2 The Context

During the Renaissance many works from Greek and Roman antiquity were recovered, studied, and expanded with commentaries. At first, this process of acquisition and assimilation of the classical cultural heritage concerned mainly the fields of literature, history and philosophy. Later, in the sixteenth century, this process began to include works on scientific and technical subjects as well. The study of ancient Greek texts in the original language and their new translations directly into Latin – replacing the much criticized extant Arabic translations – as well as the comparative study of different sources became one of the main activities of mathematicians and astronomers.

In this general movement a number of ancient texts on mechanics were also discovered and studied: the Mechanical Problems ascribed to Aristotle; a work by Archimedes entitled On the Equilibrium of Planes, which according to Baldi's testimony was regarded by Guidobaldo as “the book of Elements of the whole field of mechanics”¹ and which he published in 1588 under the title In duos Archimedis aequeponderantium libros Paraphrasis; Pneumatics by Hero of Alexandria; and Book Eight of Pappus' Mathematical Collections.² Among these works, it was the text ascribed to Aristotle which received the greatest attention of the scholars: the studies and commentaries of this work made during the Renaissance were essential for the development of mechanics before Galileo and Descartes.³

In Mechanical Problems, which was generally considered to be a work by Aristotle himself, though it is now thought to have been produced within the Peripatetic School,⁴ the author tried for the first time to base the explanation of the working of `simple machines' (such as the lever, the windlass, the wedge and the pulley) on a single mathematical principle, and to solve a series of questions which are answered by referring to the same mathematical model. The starting point of the whole treatise was the astonishment caused by operations carried out by means of a lever, such as the lifting of great weights which man was unable to move without that instrument. But even greater astonishment was caused by the fact that, by adding weight to weight, that is, by adding the weight of the lever to the weight to be lifted, the whole thing could be moved more easily. This fact seemed to challenge the obvious relation between the force needed to move a certain body and its weight. In fact, experience clearly shows that things `weighing less' are easier to move than things `weighing more.'

The author of Mechanical Problems moved on to formulate the principle that could explain this remarkable fact: this principle is directly related to the movement of the lever, so that the working of the simple machines can be reduced to the properties of a circle. He also considers it remarkable that the circle is made up of opposites, a fact that becomes obvious when the circle is generated by a rotating line fixed at one end:

1The circle is made by what is stationary, i.e., one end of the radius, and by what is moving, i.e., the other parts of the radius that move round and produce the surface of the circle.

2The circle is concave inside the circumference, as well as convex outside the circumference.

3The rotating circle moves simultaneously in opposite directions, for it moves simultaneously forwards and backwards.

4The circle is made by the movement of a line drawn as a radius from the center, but no two points on that line move at the same pace: the point which is further from the fixed center moves more rapidly.

This idea of the circle differs remarkably from that which is contained in Definition 15 and 16 of Book One of Euclid's Elements. Here the figure is already given, without any reference to its generation. In Mechanical Problems, on the other hand, the whole argument seems to be based on properties derived directly from the way in which the figure is produced: it is traced not by means of a pair of compasses, but rather by means of a line rotating around a fixed point at one end.

From the property of the circle, according to which points marked on the rotating radius move at different speeds, the author of Mechanical Problems explains why bodies placed on the radius at different distances from the center each move at different speeds, increasing with their distance from the center. He regards the motion of the points on the radius as composed of a natural downward motion along the vertical tangent, and a lateral motion against nature toward the fixed center of the rotation, and shows that this lateral component of the motion increases the closer it is to the center. This moving closer to the center was perceived as obstructing the natural motion, which in consequence is slowed down.

Having thus explained why the point more distant from the center moves more quickly than the point closer to it, though impelled by the same force, the author of Mechanical Problems moves on to explain why larger balances are more accurate than smaller ones: the extremity of the balance scale must move at a greater speed, under the influence of the same weight, the greater its distance is from the pivot upon which it turns. Consequently in a larger balance the same weight makes a more visible movement.

Many sixteenth-century authors studied the Mechanical Problems. Some, such as Niccolò Leonico Tomeo⁵ and Alessandro Piccolomini,⁶ translated this work from the original Greek into Latin and added brief linguistic comments. Others, such as Girolamo Cardano⁷ and Niccolò Tartaglia,⁸ discussed only a few questions in detail: they examined the theory of the balance presented by the author of the ancient text in light of the new concepts and methods of demonstration introduced by the medieval science of weights (scientia de ponderibus). Tomeo was one of the leading scholars to study Aristotle's works during the first decades of the sixteenth century. After translating De parva naturalia, De motu animalium and De incessu animalium, around 1525 he turned to the task of translating the Mechanical Problems. His translation was so good that it replaced a previous translation made in 1517 by the Venetian humanist Vittore Fausto and became the text used by most of the commentators of Mechanical Problems from the mid-sixteenth to the mid-seventeenth century. The first and second editions of this translation, published in Venice in 1525 and in Paris in 1530, were provided with a commentary which was not reprinted in later editions.⁹

The text of Mechanical Problems is often so obscure and compact that from early on it required explanatory notes and commentaries. Alessandro Piccolomini chose to make it more accessible by publishing a paraphrase of the work in 1547 in Rome: In mechanicas quaestiones Aristotelis, paraphrasis paulo quidem plenior shows both his erudition in the use of available manuscripts as well as his practical knowledge of the contemporary mechanical technology.¹⁰

Tomeo's translation and Piccolomini's paraphrase were the main channels through which the knowledge of the Mechanical Problems spread during the sixteenth century. Baldi himself made use of both of these works in his commentary. Other authors discussed only some of the questions of the Aristotelian text and made important criticisms of the principles and arguments presented in it.

Niccolò Tartaglia, in Book Seven of his Quesiti et inventioni diverse, published in 1546, analyzes the general principles stated in Mechanical Problems and argues that they were not adequate for the correct solution to the problem of the equilibrium of the balance in the first two questions of that work.¹¹ Tartaglia's criticism preluded a new approach to the problem of the balance, which was developed in Book Eight of Quesiti et inventioni diverse on the basis of the concept of gravitas secundum situm formulated in several writings ascribed to the medieval author Jordanus Nemorarius, dating back to the thirteenth century.¹²

A similar discussion of the Aristotelian text can be found in De subtilitate by Girolamo Cardano,¹³ who also favorably considered the medieval science of weights. For Cardano, the study of the balance was part of a general discussion on motion in connection with technical contrivances: the analysis of the working of a machine was for him an important means for understanding natural principles. It was within the context of a general theory of motion and rest that Cardano studied heavy bodies in motion and at rest in the balance and in other machines. Though he considered many arguments in Mechanical Problems as inadequate and superseded by the medieval science of weights and by Archimedes' work, the Aristotelian text exerted a remarkable influence on Cardano, and supplied a considerable amount of topics for discussion in his Opus novum de proportionibus.¹⁴

Compared to these approaches, Baldi's Exercitationes represent a different way of discussing the contents of Mechanical Problems: the Exercitationes contain a systematic analysis of this work in light of Archimedean mechanical principles such as the concept of center of gravity. In his discussion of the Aristotelian text, Baldi makes frequent use of Guidobaldo's Mechanicorum liber,¹⁵ which offers a new systematic theory of the simple machines. In many important digressions from the text, Baldi extends his study to include new mechanical problems similar to those presented in the ancient work.

This new approach was made possible by the recovery and study of Greek geometry and mechanics by Federico Commandino and Guidobaldo del Monte. In the mid-sixteenth century Commandino started a systematic program of Latin translation of works by great Greek mathematicians such as Euclid, Archimedes, Apollonius and Pappus. These translations were supplemented with important notes, but did not add anything new to the original texts. In some cases, however, the lack of a textual tradition resulted in the production of a new work. The most important example is Liber de centro gravitatis solidorum, conceived as a reconstruction of a lost ancient text containing a theory of the center of gravity presupposed by some theorems in other works of Archimedes.¹⁶

After this reconstruction Commandino did not continue with mechanical topics, apart from his translation of Pappus' Mathematical Collections, which in Book Eight contains a summary of Hero's Mechanics. But he did not work on the main ancient text concerning the theory of the center of gravity, i.e., Archimedes' On the Equilibrium of Planes, which was not included in his 1558 edition of Archimedes' works.¹⁷ The task of studying and commenting this text was left to the most famous of Commandino's pupils, Guidobaldo del Monte, who first used the law of the equilibrium of the balance given in this text as the foundation of a rigorous theory of the simple machines in Mechanicorum liber, and then expounded and explained Archimedes' work in the form of a paraphrase.¹⁸

These works established the Archimedean method of dealing with static problems as superior to those used in Mechanical Problems and in the scientia de ponderibus. Though the tradition of the medieval science of weights followed by Tartaglia and Cardano was sharply criticized by Guidobaldo in the treatise on the balance inserted in Mechanicorum liber, the Aristotelian work was still considered to be relevant to the study of mechanical problems, and the principle of equilibrium formulated in this work was regarded as true and fundamental, and simply in need of a better explanation.

This explanation was given by Guidobaldo within the context of the general development of mechanics described in the preface to Book One of Paraphrasis,¹⁹ where Archimedes' work is considered as laying the true foundation of mechanical science, but at the same time as being closely connected to the Aristotelian work. The general theoretical principle presented in Mechanical Problems needed to be specified by determining the exact proportion between weights and distances in the lever. This idea was clearly expressed by Baldi in his biography of Archimedes, included in Vite de' matematici:

Since Archimedes (as is probable and as Guidobaldo himself guessed in the preface to Book One of On the Equilibrium of Planes) regarded this Aristotelian work as based on solid principles, but not very clearly explained, he wanted to make it more explicit and more easily understandable by adding mathematical demonstrations to physical principles. Aristotle solved the problem concerning the reason why the longer the lever, the easier it is to move the weight, by saying that this happens because of the greater length on the side of the moving power; this was true according to his principle, in which he supposed that things which are at a greater distance from the center move more easily and with greater force; the cause of which he saw in the greater speed with which the larger circle moves compared to the smaller circle. This cause is indeed true, but lacks precision; for given a weight, a lever and a power, I do not know how I should divide the lever at the point where it turns so that the given power balances the given weight. Archimedes accepted Aristotle's principle but went further; he was not satisfied with saying that the force would be greater on the longer side of the lever, but he determined how much longer it should be, that is, what proportion it should have with the shorter side, so that the given power would balance the given weight. [...] He established this with a brilliant demonstration in Book One of On the Equilibrium of Planes, which, as Guidobaldo pointed out, was the book of Elements of the whole field of mechanics. In the preface of his paraphrase of Archimedes' work, Guidobaldo showed that Archimedes had followed Aristotle entirely, as far as the principles were concerned, but to them had added his own exquisite demonstrations.²⁰

This is the program of Baldi's Exercitationes: to supply the Aristotelian Mechanical Problems with Archimedes' principles and demonstrations.

Footnotes