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Monte, Guidobaldo, Marchese Del


(b. Pesaro, Italy, 11 January 1545, d. Pesaro, 6 January 1607),

mechanics, perspective, astronomy, mathematics. For the original article on Guidobaldo del Monte see DSB, vol. 9.

Guidobaldo has often been viewed as a minor figure in the history of mechanics; Duhem described him as “sometimes in error, always mediocre” (Van Dyck, p. 373). Recent research has delved into his writings in greater detail and revealed his frequently ingenious approaches to contemporary problems in physical science and elucidated the historical importance of his work on perspective. This article is divided into two parts. The first emphasizes his work in mechanics and cosmology; the second deals primarily with his writings on perspective.

Mechanics . “Mechanics is no longer mechanics if it is separated from machines”; in current terms mechanics deals with constrained material systems. This is the program that Guidobaldo announces in the preface to his Mechanicorum Liber (Treatise on Mechanics), in which machines are the five simple machines of ancient engineering tradition, namely, the lever, the pulley, the wheel and axle, the wedge, and the screw.

The Mechanicorum Liber gives the appearance of an unbalanced text because 80 percent of its pages deal solely with levers and pulleys; this circumstance can be explained by the fact that Guidobaldo performed many experiments with levers and pulleys, constructing for this purpose true experimental apparatuses: special pulley systems “that turned with a puff of air,” equal-armed levers with coinciding centers of gravity and points of suspension. For this particular type of lever Guidobaldo anticipates neutral equilibrium, and he demonstrates it on the grounds that whatever may be the position of the lever, the center of gravity always remains at the same height.

This subject occasioned a debate between Guidobaldo on one side and, on the other, Giordano, Niccolò Tartaglia, and Girolamo Cardano, who denied the state of neutral equilibrium and maintained that the lever would return spontaneously to the horizontal position. The dispute is a good example of scientific “rhetoric” in that Guidobaldo turns his opponents’ own arguments against them. For example, in order for the lever to return spontaneously to the horizontal position, the highest part of the lever must “gravitate” more than the lower part, but this implies a displacement of the center of gravity, which is absurd because the position of the center of gravity does not change according to the inclination of the bodies. Therefore it is not true, as is commonly stated, that for Guidobaldo one must take into account the convergence of the weights on the balance toward the center of Earth: his thesis appears only as a rhetorical device in this dispute on equilibrium, and in fact Guidobaldo does not take it into account when he studies the problem in the early propositions of the Mechanicorum Liber. Guidobaldo shows that the properties of simple machines are reducible to the properties of the lever. For these simple machines he describes the relations among the weights, hoisting heights of the weights, the time elapsed, and the velocity, along with a physical magnitude that he calls variously “stress,” “power,” or “force,” which in any case indicates something capable of producing or inhibiting the motion of a mechanical system. Guidobaldo enunciates a principle that in current terms can be expressed thus: in machines there is no saving of labor. From his technical experiments Guidobaldo extracted important data such as the clear identification of string tension and of the constrained reaction of the supports that govern the pulley systems.

One should note the parallel sociocultural action, so to speak, that Guidobaldo undertakes. First, on the philosophical plane, he joins the debate about the relation between art and nature; he shows clearly that machines neither deceive nor surpass nature, but they produce the same effects that nature itself would produce under the same conditions. Second, following the line of thought begun by his mentor Federico Commandino, Guidobaldo continues in the work of reestimation of both theoretical and applied mathematical disciplines within a cultural context that traditionally privileged philosophical, theological, juridical, and medical studies. Thus, Guidobaldo promotes mechanical science around the idea of a broad and highly esteemed presence of machines in the ancient world. He exalts as great “mechanicians” Hero of Alexandria, Ctesibius of Alexandria, and Pappus of Alexandria, and he contests Plato’s criticisms of Eudoxus of Cnidus and Archytas of Tarentum for their conceptions of mechanics, which were more applicative than speculative. The highest place of all is occupied by Archimedes, regarded as the ideal figure of the technologist capable of fusing theory with practice, speculation with action.

As far as mechanics is concerned, Guidobaldo perceives an absolute continuity between Aristotle and Archimedes. The Aristotle that he examines is the Aristotle of the Mechanical Problems, in which Guidobaldo claims to have found in implicit form principles that Archimedes later formulated in a rigorous manner. His appreciation of Aristotle is not simply formal; the Mechanicorum contains not only the Archimedean approach to problems, which considers weights, centers of gravity, and distances from the fulcrum, but also the Aristotelean approach, which deals with weights, displacements, and “virtual” velocities, which today we would call the static and dynamic aspects of mechanics.

The above-mentioned debate over the neutral equilibrium of the lever, besides its polemic purposes, assumed for Guidobaldo a theoretical importance. In the context of his studies, the neutral equilibrium marks the delicate passage from immobility to movement. In machines, says Guidobaldo, the disruption of equilibrium, or rather the passage from supporting weights to moving them, occurs when the relation between resistance and power becomes less than the relation between the “virtual” displacement of power and that of resistance. But how much less? Guidobaldo speaks of this in a letter written in 1580 to Giacomo Contarini (1536–1595), a Venetian patrician and an expert in fortifications. Guidobaldo concludes that “matter creates some resistance” when weights move, not when they are supported.

The first printed text on mechanics, the Mechanicorum Liber was a great success as evidenced by its translation into Italian four years later, which indicated an active interest on the part of technicians who did not know Latin but who needed to understand the principles of their end products in order to be able to improve them.

Cosmology . In the writings of Guidobaldo the subject of cosmology occupies a somewhat secondary position, which is in keeping with the general orientation of the scientific environment in Urbino, where there was scant interest in such questions. Guidobaldo, however, finds it natural to accept the geocentric image of the universe and he justifies it mechanically. In his commentary on the Equiponderanti (Plane Equilibrium) of Archimedes he establishes that the Earth, since it is a sphere, has a single center that exhibits both the geometric properties of symmetry and the mechanical properties of the center of gravity. Since according to Aristotle the Earth lies at the center of the universe, the center of gravity will also be at that point. To the objection that the Earth is formed of water and earth, “elements” that have different specific gravities, Guidobaldo responds by citing a proposition in On Floating Bodies in which Archimedes demonstrates that the surface of the waters has a spherical form with its center coincident with the center of the Earth. In a page of the Meditatiunculae (Little Meditations) he again approaches the question from a mechanical point of view; in fact, he recognizes that the displacement of bodies on the surface of the Earth makes the distribution of weights change, thus causing a displacement of the terrestrial center of gravity and consequently of the entire Earth, an old idea that can be found in Giovanni Buridano. Guidobaldo’s booklet De Motu Terrae (Concerning the Earth’s Motion) has been lost, but given these presuppositions it probably did not contain any different ideas.

The Problemi astronomici (Problems of Astronomy), published posthumously by his son Orazio, is a text dedicated exclusively to mathematical astronomy; not even in the chapter on comets does Guidobaldo give any opinions on the terrestrial or celestial nature of those bodies.

He was, however, compelled to speak out in 1604, when there appeared a supernova that called into question the physical doctrine that the heavens are incorruptible. The easiest solution was to classify it as a comet, but unlike comets, the supernova did not show an inherent movement with respect to the fixed stars. This fact was indeed confirmed by Johannes Kepler’s observations, which were known to Guidobaldo very probably through Father Christopher Clavius. Guidobaldo accepted these observations insofar as they conformed to his own. Nevertheless, he remained in doubt about the star-comet alternative and was unable to decide whether the heavens could be “corruptible.”

Guidobaldo on Perspective . In 1600 Guidobaldo published Perspectivae Libri Sex (Six Books on Perspective), which became a turning point in the history of the mathematical theory of perspective. Before Guidobaldo, Commandino, Egnazio Danti, and Giovanni Battista Benedetti had sought to understand the geometry behind perspective, and they had been successful in proving the correctness of certain perspective constructions. Guidobaldo, however, took a different approach, in which he based his considerations on general geometrical laws. He was the first to realize the importance of the perspective images of sets of parallel lines as the basis of constructions, and he created the concept of a general vanishing point. His accomplishments were so fruitful that it is appropriate to designate him the father of the mathematical theory of perspective.

Guidobaldo’s inspiration to take up perspective most likely came from his teacher Commandino. Thus, of the two manuscripts on perspective that Guidobaldo left— presumably dating from the period 1588–1592—the oldest one reflects many of Commandino’s ideas. The younger one, by contrast, contains new ideas that resulted in Guidobaldo’s innovative treatment of perspective.

Before Guidobaldo, it was common knowledge among mathematicians and practitioners of perspective that the images of lines perpendicular to the picture plane converge in one point—later called the principal vanishing point—which is the orthogonal projection of the eye point upon the picture plane. Similarly, some writers were aware that the images of horizontal lines forming an angle of 45° with the picture plane converge at a point on the horizon—later called a distance point. Guidobaldo realized, and proved, that the images of a set of parallel lines that cut the picture plane φ (Figure 1) all meet in a point, say V, which he called their punctum concursus. He proved that this convergence point, later called a vanishing point, is the point of intersection of the picture plane and the line among the parallel lines that passes through the eye point O. This insight gave Guidobaldo a means to determine the image of a line l that cuts the picture plane in a point A (Figure 1): Since the point A is situated in the picture plane it is its own image and hence lies on the image of l; furthermore the image of l, prolonged, passes through its vanishing point V; in other words the image of l is determined by the points A and V.

From the image of a line, Guidobaldo turned to determining the image of a given point; he did this by constructing the images of two lines passing through the

given point. He was so taken by this possibility that he presented no fewer than twenty-three different methods of constructing the image of a point.

All Guidobaldo’s successors took over his concept of a vanishing point either directly from his work or from some of the authors inspired by him, among whom Simon Stevin and Samuel Marolois presumably were the most influential. A great part of the further development of the theory of perspective consisted of generalizations of Guidobaldo’s ideas. Thus, later mathematicians introduced the concept of a vanishing line for a set of parallel planes cutting the picture plane. This line consists of the vanishing points of all the lines in the parallel planes—the horizon being a noticeable example of a vanishing line, namely of horizontal planes. Guidobaldo did not single out the concept of a vanishing line, but it occurs implicitly in his work. After Guidobaldo, inverse problems of perspective caught the interest of several of the leading mathematicians in the field of geometrical perspective. Guidobaldo had also touched on this topic, and he opened up a few other topics.

It is impressive how much Guidobaldo obtained by combining classical Greek geometry with his concept of vanishing points. His style of presentation, however, is remarkably inept because he included a lot of unnecessary theorems (for more details on Perspectivae Libri Sex, see Andersen, 2007).

Guidobaldo on Euclid and Proportions . As Paul Lawrence Rose wrote in the original DSB article, three manuscripts by Guidobaldo on proportions and on Euclid’s Elements have been identified; however, at present the locations of only two of them are known. Guidobaldo’s work on the theory of proportion, and in particular his generalization of Euclid’s concept of composition of ratios, were treated by some scholars at the end of the last century.



I sei libri della prospettiva di Guidobaldo dei marchesi del Monte dal latino tradotti, interpretati e commentati [The six books on perspective by Marquis Guidobaldo del Monte, translated from the Latin with an interpretation and a commentary]. Edited by Rocco Sinisgalli. Rome: Bretschneider, 1984.

La teoria sui planisferi universali di Guidobaldo Del Monte [Theory on the planispheres of the universe by Guidobaldo del Monte]. Edited by Rocco Sinisgalli and Salvatore Vastola. Florence: Cadmo, 1994.


Andersen, Kirsti. The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge. New York: Springer, 2007. See pages 237–265.

Bertoloni Meli, Domenico. “Guidobaldo Dal Monte and the Archimedean Revival.” Nuncius: Annali di storia della scienza 7, no. 1 (1992): 3–34.

_____. Thinking with Objects: The Transformation of Mechanics in the Seventeenth Century. Baltimore: Johns Hopkins University Press, 2006.

Biagioli, M. “The Social Status of Italian Mathematicians, 1450–1600.” History of Science 27 (1989): 41–95.

Field, J. V. The Invention of Infinity: Mathematics and Art in the Renaissance. Oxford: Oxford University Press, 1997. See pages 171–177.

Galluzzi, Paulo. Momento: Studi galileiani [Moment: Galilean studies]. Rome: Edizioni dell’Ateneo & Bizzarri, 1979.

Gamba, Enrico. “Guidobaldo Del Monte matematico e ingegnere” [Guidobaldo del Monte, mathematician and engineer]. In Giambattista Aleotti e gli ingegneri del Rinascimento [Giambattista Aleotti and the engineers of the Renaissance], edited by A. Fiocca. Florence: Olschki, 1998.

_____, and Vico Montebelli. Le scienze a Urbino nel tardo Rinascimento [The sciences in Urbino in the High Renaissance]. Urbino: QuattroVenti, 1988.

Giusti, Enrico, ed. Euclides Reformatus: La teoria delle proporzioni nella scuola galileiana [Euclid reformed: The theory of proportions in the Galilean school]. Turin: Bollati Boringhieri 1993. Includes the two booklets In Quintum Euclidis Elementorum Librum (On the fifth book of Euclid’s Elements) and the De Proportione Composita (On compound proportion).

Marchi, Paola. L’invenzione del punto di fuga nell’opera prospettiva di Guidobaldo dal Monte. PhD diss., Università degli studi di Pisa, Italy, 1998.

Micheli, Gianni. “Guidobaldo del Monte e la meccanica” [Guidobaldo del Monte and mechanics]. In La matematizzazione dell’universo: Momenti della cultura matematica tra '500 e '600 [The mathematicization of the universe: Episodes in mathematical culture between the sixteenth and seventeenth centuries], edited by Lino Conti. Assisi: Porziuncola, 1992.

Naylor, R. “The Evolution of an Experiment: Guidobaldo del Monte and Galileo’s Discorsi Demonstration of Parabolis Trajectory.” Physis 16 (1974): 323–346.

Renn, Jürgen, ed. Galileo in Context. Cambridge, U.K.: Cambridge University Press, 2001.

Van Dyck, M. “Gravitating towards Stability: Guidobaldo’s Archimedean-Aristotelian Synthesis.” History of Science 44 (2006): 373–407.

Enrico Gamba

Kirsti Andersen

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Monte, Guidobaldo, Marchese Del


(b. Pesaro, Italy, 11 January 1545; d. Montebaroccio, 6 January 1607)

mechanics, mathematics, astronomy.

[He is known as Guidobaldo del Monte, although his signature reads Guidobaldo dal Monte, The form Guido Ubaldo (from the Latinized version) is often used, Ubaldo being taken incorrectly as the family name.]

Guidobaldo was born into a noble family in the territory of the dukes of Urbino. While at the University of Padua in 1564 he studied mathematics and befriended the poet Torquato Tasso. Later Guidobaldo served in campaigns against the Turks and in 1588 was appointed visitor general of the fortresses and cities of the grand duke of Tuscany. Soon afterward Guidobaldo retired to the family castle of Montebaroccio near Urbino, where he pursued his scientific studies until his death.

Guidobaldo was a prominent figure in the renaissance of the mathematical sciences. At Urbino he was a friend and pupil of Federico Commandino and an intimate of Bernardino Baldi, the mathematical historian. In 1588 Guidobaldo saw Commandino’s Latin translation of Pappus through the press at Pesaro. The autograph transcript had initially been sent to the Venetian mathematician Barocius for publication; but Barocius, having refused to edit the work without making extensive changes, sent the manuscript to Guidobaldo, who published the text exactly as he found it. Concerning Pappus,Guidobaldo also corresponded with the Venetian senator Jacomo Contarini, who helped Guidobaldo secure an appointment at Padua for Galileo. Guidobaldo’s correspondence with these and other friends is an important source for the history of the mathematics of the period.

Guidobaldo’s first book, the Liber mechanicorum (1577), was regarded by contemporaries as the greatest work on statics since the Greeks. It was intended as a return to classical Archimedean models of rigorous mathematical proof and as a rejection of the “barbaric” medieval proofs of Jordanus de Nemore (revived by Tartaglia in his Quesiti of 1546), which mixed dynamic principles with mathematical analysis.

The Liber may be seen as a forceful argument that statics and dynamics are entirely separate sciences; hence no unified science of mechanics is possible. This attitude is evident in Guidobaldo’s treatment of equilibrium in the simple machines, which he terms the case where the power sustains the weight. He stresses that a greater power is needed to move the weight than to sustain it and that the power which moves has a greater ratio to the weight moved than does the power which sustains to the weight sustained. Consequently, the same principle and proportions cannot hold good for both moving and sustaining.

Galileo overcame this objection to a unified mechanics by positing that an insensibly greater amount of power was needed to move, than to sustain, a given weight. Guidobaldo had scorned the use of insensibilia in mechanics, probably because they were not susceptible of precise mathematical definition. Like his contemporary Benedetti, Guidobaldo attacked Jordunus, Cardano, and Tartaglia for assuming that the lines of descent of heavy bodies were parallel rather than convergent to the center of the earth. The answer of both Tartaglia and Galileo to this demand for unreasonable exactitude in mechanics was that, at a great distance from the center, the difference between the parallel and convergent descents was insensible.

This extreme concern for precision led Guidobaldo to reject the valid inclined-plane theorem of Jordanus in favor of the erroneous theorem of Pappus. Pappus’ premise that a definite amount of force was needed to move a body horizontally was in accord with the view of Guidobaldo that more power was required to move than to sustain the body. Moreover, Jordanus’ theorem seemed vitiated by its neglect of the angle of convergence of the descents. By supposing against Pappus (whom he named) and Guidobaldo (whom he did not name) that an insensible amount of power was required to move a body horizontally, Galileo was able to apply the principle of virtual displacements to both static and dynamic cases and was able to frame useful principles of virtual work and inertia. Guidobaldo’s quest for mathematical rigor may have barred such imaginative concepts from his mind.

The most fruitful section of the Liber mechanicorum deals with pulleys, reducing them to the lever. This analysis—which is far superior to that of Benedetti— was adopted by Galileo. In two subsequent mechanical works Guidobaldo developed other ideas of this first book. These works were the Paraphrase of Archimedes: Equilibrium of Planes (1588), a copy of which was sent to Galileo, and the posthumous De cochlea (1615).

Guidobaldo was Galileo’s patron and friend for twenty years and was possibly the greatest single influence on the mechanics of Galileo. In addition to giving Galileo advice on statics, Guidobaldo discussed projectile motion with him, and both scientists reportedly conducted experiments together on the trajectories of cannonballs. In Guidobaldo’s notebook (Paris MS 10246), written before 1607, it is asserted that projectiles follow parabolic paths; that this path is similar to the inverted parabola (actually a catenary) which is formed by the slack of a rope held horizontally; and that an inked ball that is rolled sideways over a near perpendicular plane will mark out such a parabola. Remarkably the same two examples are cited by Galileo at the end of the Two New Sciences, although only as postscripts to his main proof—which is based on the law of free fall—of the parabolic trajectory.

Among Guidobaldo’s nonmechanical works are three manuscript treatises on proportion and Euclid; two astronomical books, the Planisphaeriorum (1579) and the posthumous Problematum astronomicorum (1609); and the best Renaissance study of perspective (1600).

Guidobaldo helped to develop a number of mathematical instruments, including the proportional compass, the elliptical compass, and a device for dividing the circle into degrees, minutes, and seconds.


I. Original Wokks. Guidobaldo’s published works are Liber mechanicorum (Pesaro, 1577; repr. Venice, 1615); Italian trans. by Filippo Pigafetta, Le mechanice (Venice, 1581; repr. Venice, 1615); Planisphaeriorum universalium thearica(Pesaro, 1579; repr. Cologne, 1581); De ecclesiastici kalendarii restitutione opusculum (Pesaro, 1580); In duos Archimedis aequeponderantium libros paraphrasis (Pesaro, 1588); Perspectivae libri sex (Pesaro, 1600); Problematum astronomieorum libri septem (Venice, 1609); and De cochlea libri quatuor (Venice, 1615).

MS works of Guidobaldo are the Meditatiunculae, Bibliothéque Nationale (Paris), MS Lat. 10246; In quintum Euclidis elementorum commentarius and De proportione composita opusculum, Biblioteca Oliveriana (Pesaro), respectively MSS 630 and 631; and a treatise on the reform of the calendar, Biblioteca Vaticana, MS Vat. Lat. 7058. A collection of drawings of machines by Francesco di Giorgio Martini in the Biblioteca Marcianu (Venice), MS Lat. VIII 87(3048), was formerly owned by Guidobaldo. The present location of the MS In nonnulla Euclidis elementorum expositiones (item 194 bis in the Boncompagni Sale Catalogue of 1898) is not known.

Guidobaldo’s letters (some are copies) are scattered: Biblioteca Nazionale Centrale (Florence), MSS Galileo 15, 16, 88; Biblioteca Comunale “A. Saffi” (Forli), MSS Autografi Piancastelli Nos. 755, 1508; Archivio di Stato (Mantua), Corrispondenza Estera, E.XXVIII, 3; Biblioteca Ambrosiana (Milan), MSS D.34 inf., J.231 inf., R.121 sup.; Bodleian Library (Oxford), MS Canon. Ital. 145; Bibliothèque Nationale (Paris), MS 7218 Lat.; Biblioteca Oliveriana (Pesaro), MSS 193 Ter.; 211/ii; 426; 1580 (MS 1538 = Tasso to Guidobaldo); Archivum Pontificiae Universitatis Gregorianae (Rome), Cassetta 1, MSS 529–530; Biblioteca Comunale degli Intronati (Siena), MS K.XI.52; Biblioteca Universitaria (Urbino), MS Carità Busta 47, Fasc. 6; and Biblioteca Nazionale Marciana (Venice), MS Ital. IV, 63 (Rari V.259).

Favaro has printed the Galileo correspondence in the Opere of Galileo, vol. X; and the two Marciana letters in Due Letter. Rose, Origins, prints Ambrosiana MS J.231 inf., and Arrighi, Un grande, has six letters from Oliveriana MS 426, with the prefaces of MSS 630 and 631. Most of Le mechanice is translated in Drake and Drabkin, Mechanics. Important pages from Paris MS 10246 are in Libri, Histoire, IV, 369–398.

II. Secondary Literature. A bibliog. is in Paul Lawrence Rose, “Materials for a Scientific Biography of Guidobaldo del Monte,” in Actes du Xlle congrès international d’histoire des sciences, Paris, 1968, 12 (1971), 69–72. The earliest biography is the short note by Guidobaldo’s friend Bernardino Baldi, Cronica de’ matematici (Urbino, 1707), 145–147. Baldi’s full Vita has disappeared. Giuseppe Mamiani, Elogi storici di Federico Commandino, G. Ubaldo del Monte… (Pesaro, 1828), is informative, although few references are given. The Guidobaldo section was earlier published in the Giornale arcadico, vols. IX, X (Senigallia, 1821). The 1828 ed. is reprinted in Mamiani, Opuscoli scientifici (Florence, 1845).

On Guidobaldo’s mechanics see Antonio Favaro, “Due lettere inedite di Guidobaldo del Monte a Giacomo Conlarini,” in Atti del Istituto veneto di scienze, lettere ed arti, 59 (1899– 1900), 303–312. Pierre Duhem, Les origines de la statique, I (Paris, 1905), 209–226, was very critical of Guidobaldo. Stillman Drake and I. E. Drabkin, Mechanics in Sixteenth Century Italy (Madison, Wis., 1969), 44–52 and passim, are more favorably disposed.

Guidobaldo’s astronomical interests are illustrated in Gino Arrighi, “Un grande scienziato italiano; Guidobaldo dal Monte…,” in Atti dell’ Accademia lucchese di scienze, lettere ed arti, n.s. 12 (1965), 183–199.

For mathematical instruments see Paul Lawrence Rose, “The Origins of the Proportional Compass,” in Physis, 10 (1968), 54–69, and “Renaissance Italian Methods of Drawing the Ellipse and Related Curves,” in Physis, 12 (1970), 371–404.

See also Guillaume Libri, Histoire des sciences mathématiques en Italie, IV (Paris, 1841), 79–84, 369–398; and Antonio Favaro, “Galileo e Guidobaldo del Monte,” (Scampoli Galileani 146), in Atti dell’ Accadetnia di scienze, lettere ed arti di Padova, 30 (1914), 54–61.

Paul Lawrence Rose

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