In duos Archimedis Aequeponderantium libros paraphrasis scholijs illustrata. Pesaro: apud Hieronymum Concordiam, 1588 [colophon, 1587]. [Bound with:] SCALETTI, Carlo Cesare. Scuola mecanico-speculativo-pratica in cui si esamina la proporzione, che hà la potenza alla resistenza del corpo grave, e la causa per la quale la suddetta potenza si estenda a maggior’attività mediante la machina; opera utile all’uso civile, e militare necessaria ad ogni matematico, ingegniero, architetto, machinista, e bombardiere. Bologna: Costantino Pisarri, 1711.

Pesaro; Bologna: apud Hieronymum Concordiam; Costantino Pisarri, 1588 [colophon, 1587]; 1711.

First edition, the beautiful Macclesfield copy, of Guidobaldo’s Paraphrasis, the important companion to his Mechanicorum liber (1577), regarded as the greatest work on statics since the Greeks, which employed the mathematically rigorous proofs of Archimedes to the investigation of problems in mechanics. “This work complements the Mechanicorum liber, and together they represent the greatest opus of 16th century mechanics” (Roberts & Trent, p. 13). They had a profound and lasting impact on the methodology adopted by contemporary century Italian scientists, most notably Galileo: “Guidobaldo was Galileo’s patron and friend and was possibly the greatest single influence on the mechanics of Galileo” (DSB). The Mechanicorum liber had reduced the study of simple machines to the law of the lever, according to which bodies balance on the ends of a lever when their distances from the fulcrum are inversely proportional to their weights. Guidobaldo’s 1588 work is a paraphrase of Archimedes’ ‘On the equilibrium of plane figures,’ which included a geometrical proof of the law of the lever. This is Archimedes’ most important surviving work in mechanics, but much of it “is undoubtedly not authentic, consisting as it does of inept later additions or reworkings” (DSB). Guidobaldo’s purpose in the Paraphrasis is to explicate and correct the Archimedean text, thereby providing a secure geometrical foundation for the law of the lever, and hence for the whole of the science of statics. The Law of the Lever was among the first laws of nature to be formulated in quantitative terms. It dates back at least to Archimedes’ On the Equilibrium of Planes and possibly even to Aristotle’s Mechanics. Shortly after its first formulation, scholars like Archimedes and Euclid were already seeking to prove it by means of deduction from general axioms and postulates. The Law of the Lever thus comprises the very core of rational mechanics” (Schlaudt, p. 93). Like Archimedes, Guidobaldo also considers the application of the law of the lever to the determination of centres of gravity: of plane figures bounded by straight lines in the first book, and in the second book segments of conic sections (treated by approximating these curved figures by inscribed polygons). Monte’s work is here bound with the rare first edition of Scaletti’s treatise on civil engineering, with chapters on statics, mechanics, on the practical construction of fountains, buildings, bridges, etc. (BL only on COPAC; only three other copies located in auction records).

“Guido Ubaldo, Marquis del Monte, was born at Pesaro on 11 January 1545. He entered the University of Padua in 1564, where one of his companions in study was the poet Torquato Tasso. On his return from the university, he continued his studies in mathematics under Federico Commandino at Urbino” (Drake, p. 44). “Writing to a friend in Paris in 1633, Galileo declared that ‘at the age of twenty-one, after studying geometry for two years he worked out a number of propositions about the center of gravity of solids.’ Galileo had become acquainted with Commandino’s Liber de centro gravitatis solidorum that had been published in 1565 and had opened, or rather reopened, a new field of research but suffered from what Galileo called ‘some imperfections.’ These he sought to set right by following the example of ‘that very great mathematician,’ Guidobaldo del Monte, to whom he sent his demonstrations” (Shea, pp. 97-98). “In 1588 [Guido Ubaldo] received from Galileo some theorems on centers of gravity with a request for his opinion. In this way a correspondence was opened which continued until his death in 1607. Guido Ubaldo was favourably impressed with Galileo’s talents, and sent to him a copy of his second important contribution to mechanics, a paraphrase of and commentary on the work of Archimedes on plane equilibrium [the offered work] … In appraising probable influences on Galileo, one should remember that, before Galileo wrote anything on motion, he had received this book from his most valued patron, [and] that it was a book on Archimedes (whom Galileo admired above all other writers)” (Drake, pp. 45-46).

“Guido Ubaldo’s two chief works on theoretical mechanics make clear his devotion to the idea of mathematical rigor of treatment and his repugnance for medieval writings on the science of weights and for Tartaglia’s adherence to that tradition. The Mechanics contains numerous criticisms of these writers, and in the Paraphrasis of 1588 Guido Ubaldo wrote: ‘And however much Jordanus Nemorarius (whose followers include Niccolo Tartaglia and others) struggled in his book De ponderibus to prove this same proportion of the general lever by many means, yet not any of the proofs were worthy to be called demonstrations, and were scarcely to be credited. For he put things together which in no way command conviction and perhaps do not even persuade anyone by probability, when in mathematical demonstrations the most precise reasons are required. And on that account it never seemed to me that this Jordanus should even be reckoned among writers on mechanics’” (ibid.).

“With the Paraphrasis, Guidobaldo attended to restore the integrity of the Equilibrium of Planes, Archimedes’s principal work of mechanics. The corrupted text presented Guidobaldo with problems of essentially three kinds: minor technical problems, like missing argumentative steps in the demonstrations; completely inconclusive demonstrations that requested a massive intervention in the text with lemmata or auxiliary propositions; and, most seriously, obscurities regarding the key notions of Archimedean mechanics. Approaching this challenge, Guidobaldo adopted a quite ‘philological’ modus operandi: firstly, to establish a correct text, he had recourse to the Greek version of the editio princeps (1544), which appeared to him less corrupt than the existing Latin translations. In the course of the Paraphrasis, he indicated Greek passages which did not seem to make sense and recommended more reasonable wordings in these situations … Guidobaldo reports every single word of the Archimedean treatise, clearly distinguishing them, by using different fonts, from his own additions and explications. These are either inserted in the text of the demonstrations (if necessary for understanding), or reported after the propositions: in fact, each proposition is followed by a scholium containing explications of linguistic, mathematical or conceptual nature. If a demonstration lacks a crucial element or does not motivate a relevant step, Guidobaldo inserts lemmata to prove the missing assertions. With this system, the most interesting information obviously is contained, except for the preface, in the scholia: it is there that Guidobaldo can give utterance to his ideas and remarks regarding both Archimedes’s and his own ideas” (Frank, pp. 185-187).

“As Guidobaldo explains in his dedicatory letter, Paraphrasis is meant to answer criticisms that were made of his earlier Mechanicorum liber by people who were maybe not so adept in ‘the mechanical way of investigating the causes of things’ (page 1 of the unnumbered dedicatory letter). In this way, he immediately introduces one of the running themes of his commentary—if not the most important message of the whole book—that the mechanical science has a special way of demonstrating its propositions, which must be grasped before one can truly understand any of its claims. In Mechanicorum liber Guidobaldo had moreover assumed the validity of the law of the lever, whereas this law actually contains the true foundation of the science of mechanics (as he never tires of stressing in this work devoted to the demonstration of its validity). In Paraphrasis, Guidobaldo accordingly shows that the validity of the law of the lever is indeed grounded in a special method of demonstration—in the ‘argumentandi modus huius scientiae maximè proprius’ (p. 44) …

“Before entering into the propositions and their proofs, Guidobaldo deems it necessary to explain two further things in his preface. First it needs to be understood what the proper definition of center of gravity is, and secondly it needs to be explained how Archimedes can treat the center of gravity of plane figures. The absence of a definition of center of gravity in Archimedes’s treatise is solved in exactly the same way as in the earlier Mechanicorum liber by introducing the definition that Pappus had given to the notion (which is obviously a further token of the great importance of Pappus’s treatise in shaping late sixteenth-century views on the scope of mechanics). This definition reads: ‘The centre of gravity of any body is a certain point within it, from which, if it is imagined to be suspended and carried, it remains stable and maintains the position which it had at the beginning, and is not set to rotating around that point’ (p. 9). Following the example of Pappus himself, Guidobaldo immediately links the existence of such a point within any body with the natural propensity that all bodies have to go to the center of the world. This essential connection is understood if we consider what happens with a body that is hypothetically placed at the center of the world: since it will be absolutely at rest, there must be a point in the body around which all parts of the body have ‘equal moment’ (in phrasing the condition in this terminology, Guidobaldo refers to the alternative definition as given by Commandino, the first part of which states that ‘the centre of gravity of any solid shape is that point within it around which are disposed on all sides parts of equal moments’) (p. 9). This is actually the only place where Guidobaldo actually uses the notion of ‘moment’ …

“Maybe the most important of Guidobaldo’s considerations, however, is that the first eight propositions, which contain the true foundation of the science of mechanics, need not be limited to plane figures (p. 19). Guidobaldo thus stresses that Archimedes in these propositions refers to ‘magnitudes’ in general, as this name is common both to solid and plane figures (Guidobaldo even goes as far as having his figures accompanying these propositions alternatively represent solid and plane figures). The restriction to plane figures is thus actually only relevant to the further propositions, introduced to square the parabola, and need not concern anyone who is primarily interested in understanding the foundations of the science of mechanics.

“A more interesting perspective on Guidobaldo’s exposition of Archimedes’s proof procedure comes to light if we connect it with Mach’s well-known criticism of the same proof. Mach famously questioned how Archimedes could possibly determine the conditions for equilibrium of unequal bodies assuming as only input that equal bodies at equal distances are in equilibrium. According to him, the only way in which the inverse proportionality encoded in the law of the lever could be derived from the symmetrical situation was by actually presupposing its validity in the proof itself. The crucial point that he singled out for his criticism was the following: Archimedes makes the action of two equal weights to be the same under all circumstances as that of the combined weights acting at the middle point of their line of junction … Guidobaldo notes that Archimedes makes a very important move in considering the center of gravity of the magnitude composed from the two equal magnitudes. Moreover, and this is absolutely crucial, he stresses that since this is one magnitude, it also has one unique center of gravity—and this must be completely independent of the form of the composing magnitudes (p. 43). But as can be seen from Guidobaldo’s comments in a scholium preceding the actual proof of the law of the lever, this insight is enough to undercut Mach’s criticism of Archimedes’s proof (pp. 55–58). Guidobaldo had already interpreted the definition of a body’s center of gravity by stating that it is in this point that the tendency toward motion of the body is concentrated. We can thus validly assume that the equilibrium that subsists between two bodies will not be disturbed if we replace one of these bodies with two equal bodies, both half its weight, and placed such that their centers of gravity are equally far from its center of gravity …

“The proof of the law of the lever [for commensurable magnitudes] comes down to showing that if the weights of two magnitudes are inversely as the distances from which they are suspended, then these weights can be distributed over different smaller magnitudes along the line connecting the original magnitudes’ centers of gravity in such a way that it follows directly from this fifth proposition that the common center of gravity of all these smaller magnitudes taken together—and thus also of the original magnitudes—coincides with the point of suspension (i.e., we can transform the asymmetric case into a symmetric one). This proof involves two essential ingredients: purely geometrical facts about the relations holding between the distances and the smaller magnitudes in which the original magnitudes are divided (facts which show that it is possible to distribute the weights along the line in such a way that the conditions of the fifth proposition will be satisfied); and the assumption that there is a mechanical equivalence between the original situation and the one in which the weights are divided and distributed along the line according to the scheme first made clear in the fourth proposition (but now extended to an arbitrary number of parts). Guidobaldo’s care in laying out the conditions which underwrite the validity of this proof is brought out nicely in some of the editorial interpolations which he interjects in the Archimedean text of the proof” (Van Dyck, pp. 17-27).

Having explicated the proof of the law of the lever in the commensurable case (Proposition VI), Guidobaldo goes on to consider the incommensurable case (Proposition VII), which is defective and incomplete in the original text. “After the fifth proposition and its corollaries, which make statements about systems of equal weights and their barycentre, Guidobaldo prepares the demonstrations of Propositions VI and VII, i.e. the law of the lever for commensurable and incommensurable magnitudes. This required an intensive intervention, since particularly the seventh proposition is heavily damaged … To restore the mathematical integrity of the seventh proposition, Guidobaldo has to prove another lemma and an auxiliary proposition. The lemma shows that, given two incommensurable magnitudes, it is possible to take away from the bigger one such a part, that it remains bigger than the smaller, but results commensurable to it … With this preparation, Guidobaldo is able to fill the logical lacunae of the demonstration of Proposition VII” (Frank, pp. 189-190).

In the remainder of the first book, Guidobaldo determines the centres of gravity of the triangle, parallelogram, and trapezoid. In the second book he applies these results to find the centre of gravity of the segment of a parabola.

“The conclusion of the first book is constituted by a series of problems and propositions that Guidobaldo adds on his own. First, he shows that with Archimedes’s results the barycentre [centre of gravity] position of every rectilinear figure can theoretically be found, by decomposing it into triangles. Then he adds two propositions on the division of figures by lines passing through their centres of gravity: firstly, he shows that there are figures, like the parallelogram, which are divided always in two equal parts by any line passing trough the barycentre. Then, he proves that certain other figures, like the triangle, are not necessarily divided in two parts of equal area … Finally, Guidobaldo revisits a topic delineated in the preface: the knowledge about the barycentres of plane figures can be used for the determination of the centres of gravity of prisms having the basis and the top constituted by a plane figure with known barycentre: it is sufficient to divide the connection line of these barycentres in two halves, and the division point is the centre of gravity of the prism” (Frank, pp. 191-192).

Contrary to what is sometimes stated about the Paraphrasis, this book does not treat hydrostatics.

1. BM STC It. p. 37; Bib. Mat. It. I, 179-80; Graesse I, 180; Macclesfield 1436 (this copy); Riccardi II.179; DSB IX.487-489; Bibliotheca Chemico-Mathematica, 12033. Drake, Introduction to: Drake & Drabkin, Mechanics in Sixteenth-Century Italy, 1969.

Frank, Guidobaldo dal Monte’s Mechanics in Context, Ph.D. Thesis, 2011 ( Schlaudt, ‘Hölder, Mach, and the Law of the Lever,’ Philosophia Scientiae (2013), pp. 93-116. Shea, ‘Guidobaldo del Monte: Galileo’s Patron, Mentor and Friend,’ Chap. 4 in Guidobaldo del Monte (1545–1607): Theory and Practice of the Mathematical Disciplines from Urbino to Europe (Becchi, Meli & Gamba (eds.)), 2013. Van Dyck, ‘Argumentandi modus huius scientiae maxime proprius. Guidobaldo’s Mechanics and the Question of Mathematical Principles,’ Chap. 1 in ibid. II. Cicognara 958; Graesse VI, 288; Riccardi I.427. Montanari, A Gli uomini illustridi Faenza, 1883, p. 193.

Two works bound in one vol., small folio (290 x 196mm). [Monte:] pp. [iv], 202, [2], title with woodcut vignette of lever and motto ‘Mechanicorum machina’, historiated initials, woodcut illustrations and text within ruled borders; [Scaletti:] pp. [xvi], 188 with 12 folding plates by Moretti after Scaletti, including engraved frontispiece showing the erection of an obelisk. Eighteenth-century mottled calf, Macclesfield bookplate on front paste-down and their blind-stamp on first three leaves. A fine copy.

Item #4716

Price: $12,500.00