This paper examines geometrical arguments from Galileo's Mechanics and Two New Sciences to discern the influence of the Aristotelian Mechanical Problems on Galileo's dynamics. A common scientific procedure is found in the Aristotelian author's treatment of the balance and lever and in Galileo's rules concerning motion along inclined planes. This scientific procedure is understood as a development of Eudoxan proportional reasoning, as it was used in Eudoxan astronomy rather than simply as it appears in Euclid's Elements. Topics treated include the (...) significance of the circle in Galileo's demonstrations, the substitution of rectilinear elements for heterogeneous factors like weight and curvilinear distance, and the way in which elements of a motion are used to measure other elements of the same motion. The indirectness of Galileo's proofs, his conception of speed as relative and comparative, and the meaning of his concept of moment all come into clearer focus. Conclusions are drawn about Galilean idealization, and also about the contrast of literal versus figural modes of explanation in Galileo's science. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

In On Local Motion in the Two New Sciences, Galileo distinguishes between ‘time’ and ‘quanto time’ to justify why a variation in speed has the same properties as an interval of time. In this essay, I trace the occurrences of the word quanto to define its role and specific meaning. The analysis shows that quanto is essential to Galileo’s mathematical study of infinitesimal quantities and that it is technically defined. In the light of this interpretation of the word quanto, Evangelista (...) Torricelli’s theory of indivisibles can be regarded as a natural development of Galileo’s insights about infinitesimal magnitudes, transformed into a geometrical method for calculating the area of unlimited plane figures. (shrink)

The ArgumentOne of the earliest arguments for Copernicanism was a widely accepted fact: that on a horizontal plane a body subject to no external resistance can be set in motion by the smallest of all possible forces. This fact was contrary to Aristotelian physics; but it was a physical argument (by abduction) for the possibility of the Copernican world system. For it would be explained if that system was true or at least possible.Galileo argued: only nonviolent motions can be caused (...) by the smallest of all possible forces; hence resistance-free horizontal motions are nonviolent; this confirms Copernicanism insofar as it designates the rotations of celestial spheres (being resistance-free horizontal motions) as nonviolent.Galileo's argument was compatible with (and supportive of) the specific Copernican version of impetus mechanics; but it was also compatible with a (somewhat qualified) principle of inertia. Thus it promoted decisively the transition from impetus mechanics to classical inertial mechanics. (shrink)

This article analyzes Galileo's mathematization of motion, focusing in particular on his use of geometrical diagrams. It argues that Galileo regarded his diagrams of acceleration not just as a complement to his mathematical demonstrations, but as a powerful heuristic tool. Galileo probably abandoned the wrong assumption of the proportionality between the degree of velocity and the space traversed in accelerated motion when he realized that it was impossible, on the basis of that hypothesis, to build a diagram of the law (...) of fall. The article also shows how Galileo's discussion of the paradoxes of infinity in the First Day of the Two New Sciences is meant to provide a visual solution to problems linked to the theory of acceleration presented in Day Three of the work. Finally, it explores the reasons why Cavalieri and Gassendi, although endorsing Galileo's law of free fall, replaced Galileo's diagrams of acceleration with alternative ones. (shrink)

Summary One aim of this paper is to provide an assessment of the recent attempts to interpret the development of Galileo's theory of motion in the late Paduan period 1604?1610. In addition to this a new interpretation of this process of development is advanced. This interpretation is the first that proves able to provide a full account of all the features on folio 152r of volume 72 of the Galilean manuscripts which has been claimed to be of crucial significance. The (...) major feature of the interpretation is that it is the first to account for the relationship between Galileo's theoretical and experimental work in the period 1604?1610. (shrink)

Recent study of Galileo's surviving manuscript notes on motion has revealed that by 1609 he had developed the major part of his theory of projectile motion. During the period of these theoretical advances Galileo was engaged in important related experimental investigations; this has become clear from the study of folios 114r and 116v of the manuscript on motion. This paper provides an interpretation of a manuscript not previously discussed—folio 81r. The analysis provided indicates that it is evidence of an important (...) early experiment of the series in which Galileo studied the projectile trajectory. The reconstruction of the experiment concerned reveals relationships with the enquiries suggested by folio 114r and 116v and indicates that Galileo examined the form of projectile trajectories with some care. (shrink)

Galileo did not develop a systematic methodology but rather a methodical form which represents an essential part of the development of modern scientific thought. Keywords to the methodical form of Galileo's thought are: 1) The geometrization of the sciences - this refers especially to the explication of the methodological priority of a theory of measurement. 2) Argomento ex suppositione, that is, the coupling of the originally proof-theoretical distinction between analysis and synthesis to elements of a methodology of empirical science. 3) (...) The axiomatic structure of mechanics that corresponds to the modern semantic theory conception and to constructivist conceptions of the philosophy of science. 4) A constructive concept of experience, which replaces the Aristotelian concept of a phenomenal experience in the construction of physics. (shrink)

One of the distinctive features of modern science is a commitment to empiricism—a fundamental expectation that theoretical hypotheses will survive encounters with observations. Those that comport with the theory's explanations and predictions confirm the theory. Anomalous observations that do not fit theoretical expectations disconfirm it. Moreover, experiments can be contrived to generate observations that might serve to confirm or disconfirm a theory. Philosophers of science may disagree as to how exactly all of this is supposed to work, but the basic (...) empiricist expectation almost goes without saying. To deny it is to rule one's self out of the bounds of the scientificenterprise.Strange as it might seem to modern... (shrink)

The aim of this paper is to take Galileo's mathematization of nature as a springboard for contrasting the time-honoured empiricist conception of phenomena, exemplified by Pierre Duhem's analysis in To Save the Phenomena , with Immanuel Kant's. Hence the purpose of this paper is twofold. I) On the philosophical side, I want to draw attention to Kant's more robust conception of phenomena compared to the one we have inherited from Duhem and contemporary empiricism. II) On the historical side, I want (...) to show what particular aspects of Galileo's mathematization of nature find a counterpart in Kant's conception of phenomena.------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- --------------------------------------------------. (shrink)

Galileo proposed what has been called a proto-inertial principle, according to which a body un horizontal motion will conserve its motion. This statement is only true in counterfactual circumstances where no impediments are present. This paper analyzes how Galileo could have been justified in ascribing definite properties to this idealized motion. This analysis is then used to better understand the relation of Galileo’s proto-inertial principle to the classical inertial principle.

Quel est le rôle de Galilée dans la naissance séculaire de la science moderne? Je réponds à la question ci-dessus à la lumière de deux nouveaux éléments. Dès le xvie siècle, Nicolas de Cues, quoiqu’il ne pratiquât pas la science expérimentale, a anticipé une partie substantielle de la révolution copernicienne et de la naissance de la méthodologie galiléenne. Le deuxième élément est l’introduction d’une nouvelle conception des fondements de la science ; ils sont définis comme constitués de trois dialectiques. Après (...) cette définition, la naissance de la science moderne correspond à un très long processus historique qui se conclut à notre époque. Sur cette longue période, Galilée ne fut pas seulement le premier à recourir à une méthodologie scientifique, mais aussi presque le seul à jamais avoir été conscient de l’ampleur des enjeux intellectuels de l’entreprise scientifique. (shrink)

Peter Dear ha proporcionado recientemente un análisis de la transformación que sufrió el recurso a la experiencia en la filosofía natural del siglo XVll. De la experiencia de lo cotidiano se pasa a la descripción detallada de una experiencia artificial irrepetible, localizada espacio-temporalmente y producida por instrumentos más o menos complejos. EI artículo explora dicha interpretación en referencia a la construcción de la ciencia del movimiento de Galileo, mediante un análisis dcl experimento del plano inclinado que se describe en los (...) Discorsi y un manuscrito, y concluye que la interpretacion de Dear dificulta considerablemente la caracterización de la práctica de Galileo.Peter Dear has recently put forward an analysis of th transformation underwent by the appeal to experience in Seventeenth-Century natural philosophy. According to Dear, this transformation lies in the change from common experience to the detailed description of an unrepeatable artificial experience space temporally located and produced by more or less sophisticated instruments. This paper explores Dear’s interpretation with regard to the construction of Galileo’s science of motion, by analyzing the celebrated inclined plane experiment described in the Discorsi as well as one of Galileo's manuscripts and concludes that Dear’s interpretation makes very difficult the characterization of Galileo’s practice. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion that was first (...) conceived by ancient Greek mathematicians. (shrink)