geometry, time and force in the diagrams of descartes, galileo, torricelli and newton - emily r....

7/28/2019 Geometry, Time and Force in the Diagrams of Descartes, Galileo, Torricelli and Newton - Emily R. Grosholz

1/13

Geometry, Time and Force in the Diagrams of Descartes, Galileo, Torricelli and NewtonAuthor(s): Emily R. Grosholz

Reviewed work(s):Source: PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association,Vol. 1988, Volume Two: Symposia and Invited Papers (1988), pp. 237-248Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/192887 .

Accessed: 19/05/2012 12:39

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

The University of Chicago Press and Philosophy of Science Association are collaborating with JSTOR to

digitize, preserve and extend access to PSA: Proceedings of the Biennial Meeting of the Philosophy of ScienceAssociation.

http://www.jstor.org
http://www.jstor.org/action/showPublisher?publisherCode=ucpresshttp://www.jstor.org/action/showPublisher?publisherCode=psahttp://www.jstor.org/stable/192887?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/192887?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=psahttp://www.jstor.org/action/showPublisher?publisherCode=ucpress


2/13

Geometry, Time and Force in the Diagrams ofDescartes, Galileo, Torricelli and Newton

Emily R. Grosholz

PennsylvaniaStateUniversityMathematicsplays a centralrole in thedescription,explanationandmanipulationofnaturalphenomena.To whatextent,and how andwhy mathematicsappliesto nature s aproblemthathas long occupied philosophers.Descartes,Leibniz, Kant,Mach andPoincar6,to mention some of the most distinguishednames,offer global solutions to thisproblemthatarebased on deep-lying metaphysicalassumptions.In this essay, I wouldlike to suggest an alternativeapproach,which is piecemealrather hanglobal, andhistori-cal before it is metaphysical.I want to propose,first, thatthe questionof appliedmathematicsbe recastas a ques-tion abouthow mathematicsandphysics, a physics "alwaysalready"mathematized,arepartiallyunified at variouspoints in history, n such a way thatthey can share certainitems, problemsand methods while nonethelessremainingquitedistinct.And, second, Isuggest that these unificationsmay be quite heterogeneousandvariable over time. If we

consider Archimedes'combinationof geometryandstatics,the Bernoulli'sdevelopmentof the theoryof differentialequations n the serviceof mechanics,andthe twentieth cen-turymarriageof logic andcomputer echnologyin all theirrich historicaldetail,we maydecide that the factors thatdistinguishthem arephilosophicallymoreinteresting hanthose they have in common.In short,perhapsphilosophersoughtto reasonupwards romcase studies of the multifariousways in which mathematicalandphysicaldomains can bejoined, beforethey attempt o makeglobal pronouncementsabout that union.Thepresentessay is one such case study.Its focus is theprojectof a geometricalphysicspresented n Descartes'Principlesof Philosophy,andwhich also apparently ependson hisGeometry.My arguments thatDescartes'conceptionof the"order f reasons"bothorga-nizes andimpoverisheshis mathematicsandhis physics,andmoreover nterfereswith his

own intention o unifythemin a novel andmorethoroughgoingway.ThenI will show howhis contemporariesGalileo andTorricelliprofitfrompossibilitiesthatDescartes'stronglyreductionistmethodologyhas excluded,and so manageto achieve a deeperunificationofmathematicsandphysics,specificallywithrespectto theparameters f timeand force.In a famouspassagein thePrinciples of Philosophy,PartII, section 11,Descartesannounces a kind of identitybetween the objectof geometricalstudy,space, andtheobjectof physics, res extensa,matter.He writes:"Ifwe concentrateon the idea which wehave of some body,for examplea stone, andremove from thatidea everythingwhich we

PSA 1988, Volume2, pp. 237-248Copyright? 1989 by thePhilosophyof Science Association


3/13

238know is not essential to the natureof body;we shalleasily understand hat the sameextension which constitutes the natureof body also constitutesthe natureof space, andthat these two thingsdifferonly in the way that the natureof genusor species differsfrom that of the individual." MillerandMiller 1983/4,p. 34). A physical object is thusprecisely andmerelyan instantiationof a regionof three-dimensionalEuclideanspace.And he reiterates his identificationof the subjectmatterof physics and thatof geometryquite stronglyin the last section of PartII of thePrinciples:"ForI openly acknowledgethatI know of no kind of materialsubstanceother than thatwhich can be divided, shaped,and moved in every possible way, andwhich Geometerscall quantityand take as theobjectof their demonstrations."Millerand Miller1983/4,p. 77).

Behind this conflation of physicalwithmathematicalobjects lies Descartes' desire topurify physics of the anthropomorphic,ntentionalandpsychological qualitiesandexpla-nations of late Renaissancescience, of the ironfilings which long for the loadstone andthe planetswhichkeep turning hemselves to avoid a sunburnon one side. And thereinlies the originof the austereand nobleprojectof modernscience, to know natureapartfrom the accidentsof humanperceptionandperhapseven ourconceptualcategories,toknow naturewithoutprojectinga humanface on it. Matter,accordingto Descartes,hasno attributesbesides thequantifiableones thatstem from its extendedness.The essenceof matter s thereforealso mathematical;matterhas an inherentstructurearticulableasEuclideangeometry.And since Descartes'greatmathematicalwork,the Geometry, sdesignedto reformulateandrationalizeEuclideangeometry,his projectof mathematizingnaturewould seem here to find its appropriate rounding.

Philosophicalhistoriansof the seventeenthcenturyhowever have not failed to noticethat Descartes' successes in mathematizingphysics are few andfarbetween: he enunci-ates a theorem n optics, a characterization f inertialmotion andsomething ike a con-servationof momentumprinciplefor impact.The rest of his physics, expoundedat lengthin thePrinciples, the Worldand theTreatiseof Man(forDescartes,biology was a partofphysics), is surprisinglyqualitativeandinexact; t advancesby loose analogyand a quiteimaginative arrayof "mechanisms".I would like to explainthispuzzlingincongruityat the heartof Descartes' projectas aconsequenceof the way his method leads himto organizegeometryandphysics, for hisstronglyreductionistandthereforehomogenizingway of arranginga subjectmatterimpedesthedevelopmentof his mathematicsandgeneratessevereconceptualproblemsfor his physics. And since it also leads himto conflate the domains of geometryandphysics, it ultimatelytends to block theirunification.Descartes holds that a subjectmattercan and should be organizedaccordingto "theorderof reasons,"as a linearprogression romsimplesto complexes, such thateach itemin the chain is knownwithout the aid of succeedingitems and all items are known solelyon the basis of those thatprecedethem.Thus a subjectmatterbegins with items thatareknownin themselves,and becomes a progressionof successively morecomplex entitiesthat aresimples in some kind of association. The simplesfor Cartesiangeometry,as heannounceson the firstpages of the Geometry,are rectilinear ine segments,and theirform of association is proportions Smithand Latham1954, pp. 2-5). The complexes arethenproblems(like the trisectionof theangle,or the instances of Pappus' problemdis-cussed below) andhigher algebraiccurves (like the conic sections, and some cubics),which Descartesrangesinto hierarchies n Books II andIIIof the Geometry.Descartes' choice of startingpoints,straight ine segmentswhich alone can stand astermsin relations of proportionality, elps to streamlineandreorganizegeometry.Theclosed algebraof line lengthsthatopens the Geometryallows himto use algebra n thesolution of geometricalproblems,and to define themultiplicationof line segmentsforany numberof factors,where classical Greek mathematics imited the numberof factors


4/13


5/13

240Theproblem s to findthepointsP whichsatisfythefollowingconditions.If anevennumber 2n)of lines Li aregiven in position,the ratio of theproductof the firstn of thedito theproductof theremainingn di should be equalto thegivenratioo/[P,wherea andParearbitraryine segments.If anodd number 2n- 1)of linesLi aregiven in position,the

ratioof theproductof the firstn of thedi to theproductof theremaining n-1)di times ashouldbe equalto thegivenratio at/p. (Thecase of three ines is anexception,since it aris-es when two lines coincidein the fourlineproblem.)Thereare n factpointswhichsatisfyeach suchcondition,andtheywill forma locus on theplane.Since theGreeks nterpretedtheproductsof two andthree ines respectivelyas areasandvolumes,Pappus,reportingonthe work of Apollonius,hesitated o generalizebeyondthe case of six fixed lines.In the middle of Book I, Descartesdescribeshis attackon theproblemandthenproudlyannounces,"Ibelieve thatI havein thisway completelyaccomplishedwhatPappus ells usthe ancientssoughtto do,"(Smithand Latham1954,pp.26-7) as if he had solved theprob-lem in a thoroughgoingway foranynumberof lines. While it is truethathis combinationof algebraic-arithmeticalndgeometrical esultsproducesanimportant dvance n the solu-

tion of theproblem,his treatment f theproblem n theGeometrys hardlycomplete,for headdsonly one new locus, thesolutionto a five-line version of theproblem, o thosealreadyknown,that s, theconic sections,whichcorrespondo four-lineversionsof it.Descartes'explanation f how he proposes o solve thisproblemoccurs at theend ofBook I, accompaniedby a diagramof its four lineversion(Smithand Latham1954,pp. 26-35). (Diagram2) Hechoosesy equalto BC (which s dl) andx equalto AB, and shows howall the otherdican be expressed inearly n x andy. Then theproportions efiningthe condi-tions for thecases of 2n and(2n- 1) linesgiven abovecanbe rewritten s equations n x andy. For2n lines, theequationwill be of degreeat mostn;for(2n-1)lines, it will be of degreeatmostn, but thehighestpowerof x will be at most(n-1). (For2n and(2n-1)parallel ines,wherey is the sole variable nvolved,theresult s anequationn y of degreeat mostn.)

Tsr,RE /A " B \

/ D--- '- ------------

DiagramThe point-wiseconstructionof the locus is thenundertaken s follows. One chooses avalue for y andplugs it into the equation,thusproducinganequation n one unknown,x.

For the case of 2n lines, theequation s of degreeat mostn and for (2n-1) lines, it is ofdegreeat most (n-1). The roots of this equationcan then be constructedby means ofintersectingcurves which must be decidedupon.This procedure, nfinitely iterated,gen-


6/13

241eratesthe curvepointby point (Bos 1981). Thus it seems thatPappus' problemhas beenreducedto the geometricalconstructionof rootsof equations n one unknown: he con-structionof line segmentson the basis of rationalrelationsamongother line segments.

The combinationof algebraandgeometry s worthyof note. By rewriting hecondi-tions of theproblemas anequation,Descarteshas converted t froma proportionalityinvolving lines, areasor volumes as terms(as it was in the classical formulation) o anequationabout ine segments.It has become analgebraicproblem o whichtechniques orsimplifyingandsolving equationscan be applied.Yet it has not ceased to be a geometricproblemas well, thoughthealgebraicconversion has altered he geometry.Thediagram sstill centrallypresent, hough t only involves rectilinear ine segments;no areasorvol-umes intervenehere or as thefocus of auxiliaryconstructions.The auxiliaryconstructionswill be instead the constructionof each x fora given y, usingcertainchosen constructingcurvesas well as variousgeometrical heorems.Because theproblemcan be viewedsimultaneouslyas algebraicandgeometrical,results frombothdomainscan be brought obearuponit, thusorganizingandfacilitating ts solution.Also, Descartes' abstract tate-ment of theprocedure eems to imposeno limits on the numberof lines initially "giveninposition,"andthus to escape the strictures f theGreekformulation ntirely.

And yet Descartes' solution to Pappus'problemdoes notresultdirectlyin the discov-ery andinvestigationof a richcollection of new algebraiccurves. For one thing, the con-structionof rootsof higheralgebraicequations,and thereforepointson the relevantloci,is not as easy as Descartes' naive faithin the step-wise advance of reasonenvisages. Andsecondly,the stronglyreductionistdriftof Cartesianmethodkeeps deflectinghis interestfromcurves backto nexuses of line segments.The very firstthingthat Descartessaysabout his approach o Pappus'problemmay seem oddif we expect him to be primarilyinterested n the loci which the problemgenerates(SmithandLatham1954,pp. 24-5). (Ihave correctedthe translationof the word"degre".)First,I discoveredthatif thequestionbe proposedfor only three, four,or fivelines, the requiredpointscan be foundby elementarygeometry,thatis, by the useof the ruler andcompassesonly, andtheapplicationof thoseprincipleswhich Ihave alreadyexplained,except in thecase of five parallel ines. In thiscase, andinthe cases wheretherearesix, seven, eight, or nine given lines, therequiredpointscan always be foundby means of thegeometryof solid loci, that s, by using someone of the threeconic sections.Here, again,there s an exception in the case ofnine parallel ines. For this and thecases of ten, eleven, twelve, or thirteengivenlines, therequiredpoints may be foundby means of a curve of level next higher(degreplus compose) thanthat of the conic sections. Again, thecase of thirteenparallellines must be excluded,for which, as well as for the cases of fourteen,fif-teen, sixteen, andseventeenlines, a curve of level nexthigher(degre plus com-pose) thanthe precedingmust be used;and so on indefinitely.Forin this passage, he is classifying cases of theproblemnotby some feature of thelocus generated,butratherby what kindof curvecan be chosen in thepoint-wisecon-structionof the locus, thatis, in theconstructionof the line segmentx given therelevantequationin x andy and a definite value for y. He iteratesthis classification of cases at thevery end of Book I in moreexplicitly algebraicterms.Otherwisestated,this classificato-ry scheme does not pertain o curves (describableby indeterminate quationsin twounknowns)butto problems(describableby determinate quations n one unknown).Curvesintervenein this passageonly as constructingcurves;each higherlevel of prob-lem will requirea constructingcurve of higherlevel (degreplus compose).The diagram ust given containsno hint of the locus, only the nexus of line segmentswith theirspecified relations to an arbitrary ointC of the locus. The implied auxiliaryconstructionwould be the determinationof the line segmentx (for a given value of y) by


7/13


8/13


9/13


10/13

245tides bumpinto each other,does notbring any furthermathematics nto play,whichmight illuminatethe physical situation. So far,the links between Descartes' geometryandphysics seem to be missing or trivial.And yet, having stipulated hat all materialpar-ticles are shapedvolumes and that all interaction s collision and thus covered by theseven rules,Descartes appearssatisfiedthat n principlethe work of mathematizingphysics is complete. (However, n his (forthcoming)AlanGabbey arguesthatDescartesindeedenvisaged a morecomplete physics, that would relate the precisemicroscopicphenomenaof the Principles to the macroscopicworldof machines,free fall, projectilemotion, etc., more fully andmathematically.)

What aboutthe curves thatfigurein the Geometry?Descartes discusses curves in thePrinciplesonly as thetrajectories f bits of matter temmingnot fromthenatureof matterormotion,but fromexternalexigencies imposed by theplenum:motionin aplenumcanonly takeplace, if at all, in a circuit.Andtheboundary ondition mposed by theexistenceof theplenumis not strongenoughto determinewhatpreciselythe curvemightbe, so thatthen thepeculiargeometricalpropertiesof thatcurvemightbe exploitedin the serviceofphysics, as Newton exploits thepropertiesof theellipse in PropositionXI, Book I of thePrincipia,wherehe derives the inversesquare aw (MotteandCajori1934,pp. 40-2).

Moreover,Descartes' inabilityto focus on curves as algebraic-geometric-numericalhybridscontributes o his inabilityto regardcurves as representativeof the relationsamongcontinuouslyvarying parameters.Nothingin Descartes'Principles is comparableto Galileo's famous analysisof projectilemotion(CrewanddeSalvio 1954, pp. 248-50),which takes the paraboliccurve of a projectile'strajectory o expressrelationsamongtime, distance,velocity and the accelerationof gravity,or to Newton's PropositionXI,where the elliptical trajectoryof a pointmass circlinga center of force does much thesame. The most significant employmentof curves in earlymodem physics doesn't occurto Descartes. Andof course some of the most important uch curves were transcendental,curves which he hadexcluded from mathematicsaltogether.Descartes'organization f physicsexcludes theinvestigationof acceleratedinear andcurvilinearmotion.Theproblemsconcerningcontinuouslyvaryingforceswhichpose suchthornyandfruitfulproblems orhis contemporaries ndsuccessors s simplyavoided.Significantly,Descartes' most mathematically ophisticated ttempt o quantifyphysicsoccursin a context where the temporalanddynamicdimensionsof the subjectmatterareirrelevant; pticsis veryclose to a purephysical geometry,withlightraysplayingtherolesof lines. Ina sense thenDescartesnevermakesthetransition romkinematics o dynamics,as his contemporariesGalileoand Torricelli ucceedin doing.ForDescartes,no physicalparameter,ncludingof course what he calls"force,"varies n anyessential or interestingway withtime,andstrictlyspeakingbodies never accelerate. Theaccountof theaccelera-tiondue to gravitynearthe surfaceof theearth hathis vortextheoryprovidesformacro-scopic phenomena s too complicated o be quantifiable, s Descarteshimselfadmits.)Torricelli earned from his masterGalileo the importanceof the parameter ime in theanalysisof physical situations,anddevelops a more sophisticatedaccount of percussion.He arguesthat the percussionof a falling objectexercises an infiniteforce accumulated nthe intervalof timerequired or its fall, becauseany such finite intervalcontains an infin-ity of instants, n each of which the bodyexertsthe simple impulsionof its weight; thisinfinite accumulation s extinguishedas theobject struckabsorbsthe shock in a finiteinterval of time, containinganinfinityof instantsas well. Thusequilibrium s reestab-lished. Accordingto Torricelli'sanalysis,then,each momentorequiresan instantof timefor its generationor its extinction.Thoughhe supposesthatmomentiare finite and thustheirsummation s infinite,he reasonsabout force as a continuouslyvarying magnitude,

somethinglike an integraltaken withrespectto time. Moreover,as Galileo also does onotheroccasions, Torricellistudies thecontinuousaccumulationof impulsionsas a limit-ing case of a finite numberof successive smallblows. (My exposition here is indebted to


11/13

246DeGandt(forthcoming).)This is just the kind of reasoningNewton uses in PropositionI,Book I of thePrincipia, which is Newton's versionof Kepler'sLaw of Areas (MotteandCajori1934, pp. 40-2). (Diagram6)

f eE d"! ""-.. E .. ". dF , . C

i '// ...5 XZ:?.

S ADiagramThe claim is: "The areas which revolvingbodies describeby radiidrawnto animmoveablecenter of force do lie in the same immovableplanes,andareproportional othe times in which they aredescribed."Newton'sproofis illustratedby the figure.S isthe centerof force. A body proceedson an inertialpathfromA to B in an intervaloftime;if not deflected, it would continue on in a second,equalintervalof time along thevirtualpathBc. However,Newton continues,"whenthebody is arrivedatB, supposethata centripetal orce acts at once with a great mpulse,"so thatthe body does not arriveatc, but at C. Then cC ( =BV) represents hedeflection of the body due to the force;indeed,cC = BV becomes the geometricalrepresentative f the force. TheperimeterABCDEF.. represents he trajectoryof the bodyas it is deflected at thebeginningof eachequal intervalof time by discrete andinstantaneousmpulsionsfrom S. Newton thenusesthe Euclidean theoremthattriangleswith equalbases andequalelevationshaveequalareas,to show that the area of triangleSAB = the area of triangleSBc = the area of trian-gle SBC; this equalityextends to trianglesSDC, SED, SFE... by the samereasoning,sothatequalareas are described n equaltimes.We haveonly, Newton concludes, "to let thenumberof those trianglesbe augmented,and their breadthdiminished n infinitum" orthis result to applyto a continuously actingforce and a curvedtrajectory.

Descartes' diagramof Pappus' problem s ambiguousbecause his treatmentof itallows it to representnot only a geometriclocus, butalso an algebraicequation n twounknowns.Because it can be read in bothways, bothgeometricandalgebraicresults canbe brought o bear on the problem,and theircombination s the key to Descartes' success.HereI would like to urgeaninterestingparallelbetween the partialunificationof twomathematicaldomains,and of a mathematicaldomain and a physicaldomain.


12/13


13/13

geometry, time and force in the diagrams of descartes, galileo, torricelli and newton - emily r....

Documents