e Geometrization of Motion: Galileo’s Triangle of Speed and its Various Transformations

8/10/2019 e Geometrization of Motion: Galileos Triangle of Speed and its Various Transformations

1/38


2/38

411C.R. Palmerino / Early Science and Medicine 15 (2010) 410-447

at the same time essential for the latters comprehension. This mightexplain why, in spite of their meager aesthetic appeal, diagrams havefor a long time been more respected than other scientific illustrations.

Contrary to emblematic, symbolic or allegorical figures, which wereusually dismissed as decorative, diagrams were regarded as integralelements of a scientific text and were hence reproduced in moderneditions and in the secondary literature. Terms such as eccentric,epicycle, mean speed theorem or law of refraction cannot fail toevoke, in the historian of science, precise mental images due to the

acquaintance with the respective diagrams.It is however only in recent years that scholars have begun studyingscientific images, and hence also diagrams, in their own right. In animportant article published in 1985, Samuel Edgerton invited histori-ans of science to treat illustrations not as afterimages of verbal ideas,but as a unique form of pictorial language with its own grammar andsyntax.1Edgertons appeal did not go unheard.2Over the last 25 years

scholars have devoted increasing attention to scientific illustrations,which they have studied in relation to their underlying pictorial tech-niques, their typology, their relation to the text and their explanatoryfunction.3In the case of historical diagrams, scholars have, for example,

1) Samuel Y. Edgerton, e Renaissance Development of Scientific Illustrations, in

Science and the Arts in the Renaissance, ed. John William Shirley and F. David Hoeniger(Washington, D.C., 1985), 168-197, here at 168.2) Edgerton was, however, not the first one to study scientific illustrations. Amongthe pioneering studies on the subject one should mention John E. Murdoch,Albumof Science: Antiquity and the Middle Ages (New York, 1984).3) It is impossible in the space of a footnote to do justice to the immense bibliographyabout the subject. I shall therefore limit myself to mentioning some selected studies:Michael Lynch & Steve Woolgar (eds.), Representation in Scientific Practice (Cam-

bridge, Mass., 1990); David R. Topper, Natural Science and Visual Art: Reflectionson the Interface, in Beyond History of Science: Essays in Honor of Robert E. Schofield,ed. Elizabeth Garber (Bethlehem, 1990), 296-310; Brian J. Ford, Images of Science:

A History of Scientific Illustration (London, 1992); omas Da Costa Kaufmann, eMastery of Nature: Aspects of Art, Science, and Humanism in the Renaissance (Princeton,1993); Renato G. Mazzolini (ed.), Non-Verbal Communication in Science Prior to 1900(Florence, 1993); Brian S. Baigrie (ed.), Picturing Knowledge: Historical and Philosophi-cal Problems Concerning the Use of Art in Science (Toronto, 1996); Pamela O. Long,Objects of Art/Objects of Nature: Visual Representation and the Investigation ofNature, inMerchants & Marvels: Commerce, Science and Art in Early Modern Europe,


3/38

412 C.R. Palmerino / Early Science and Medicine 15 (2010) 410-447

analyzed the different ways in which they could complement, sum-marize or substitute for the texts that they accompanied; they havedrawn a distinction between synoptic, memorative, and functional

roles; and they have tried to determine whether the mode of represen-tation of the mathematical sciences could be clearly distinguished fromthe mode of representation of the descriptive sciences.4

In his article, however, Edgerton formulated a controversial thesisconcerning the influence of Renaissance art on science. In his view, not

ed. Pamela Smith and Paula Findlen(New York, 2002), 63-82; Wolfgang Lefvre,Jrgen Renn and Urs Schoepflin (eds.), e Power of Images in Early Modern Science(Basel, 2003); Sachiko Kusukawa & Ian Maclean (eds.), Transmitting Knowledge. Words,Images, and Instruments in Early Modern Europe (Oxford, 2006).4) An attempt to classify diagrams according to their function and their relation tothe text has been made by Andreas Gormans, Imaginationen des Unsichtbaren. ZurGattungstheorie des wissenschaftlichen Diagramms, in ErkenntnisErfindungKon-struktion. Studien zur Bildgeschichte von Naturwissenschaften und Technik vom 16. biszum 19. Jahrhundert, ed. Hans Hollnder (Berlin 2000), 51-71; the question of howthe iconography of descriptive sciences differs from the iconography of mathematicalsciences has been addressed, among others, by Martin Kemp, Temples of the Bodyand Temples of the Cosmos: Vision and Visualization in the Vesalian and Coperni-can Revolutions, in Baigrie (ed.), Picturing Knowledge, 40-85; Sachiko Kusukawa,Illustrating Nature, in Books and the Sciences in History, ed. Marina Fransca-Spadaand Nick Jardine (Cambridge, 2000), 90-113; Sven Dupr, Visualization in Renais-

sance Optics: e Function of Geometrical Diagrams and Pictures in the Transmissionof Practical Knowledge, in Kusukawa and Maclean (eds.), Transmitting Knowledge,11-39. Among the many studies concerning the use of diagrams in the mathemati-cal sciences, I want to mention here John J. Roche, e Semantics of Graphs inMathematical Natural Philosophy, in Mazzolini (ed.), Non-Verbal Communication,197-233; Reviel Netz, e Shaping of Deduction in Greek Mathematics: A Study inCognitive History (Cambridge, 1999); omas L. Hankins, Blood, Dirt, and Nomo-grams: A Particular History of Graphs, Isis, 90 (1999), 50-80; idem, A Large and

Graceful Sinuosity: John Herschels Graphical Method, Isis97 (2006), 606-633;Wolfgang Lefvre, e Limits of Pictures: Cognitive Function of Images in Practi-cal Mechanics1400 to 1600, in Lefvre, Renn and Schoepflin (eds.), e Power ofImages, 69-88;Judith V. Field, Renaissance Mathematics: Diagrams for Geometry,Astronomy and Music, Interdisciplinary Science Reviews, 29 (2004), 259-277; MichaelMahoney, Drawing Mechanics, in Picturing Machines 1400-1700, ed. WolfgangLefvre (Cambridge, 2004), 281-306; Christoph Lthy and Alexis Smets, Words,Lines, Diagrams, Images: Towards a History of Scientific Imagery in Evidence andInterpretation in Studies on Early Science and Medicine, ed. William R. Newman &Edith Dudley Sylla (Leiden, 2009), 398-439.
http://www.ingentaconnect.com/content/external-references?article=0308-0188(2004)29L.259[aid=9282574]http://www.ingentaconnect.com/content/external-references?article=0308-0188(2004)29L.259[aid=9282574]http://www.ingentaconnect.com/content/external-references?article=0308-0188(2004)29L.259[aid=9282574]


4/38


only did the Renaissance scientific picture give precise informationabout the physical world () without need of explanatory texts andwithout the need for the viewer to refer to the actual objects depicted,

but new pictorial techniques such as linear perspective and chiaroscuroactually boosted the scientific revolution by making it possible to inventpractical workable machines.5

Edgertons thesis was strongly criticized by Michael Mahoney, whoobserved, first of all, that however ingenious and intricate, the devicesdrawn in the Renaissance were not intrinsically different from ancient

and medieval prototypes.6

Secondly, Renaissance drawings could neitherdisplay the workings of a machine, nor distinguish between the feasibleand the fantastical. Thirdly, the science of mechanics underwent aradical transformation in the seventeenth-century as it came to treatthe machine as an abstract, general system of quantitative parameterslinked by mathematic relations (). The defining terms of the systemslay in conceptual realms ever farther removed from the physical space

the artists had become so adept at depicting. Those terms could not bedrawn; at best, they could be diagramed.7Mahoney showed how theevolution of the science of mechanics could be reconstructed by fol-lowing the changing nature of the diagrams. He began his analysis withGalileo, who gradually came to produce diagrams of motion represent-ing a mathematical space wholly divorced from the physical space in

which the motion itself is taking place; then examined the graphs byHuygens, which represent the traces of a mathematical argument tak-ing place in another conceptual realm altogether, and Newton, whichform intricate patterns of lines analyzable only by a body of sophisti-cated, at times even counterintuitive concepts. He then proceeded toVarignon, who recast Newtons mechanics in the language of Leibnizscalculus; and ended with Euler, in whose view a diagram, however

good, formed a curtain hiding the essence of mechanics.8The very conceptual development that forms the subject of Mahoneys

article has been explained by Michel Blay in terms of a shift from

5) Edgerton, e Renaissance Development, 169.6) Michael Mahoney, Diagrams and Dynamics: Mathematical Perspectives on Edger-tons esis, in J.W. Shirley and F.D. Hoeniger (eds.), Science and the Arts, 198-220.7) Ibid.,200.8) Ibid.,203-217.


5/38


6/38


s = gt,

stated, in its original formulation, that the spaces traversed by a falling

bodies are to each other as the squares of the times elapsed (s 1: s2 =t

1: t

2

).The fact that Galileo relied on the Eudoxian theory of proportions,

which only permits the expression of ratios between homogeneous mag-nitudes, explains why his diagrams of motion display a greater resem-blance to the configurations used by fourteenth-century Calculatores,

and in particular by Oresme, than to Newtons diagrams.

Oresmes diagram (figure 1) is meant to compare a motion of uni-formly changing speed (CAB) with a uniform motion (FGBA). Thehorizontal line AB stands for the duration of the two motions, whereas

the vertical lines (perpendicular to AB) represent the intensity of thevelocity at the various instants of time. Oresme observes that

if in all the instants of time EG, the velocity is equally intense, then on every pointof line AB there will be an altitude everywhere equal, and the figure will be uni-formly high, i.e. a rectangle designating this velocity that is simply uniform. Butif in the first instant of the time there is a velocity of a certain amount and in themiddle instant of the whole time there is a velocity half [of that of the first instant]

and in the middle instant of the last half [of the time] there is a velocity one quar-ter and so on proportionally for all other instants (and consequently there will bezero velocity in the last instant), then () there will be the figure of a right trian-gle designating the velocity; this velocity was in fact one uniformly difform ter-minated at no-degree in its last instant.10

10) Marshall Clagett (ed. and transl.), Nicole Oresme and the Medieval Geometry ofQualities and Motions: A Treatise of the Uniformity and Difformity of Intensities Knownas Tractatus de configurationibus qualitatum et motuum(Madison, 1968), 291.

Figure 1: Oresmes representation of the Mean Speed eorem (Clagett, NicoleOresme , 99).


7/38


Later on in his text Oresme observes that the two small triangles EFCand EGB being equal, the qualities imaginable by a triangle and arectangle of this kind are equal.11Given that in the case under scrutiny

the quality is represented by the total velocity, which is to say thepunctual velocity enduring in time,12and given that in rectilinearmotion, as in motion of descent, the velocity of motion is attendedwith [i.e. is measured by] the space,13it is clear that the two motionsunder comparison will traverse equal spaces in equal times.

Figure 2, which is taken from Galileos Discorsi and will be discussed

in detail below, also compares a uniform motion and a uniformly accel-erated motion taking place in equal times. Here the vertical line ABrepresents the time of fall, the horizontal lines EB and FB the final

speeds of the accelerated and uniform motion respectively, and the linesparallel to it the instantaneous speeds. The areas of the triangle and ofthe rectangle are identified by Galileo with the total speed, which he

11) Ibid., 411.12) Ibid., 293.13) Ibid., 279.

Figure 2: Galileos Representation the Mean Speed eorem (Galilei, Opere, 8: 208).


8/38


asserts to be proportional to the space traversed, represented by theseparate line CD.

In her contribution to this fascicle, Edith Sylla draws attention to

the fact that in his De configurationibus, Oresme explicitly asserts thatif a uniformly difform quality is divided in equal parts, then the ratioof the partial qualities, i.e., their mutual relationship, is as the series ofodd numbers.14It is not my intention here to discuss whether Oresmeslaw of a uniformly difform quality can really be labeled law of free fallor to investigate if and how the Oresmian diagrams exerted an influence

on Galileo.15

Rather, I want to use the comparison between the twodiagrams to show that, in his attempt to geometrize motion, Galileohit upon two difficulties that Oresme had not encountered.

The first one concerns the interpretation of the line AB. As JohnRoche has noticed, Oresmes diagram may be considered the firsttotally conventional graphical representation of a law in physics, asneither the vertical nor the horizontal lines represent a spatial exten-sion.16The reason for this has certainly to be sought in the fact that thedoctrine of the configuration was not exclusively applied to thedescription of motion, but was rather meant to describe and representthe alteration over time of all intensive qualities, velocity being onlyone of them. Figure 1 could, for example, be used to compare a bodyat constant temperature with a body cooling down. In this case the

horizontal line AB would still stand for the time elapsed, whereas thevertical lines perpendicular to it would represent the intensity of thetemperature of the two bodies in successive instants of time.

As Matthias Schemmel has recently observed, in the early modernperiod, the diagrams were applied to motion without being embeddedin a general theory of change. There was, therefore, no particular pref-

14) Ibid., 563.15) For a discussion of a possibile influence of the Oxford Calculatoresand/or Oresmeon Galileo, see Anneliese Maier, Die Vorlufer Galileis im 14. Jahrhundert (Rome,1949); ead.,An der Grenze von Scholastik und Naturwissenschaft (Rome, 1952); EdithSylla, Galileo and the Oxford Calculatores: Analytical Languages and the Mean-Speedeorem for Accelerated Motion, in Reinterpreting Galileo, ed. William A. Wallace(Washington, 1986), 53-108; Antonio Nardi, La quadratura della velocit. Galileo,Mersenne, la tradizione, Nuncius 3 (1988), 2764.16) Roche, e Semantics of Graphics, 207.


9/38


erence to interpret the line of extension as representing time, and itsinterpretation as space traversed () became equally plausible.17As amatter of fact, the spatial interpretation initially appeared to Galileo

more plausible than the temporal interpretation. As we shall see below,in his first diagrams of motion Galileo took the vertical line to representthe space traversed by the freely falling body and assumed the speed offall to grow in proportion to that space. It was only after he realized thefalsity of this assumption that he changed the interpretation of thevertical line of his triangle of speed. I therefore think that Schemmel is

perfectly right in claiming that Galileo used the temporal interpreta-tion of extension not simply because this was the canonical way todescribe change, but because he had arrived at the insight that thespatial interpretation contradicted his other findings.18

Another difference between the Oresmian and the Galilean diagramconcerns the interpretation of the surface of the triangle. As we havejust seen, Oresme identified this area with the space traversed, on the

account that space is the measure of total velocity. For Galileo, whointerprets the figure in the light of Cavalieris theory of indivisibles, thisidentification is far more problematic; for the addition of an infinitenumber of degrees of speed cannot engender space, but only totalspeed.19Moreover, Galileo seems to be aware of the difficulties linkedto the comparison of two infinite sets of indivisibles; that is to say, of

all the degrees of speed contained in the triangle EAB and all the degreesof speed contained in the rectangle AGFB.20The main goal of the following pages is to show how the difficulties

encountered in the construction of his diagrams of motion helpedGalileo to detect some important conceptual mistakes and how he tried,by means of new diagrams, to solve, or sometimes simply to hide, thesemistakes.

17) Matthias Schemmel, e English Galileo: omas Harriots Work on Motion as anExample of Precassical Mechanics, 2 vols. (Dordrecht, 2008), 1: 57.18) Matthias Schemmel, Medieval Representations of Change and their Early ModernApplication (Berlin, Max Plank Insitute for the History of Science, preprint series,forthcoming), 24.19) Blay and Festa, Mouvement, continu, 96.20) Cf. Paolo Galluzzi,Momento. Studi Galileiani (Rome, 1979), 354.


10/38


11/38


The question of how Galileo managed to derive a correct law from afalse principle has been widely debated by scholars and I do not wantto dwell upon it here.22What interests me, rather, is the possible origin

of Galileos mistake. As observed by Galluzzi 30 years ago, Galileoarrived at his evident proposition by extending to the study of accel-erated motion a principle used in his mechanical speculations.23Gal-luzzis hypothesis finds confirmation in a manuscript fragment (Ms 72,fol. 128r), probably redacted shortly after the letter to Sarpi, in whichGalileo writes that his principle corresponds to all the experiences that

we see in the instruments and machines that work by striking, in whichthe percussent makes so much the greater effect, the greater the heightfrom which it falls.24

A different explanation of Galileos mistake, which is in fact notincompatible with Galluzzis, has been offered by Michael Mahoney,according to whom spatial intuition must take some share of theblame. Galileos first diagrams were still Archimedean in kind, which

means that they depicted the apparatus as a spatial object and locatedthe operative parameters in its constituent elements. In the case of thelaw of fall, the apparatus was clearly an inclined plane on which Galileotried to locate the parameters of acceleration.25Another manuscriptfragment (Ms 72, fol. 163v), which was probably written before theletter to Sarpi, clearly shows how the inclined plane figured in Galileos

early speculations about motion.26

In this folio Galileo tries, by means

22) See, e.g., Stillman Drake, Galileos Work on Free Fall in 1604, Physis, 16 (1974),309-322; Enrico Giusti, Galileo e le leggi del moto, in Galileo Galilei, Discorsi edimostrazioni matematiche intorno a due nuove scienze, ed. E. Giusti (Turin, 1990),ix-lx, esp. xxv-xxxvi; Blay and Festa, Mouvement, continu, 65-72 ; Damerow et al.,Exploring the Limits, 165-179.23) Galluzzi,Momento, 272.24) Galilei, Opere, 8: 373, translated in Damerow et al., Exploring the Limits,360-361.For an analysis of this fragment, see also Stillman Drake, Galileos 1604 Fragmenton Falling Bodies (Galileo Gleanings XVIII), e British Journal for the History of Science, 4 (1969), 340-358.25) Mahoney, Diagrams and Dynamics, 206-207.26) In his Ph. D. thesis Jochen Bttner provides convincing evidence of the fact thatfolios 163, 164 and 172 of Ms. 72 must have been drafted around the end of 1602,that is to say shortly after Galileoe letter to Guidobaldo of 29 November 1602 (seeJochen Bttner, Galileos Challenges: e Origin and Early Conceptual Development of
http://www.ingentaconnect.com/content/external-references?article=0007-0874(1969)4L.340[aid=9283559]http://www.ingentaconnect.com/content/external-references?article=0007-0874(1969)4L.340[aid=9283559]http://www.ingentaconnect.com/content/external-references?article=0007-0874(1969)4L.340[aid=9283559]http://www.ingentaconnect.com/content/external-references?article=0007-0874(1969)4L.340[aid=9283559]http://www.ingentaconnect.com/content/external-references?article=0007-0874(1969)4L.340[aid=9283559]


12/38


of figures 4 and 5, to offer a demonstration of the so-called DoubleDistance Rule, which states that the space traversed in a given time bya body in uniform rectilinear motion is twice the space traversed, inthe same time, by a body in naturally accelerated motion having a finalspeed equal to the constant speed of the uniform motion.

In figure 4 one sees that the area of triangle abc, which representsall the velocities acquired by a body falling vertically from ato b, ishalf the area of the rectangle abcd,which represents all the velocities ofa motion made across the same space abwith a uniform speed bc. Fig-ure 5 is used by Galileo to show that a body first rolling down theinclined plane ab and then along the horizontal line bcwould take thesame time to cover the distances ab and bc (double than ab). As Mat-

thias Schemmel recently observed, although Galileo derived both thelaw of fall and the Double Distance Rule from the assumption of aproportionality between velocity and space, he later recognized that,taken together, the two theorems contradicted precisely that assump-tion. In fol. 91v of Ms. 72 Galileo in fact used the Double Distance

Galileos eory of Naturally Accelerated Motion on Inclined Planes(Ph. D. thesis, Berlin,Humboldt-Universitt, 2009), 401-406, 455.

Figure 5: Ms. 72, fol. 163v: An accelerated motionturns into a uniform motion (Galilei, Opere, 8: 384).

Figure 4: Ms. 72, fol.163v: Galileos earlyattempt to demonstratethe double distance rule(Galilei, Opere, 8: 384).


13/38


Rule to prove that the speed of fall increases in proportion to the timeelapsed.27To me it seems plausible to assume that Galileo realized theshortcomings of his demonstration by reasoning on figure 4, for the

main problem of this diagram is that it offers two mutually contradic-tory representations of the space traversed by the falling body. Thelatter is in fact identified both with the line ab,which is common tothe triangle abcand the rectangle adbc, and with the area abc,which isinstead half the area of adbc.

If we return to the letter to Sarpi, we see that the diagram accompa-

nying it neither depicts a spatial object, nor represents the variousparameters of acceleration. Only one physical magnitude finds a rep-resentation i2n the figure, namely the vertical line of fall, which isarbitrarily divided in three segments that do not need to be equal.Galileo tells us that the degrees of speed the body has at the points b,cand dare to each other as the lines ab, acand ad, but he does not drawthe segments corresponding to the degrees (cf. fig. 3, above). This hap-

pens instead in the manuscript note mentioned above (fol. 128 r) whereGalileo starts, as in the letter to Sarpi, by enunciating the principleaccording to which

the naturally falling heavy body goes continually increasing its velocity accordingas the distance increases from the terminus from which it parted, as, for example,the heavy body departing from the point a and falling through the line ab. I sup-pose that the degree of velocity at point dis as much greater than the degree ofvelocity at the point c as the distance da is greater than ca () ; and this princi-ple assumed I shall demonstrate the rest.28

The rest is demonstrated by means of figure 6.

27) Schemmel, Medieval Representations. For an analysis of the demonstration con-tained in fol. 163 v, see also Winfred L. Wisan, e New Science of Motion: A Studyof Galileos De Motu Locali,Archive for History of Exact Sciences, 13 (1974), 103-306,204-207; Damerow et al., Exploring the Limits, 176-179.28) Galilei, Opere, 8: 373-374, translated in Damerow et al., Exploring the Limits, 360.
http://www.ingentaconnect.com/content/external-references?article=0003-9519(1974)13L.103[aid=6968374]http://www.ingentaconnect.com/content/external-references?article=0003-9519(1974)13L.103[aid=6968374]http://www.ingentaconnect.com/content/external-references?article=0003-9519(1974)13L.103[aid=6968374]


14/38


Figure 6: Ms. 72, fol. 128 r: e increase of speed in accelerated motion (Galilei,Opere, 8: 373).

After having drawn the line ab,which stands for the distance fallen,

Galileo represents the degrees of speed acquired at the points c,d,e andf by means of segments proportional to ac,ad,ae andaf. The fact thatthese segmentswhich he draws perpendicular to abterminate inthe lineak,reveals that the increase of speedis continuous. Given thatwhat is true for these degrees of speed must be true for all the degreesof speed acquired in the fall, the overall velocity with which the body

has passed the line ad is to the overall velocity with which it has passedthe line acas the triangle adhis to the triangle acg. The same thing isasserted by Galileo in folio 85 v of Ms. 72, where he explicitly statesthat the two triangles are constituted by an infinite number of lines.The conclusion Galileo draws from this is that the velocity along adand the velocity along ac are to each other as the squares of the spacestraversed.

There is one magnitude that is absent from Galileos diagram; namely,the time of fall. The latter makes its appearance only in the last part ofthe demonstration, where we read that

Since velocity to velocity has contrary proportion of that which time has to time(for it is the same thing to increase the velocity as to decrease the time), thereforethe time of the motion along adto the time of the motion on achas half theproportion that the distance adhas to the distance ac. e distances, then, from


15/38


the beginning of the motion are as the squares of the times, and, dividing, thespaces passed in equal times are as the odd numbers from unity.29

As already observed by scholars, this passage contains two mistakes:first, the inverse proportionality between velocity and time only holdswhen the spaces traversed are equal, which is clearly not the case here.Secondly, from this inverse proportionality, Galileo erroneously infersthat times are to each other as the square roots of the spaces.30

But what is relevant from our point of view is the fact that none ofthe diagrams drawn by Galileo in the manuscript notes redacted around1604 provide a representation of the law of fall. Galileo enunciates andrepresents the double distance rule and the length-time proportion-ality, but does not manage to produce a diagram showing the growthof spaces in successive intervals of time.31

I am inclined to believe that it was precisely this factthat the prin-ciple of the space/speed proportionality did not yield a diagrammatic

representation of the odd number lawthat helped Galileo to realizethe falsity of this principle. The manuscript notes clearly reveal thatGalileo did struggle to produce a diagram into which the three param-eters of acceleration, speed, time and space, could be made to fit. As Imentioned above, in folio 91v Galileo makes use of the Double Dis-tance Rule to demonstrate that the speed of fall must increase in pro-portion to the time elapsed, but produces a hybrid diagram (figure 7)which conflates the representation of speed with that of space. Whilestating that the moments of speed cd and be are in the same proportionas the intervals of time ac and cb, Galileo takes the segments ac and

29) Galilei, Opere, 8: 374, transl. in Damerow et al., Exploring the Limits,360.30) Giusti, Galileo e le leggi del moto, xxxv; Blay and Festa, Mouvement, continu,69, Damerow et al., Exploring the Limits,173.31) e Double Distance Rule is demonstrated in fol. 163v discussed above, whereasthe principle of the lenth-time-speed proportionality, according to which the timesof descent of a body along inclined planes of different length but equal elevation areproportional to the length of those planes, is demonstrated in fol. 179r. e diagramaccompanying this demonstration is also interesting, as it merges the representationof the principle of speed/space proportionality with the representation of the inclinedplane.


16/38


cb to represent at the same time the spaces traversed and the timeselapsed.32

Figure 7: Ms. 72 91v.

As is well known, the first published formulation of Galileos law of freefall is to be found in the Dialogue Concerning the Two Chief World Sys-tems.After having stated, in the First Day, that a falling body startingfrom rest passes through an infinite number of degrees of speed, Sal-viati states, in the Second Day, that the fact that free fall is continually

accelerated

is of no value unless one knows the ratio according to which the increase in speedtakes place, something which () was first discovered by our friend the Acade-mician, who in some of his yet unpublished writings () proves the following.e acceleration of straight motion in heavy bodies proceeds according from theodd numbers beginning from one (). is is the same as to say that the spaces

passed over by a body starting from rest have to each other the ratios of the squaresof the times in which such spaces were traversed. Or we may say that the spacespassed over are to each other as the squares of the times.33

Although Salviati assures his friend that there is a most purely math-ematical proof of this statement, he does not produce it and neitherdoes he draw a diagram representing the law. A few pages later, however,

he provides a mathematical demonstration of another proposition;namely, the Double Distance Rule, this time using a diagram (figure

32) A careful analysis of fol. 91 v and fol. 152 r has enabled scholars to reconstructhow Galileo discovered the incompatibility of the proportionality between increaseof velocity and growth of space with the law of fall. See Damerow et al., Exploringthe Limits, 180-188.33)Galileo Galilei, Dialogue Concerning the Two Chief World Systems, tr. S. Drake, 2nded. (Berkeley, 1967), 221-222 (= Opere, 7: 248).


17/38


8). Salviati considers the triangle ABC, whose side AC is divided in anynumber of equal parts AD, DE, EF and FG. He asks us to imagine the

sections marked along side AC to be equal times. Then the parallelsdrawn through the points D, E, F and G are to represent the degreesof speed, accelerated and increasing equally in equal times.

But since the acceleration is made continuously from moment to moment, andnot discretely from one time to another, and the point A is assumed as the instantof minimum speed (that is, the state of rest and the first instant of the subsequent

time AD), it is obvious that before the degree of speed DH was acquired in thetime AD, infinite others of lesser and lesser degree have been passed through. esewere achieved during the infinite instants that there are in the time DA corre-sponding to the infinite points on the line DA.34

While in folio 128 r, discussed above, Galileo inferred the continuityof acceleration from the diagram (that is to say from the fact that the

segments representing the moments of speed all terminated on the sameline), here he constructs the diagram on the assumption that accelera-tion is continuous. He assumes, in other words, that if a falling bodystarting from rest acquires a degree of speed BC in the time intervalAC, then all the degrees of speed acquired in the time AC must fit in

34) Ibid., 228-229 (= Opere, 7: 255).

Figure 8: e demonstration of the Double Distance Rule in the Dialogue (Galilei,Opere, 7: 255).


18/38


the triangle ABC. This assumption rests, in turn, on the hypothesis thattime is composed of an infinite number of instants (just like a geo-metrical line is composed of an infinite number of points), and that

the total speed of fall is composed of an infinite number of degrees ofspeed (just as a triangle is composed of an infinite number of lines).Although the diagram clearly shows that the lines HD, IE, KF and LGcut the triangle in areas that grow according to the odd numbers start-ing from one, Galileo is careful not to identify those areas directly withthe spaces traversed. Rather than referring directly to the previously

enunciated law of fall, he prefers to establish a link between space andtime by means of the total speed. For he states that whatever space istraversed by the moving body with a motion which begins from restand continues uniformly accelerating, it has consumed and made useof infinite degrees of increasing speed corresponding to the infinite lineswhich, starting from point A, are understood as drawn parallel to lineHD and to IE, KF, LG and BC.35

Galileo proceeds in his demonstration by drawing the parallelogramAMBC, which represents a uniform rectilinear motion made in thetime AC with a speed equal to the final speed BC of the uniformlyaccelerated motion. He notices that the aggregate of degrees of speedrepresented by the parallelogram

is double that of the total of the increasing speeds in the triangle, just as the par-allelogram is double the triangle. And therefore if the falling body makes use ofthe accelerated degrees of speed conforming to the triangle ABC and has passedover a certain space in a certain time, it is reasonable and probable that by mak-ing use of the uniform velocities corresponding to the parallelogram it would passwith uniform motion during the same time through double the space which itpassed with the accelerated motion.36

A comparison between the diagram of the Dialogue (fig. 8) and thatused in fol. 163v to prove the Double Distance Rule(fig. 4) makes itclear why one should prefer a temporal interpretation over a spatialinterpretation of the vertical line. While figure 4 contained two mutu-ally inconsistent representations of the space of fall (i.e. the line ab and

35) Ibid., 229 (= Opere, 7: 255).36) Ibid., 229 (= Opere, 7: 256).


19/38


the surface abc), and no representation of the time elapsed, figure 8clearly shows that the two motions under comparison take place in thesame time (AC) but cover two distances, one of which is twice as large

as the other (the area of AMBC being double the area of ACB). Anotherimportant difference between the two diagrams lies in the fact that infigure 8, the triangle ABC is subdivided into areas which visibly growaccording to the series of the odd numbers starting from 1. However,the fact that Galileo doesnt label his conclusion as certain but asreasonable and probable is a clear indication that he finds it problem-

atic to infer the relation between spaces traversed from the relationbetween total velocities.37This is why he does not explicitly comparethe spaces traversed by the falling body in the equal and successive timesAD, DE, EF, FG and GC, leaving to the careful reader the task ofintuiting the odd number law in the triangle of speed.

4. Cavalieris Imagined Diagram of Acceleration and its Influenceon Galileo

In 1632, the year in which the Dialogue came out, Galileos pupilBonaventura Cavalieri also published a book, the Specchio ustorio. Inchapter XXXIX, which is devoted to the motion of heavy bodies, Cav-alieri reports that after having learned from Galileo that the acceleration

of fall happens according to the odd number ratio, he has tried to arriveat the same conclusion through a different root. His idea was to com-pare all the degrees of speed traversed in an accelerated motion to allthe concentric circumferences that can be found in a circle, with thecenter representing the state of rest. Cavalieri however does not draw adiagram, but only imagines replacing Galileos triangle of speed with acircle of speed.

Given that it is impossible to sum up an infinite number of circumferences, I usethe area of the circle, and from that I deduce the proportions of the aggregatedspeeds, starting from the centre, which is rest, and proceeding to the outmost cir-cumference, that is to say to the maximum; for I have demonstrated in my Geom-etry that the ratio which is found between two circles is the same as the ratio which

37) Blay and Festa, Mouvement, continu, 76; Giusti, Galileo e le leggi del moto, liv.


20/38


is found between all the circumferences described around the centre of the oneand those described around the centre of the other (). And given that the areasof different circles are to each other as the squares of their respective semi-diam-

eters, and that the ratio with which the speed of a body grows is the same as theratio with which the spaces traversed grow (), the spaces traversed by the body() are to each other as the squares of the radii of the circles representing thosevelocities, and hence as the squares of the times, which we identify with the radii(). Hence the spaces traversed by a falling body in equal and successive inter-vals of time grow according to the odd numbers starting form one (). I havesaid all this in passing, which is why I have not explained myself with a figure, norwith the clarity that would be necessary, and I refer the reader to what Galileos

subtlety will explain in the work on motion that he announces in his Dialogues.38

The fact that Cavalieri refers the reader to Galileos forthcoming workon motion, means that he does not consider the Dialogue to contain aclear proof of the law of fall. Moreover, he seems to indicate that inorder to be sufficiently clear such a proof should be accompanied by adiagram, which, as we have seen, was not the case in the Dialogue.Ashas already been observed, Cavalieris proof does not solve any of theproblems implicit in Galileos demonstration. On the contrary, the factthat Cavalieri compares motions taking place in different times rendersthe equation of total speed and space traversed even more problematicthan it was the case in the Dialogue.39

But what about the diagram which Cavalieri imagines to substitute

for the triangle of speed? According to Festa and Blay the detour throughthe geometry of the circle can only find its reason in the need to use adiagram differing from Galileos: En fait, cela naura rien apport, carltude du cercle au moyen des indivisibles passe par une transformationcercle-triangle.40Blay and Festa think here of a theorem of the Geome-tria indivisibilibus, in which Cavalieri transforms the circles into tri-angles in order to prove that their areas are to each other as all theircircumferences. I think that Cavalieris choice instead finds its reasonin the need to eliminate from the diagram a vertical line which couldbe mistaken for the space of fall. As the manuscript notes analyzed above

38) Bonaventura Cavalieri, Lo specchio ustorio, overo, Trattato delle settioni coniche (Bolo-gna,1632), 159-162, translation mine.39) Giusti, A Master and His Pupils, 128.40) Blay and Festa, Mouvement, continu, 79.


21/38


clearly show, the fact that Galileo used a right triangle to represent boththe acceleration of fall and the motion along an inclined plane, madeit appear natural to interpret the vertical lines of both diagrams as the

distance traversed by a falling body. Cavalieri might also have thoughtthat the image of a circle expanding progressively from its center to theperiphery was more suited to describe the passage of a body throughinfinite degrees of speed than that of a triangle which is first drawn andthan divided, in thought, into an infinite number of parallel lines.

I am convinced that Cavalieris alternative representation of motion

left a trace in Galileos work, notably in the First Day of the Two NewSciences. In the context of a discussion about the strength of materials,Galileo introduces a digression about the paradoxes of infinity. One ofthem is the famous problem of the Rota Aristotelis (discussed in PhilippeBouliers contribution to this fascicle), which Galileo uses to providean indirect confirmation of the hypothesis that material bodies arecomposed of an infinite number of non-extended atoms, among which

infinite non-extended voids are interspersed. As I have argued else-where, the only way to understand the paradoxical atomism proposedby Galileo in the First Day of Two New Sciences is to link it to thetheory of acceleration proposed in Day Three.41 The analysis ofthe paradox enables Galileo to show, first of all, that space, time andmotion are all composed of unextended indivisibles; for given that a

circumference touches a plane in one point, during a complete rotationits infinite points will successively touch the plane without resting onit for more than one instant. The main challenge posed by the paradoxwas however to explain what happens to the internal circumferencewhen the external one accomplishes a revolution on its tangent.

Galileos answer consists in claiming that while each of the points ofline BF is touched by a point of the external circumference, only half

of the points of line CE are touched by the internal circumference, the

41) Carla Rita Palmerino, Una nuova scienza della materia per la scienza novadel moto.La discussione dei paradossi dellinfinito nella prima giornata dei Discorsi galileiani,inAtti del Convegno su Atomisme et continuum au XVIIe sicle, ed. Egidio Festa andRomano Gatto (Naples, 2000), 275-319; ead., Galileos and Gassendis Solutionsto the Rota Aristotelis Paradox. A Bridge between Matter and Motion eories, in

Medieval and Early Modern Corpuscular Matter eories, ed. Christoph Lthy, JohnE. Murdoch, and William Newman (Leiden, 2001), 381-422.


22/38


other half being left empty (cf. figure 9). Transferring his conclusionfrom geometry to physics, Galileo observes that a rarefied body, justlike the line traced out by the smaller circle, has to be considered as

being composed of an infinite number of non-extended points, part ofwhich are filled with matter and part of which are void.42

Figure 9: e Paradox of the Wheel in the Two New Sciences (Galilei, 7: 68).

Although the official goal of Galileos analysis of the paradox was toexplain the difficult problem of the rarefaction and condensation ofmatter, I believe that it was also meant to solve some difficulties linkedto the theory of acceleration. Take, for example, the case of a body thatis carried from rest on an inclined plane, and also along a vertical ofthe same height. Galileo tells us that the degrees of speed acquired inthe two descents are the same, but that the times of the movementswill be to one another as the lengths of the plane and of the vertical.43How can these claims be reconciled with the contention that a fallingbody acquires a new degree of speed in each instant of time? Galileomust have raised this problem in a letter to Bonaventura Cavalieri,which is unfortunately lost to us. In his reply Cavalieri wrote:

I will not deny that parallels intercept the same quantity of points on the perpen-dicular and on the oblique line, just as it happens with the concentriccircumfer-ences, but this does not mean that the two lines are equally long. For if we wantto compose the lines out of points, I believe that the difference between the two

42) Galileo Galilei, Two New Sciences, transl. and ed. Stillman Drake, 2nded. (Toronto,1989), 33 (= Opere, 8: 71).43) Ibid., 175 (= Opere, 8: 215)


23/38


24/38


claim that there is no motion without speed would be the same as toclaim that there is no line without length, he came to the conclusionthat

Just as it is impossible, starting from a point without length to draw a line with-out passing through infinite shorter and shorter lines, which are comprisedbetween the line and the point, so a mobile starting from rest before acquiringany degree of speed must pass through the infinite degrees of speed occurringbetween any degree of speed and infinite slowness.47

Galileo illustrated his conclusion by means of figure 10, which showshow the intersections between the angle bcand the line degrow as dedescends towardsfg.

Figure 10: Galileos letter to Carcavy, 5 June 1637: e passage of the body throughinfinite degrees of speed (Galilei, Opere, 17: 92).

What is true for lines is also true for time and motion. Just as it isimpossible to draw a line, however short, without an infinite numberof shorter and shorter lines occurring between it and point a; so too isit impossible, in the free fall of a body, to assign an interval of time,however small, without an infinite number of smaller times being com-

prised between it and the first interval, or to assign a degree of speed

47) E s come partirsi dal punto, che manca di lunghezza, non si pu entrare nellalinea senza passare per tutte le infinite linee, minori e minori, che si comprendono traqualsivoglia linea segnata e `l punto, cos il mobile che si parte dalla quiete, che nonha velocit alcuna, per conseguire qualsivoglia grado di velocit deve passare per glin-finiti gradi di tardit compresi tra qual si sia velocit e laltissima et infinita tardit,Opere, 17: 92, translation mine.


25/38


such that the falling body has not passed through an infinite numberof smaller ones.48

The letter to Carcavy is particularly important for us because it sheds

light on Galileos view about the meaning and function of diagrams.The choice of using geometrical lines to represent time and velocity isnot arbitrary, but is rooted in the idea that there is an isomorphismbetween the magnitudes being represented and the elements represent-ing them: time, speed and lines are all actually composed of an infinitenumber of indivisibles.

In the Two New Sciences, Galileo was to use not a geometrical, but aphysical argument, based on the phenomenon of percussion, to dem-onstrate that falling bodies pass through an infinite number of degreesof speed. However, the answer to Fermat is implicitly alluded to or,better said, graphically reproduced, in the First Day by means of anothergeometrical paradox: the so-called paradox of the bowl.

With the help of figure 11, Galileo demonstrates that a plane moving

from DE up to AB intersects equal areas on the cone CDE and on thebowl ADFEB. But this equality seems to disappear when the planereaches AB, for here the two always-equal solids, as well as their alwaysequal bases, finally vanishthe one pair in the circumference of a

circle, and the other pair in a single point.49The details of Galileosreasoning (which Philippe Boulier reconstructs in his contribution tothis fascicle) need not concern us here. What I want to stress is thatGalileo seems to use the diagram to show that the passage from rest

48) Ibid., 92-93.49) Galilei, Two New Sciences, 36 (= Opere, 8: 75).

Figure 11: e paradox of the bowl in the Two New Sciences (Galilei, Opere, 8: 74).


26/38


to motion is a jump from the one to the infinity. For however closethe plane may come to the line AB, and hence however small the lineCF may be, there will always be an infinite number of shorter lines.

Moreover, figure 11 can also be seen as a representation of the DoubleDistance Rule, as Galileo reminds us that the aggregate of lines con-tained in the parallelogram ABED (which can be taken to representa uniform motion) is twice the aggregate of lines contained in thetriangle DCE (representing a uniformly accelerated motion). Finally,Galileos analysis of the paradox also provides an explanation of the

puzzling fact that in the accelerated motion the first instant is repre-sented by a point, whereas in the uniform motion it is represented bya line (cf. fig. 8).

As a final comment on the paradox, Salviati observes that the infiniteis inherently incomprehensible to us, as indivisibles are likewise; so justthink what they will be when taken together! If we want to compose aline of indivisible points, we shall have to make these infinitely many,

and so it is necessary to understand simultaneously the infinite and theindivisible.50To understand simultaneously the infinite and the indivis-ible is precisely what Galileo tries to do in the Third Day of the Discorsi,where he introduces a number of new diagrams of acceleration.

6. e Triangles of Speeds in the ird Day of the DiscorsiIn Day three of the Two New Sciences Salviati reads aloud a Latin trea-tise On Local Motionwritten by his friend Academician. The sectionOnNaturally Accelerated Motion opens with a definitionwhich statesthat a uniformly accelerated motion is that which, abandoning rest,adds on to itself equal momentaof swiftness in equal timesfollowedby a postulatewhich states that the degrees of speed acquired by thesame moveable over different inclinations of planes are equal wheneverthe heights of those planes are equaland by a number propositionsthat are proven demonstratively.51

In Proposition I, Theorem I, Galileo demonstrates the so-calledmean speed theorem according to which the time in which a certain

50) Ibid., 38 (=Opere, 8: 76-77).51) Ibid., 162 (= Galilei, Opere, 8: 205).


27/38


space is traversed by a body in uniformly accelerated motion from restis equal to the time in which the same space would be traversed by thesame body with a uniform motion whose degree of speed is one-half

the final degree of speed of the previous motion. As in the Dialogue,Galileo makes use of a diagram comparing the uniform motion withthe uniformly accelerated one.

Figure 12: Galileos Representation the Mean Speed eorem (Galilei, Opere, 8:208).

In figure 12, as in figure 8 above, the vertical line represents the timeof fall, whereas the lines parallel to the base of the triangle represent thedegrees of speed acquired by the falling body. One obvious differencebetween the two figures is that in figure 12 the parallelogram is notdouble, but equal to the triangle. By using a typically Cavalerianlanguage,52Galileo observes that

If the parallels in triangle AEB are extended as far as IG, we shall have the aggre-gate of all parallels contained in the quadrilateral equal to the aggregate of thoseincluded in the triangle AEB, for those in triangle IEF are matched by those con-tained in triangle GIA, while those which are in the trapezium AIFB are common.Since each instant and all instants of time AB correspond to each point and allpoints of line AB () it appears that there are just as many momentaof speedconsumed in the accelerated motion according to the increasing parallels of tri-

52) Blay and Festa, Mouvement, continu, 94.


28/38


angle AEB, as in the equable motion according to the parallels of the parallelo-gram GB. For the deficit of momentain the first half of the accelerated motion(.) is made up by the momentarepresented by the parallels of triangle IEF. It is

therefore evident that equal spaces will be run through in the same time by thetwo moveables (); which was the proposition intended.53

It is interesting to see that this time Galileo presents his conclusion notas reasonable and probable, as he had done in the Dialogue, but asevident. This self-confidence probably stems from the fact that in thediagram of the Two New Sciences,space is no longer conflated with the

total speed, but finds an independent representation in the line CD.A fact which has so far been neglected by scholars, but which in myview is important, is that Galileo begins building his diagram preciselyfrom line CD. While in the Dialogue and in the Specchio ustorio spacewas introduced only at the end of the demonstration, in the Two NewSciences Galileo takes the line AB to represent the time in which thespace CD is traversed by a moveable in uniformly accelerated movementfrom rest A. Put differently: while in the Dialogue Galileo comparedtwo motions taking place in equal times and then established the rela-tion between the spaces traversed, here he first postulates that a bodyin uniformly accelerated motion traverses a given space in a given time,and then tries to find out which speed a body in uniform motion shouldpossess in order to traverse the same space in the same time.

The same procedure is used in the demonstration of Proposition II,Theorem II, in which Galileo proves that if a moveable descends fromrest in uniformly accelerated motion, the spaces run through in anytimes whatever are to each other as the duplicate ratio of their times;that is, are as the squares of those times.54

Also in this case, Galileo first draws the two vertical lines AB and HI

and postulates that HL is the space traversed in time AD (cf. figure 13).Subsequently he adds to the diagram the oblique line AC and thehorizontal lines OD and PE. By comparing the degrees of speed in Dand E, he manages to prove that in time AE, double of AD, the bodyshall traverse space HM, which is four times HL. From this Galileo

53) Galilei, Two New Sciences, 165-166 (= Galilei, Opere, 8: 208-209).54) Ibid., 166(= Galilei, Opere, 8: 209).


29/38


30/38


Figure 14: Sagredos representation of the odd-number-law in the Two New Sciences

(Galilei, Opere, 8: 211).

Sagredos reasoning is appreciated by Simplicio, who comments:

Really I have taken more pleasure from this simple and clear reasoning of Sagre-dos than from the (for me) more obscure demonstration of the Author, so that Iam better able to see why the matter must proceed in this way.56

It would seem as if Galileo considered the explanatory power of thediagrams as inversely proportional to the rigor of the proofs accompa-nying them: the demonstrations contained in the Latin treatise, whichare here labeled as obscure by Simplicio, engender the bipartite dia-grams of figure 12 and figure 13, in which the left part, offering arepresentation of the time/speed proportionality, appears disconnectedfrom the right part, representing the odd-number law. Sagredos reason-ing, which in Galileos eyes lacks the essential features of a rigorousdemonstration as it conflates the aggregates of speed with space tra-versed, has the advantage of producing a clear visual representation ofthe odd number law by integrating it in the triangle of speeds.

56) Ibid., 169 (= Galilei, Opere, 8:212).


31/38


7. Pierre Gassendis Version of Galileos Triangle of Speed.

As I mentioned above, Cavalieris idea to replace Galileos triangle of

acceleration with a circle was probably motivated by the need to elim-inate from the diagram a vertical line that could have been mistakenfor the time of fall. A similar choice was made by Pierre Gassendi who,in his Epistolae duae de motu impresso a motore translato,published in1642, represented the acceleration of fall by means of an isosceles tri-angle (figure 15), in which time is represented by the oblique lines AKand AL.

Figure 15: Gassendis triangle of speed (Gassendi, Opera, 3:498a).

It is important to recall, in this context, that, in the Epistolae,Gassendipresents the acceleration of fall as the result of the joint action of twoforces: the attractive force of gravity, and the impelling force of the air;both of which are supposed to impart to the body a new degree of speedin each successive moment of time; but while the vis attrahens is oper-ative from the first moment of time, the vis impellens comes into playonly in the second moment.

When I speak of first moment I mean the minimum, in which one simple ictusis impressed by attraction, and a minimum space is traversed with a simplemotion, and to which subsequently degrees of speed can be added by repeatedictus.57

57) Igitur cum primum momentum accipio, minimum intelligo, in quo unus, etsimplex ictus per attractionem imprimatur, peragaturque minimum spatium, motu


32/38


Although Gassendi defines acceleration as uniform and continuous, itis clear from his words that he conceives of it as a discrete process. Inthe lines just quoted he in fact describes the moment of time as an

extended minimum during which the body traverses a minimum ofspace with a uniform speed. This means that if only one force acted onthe falling body, the spaces traversed by it would grow not accordingto the series of the odd-numbers, but according to that of natural num-bers.58While Galileo had demonstrated that when the degrees of speedare increased in equal times according to the simple series of natural

numbers, the spaces run through in the same times undergo increasesaccording with the series of odd numbers from unity,59 Gassendibelieves that, under the action of a single accelerating force, the spacestraversed in successive moments of time would grow according to theseries of natural numbers, just like the degrees of speed themselves. Andthis is why he thinks that only a collaboration between vis attrahens andvis impellens can bring about an acceleration according to the odd-

number law.The difference between Gassendis and Galileos diagrams of accel-

eration lies, however, not only in the choice of the signifiers, but also,and more importantly, in the selection of the signified. Gassendi attri-butes no meaning to the bases of the triangles, that is to say to thesegments KO, ON, NM and ML, which Galileo has identified withthe degrees of speed. The latter are represented instead through thetriangles ADE, HKO, ILM, etc., the number of which corresponds tothe spaces traversed. In the trapezium DEFG, the central triangle rep-resents the degree of speed ADE acquired by the falling body in thefirst moment of time and preserved in the second moment, while thetwo external triangles stand for the two new degrees of speed which the

exsistente simplici, et cui deinceps accedere, ex repetitis ictibus, gradus celeritatis pos-sint, Pierre Gassendi, Opera Omnia, 6 vols. (Lyon, 1658), 3: 497b, translation mine.58) Nam fac unicam esse causam, exempli gratia attractionem; concipies quidem . . .radij magnetici . . . motum, sive impetum lapidi imprimunt . . . in primo momento,qui non deleatur, sed perseveret in secundo, in quo alius similis imprimitur . . . adeo utimpetus ex continua illa adiectione continuo increscat, motusque semper velocior fiat.Verum facile erit pervidere consequi ex hac adiectione incrementuum celeritas secun-dum unitatum seriem; nempe ita ut in primo momento sit unus velocitatis gradus, insecundo sint duo, in tertio tres, in quarto quatuor, ibid., 3: 497a, translation mine.59) Galilei, Two New Sciences,167 (= Galilei, Opere,8: 210).


33/38


vis attrahensand the vis impellenshave imparted upon the body. In thesame way, in the trapezium FGIH, the three central triangles representthe degrees of speed which the body acquires in the second moment of

time, while the two external triangles stand for the two newly acquireddegrees of speed.

In the years to come Gassendi realized, however, that his interpreta-tion of the diagram of speed stood in contradiction to Galileos theoryof accelerated motion. This happened in the course of a polemic withthe Jesuit Pierre Le Cazre, who in 1645 published a booklet, the Physica

Demonstratio,which asserted against Galileo: 1) that the speed of fallgrows in proportion with the space, rather than with time, and, 2) thatthe spaces traversed by a falling body in successive equal times growaccording to the series of ever doubling numbers, thus as 1, 2, 4, 8,etc.60Interestingly, the diagram accompanying Cazres demonstrationwas similar to the one used by Galileo in the letter to Sarpi, for theJesuit limited himself to drawing a vertical line representing space, and

noted on it the points at which new degrees of speed were acquired.In the same year, 1645, Gassendi sent a letter to Cazre, in which he

highlighted the main contradictions of the Physica Demonstratio.Whatis relevant to the present discussion is that passage of the letter in whichGassendi tried to prove that Cazres definition of naturally acceleratedmotion was incompatible with a diagrammatic representation of the

law of fall.Gassendi began by drawing the line AB, representing the space offall, and then divided it into nine equal parts AC, CD, DE, etc., at theend of each of which the body was supposed to acquire a new degreeof speed (figure 16). Now, if one represented these degrees of speed bymeans of small triangles, like ALC, CMD, LCM, and so on, one couldsee that

from C to D, the speed has not grown uniformly and with the same ratio withwhich it had begun and had continued as far as D; for if it had, it would not havedescribed the rectangleLD, which is composed of two triangles, but the trape-

60) Pierre Le Cazre, Physica demonstratio (Paris, 1645). For an analysis of Cazres theoryof acceleration, see Carla Rita Palmerino, Two Aristotelian Responses to GalileisScience of Motion: Honor Fabri and Pierre Le Cazre, in e New Science and JesuitScience, ed. Mordechai Feingold (Dodrecht, 2003), 187-227, esp. 206-208.


34/38


zium CN, which is made of three of them. For the same reason it is clear that ifthree triangles were adapted to DE, two would still be missing []. We thereforeunderstand that the degrees of speed which are missing in order to obtainthe uni-

formity of acceleration are as many as the triangles which we count on the left incompleting the sum of the triangles contained in APB. It is therefore clear thatone cannot define as uniformly accelerated a motion that acquires equal incrementsof speed in equal spaces, but instead one which acquires equal increments [of speed]in equal times.61

To paraphrase Gassendis reasoning: if the speed of free fall augmentsin a uniform manner, then it must be possible to represent the accel-

eration of a body in various intervals of space or time by means ofsimilar geometrical figures. These figures cannot be but triangles, giventhat only the areas of triangles possess the property of growing uni-formly and with the same ratio.And finally, since only the hypothesisof a direct proportionality between the degree of speed attained andthe time elapsed will result in a triangular representation of the accel-

eration of fall, the definition of Galileo must be preferred to Cazres.62

It is therefore as if Gassendi interpreted the absence of any bi-dimen-sional diagram representing the joint growth of space and speed fromCazres Physica demonstratioas a sign of the fallacy of the law of the everdoubling numbers.

By reasoning on diagram 16, Gassendi also realized that his owntheory of acceleration, as presented in the Epistolae de motu,relied on

the non-Galilean assumption that the speeds were in the same propor-tion as the spaces.63Instead of drawing a new diagram of acceleration,

61) Nihil est opus, ut desudem ad ostendendum non increvisse velocitatem aequabili-ter, eodemve tenore ex C in D, quo incoeperat, perrexeratque usque in D; ut fecissetenim, oporteret descriptum esse non quadrangulum LD constans ex duobus triangulis;

sed trapezion CN constitutum ex tribus. Eadem autem ratione manifestum est, si adDE aptentur tria triangula, defutura duo; si ad EF quatuor, defutura tria, et ita deinceps. . . ut proinde intelligamus totidem deesse ad accelerationis aequabilitatem velocitatisgradus, quot numerare licet triangulos ad laevam e regione cuiusque partis, complendosummam traingulorum APB. Constare ergo videtur Motum aequabiliter acceleratumdefiniri non posse illum Qui aequabilibus spatiis aequalia celeritatis augmenta acquirat;sed potius illum, Qui acquirat aequalia aequalibus temporibus, Gassendi, Opera, 3:567b, translation mine.62) Ibid., 3: 567b-568a.63) Ibid., 3: 621b.


35/38


36/38


37/38


of accelerated motion. In introducing his second new science, Galileoobserves that

ere is nothing wrong with inventing at pleasure some kind of motion and the-orizing about its consequent properties, in the way that some men have derivedspiral and conchoidal lines from certain motions, though nature makes no use ofthese paths (). But since nature does employ a certain kind of acceleration fordescending heavy things, we decided to look into their properties so that we mightbe sure that the definition of accelerated motion which we are about to adduceagrees with the essence of naturally accelerated motion. And at length, after con-

tinual agitation of mind, we are confident that this has been found, chiefly for thevery powerful reason that the essentials successively demonstrated by us corre-spond to, and are seen to be in agreement with, that which physical experimentsshow forth to the senses.66

In his derivation of the law of fall, Galileo allowed himself to be guidedby two principles. The first is that of the simplicity of nature, which

led him to conclude that no simpler addition and increase can bediscovered other than that which is added on always in the same way.The second principle is that of the continuity of acceleration, whichstates that speed may be increased or diminished in infinitum, accord-ing to the extension of time.

These two principles, taken together, can be translated in what Iwould call the principle of representability of acceleration. If it is true

that the speed of fall increases continuously and that in each successiveinstant of time a falling body acquires a new degree of speed, then itmust be possible to represent intervals of time and degrees of speed bymeans of proportional lines, and to represent acceleration by means ofa continuously growing surface like that of a triangle. This is why, ratherthan regarding the diagrams of motion as a static complement to

mathematical demonstrations, one should try to follow in parallel thesuccessive steps in the reasoning and in the construction of the figure.As we have seen, the fact that Galileo starts building figures 12 and 13from the line representing space is essential to understand the unfoldingof his demonstration.

66) Galilei, Two New Sciences, 153 (= Opere, 8: 197).


38/38


It would seem, moreover, as if the difficulties encountered in theconstruction of the diagrams helped Galileo and his followers to detectfallacies in their own reasoning. It is, I would argue, not too daring to

assume that Galileo abandoned the hypothesis of the proportionalitybetween the degree of velocity and the space traversed when he under-stood that it was impossible, on the basis of that hypothesis, to build adiagram representing all the essential parameters of acceleration; or thatGassendi changed his mind about the double cause of free fall when herealized that in his first diagram the representation of space was con-

flated with the representation of speed.Finally, it should have become clear that in their attempts to geom-etrize motion, Galileo, Cavalieri and Gassendi regarded their respectivediagrams of acceleration not as a conventional representation of the lawof fall, but as a way to mimic the behavior of falling bodies. We haveseen, for example, that Cavalieri compared the acceleration of a fallingbody to the expansion of a circle from the centre to the periphery, and

that he resorted to the example of two rolling circles to solve a difficultyconcerning the descent of bodies along different inclined planes. It wasalso interesting to see that when the Sagredo of the Two New Sciencesproduces his fanciful diagram showing the simultaneous growth ofspace, time and speed, Simplicio expresses his satisfaction for beingfinally able to see with clarity why acceleration must happen this way.

However, Galileos scruples about the legitimacy of identifying theaggregate of speed with the spaces traversed prevented him fromusing Sagredos diagram as an official representation of free fall.Gassendi, who did not share Galileos scruples, and quite probablybecause he did not even understand them, chose, like Sagredo, for aclear diagram, which would allow the reader to visualize the odd-number law immediately. More strongly, he interpreted the fact that

the odd number law could be inscribed into a triangle in which space,time and speed found a simultaneous representation, as the ultimateproof of its validity.

Documents

e Geometrization of Motion: Galileo’s Triangle of Speed and its Various Transformations