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NATURES DRAWING: PROBLEMS AND RESOLUTIONS IN THE MATHEMATIZATION OF MOTIONAbstractThe mathematization of motion around the turn of the 17th century involved the ascription of new ontological status to both motion and its mathematical depiction. For Kepler and Galileo geometrical curves were not ideal representations of motion, but traces left by nature itself, which the mathematician is called upon to analyze. In contrast to their Renaissance predecessors, for whom mathematical order was alien to motion, Kepler and Galileo construed nature in terms of pure motion, essentially mathematical, hence orderly. Their later century successors no longer conceived the application of mathematics to nature as requiring difficult metaphysical legitimation. For Huygens and Hooke the mathematical structure of nature resides in its malleability to the manipulations of the mathematician.

The ResolutionAnd I have not satisfied my soul with speculations of abstract Geometry, namely with pictures, of what there is and what is not to which the most famous geometers of today devote almost their entire time. But I have investigated the geometry that, by itself, expresses the body of the world following the traces of the Creator with sweat and heavy breath. (Kepler, Ad Vitellionem, Dedication to the Emperor)" 1

These are the words in which Kepler expresses his ideal of mathematical investigation of nature and these are the words with which Galileo introduces his readers to his great achievement in mathematicalNeque animum explevi speculationibus Geometriae abstractae, picturis scilicet , in quibus pene solis hodie celeberrimi Geometrarum aetatem transigunt: sed Geometriam per ipsa expressa Mundi corpora, Creatoris vestigia cum sudore et anhelitu secutus, indagaui (GW Vol. 2, 10)1

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mechanics, the analysis of accelerated motion:And first of all it seems desirable to find and explain a definition best fitting natural phenomena. For anyone may invent an arbitrary type of motion and discuss its properties; thus, for instance some have imagined helices and conchoids as described by certain motions which are not met with in nature, and have very commendably established the properties which these curves possess in virtue of their definitions; but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions. (Galileo, Two New Sciences, Third Day [197], 160)

The near-identity of phrasing is striking. Galileo and Kepler are not usually considered to have shared epistemological ideals. But here they both apply the same contrast and aver the same choice: between abstract geometry produced by motions which are not met with in nature on the one hand and geometry that exhibits observed motions or expresses the body of the world on the other, Kepler and Galileo choose to follow real traces. Moreover, this phrasing seems to resonate, even echo, Leon Battista Albertis recommendation thatone should avoid the habit of those who aspire to extol in the art of painting after their own ingenuity with no natural forms present in front of their mind or eyes (Alberti, De Pictura, 98.) 2

Again, Alberti and the Perspectiva tradition he represents are not often considered an important chapter in the history of the mathematization of nature outside optics 3 . Yet this nexus of mathematics, motion and drawing is too telling to ignore. It offers another dimension to the always relevant theme of the mathematization of nature in early modern science.GALILEO

One important achievement of recent scholarship was to reformulate the grand themes concerning the emergence of Galilean mechanics in

2 Fugienda est consuetudo non nullorum qui suoppte ingenio ad picturae laudem contendunt, nullam naturalem faciem eius rei oculis aut mente coram sequentes. 3 C.f. Kemp, The Science of Art.

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clear and well defined terms. Thus, instead of new science vs. Church establishment, we have the differing intellectual agendas of, e.g., Dominicans and Jesuits4

; instead of metaphysical persuasion vs.

experimental evidence we nave the complex of artisan and practicalmechanical traditions from which evolve new ways of matching theoretical and practical forms of knowledge 5 ; instead of neo Platonism vs. internal mathematical development we have Aristotelian mixed sciences 6 and mechanical Archimedean modeling 7 . These contributions and the substance they add to traditional historiography do not offer a compromise between the oppositions in which this tradition was cast. Rather, they reveal that these epistemological and ontological categories were themselves a result of the emergent scientific practices. In particular, works like Sheas 8 , Naylors 9 or Renn and Damerow 10 make clear that neither mathematization nor nature can be taken as self explanatory; that the metaphysical and the technical aspects of Galileos work were too fluid and mutually dependent for either to be considered as the real engine behind the mechanical philosophy of nature which we take as his legacy, and that both the mechanical phenomena and the mathematical apparatus were reshaped in the process of its evolution. The surprising affinity between Alberti, Kepler and Galileo suggests another perspective on this complexity. It highlights the ontological difficulties at the interface of mathematics and experience and the4 5 6 7

C.f. Feldhay, Galileo and Church; Dear, Discipline and Experience. C.f. Renn and Daemrow, Lefevre. C.f. Laird, Galileo and the Mixed Sciences, C.f. Bertoloni Meli, Guidobaldo dal Monte and the Archimedean revival; Machamer, Galileo's machines, his mathematics and his experiments, in: Machamer, Cambridge Companion to Galileo.

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Shea, The Galilean Geometrization of Motion in: Shea (ed.), 51-60. Naylor, Galileo, Copernicanism and the origins of the new science of motion. Renn and Daemrow, Hunting the White Elephant.

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flexible and tentative ways in which these difficulties were confronted: what exactly does motion need to be mathematically treatable; and what does the ontological status of mathematical entities need to be so they can legitimately represent motion? The following is not a study in the technical solutions to the mathematical challenges 11 or in the new scientific concepts of motion and their development. points in the process. The paragraph from Galileos Two New Sciences is important for many reasons, not the least of which is the one noted already by Koyr (Galileo Studies, 67), that it implies that Galileo already knows that the shape of projectiles trajectories is a parabola when he seeks for a definition of accelerated motion. What has hardly been noted, however, is what it is that Galileo thinks he knows in the parabola: it is neither a mathematical abstraction of a natural phenomenon nor a reflection of some ideal infrastructure, but a curve drawn by Nature herself. Mathematics has been applied to the study of motion before, but a very different ontological status was ascribed to this application: indivisible points, or lines, are non existent, stresses Nicole Oresme. Indeed, it is necessary to feign them mathematically (mathematice fingere) for the measures of things and for the understanding of their ratios, but Of course, the line of intensity of which we have just spoken is not actually extended but is only so extended in the imagination (Oresme, Nicole Oresme and the medieval Geometry, 165-169). Imaginary lines and artificially produced curves are exactly what Galileo vows not to waste time on. His investigation would be of those met in nature. Not because the imaginary lines fail to properly represent nature Galileos distinction is not between curves succeeding or failing in representing nature, but between types of motion.11

Rather, it is a

presentation of the forms this ontological dilemma takes at a few crucial

Some

C.f. Gal and Chen.

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motions are artificial, some met in nature. But even this distinction is secondary. investigate. All motions leave traces which can be studied mathematicallyit is up to the mathematician to decide which to The relation between drawing and nature proclaimed by For Alberti, the painter draws natural forms. Galileo is stronger than the one assumed by Alberti, whose 1435 De Pictura is cited above. For Galileo, natural motions themselves draw the curves to be studied. Moreover, motions produced by nature can be harnessed to artificially draw curves 12 :Salviati: there are many ways of tracing these curves; I will mention merely there two which are the quickest of all. One of these is really remarkable; because by it I can trace thirty or forty parabolic curves with no less neatness and precision and in a shorter time than another man can, by the aid of a compass, neatly draw four or six circles of different sizes upon paper. I take a perfectly round brass ball about the size of walnut and project it along the surface of a metallic mirror held in a nearly upright position, so that the ball in its motion will press slightly upon the mirror and trace out a fine sharp parabolic line; this parabola will grow longer and narrower as the angle of elevation increases. The above experiment furnishes clear and tangible evidence that the path of a projectile is a parabola; a fact first observed by our friend and demonstrated by him in his book on motion which we shall take up at our next meeting in the execution of this method, it is advisable to slightly heat and moisten the ball by rolling in the hand in order that its trace upon the mirror may be more distinct. The other method of drawing the desired curve upon the face of the prism is the following: Drive two nails into a wall at a conve