592). One link between them was Galileo's Pisan friend and colleague Jacopo Mazzoni in whose work Guidobaldo was interested.^{59} Galileo's initial scientific interests, concerning problems of static equilibrium analyzed in the style of Archimedes, were well matched with those of Guidobaldo. Later Galileo also emulated Guidobaldo's activities as an engineer-scientist, setting up a workshop for producing scientific instruments and writing treatises on fortification and mechanics.^{60} However, in the course of time, significant differences emerged in their approach to the developing mathematical science of nature in which Galileo took a position closer to that of Benedetti. In contrast to Guidobaldo, Galileo was convinced, in particular, that also phenomena of motion such as projectile motion, the oscillations of a pendulum, or motion along an inclined plane were amenable to an exact mathematical treatment. Like Benedetti, but not Guidobaldo, he furthermore developed a keen interest in the Copernican world system. In 1592, the year of Galileo's move to Padua, Guidobaldo was visited at Montebaroccio by Galileo with whom he performed the experiments on projectile motion that led to the discovery of the law of fall.^{61} On that occasion, and probably even earlier, they must have discussed foundational issues of mechanics as well, including the relation between Guidobaldo's and Benedetti's approach, possibly using the very copy of Benedetti's book, parts of which are reproduced here. Galileo's early intellectual career thus unfolded in the midst of the tension between Guidobaldo and the *classicist* Urbino school, on the one hand, and Benedetti's more open–minded attitude to tradition, on the other.

## Footnotes

Richard de Fournival, ca.1201–1260

For the following, see Renn and Damerow 2010.

Albertus Magnus, ca.1200–1280

For an overview, see Abattouy et.al. 2001 and Speer and Wegener 2006.

See also the discussion in Høyrup 1988.

Thābit ibn Qurra, died in 901.

Gerard of Cremona, 1114–1187.

See the edition of the manuscripts and the commentaries by Moody and Clagett 1960; see also Brown 1967.

Compare the propositions in Nemore 1533 with the corresponding propositions in Moody and Clagett 1960, 119–142.

Nemore 1533, D i verso, page 312 in the present edition.

Moody speculated that the theorems were transmitted independently from the proofs and traditionally ascribed to Euclid. From this view, Jordanus can neither be the author of the concept of *positional heaviness* nor of the theorems, but rather emerges as a commentator who developed the technical proofs found in the *Elementa super demonstrationem ponderum* Moody and Clagett 1960, 146–147 as well as those of the improved and extended version of this treatise, the *De ratione ponderis* Moody and Clagett 1960, 167–227.

While this still leaves the speculative possibility that such proofs once existed, were then lost, and finally reconstructed by Jordanus, such a reconstruction remains without any specific historical evidence in the sources.

There is little historical evidence concerning Benedetti's personal life. For biographical accounts, see the pioneering study from 1926 by Giovanni Bordiga, with a commented bibliography by Pasquale Ventrice of 1985 Bordiga 1985. See also Drake and Drabkin 1969, 31–41, Cappelletti 1996, and Drake 2008. A comprehensive review of his work and historical context may be found in Manno 1987. For a reconstruction of Benedetti's European network of correspondents, see Cecchini 2002. For a detailed presentation of Benedetti's mechanical theories we benefitted from Maccagni's studies, in particular, Maccagni 1967.

Luca Gaurico, 1476–1558, in *Tractatus astrologicus* Gaurico 1552.

Niccolò Tartaglia, 1500?–1557.

Lodovico Ferrari, 1522–1565.

In the letter to the reader of Benedetti's *Diversarum speculationum mathematicarum et physicarum liber* Benedetti 1585, first page of the author referred to Tartaglia as his main mathematical source.

Nemore 1565. See the discussion in chapter 3.9.

Benedetti 1555; see Benedetti 1985.

Jean Taisnier, 1508–1562.

Taisnier 1562, see the discussion in Drake 2008.

Ottavio Farnese, 1524–1586.

Benedetti 1574. Drake 2008 and others following him suggest the year 1573. This seems to be an error.

Emanuele Filiberto, 1528–1580.

See Ricuperati 1998.

See Mamino 1989 and Roero 1997.

See Seidengart 2006 and Omodeo 2009.

For the following short biography, see the volume on Guidobaldo's *Mechanicorum liber* in this series; see also Rose 2008 and Gamba and Andersen 2008. For extensive discussions of Guidobaldo's science and historical context, see Gamba and Montebelli 1988, Biagioli 1990, Bertoloni Meli 1992, Gamba 1998, Micheli 1992, Henninger-Voss 2000, Bertoloni Meli 2006, Dyck 2006a, Dyck 2006b and Becchi et.al. 2012.

Maximilian II, 1527–1576.

Federico Commandino, 1509–1575.

Pappus of Alexandria, ca. 290–350.

Heron (or Hero) of Alexandria, ca. 10–70.

DelMonte 1581; see the discussion in Henninger-Voss 2000.

Filippo Pigafetta, 1533–1604.

DelMonte 1588, see the discussion in Frank 2007.

DelMonte 1600, see the discussion in Gamba and Andersen 2008 and Marr 2011.

Bernardino Baldi, 1553–1617.

Baldi 1707. The manuscript version is preserved at University of Oklahoma Libraries, History of Science Collections. See also Nenci 1998.

Baldi 1621; see volumes 3 and 4 on Baldi's treatise in this series Baldi 1621.

Gerolamo Cardano, 1501–1576.

Baldi 1707, 140. Baldi himself was keenly interested in gnomonics on which he wrote an extensive manuscript that remained, however, unpublished.

This information is based on recent studies by Francesco Menchetti, subsequently extended by Martin Frank, see Menchetti 2012.

Jacopo Mazzoni, 1548–1598. Guidobaldo's interest in Mazzoni has recently been stressed by Martin Frank (personal communication).

For an extensive historical discussion, see Valleriani 2010.

See Renn et.al. 2001.