TheRoleofNumerical TablesinGalileoand Mersenne
DomenicoBertoloniMeli Indiana University
Numerical tables are important objects of study in arange of elds, yet they have been largely ignored by historians of science. This paper contrasts and compares ways in which numerical tables were used by Galileo and Mersenne, especially in the Dialogo andHarmonie Universelle. I argue that Galileo and Mersenne used tables in radically different ways, though rarely to present experimental data. Galileo relied on tables in his workon error theory in day three ofthe Dialogo and also used them in avery differ- ent setting in the last day of the Discorsi. In Mersenne’s case they represent an important but so far unrecognized feature ofhis notion of universal har- mony.I conclude by presenting aclassication of different ways in which ta- bles were used within the well-dened disciplinary and temporal boundaries of my research. In doing so, however,Iprovide auseful tool for extending similar investigations to broader domains.
1.Thevaried role oftables Thetopic of this workshop,“ Galileo in Paris,”provides an ideal opportu- nity for exploring anumberof themes linked to the occurrence androle of numerical tables. Aspointedout in aclassic essay byThomas Kuhn, nu- merical tables are animportant if understudiedfeature of many worksin the history of science. Of course, it wouldbe inappropriate here to at- tempt even an outline of the history of avisual tool spanningover half a millennium. Tables invarious forms probablypredate the GutenbergBi- ble, since almanacs, calendars, andprognostications were printedin all
Iwishto thankfor helpful discussions and suggestions Roger Ariew, Mario Biagioli, Jordi Cat,Dan Garber, Len Rosenband, George Smith, and T edPorter.Special thanks to Roger forhis warmhospitality and to Peter Dear for his suggestions on apreliminarydraft of this essay.
Perspectives onScience 2004,vol. 12,no. 2 ©2004by TheMassachusetts Institute of Technology
164 PerspectivesonScience165 likelihood in ephemeral publications before the 1450s.T ables come in many different forms andserve avariety of purposesin many disciplines, from accounting to optics andfrom the world ofinsurance toastronomy. Mypurposeshere are largely to dowith the mathematical disciplines and especially hydrostatics, astronomy,the science ofmotion andimmediately related elds. Ihave identied several types of tables, rangingfrom those usedas acomputer givingin convenient form the results of tedious com- putations,to those providingthe result of extensive experimental pro- grams.1 Thefollowing section deals with hydrostatics andshows the care and skill Galileo devoted inspeci c instances to handlingexperimental and numerical data. Inconnection with Galileo and,as it turnsout, Mersenne, Iexamine the workby the mathematician Marino Ghetaldi, whopub- lished celebrated tables of the weights of unitsamples of several sub- stances. Section (3) deals with Galileo’s usage of numerical tables in his refutation ofScipione Chiaramonti’s analysis of observational data of the 1572nova. Galileo’ s usage of numerical tables in astronomy was linked to arather sophisticated attempt to use error theory inorder to draw plausi- ble conclusions from aconicting set of data. Section (4) focuses onGali- leo’s presentation ofhis science of motion inthe Dialogo and Discorsi, and Mersenne’s reactions to it. Especially in Harmonie Universelle, Mersenne re- lied extensively onnumerical tables bothto examine Galileo’s claims and to present his ownviews. Lastly,Ipresent some tentative conclusions on the different roles andpurposes of numerical tables anda conceptual classication ofthe tables encountered in this work.I hopethat scholars will ndit auseful starting pointfor further research.
2.In the wakeofArc himedes’cr ownproblem The rst theme Iinvestigate is the determination ofthe weights of equal volumes of different bodies.W etendalmost automatically totalk of spe- cic weight or,to follow medieval terminology, gravitas in specie. Several mathematicians suchas Galileo andGhetaldi rigorously adoptingthe somewhat restrictive theory ofproportions,however, hadreservations in talking abouta ratio between two inhomogeneousmagnitudes, such as weight andvolume, andpreferred to avoid the medieval notionalto - gether.2 Galileo devoted one of his earliest worksto the crown problem,his 1.T.Kuhn(1977c); see also Kuhn (1977b) and Dear (1991). Relevant discussions are alsoin Dear (1988) and (1995). A relatedarea is thehistory of graphs, for which see T .L. Hankins(1990) and Campbell-Kelly (2003). 2.Napolitani(1988). 166The Roleof Numerical Tables inGalileoand Mersenne
1586 La Bilancetta, where he tried to nda more plausible methodfor solving Archimedes’crown problem thanthat providedby Vitruvius.Ga- lileo claimed that measuring the water that overowed from thecontainer, as claimed byVitruvius, was arather crude method,or “alla grossa.” Rather, hearguedthat amuchmore precise measurement of the ratio of silver to goldcould beattained bymeans of ahydrostatic balance andAr- chimedes’propositions on buoyancy .Thehydrostatic balance relies onthe difference in abody’s weight in air andin water. Fromthis difference, compared to those for bodies made ofpuregold and pure silver, one ob- tains the desired ratio of silver togoldin the crown.In order to increase the precision ofhis measurements, Galileo coiled athinwire aroundone arm of the balance. Countingthe numberof coils allowed avery accurate measure of the distance where the counterweight oughtto behung.Since the coils were too small to be easily counted,Galileo suggestedusing the soundproduced by a pointedprobe slowly sliding against them.Therefore soundwas usedto measure adistance. 3 Thusone of Galileo’s earliest worksspeaks for his considerable concerns for precision measurement. Thecontext is neither that of establishing a newtheory ,norof verifying its conclusions, butrather toprovide amore convincing philological account of howArchimedes may have solved the crown problem.Thus in LaBilancetta Galileo didnot need toprovide any numerical data. Althoughthe hydrostatic balance was nothis invention, the coiled wire was, andin all probability heusedit again. Amonghis un- publishedpapers AntonioFavaro foundtables ofexperimental data about precious metals andstones weighed in air andin water.Theprocedure for measuring the weights of equal volumes of substances appears to beparal- lel to that of La Bilancetta, thusmost likely Galileo usedthe same type of balance hehaddescribed there. Thenature ofthematerials studiedby Ga- lileo suggeststhat he hada utilitarian perspective inmind, with gold- smiths andjewelers as an obviousaudience for his work.Here wenda concrete example ofhis interest in numerical data. 4 Althoughhis tables remained unpublishedand, to myknowledge,did notcirculate, Galileo foundanother way of publicizing similar empirical data onthis topic bymeans ofthe geometric andmilitary compass. While the compass is primarily acalculation device, Galileo included data to de- termine the size ofspheres of equal weight butdifferent materials, suchas gold,lead, silver, copper,iron, tin, marble, andstone. Galileo’ s compass hada practical andutilitarian aim for arange of people includingmetal
3.Galileo(1890– 190), 1:215– 20; Drake (1978), pp. 6– 7. 4.Galileo(1890– 1909), 1:221– 8. Onexperiments in conversationwith the classics in aslightlylater period see T ribby(1991). PerspectivesonScience167
Figure 1: Galileo’s Tablesof metals andprecious stones Thistable, based on Favaro’s editionof Galileo’ s manuscript,shows the weights of differentmetals andprecious stones measured inair andwater .Fromtheir differ- enceit was possibleto determinethe weight of a unitvolume of a givensub- stance. 168The Roleof Numerical Tables inGalileoand Mersenne workers andgunners, rather thana foundational one for the newscience, yet it documentshis interest in quantitative empirical data, one Mersenne wouldhave muchappreciated. It seems appropriate in dealing with atopic like Galileo inParis to at- tempt atriangulation with another mathematician whoworked on asimi- lar topic, namely Marino Ghetaldi. There are strongreasons for mention- inghim here. Ghetaldi, agentleman from Ragusa,now Dubrovnik, was a friend andcorrespondent of Galileo, whomhe met at Padua in the circle of Gian Vincenzo Pinelli. He too hadrecourse to the hydrostatic balance, thoughhe does notappear to have usedthe coiled wire devised byGalileo. In his Promotus Archimed [e]s Ghetaldi producedcelebrated tables of the weights of several substances that Mersenne greatly admired. He repro- ducedthem in the 1623 Quaestiones in Genesim, amonumental workwhere numerical tables are largely absent,and in the 1644 Cogitata physico- mathematica. 5 Ghetaldi included several pages of tables. ThoseI showhere give the ratio between equal volumes oftwelve substances,gold, quick- silver, lead, silver, copper,iron, tin, honey ,water, wine, wax, andoil. 6 Ghetaldi didnot expand on the time, location, andcircumstances ofhis experimental work,or onthe nature of the samples examined, buthe pro- vided extensive information onthe accuracy of his procedures.He ex- plained that he hadtrouble obtainingperfect spheres,thus he worked with cylinders withthe heightequal to the diameter,andthen calculated with Archimedes the weight of the inscribed sphere.He printedthe unit lengthof the old roman foot, butin the Errata he specied that the paper 1 hadshrank, so the unithad to be increased by /40.He highlightedhis us- age of horse’s hairs tohangthe weights,since they weigh approximately as an equal volume of water.Ghetaldi relied onspecial techniques for weigh- ingbodies, such as coating apiece of goldwith athinlayer of wax to weighit inmercury ,so it does notamalgamate. He also relied onwax balls attached to lead weights,an idea with some analogies tolater usedby Galileo in adifferent context inhis disputewith the Aristotelians. 7 In conclusion, the care Ghetaldi claimed hehaddevoted to weighingproce- dureswas meant towarrant for the reliability of his tables. Butof course the tables could be read andused in different ways, as witnesses to the reli- ability of the data, as convenient unproblematic reference tools torely on, andalso as opento scrutiny andas an invitation toothers to check andim- prove onthe data.
5.Mersenne(1623), cols 1155– 6 and1159– 60; (1644), pp. 188– 92. 6.Ghetaldi(1603), pp. 32ff. 7.Ghetaldi(1603), pp. 9, 24, 34 and72. On Ghetaldi’ s worksee Napolitani (1988). PerspectivesonScience169
Figure 2: Galileo’s geometricand military compass Galileo’s geometricand military compass includesa set ofpointsfor determining thesizes ofspheres ofdifferent materials. Givena sphereof agivenmaterial, the compass enablesone to determinethe diameter ofa sphereof another material amongthose tabulated. 170The Roleof Numerical Tables inGalileoand Mersenne
Figure 3: Tablesfrom Ghetaldi’ s Promotus Archimedes Ghetaldiincluded in his work several numericaltables based on empirical data anddirect elaborations on them. The table above enables one to determinethe weightsof equalvolumes of differentsubstances. For example, from the rst col- 26 umn,the weights of equal volumes of silverand gold are intheratio of1 /31 to 100.Other tableswere especiallygeared to goldsmithsand all those interested in determiningthe ratio ofsilver to goldof analloy. PerspectivesonScience171
3.Handling conflictingastronomicald ata It is common amongGalileo scholars to contrast andcompare his method- ology andhandling of empirical data in his two principal areas ofresearch, the science of motion andastronomy .Winifred Wisan followed this ap- proach inher important studyof Galileo’s scientic methodin a paper originally delivered here in Blacksburgand subsequently published two anda half decades ago.T raditionally issues like Copernicanism, the ephemeredes ofthe Medicean planets, andthe nature andposition ofsun- spotshave attracted the lion’s share ofattention. 8 Here Ifocus onGalileo’ s handlingand presentation in tables of numeri- cal observational data onthe location of the 1572nova that appeared in Cassiopeia. Thisis the only portionof the Dialogo where Galileo hadre- course to numerical tables. Galileo tackled this problem at the beginning of the thirdday attacking Scipione Chiaramonti’s treatment of observa- tional data in Detribus novis stellis (1628),where hehadargued that the newstar was sublunar.While tryingto retrieve useful andreliable infor- mation from an array of partly contrasting data, Galileo displayed remark- able acumen andrhetorical skills in dealing with observational error. Al- thoughhe displayed similar skills andinterests elsewhere in his writings, his reections onthe 1572nova are possibly amongthe most sophisti- cated in the seventeenth century. 9 Thebrief account of the relevant por- tion of the Dialogo allows usto gaugeGalileo’ s sensitivity toobservational error. Theproblem consisted in determining the position ofthe star onthe basis ofthe contrasting observations of abouta dozen astronomers, notably whether it was located above or below the moon.Chiaramonti selected twelve couples of combinations between observations givingthe new star asufciently highparallax as to place it below the moon.Galileo’ s rebut- tal is quite complex andextends over dozens of pages ofcalculations and reections. Ofcourse, Chiaramonti hadselected amongpossible combina- tions those that suited him best,but other combinations leading to the opposite conclusion were also possible. Some combinations led to impossi- ble results, suchas placing the star very close to or inside the earth, or be- yondthe xed stars. Galileo admitted that observational errors were un- avoidable for all observations, buthe arguedthat they were more likely to besmall rather thanlarge. Forexample, if “one can modify an obviouser-
8.Wisan(1978). 9.Aslightlyearlier debate involved monetary estimations, namely if ahorseis worth 100but its value is estimatedby twopeople as 10and1,000, who commits the largest er- ror?Galileo’ s answerwas that they were equally wrong. Galileo (1890– 1909), 6:565– 612, dated1627. See also Gingerich (1973). 172The Roleof Numerical Tables inGalileoand Mersenne ror anda patent impossibility inone of [the astronomers’] observations byadding or subtractingtwo or three minutes,rendering it possible by sucha correction, thenone oughtnot adjust it byaddingor subtracting fteen, twenty,or fty minutes.”10 Galileo also arguedthat impossible values shouldnot be discarded, buthad to be corrected, addingthat “astronomers, in observingwith their instrumentsand seeking, for exam- ple, the degree ofelevation of the star above the horizon,may deviate from the truthby excess as well as bydefect, that is, erroneously deduce some- times that it is higherthan is correct, andsometimes lower.”11 Further, Galileo pointedout that the size of the mistakes in determining the star’s parallax is notproportional to the correspondingmistakes in the star’s po- sition, because tiny errors ofparallax could result in hugechanges in the star’s position,as whenparallax is small. Moreover, he attacked Chiaramonti’s presentation of data for reasons associated with orders of magnitude.In one instance Chiaramonti haddetermined astar’s distance 211 as 373,807 /4097 miles. “Now,argued Salviati/ Galileo, whenI am quite sure that what Iseek mustnecessarily differ from correctness byhundreds of miles, whyshould I vex myself with calculations lest Imiss oneinch?”12 These are quite impressive observations notfrequently foundin printat the time. Theclimax of Galileo’s attack is alengthyseries ofcalculations show- ingthe corrections required to make the twelve couples of observations se- lected byChiaramonti pointto the star’s distance from the earth being thirty-two radii. Thatvalue was selected as the closest to the moon’s orb amongthose found,or the most favorable toChiaramonti. Galileo’s calcu- lations are summarized ina table of corrections, showingthat from the sumof all the parallaxes, giving836’ , wehave tomake corrections inex- cess of756’. Bycontrast, Galileo could showthat hecould select an equal 1 numberof couples of observations requiringcorrections of 10’ /4 to make the star inthe rmament. 13 His table is based oncomputations butit re- lies ultimately onobservational data, which are subsequentlymanipulated for ostensive purposes. Galileo’s analysis reveals the care hehaddevoted toerror analysis in as- tronomyand the sophistication of his reections, byseventeenth century standards.Sagredo concludes Salviati/Galileo’s lengthydemolition of Chiaramonti’s workwith one of the most devastating passages of the
10.Galileo (1890– 1909), 7:331– 4, quotation at 314. See also Drake (1967), pp. 289–90. 11.Galileo (1890– 1909), 7:316; Drake (1967), p. 291. 12.Galileo (1890– 1909), 7:321; Drake (1967), p. 296. Galileo makes an analogous claim in the Discorsi, 8:109. 13.Galileo (1890– 1909), 7:34. PerspectivesonScience173
Figure 4: Observations ofthe1572 nova Galileotabulated twelve couple of observations of the 1572 nova, giving the par- allaxand the distance from the earth insemi diameters oftheearth. Accordingto allthe calculations the new star wouldbe sublunary ,sinceits greatest distance wouldbe thirty-two earth radii. 174The Roleof Numerical Tables inGalileoand Mersenne
Figure 5: Galileo’s tableof corrections Thisunusual table gives the corrections necessary to makeall observations agree that thenew star’ s distanceis thirty-two radii.Galileo omitted from the previous tablethe seventh value, giving exactly thirty-two radii,and also the second, giv- ingtwenty- ve radii, since the correction of only1’ 30”is negligible.His calcula- tionscontain small errors. PerspectivesonScience175
Dialogo: “Thisis as if Iwere watching some unfortunatefarmer who,after havingall his expected harvest beaten downand destroyed byatempest, goes aboutwith pallid anddowncast face, gatheringup suchpoor glean- ingsas wouldnot serve to feed achick for one day.”14
4.Testing the science ofmotion Having briey surveyed examples from hydrostatics andastronomy ,Ibe- lieve weare nowbetter placed toexamine Galileo’s presentation ofthe sci- ence of motion andhis contemporaries’reactions to it. Galileo’s manu- scripts andcorrespondence reveal his skill inanalyzing experimental data andsophistication in reducingobservational errors, yet his publishedtexts reveal little of his reections andacumen. Some ofhis Padua manuscripts andhis 1639letter to Baliani are especially signicant. 15 Ample portionsof the science of motion are discussed in the Dialogo, butthe formal presentation of the newscience occurred only inthe Discorsi, where heintroducedhis treatise in axiomatic form in Latin,inter- spersed with elucidations inItalian. Inthe Dialogo Galileo adopteda dis- cursive approach inItalian, discussingexperiments ina rather informal fashion.16 It is here, however, that hegave some numerical values that at- tracted the attention of his contemporaries. Inthe second day Galileo states that “per replicate esperienze”a ball ofone hundredpounds falls one hundred braccia in veseconds,a distance that is roughlyhalf the correct one.In this area Galileo was primarily interested in establishing propor- tions between variables suchas distances, times, andspeeds, rather than ndingnumerical values orparameters. Nosuchvalue is presented in the Discorsi. Thisis the sort ofproblem,however, that appealed to Mersenne andto the Genoese patrician Gianbattista Baliani, whoin 1632inquired with Galileo howto determine his value, since he lacked buildingsof sufcient height.At the time Galileo hadmore pressingconcerns, butin 1639,following asecond request byBaliani, hereplied that hehadused the inclined plane, thushe hadnot performed the experiment directly.As to time, with the help of some friends they countedthe numberof oscilla- tions ofapendulumof arbitrary lengthnecessary for astar to returnto its original position in twenty-four hours,and then found the numberof sec- ondsby means of the relation between periods of oscillations anddis- 14.Galileo (1890– 1909), 7:346; Drake (1967), p. 318. I havechanged chicken to chickthe Italian being pulcino. 15.Recent analyses of Galileo’s manuscriptsin volume72 oftheGalilean collection at theFlorence National Library are in P .Damerow(1992); Renn (2001). Galileo (1890–1909), 18:75– 9 and93– 5; Galileo to Baliani, August 1 andSeptember 1, 1639; Drake(1978), pp. 398– 404. 16.In the vast literature on the Dialogo seeJardine (1991). 176The Roleof Numerical Tables inGalileoand Mersenne tances. Despite givingall these details, Galileo seemed reluctant to com- mit himself to the value he hadprovided, arguing that it may need revision. Infact, ina marginal addition tohis owncopy ofthe Dialogo he didprovide amuch-improvedvalue. He hadSimplicio say that alead ball of one hundredpounds would fall more thanone hundred braccia in four pulse beats, amuchmore accurate value that wouldhave been greatly wel- come byMersenne. 17 It seems plausible that Galileo hadthe experiment tried after the 1639letter byBaliani, this time directly from ahightower as opposedto byextrapolation from an inclined plane. Galileo’s correspondence with Baliani is useful in interpreting his pub- lished data andMersenne’ s problems with them.Galileo’ s value was less thanexcellent andit was affected bya major error. Byhavingbodies roll downinclined planes, part of their speed was taken upbythe rotation. For 5 asphere the distance traveled downan inclined plane is /7 of what one may have expected byextrapolation from free fall. Theproblem does not arise, however, if one compares bodies rolling along planes with different inclinations. 18 Muchlike Baliani, Mersenne too hadtrouble with Galileo’s values. Besides the as yet unclear problem of rolling, his trouble was also dueto unitmeasures. He rst assumed that aFlorentine braccio was twenty 1 inches, thenPeiresc told him that is was 21 /2.Several years later in Rome, in alittle shopnear St.Peter’ s selling unitmeasures for the whole of Italy,Mersenne foundthat the correct value was 23inches. Suchprob- lems with unitswere of major concern to him.Even so,Galileo’ s pub- lished value was far too low andpuzzled him. 19 In1634 Mersenne publishedtwo worksdirectly related to Galileo, a Frenchtranslation of Lemecaniche anda shortpamphlet in the wake ofGa- lileo’s claims onfalling bodies inthe Dialogo, Traitédes mouvemens. Mersenne’s translation contains several additions,yet heincluded nonu- merical tables. Inthe Traité Mersenne,in aradical departure from his pre-
17.Galileo (1890– 1909), 7:250; Galileo (1890– 1909), 14:342; Baliani to Galileo,23 April1632; 18:75– 9; Galileo to Baliani, 1 August1639. For the marginal addition in Ga- lileo’s owncopy see Galileo (1998), vol. 1, p.32. 18.It haslong been a mysterywhen this problem was rstadequately conceptualized. AlthoughMersenne detected it, I suspectthat it was Huygens who became aware that the speedsof bodies of differentshapes, a ring,a cylinder,a sphere,and a prismon rollers,roll- ingdown an inclined plane were different. Huygens was also the one who found the solu- tionto theproblemof thecenter of oscillation. The two problems are related and bothrely onwhatwe call moment of inertiaof abody.SeeHuygens (1888– 1950), 19:158. 19.Mersenne (1636), p. 88; (1647), p. 192. Mersenne opened Cogitataphysico- mathematica, pp.1– 40, with an extensive discussion of unit measures of distances and weights,including the weights of coins. He compared Roman, German, Hebraic and other unitswith Parisian ones and on page2 evenestablished with the help of a microscopehow manygrains of thesand of Étaples,on the English Channel, make a Parisianfoot. PerspectivesonScience177 vious views, claimed tohave conrmed bymeans ofvery accurate experi- ments the proportionality between distances andthe square ofthe times. 1 Mersenne reported that alead ball descends 147feet in3 /2 seconds,108 feet in 3seconds,and 48 feet in 2seconds.His values were somewhat low, butbetter thanGalileo’ s publishedones. On the basis of these data hecon- structed several tables, suchas the following one. 20 Inhis tables hemade the signicant assumptionthat the proportiones- tablished is invariant underthe choice ofdifferent unitmeasures oftime. 21 Mersenne’s table is based at the very best onthree experimental data and relies onhuge extrapolations ranging,as Mersenne explains, over all the heightsand depths one could encounter.Of course, even the three experi- mental values arouse suspicion,since they showa perfect agreement down to thelast half-second andfoot. Here Mersenne was notat all interested in givingraw experimental data, butrather inproviding an ordered array of numbersloosely tied to experiment soone could determine heightsfrom the time of fall. Mersenne enjoyed presentingand discussing numerical values in general, andhe relished in particular the quasi-aesthetic pleasure of ordered columns androws. Mersenne’s brief pamphlet was later incorporated in Harmonie Universelle, asprawling treatise dealing with the problem of musical har- monybroadly conceived andcovering awide range ofsubjects, suchas the nature andproperties of soundand the features of stringinstruments. The bookis abibliographic nightmare with so many errors that it has been al- leged that notwo identical copies exist. Forthis reason it has become cus- tomary to rely onthe 1965reprint ofMersenne’s owncopy with his own annotations. 22 Thesecond book,as Mersenne explained in the introduc- tory Advertissement, is devoted to an examination of the Dialogo by Galileo, “amost excellent philosopher,”in order toestablish some principles useful in physics.23 Indeed,Mersenne scrutinized many propositionsby Galileo, bothmathematical andexperimental, suchas the problem ofextrusion of abodyon arotating earth andthe motion of pendulums.The reason for dealing with Galileo was that soundis motion andGalileo’ s Dialogo deals with motion. Readers seeking tables inMersenne’ s treatise will feel overwhelmed by
20.Dear (1988), chapters 4 and5 andpp. 136– 7. Mersenne’ s pamphlet, Traité desmouvemens, et delacheutedes corps pesans, & delaproportion de leurs differentes vitesses (1634), isextremely rare and has been reprinted in Corpus:Revue de philosophie (1986),pp. 25– 58, at 36. 21.I doubtwhether he realized the consequences of thisimplication, as several other scholarsnoticed in thesame years. This matter is explained below. 22.Mersenne (1963). For bibliographic information see pp. vii-viii. 23.Mersenne (1963), p. 84. 178The Roleof Numerical Tables inGalileoand Mersenne
Figure 6: Mersennetable of falls Mersenne’s tabledoes not provide experimental data, but rather isan extrapola- tionlinking the times offall, in half-seconds, and the distance fallen. The rst columngives the half seconds, the second column the sequence of oddnumbers, thethird column gives how many feet theball falls in eachtime interval,whereas thelast givesthe total distancefallen. PerspectivesonScience179 the richness oftheir trove. Besides the few retrieved from the 1634trea- tise, one ndsmany others dealing with several issues, includingthe prob- lem of motion.In a particularly interesting one he compared different the- oretical proportionsof fall, showingagainst hasty extrapolations that over shortdistances it is noteasy to discriminate amongthem empirically. 24 Noother place provides more revealing evidence of Mersenne’s attitude to tables thanhis criticism ofGalileo’s doctrine of falling bodies along in- clined planes. Inthe rst day ofthe Dialogo Galileo hadargued that given an inclined plane ABCwith aside CBperpendicular to the horizon,a bodytakes the same amountof time to fall along the vertical CB,as it takes to fall onthe portionCT ofthe incline from the vertex Cto the per- pendicular from Bto the hypotenuse.In all likelihood Galileo reached his conclusion from theoretical premises andextrapolations from inclined planes with low inclinations. 25 Mersenne was notone totake suchstatements at face value andexperi- mented with inclined planes of 15 o, 25o, 30o, 40o, 45o, 50o, 60o, 65o, and 75o.Two equal spheres of lead or woodwere released at the same time, while one fell vertically from aheightof vefeet, the other rolled down the inclined plane. At all elevations Mersenne foundthat the rolling sphere covered adistance noticeably shorter thanthat predicted byGali- leo. Mostvalues gave adifference of at least half afoot. However,here we donot nda table comparing expected results withexperimental data. Thiswas notone of thetasks Mersenne’s tables aimed to accomplish, yet I, andI guessmost modernreaders, wouldfeel stronglytempted to present the results of his experiment in tabular form as a“natural”layout. Indeed, while reading it Ifoundmyself constructinga table to gureout what was going on.26 In Harmonie universelle Mersenne doubtedthat Galileo hadperformed experiments withinclined planes since the ratio between rolling andfall- ingwas notas expected, butin amarginal note handwrittenin his own copy ofthe bookhe referred toGalileo’s account of the inclined plane ex- periment in Discorsi.27 Let usmove to that work. The rst two days ofthe Discorsi deal primarily withthe science ofthe
24.Mersenne (1963), p. 126. Dear (1995), p. 132,stresses Mersenne’ s concernwith ex- perience,even if itisrepeated many times. 25.Galileo (1890– 1909), 7:49– 50. 26.Mersenne (1963), pp. 111–2. For15° the sphere rolled down 12” instead of theex- pected16” ; for25° , 18”instead of 25”; for30° , 24”instead of 30”; for40° , 33”instead of 38”½ ;for45° , 36”instead of 42”; for50° , 33”instead of 46” ; for60° , 33”instead of 52” ; for65° , 36”instead of 54”; for75° , 42”instead of 58.”Dear (1995), pp. 131– 2, empha- sizesMersenne’ s narrativestyle. 27.Mersenne (1963), p. 112. 180The Roleof Numerical Tables inGalileoand Mersenne
Figure 7: Tablefrom Mersenne, Harmonie Universelle, 126 Mersenneconstructed this table as awarningagainst careless extrapolations.Sev- eral lawsof fallmay resemble eachother over ashort distance,only to divergedra- maticallylater on.Mersenne added some corrections byhand in hisown copy of the book. resistance ofmaterials, butthey also contain many digressions,including one onthe pitch ofmusical stringsin relation to their length,thickness, andtension. Unlike Mersenne,however ,Galileo didnot use numerical ta- bles, butgeometric diagrams showingthe lengthsof different strings. 28 In the Discorsi the deductive structure of Galileo’s science poses con- straints onthe role andpresentation ofexperiments. Recent workson the seventeenth century have focused onnarrative styles andconventions and onthe problems ofrelying oncontrived experiments producedat certain times andplaces tomake universal claims aboutnature. 29 However, narra- tive styles were affected byanumberof factors, suchas the specic mathe- matical discipline involved andthe particular way adiscipline was formu- lated. Ihave inmind the formulation of adiscipline inaxiomatic form
28.Galileo (1890– 1909), 8:138– 49. Mersenne’s usageof tablesto tunea stringinstru- mentis discussed below . 29. A locusclassicus isSchmitt (1969). See also, Dear (1995); Daston (1991). Of course, thespeci c featuresof how the experiment was performed, its reliability and power of per- suasionwere also highly relevant. PerspectivesonScience181
Figure 8: Fallalong an inclined plane Galileoargued that abodyfalling along the vertical CB wouldrequire the same amountof time to fallalong CT onthe incline, where BT isperpendicularto AC. However,sincea bodyon the incline CT Anormallyrotates, itwill be somewhat slowerdepending on its shapeand will not reach T. following an Archimedean pattern,as is the science of motion in Galileo’s Discorsi for example, versus alooser formulation where various results are tested andpresented piecemeal, as in the Dialogo orMersenne’s Harmonie Universelle. In the Discorsi Galileo’s science of motion was dependenton adenition anda principle orpostulate. Thede nition states that uniformly acceler- ated motion from rest is that where speeds are proportional to the times. Theprinciple states that the “gradusvelocitatis” acquired bya bodyin falling along inclined planes with different inclinations are equal when- ever the heightsof the inclined planes are equal. Galileo hadSagredo ar- guethat his “lume naturale”or innate understandingmade him accept the principle, butSalviati presented an additional experiment in its support, namely pendularoscillations. Thebob raises backto its original height even if one was to interpose anail along the vertical, forcing it tomove on adifferent path.The same wouldbe true,according to Galileo, if asphere was tofall along astraight line as onan inclined plane, rather thanan arc, as in the pendulum.In theorem 2andits corollary,Galileo provedthat distances traversed with uniformly accelerated motion are as the squares of the times, whence follows the odd-numberrule. Now follows the famous experiment with the inclined plane that attracted Mersenne’s attention in his marginal note.The experiment was meant to showthat Galileo’s mathematical construction was applicable to the real world,since falling 182The Roleof Numerical Tables inGalileoand Mersenne bodies follow the odd-numberrule. Thusthis experiment was notreally presented as havinga foundational role, since Galileo’s mathematical con- struction,as opposedto its empirical success, wouldnot be affected bya negative result. 30 It is instructive tocontrast the subsequentattitude of Galileo andhis closest associates to the postulate andthe odd-numberrule. Vincenzo Viviani pressed his master over the issue ofthe postulate until Galileo, af- ter the publication of his treatise, came upwith asolution in the form of a proof of his assumption,which was subsequentlypublished in the second edition ofthe Discorsi in 1656.As to the odd-numberrule, several mathe- maticians like Galileo himself, Torricelli, Huygens,and Le Tenneur,as well as the physician TheodoreDeschamps, defended its superiority against rival versions dueto the Jesuits LeCazre andFabri, for example, byarguing that it is invariant underthe choice of different time-units, whereas its rivals were not.Thus in bothcases there was an attempt notto provide more accurate experimental results andto present them in numer- ical tables, butrather to reduce altogether the role of experiments byin- vokingwhat wemay call aregulative orarchitectonic principle ofreason. Here the problem ofestablishing an axiomatic mathematical theory was nottied to different narrative strategy of howexperiments were per- formed, butto ndingsuitable principles. 31
30.Dear (1995), p. 127,somewhat puzzlingly claimed that the odd-number rule was “thebasic assumption” of Galileo’ s scienceof motion. This claim is nottruefor the speci c axiomaticpresentation chosen by Galileo,although here I believeDear had in mindGali- leo’s broaderconcerns rather then his speci c presentation.Similar comments apply also to p.143:“ Galileocould not achieve the requisite effect of indubitability for his rstprinci- ples,however, insofar as hecould not be sure of their universal acceptance by his expected readership.He therefore threw in his inclined plane trials ... asameansof establishingone ofhisfundamental theorems.” This statement may apply to theclaim that bodies falling alonginclined planes can rise to their original height, not to the inclined-plane experi- mentsrelevant to the odd-number rule as usedby Galileoin Discorsi. 31.Galileo (1890– 1909), pp. 214– 9. Fabri,for example, denounced the odd-number ruleand proposed instead the series of natural numbers, 1, 2, 3, 4 etc.By changing the unitof time,doubling it forexample, the odd-number rule gives the sequence 4( 1 3), 12( 5 7), 20( 9 11),etc., proportional to 1,3,5,etc.,therefore if the distance fallen inthe rsttime interval is as 1,thosefallen in subsequent intervals are as 3, 5,etc.By con- trast,changing the time units changes Fabri’ s sequencebeyond recognition. In thissense Galileo’s ruleis invariant for a changeof units of time, whereas Fabri’ s isnot. Huygens (1888–1950), 1:24– 8; Huygens to Mersenne, Leyden, 28 October 1646. Mersenne (1932–88), 12:351; Deschamps to Mersenne,1 November1643. Galileo had already out- lineda similarreasoning in his letter to Baliani of 20 February 1627, Galileo (1890–1909), 13:348– 9. T orricelli(1919), 3:326– 8, at 326– 7; T orricellito Mersenne, June1645, pp. 461– 6, at 461– 2; T orricellito Giovanni Battista Renieri, August 1647. Seealso Palmerino (1999), pp. 269– 328, at 295– 7 and319– 24 (on Le Tenneur);(2003). Archimedes, vol.6, pp.187– 227. Lenoble (1943), pp. 423– 5. Otherexamples include the PerspectivesonScience183
Figure 9: Galileo’s tablesfrom the Discorsi In Discorsi Galileoincluded tables related to parabolictrajectories. Theone above givesthe amplitude of the semi-parabola described by a bodyshot withthe same speedfor different angles of elevation. Notice that thegreatest amplitudeis for 45°, whilecomplementary angles (whose sum is 90°) givethe same amplitude. denialof perpetual motion (adopted by Stevin), Galileo’ s notionthat a pendulumbob raisesto its original height, T orricelli’s principleabout the center of gravity of combined bodieshaving to descendfor them to move,and Huygens’ relativity of motionin the in- vestigationof impact. Huygens was especially keen to adopt different versions of Torricelli’s principle.In these cases experiments had largely a privateheuristic role and a publicostensive one at best.The situation was quite different for Galileo in earlierstages oftheformulation of hisscience and for other scholars. 184The Roleof Numerical Tables inGalileoand Mersenne
Atthe endof the Discorsi and in De motu Galileo andT orricelli added numerical tables providingdata aboutparabolic trajectories with different elevations. Thosetables, however,contain noexperimental data, butre- semble standardtrigonometric tables appearing in astronomical works. Mersenne reproducedGalileo’ s tables in his 1644 Cogitata physico- mathematica. 32 Underthe narrow compass ofthis paper Ihave been able to showand discuss,albeit briey, all the tables in Galileo’s two chief publications. Whilst it is conceivable to provide acomprehensive account of numerical tables inGalileo, 33 this task is beyondmy ability for Mersenne.No man- ageable compass, however large its aperture, could have included even a representative portionof the tables in Harmonie Universelle. Thisseems to bean important andperhaps under-analyzed feature ofhis work,one re- vealing of his philosophical perspective onnature andemphasis onthe harmonyof the world.
5.Concluding reflections Mersenne’s lack of interest in axiomatic formulations andpassion for music andharmony gave hima special perspective of the mathematical disciplines. 34 In Harmonie universelle he felt the need to present atable of the weights to beattached to astringproducing the rst audible sound with aweight ofsix pounds,in order toproduceforty-two octaves, argu- ingthat the weight of the earth wouldonly sufce to reach forty-one octaves.35 ThusMersenne selected aphysically signicant value, despite the fact that one could notexperiment withit. It appears that hewas in- terested inpresenting tables covering arealm innature in afashion only partly andloosely linked to experience. Mersenne tookan aesthetic andin- tellectual pleasure bothin numbers and in their tabulations. His infatua- tion with tables was linked especially to Harmonie universelle, and to a smaller extent to La veritédes sciences, while in other worksfrom Quaestiones in Genesim to Cogitata physico-mathematica tables appear muchmore sporad- ically. In Harmonie universelle, outdoingJonathan Swift, Mersenne went as 32.Galileo (1890– 1909), 8:304– 8. Mersenne (1644), pp. 104 and 108. 33.I amnotconsidering here tables of theMedicean planets since they were presented asdiagramsrather then arrays of numbers. 34.The works of JohannesKepler are an extraordinarysource of tables,and one espe- ciallyinteresting to compare to Mersenne. Kepler titled one of hisbooks Harmonicemundi (1619).It iswellknown that the notion of harmonyhad deep theological implications for bothscholars. 35.Mersenne (1636), pp. 184– 9, at 186–7. Mersennetalks of theharmony of theseven planetsand the earth by consideringthe sound produced if theywere appended to eight equalstrings. PerspectivesonScience185 far as to present atable enabling adeaf personto tunea stringinstrument. Ironically,the table is incorrect because in talking of the grosseur of a string Mersenne shifted from its linear diameter to its cross section, buteven this quixotic attempt is revealing of the crucial role of ordered numberse- quences in understandingharmony .Thedeaf personis capable notsimply of tuningthe stringinstrument operationally ,butalso to appreciate har- monyas manifested in the table. 36 Galileo’s tables andusage ofnumerical data present different features. Galileo’s interest in the (specic) weights of different substances seems based onutilitarian aims, whereas in the case of the 1572nova observa- tional data hadmajor philosophical implications onits location. Inthe science ofmotion,however, Galileo saw proportionsas the primary math- ematical language ofnature.His science ofmotion included theorems and anumberof geometrical problems,but in this context determining nu- merical values was acuriosity devoid of deep philosophical andscienti c interest tohim. Whenwe thinkof anumerical table inthe mixed or physico- mathematical disciplines weoften have in mindone comparing theoreti- cal computations with experimental data. Suchtables were usedto argue that the discrepancies were very small, orhadan eye toexplaining second- ary effects emerging from those discrepancies. However, this is notwhat wetendto ndinthe publishedworks by Galileo andMersenne, though wemay ndit amongGalileo’ s manuscripts as aprivate heuristic device. Toremain within the science of motion,we ndextensive tables with nu- merical data of experimental results onfalling bodies with the Jesuit mathematician Gianbattista Riccioli. His enormous Almagestum novum (1651)relies extensively ontables ona remarkable range of topics and with remarkably different aims. Even within the comparatively few pages onfalling bodies,tables are usedin many ways, includingthe presentation of experimental data onbodies of various materials andsizes falling from the Asinelli tower. His data showedthat in nocase amongthe twenty-one investigated didthe spheres fall exactly at the same time. 37 Riccioli tried to provide an Aristotelian explanation for the discrepancies in the dis-
36.My favoritetable in Mersenne, La veritédes sciences (1625),is atpp. 549– 51, where one ndsthenumber of combinationsof tunesthat can be producedby an instrument with uptoftystrings. Mersenne (1636), p. 125.See also Dostrovsky (1975), pp. 169– 218, at 186–7. 37.Riccioli (1651), vol. 2, pp. 387b-388a. See also Koyré (1968), pp. 89– 117; Dear (1995),pp. 71– 85; at 82Dearreproduces one of Riccioli’s tablesshowing that the vibra- tionsof thependulum are isochronous and that a claysphere of eightounces falls in accor- dancewith the odd-number rule; on the experiment discussed in the text see pp. 83– 4; on mathematicaltables in the second half of thecentury see pp. 201– 9. 186The Roleof Numerical Tables inGalileoand Mersenne
Figure 10: Mersenne’s table,Harmonie, I 186–7 Mersenne’s remarkabletable gives the octaves producedby a stringwith different weightattached. The rst audiblesound is produced by aweightof six pounds; the rst octave requirestwenty-four pounds, and so onto theforty-second octave. Theweight of theearth isnotsuf cient to reach theforty-second octave. PerspectivesonScience187
Figure 11: Mersenne’s table,Harmonie, III 125–6 Anotherremarkable table by Mersenne.This time heprovidesdata so that evena deafmay tune a stringinstrument, based on thetension, thickness, and length of its strings.The rst tablegives the tension of strings withequal length and thick- ness. Thesecond table gives their thickness, measured as adiameter,withequal lengthand tension. The third table gives the length of strings withequal thick- ness andtension. The last tablegives their tension when both thickness and lengthare variable.Alas, the fourth table is inconsistent with the second because inthe second by thickness Mersenne means diameter,whereas thefourth makes sense onlyif thickness means cross-section. Suchare theperils of tuning instru- ments bynumericalratios alone. tances fallen bythe spheres based onthe different combinations oflight andheavy elements. After him several others,such as Boyle andNewton, hadrecourse to numerical tables to studythe properties ofcompressed air andthe resistance of uidto motion,for example. It may beuseful to attempt aclassication of the tables wehave seen thusfar, bearing in mindthat the same table may tavariety ofpurposes andbe usedby readers in different ways. The rst type presents empirical data unrelated to theory.Forexample, weights of various materials as those providedby Ghetaldi could notbe predicted andcalculated onthe basis of any theory,butcould only be measured empirically,compared, 188The Roleof Numerical Tables inGalileoand Mersenne andtabulated. Theaim of suchtables appears to have been largely practi- cal andutilitarian. Thesecond includes observational orexperimental data with theoreti- cal implications. Galileo’s analysis ofthe positions ofthe 1572nova falls into this category.Here tables were usedin various forms topresent data in asuitable form for drawingbroader conclusions. Thethird consists in calculation aids largely relying ontrigonometry . Thenumbers rely ontheory andthe table worksin away nottoo dissimi- lar from amoderncomputer. Galileo’ s tables for shootingin day four of the Discorsi andT orricelli’s in De motu fall into this category.Thiswas the most common form for astronomical tables. Thelast type seems to have adidactic, philosophical, andesthetic pur- pose at the same time. It presents tables obtained from elementary theo- retical calculations boundwithin loosely empirical ormeaningful limits, suchas the maximum heightwhence abodycan fall or the weight of the earth. Some tables showthe natural numberswith their squares,for exam- ple, orthe differences between successive squares,giving the progression of oddnumbers. Here the purposeis notto ease computation,but rather to highlightthe symmetry andregularity of certain phenomena,such as falling bodies,for example, or to ndaheightfrom the time offall, orto ponderon the regularities ofnature.It is notsurprising that suchtables gurequite prominently in aworktitled Harmonie universelle and form a characteristic feature ofMersenne’s worldview.
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