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The Role of Numerical Tables in Galileo and Mersenne 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 an important 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 in various 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 in specic 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 in order to draw plausi- ble conclusions from aconicting set of data. Section (4) focuses onGali- leo’s presentation ofhis science of motion in the 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 new theory,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 in mind,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 new science, yet it documentshis interest in quantitative empirical data, one Mersenne wouldhave muchappreciated. It seems appropriate in dealing with atopic like Galileo in Paris 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 with the 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
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