Galileo’s discovery of scaling lawsMark A. Peterson Citation: Am. J. Phys. 70, 575 (2002); doi: 10.1119/1.1475329 View online: http://dx.doi.org/10.1119/1.1475329 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v70/i6 Published by the American Association of Physics Teachers Related ArticlesMicroReviews by the Book Review Editor: Why Cats Land on Their Feet and 76 Other Physical Paradoxes andPuzzles: Mark Levi Phys. Teach. 50, 447 (2012) American Modeling Teachers Association website Modelinginstruction.org Phys. Teach. 50, 446 (2012) Another wedge issue Phys. Teach. 50, 441 (2012) Pumpkin Physics Phys. Teach. 50, 434 (2012) AAPT awards Phys. Teach. 50, 392 (2012) Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

Downloaded 23 Sep 2012 to 152.3.102.242. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission

Galileo’s discovery of scaling lawsMark A. Petersona)

Department of Physics, Mount Holyoke College, South Hadley, Massachusetts 01075

~Received 26 September 2001; accepted 7 March 2002!

Galileo’s realization that nature is not scale invariant motivated his subsequent discovery of scalinglaws. His thinking is traced to two lectures he gave on the geography of Dante’s Inferno. ©2002

American Association of Physics Teachers.

@DOI: 10.1119/1.1475329#

iooi

h

eheono

ano,Ceae

icena

viayliny

ingrdbe

nrreyoiaan

lyeb

lis

-anh

ntin

ernayi-ewe

e as

alle-ion,

nousikehisall,do.thegth

ert astofay

will

-

urend

I. INTRODUCTION

Galileo’s last book was theTwo New Sciences,1 a dialoguein four days. The third and fourth days describe his solutto the long-standing problem of projectile motion, a resultobvious importance and the birth of physics as we knowBut this was only the second of his two new sciences. Wwas the first one?

Two New Sciencesbegins in the Venetian Arsenal, thshipyard of the Republic of Venice, with a discussion of teffect of scaling up or scaling down in practical constructiprojects, like shipbuilding. The discussants include twoGalileo’s closest friends, Giovanni Francesco SagredoFilippo Salviati, and an Aristotelian philosopher Simpliciwho is perhaps Galileo’s friend and sometime adversarysare Cremonini, although he is never identified by his rname. It is unlikely, but not impossible, that these three mactually met in life, as Sagredo was a nobleman of VenCremonini was professor of philosophy at Padua, aSalviati, a nobleman of Florence and patron of Galileo, mhave been Galileo’s student at Padua. The dialogue formnot intended to represent a real meeting, but rather to proa framework for developing ideas in a lively, engaging w

The conversation on the First Day wanders into a dazzvariety of topics, but on the Second Day, following this dalong digression, it returns to a serious analysis of scalespecially in the context of the strength of materials. Accoing to the publisher’s foreword, it is this topic that shouldunderstood as the first of the two new sciences.2 It was thepublisher and not Galileo who gave the book its title, aGalileo was unhappy with it, perhaps because it seems ievant to much of the ingenious speculation of the First Da3

It is clear, though, that Galileo did assign enormous imptance to the problem of scaling and the strength of materan importance that modern readers have found more thlittle puzzling.

I will show that the key to much of what is strange inTwoNew Sciencesis to be found in two rather neglected earlectures given by Galileo on the shape, location, and sizDante’s Inferno. The text of these lectures is readily availain the standard 20-volumeOpereof Galileo among the ‘‘lit-erary’’ writings in Volume 9.4 My reconstruction of its actuasignificance, which is not at all literary, is the subject of thpaper.

II. SCALING IN THE TWO NEW SCIENCES

Two New Sciencesbegins with the subject of scaling. Galileo’s observations on scaling in general are ingeniouselegant, and entirely deserving of the prominent placegives them. These ideas are basic in physics, and are iduced in most introductory physics texts under the head

575 Am. J. Phys.70 ~6!, June 2002 http://ojps.aip.org/a

Downloaded 23 Sep 2012 to 152.3.102.242. Redistribution subject to AAPT

nft.at

fd

e-ln,

dyis

de.g-,-

dl-

.r-ls,

a

ofle

de

ro-g

of dimensional analysis. We could even say that modrenormalization group methods are just our most recent wto deal with problems of scale, still recognizably in a tradtion pioneered by Galileo. It is true that Galileo didn’t havthe algebraic notation to do dimensional analysis the waydo, but his insights within a restricted arena are the samours.

An example of such an insight is ‘‘the surface of a smsolid is comparatively greater than that of a large one’’ bcause the surface goes like the square of a linear dimensbut the volume goes like the cube.5 Thus as one scales dowmacroscopic objects, forces on their surfaces like viscdrag become relatively more important, and bulk forces lweight become relatively less important. Galileo uses tidea on the First Day in the context of resistance in free fas an explanation for why similar objects of different sizenot fall exactly together, but the smaller one lags behind

Even though the idea is completely general, most ofdiscussion on scaling is directed specifically to the strenof materials. That subject is introduced immediately inTwoNew Scienceswith the assertion that large ships out of watrisk breaking under their own weight, something that is noconcern with small ships.6 The same subject occupies moof the Second Day, which is devoted to the strengthbeams. The question is put in an especially paradoxical wby Sagredo, who says,6

‘‘Now, because mechanics has its foundation in ge-ometry, where mere size cuts no figure, I do not seethat the properties of circles, triangles, cylinders,cones and other solid figures will change with theirsize. If therefore a large machine be constructed insuch a way that its parts bear to one another thesame ratio as in a smaller one, and if the smaller issufficiently strong for the purpose for which it wasdesigned, I do not see why the larger also shouldnot be@sufficiently strong# . . . .’’

This argument is remarkable. It combines faith that~1!physics ‘‘is’’ geometry, and~2! the scale invariance of thetheorems of geometry. For the purpose of this paper wecall it Sagredo’s theory of scale invariance.

Two New Sciencesmodifies Sagredo’s theory of scale invariance. Specifically, Galileo adds material properties to~1!above, while still preserving the essential geometrical natof physics. As Salviati puts it much later, on the SecoDay,7

‘‘ . . . these forces, resistances, moments, figures,etc., may be considered in the abstract, dissociatedfrom matter, or in the concrete, associated with mat-ter. Hence the properties which belong to figuresthat are merely geometrical and non-material must

575jp/ © 2002 American Association of Physics Teachers

license or copyright; see http://ajp.aapt.org/authors/copyright_permission

dattere

iseonpom

oru

rqthteate

thpona-alete

ho

onreamab

epsiarimli

ingghneag

jusaedveam

berof tb-ntarmt

d.emthejectt istion.

as

edarhisandap-lei-ietlythehe

ies,

-ctrs,

rsy

ish to

ory

te-ow-leo

f aesswe

eoter,ssnya ofhanleo

tois

ur-ling

urss

-

be modified when we fill these figures with matterand therefore give them weight.’’

These modifications break scale invariance, but theynot change the essence of the theory: ‘‘Since I assume mto be unchangeable and always the same, it is clear thaare no less able to treat this constant and invariable propin a rigid manner than if it belonged to simple and pumathematics.’’8

A typical example of the new considerations that arwhen ‘‘we fill these figures with matter’’ starts with thbreaking of a wooden beam. If the beam breaks, all its lgitudinal fibers break, and because their number is protional to the cross-sectional area of the beam, the beastrength in withstanding a longitudinal pull should be proptional to its cross-sectional area. If a supported beam is sject to a transverse force, however, the associated tomust be balanced by the first moment of the stress infibers, which introduces an additional factor of the diameof the beam.9 Thus the beam’s strength in withstandingtransverse force should be proportional to its diamecubed.10 If the beam is made longer, the torque due tosame transverse force applied at a geometrically similarsition is proportionately greater and the beam is proportiately weaker, and so forth. There follow Galileo’s very fmous observations on why animals cannot be simply scup, but rather their bones must become proportionathicker as they get larger.11

All of this discussion is developed in a series of eigpropositions with geometrical proofs. It is clearly the gemetrical framework which makes these scaling laws‘‘science.’’12 After the eight propositions and the discussion animals, certain more detailed questions are considesuch as the problem of locating the weakest point in a bethat is, where it would break, and the problem of findingbeam that would have no unique weakest point, but wouldequally strong everywhere. These last topics are clearly srate from the original scaling theory of the eight propotions. They come after it, as applications of it, and theynot numbered, as if they had been added at a later tThese topics arise naturally when one juxtaposes the scatheory with Aristotle’sQuestions on Mechanics,13 as Galileohimself points out.14

A persistent oddity on the Second Day is a continupreoccupation with beams that break under their own weithe same phenomenon that began the whole discussion iVenetian Arsenal, except that there it was ships that brunder their own weight. Another oddity is the sudden chanof subject on the First Day. The conversation, which hadbarely gotten under way, is about the strength of ropes, mof fibers, but the question of nonfibrous materials is raissuch as metal or stone. Simplicio wants to know what gisuch materials their strength. Just before this, and perhintended to motivate it, we had read about a marble coluthat broke under its own weight.15 Although Salviati hesi-tates to take time to discuss this, Sagredo says, ‘‘But if,digressions, we can reach new truth, what harm is thermaking one now . . . ?’’16 and the conversation goes off fothe rest of the day into speculations about the atomic theof matter, cohesive forces among atoms, and questions oinfinitesimally small and infinitely large, among other sujects. The discussion is diffuse but occasionally brilliaeven if nothing definitive can be said about what holds mter together. The opening lines of the Second Day confihowever, that the intention is to apply the scaling theory

576 Am. J. Phys., Vol. 70, No. 6, June 2002

Downloaded 23 Sep 2012 to 152.3.102.242. Redistribution subject to AAPT

oterwerty

e

-r-’s

-b-ueer

reo--

dly

t-a

d,,

ea-

-ee.ng

t,theket

de,spsn

yin

ryhe

,t-,

o

all solid materials and not just to fibrous materials like wooTaking everything together, there is an odd hint of a problin the background that motivates all the discussions offirst two days, but that is never actually stated: some obof stone or metal that breaks under its own weight when iscaled up. This unspoken problem is what the conversaseems to be circling around, without ever quite saying so

In seeking to understand what theTwo New Sciencesisreally about, we recall how it came to be written. Galileo wremarkably reticent about publishing—before theStarryMessenger17 of 1610, the year he turned 46, he had publishhardly a thing. His astronomical discoveries of that yebrought him instant fame, and he parlayed this fame intoappointment as Mathematician and Philosopher to the GrDuke of Tuscany. In the negotiations leading up to thatpointment, he described the many books that he wantedsure to write, based on the research he had been qudoing at Padua, but these books did not appear. His life atTuscan court was fraught with controversy, and the booksactually wrote were reactions to events and new discoverthe Discourse on Bodies in Water,18 and the SunspotLetters.17 When the Holy Office declared the opinion of Copernicus formally heretical in 1616, Galileo’s fondest projewas closed off to him. He silenced himself for several yeaand when he finally wrote again~The Assayer!,17 after muchpressuring from his friends, it was part of a testy controvethat made him new enemies. The debacle over his greatDia-logue Concerning the Two Principal World Systems19 and thedisastrous trial of 163320 should have been the end of hcareer. Yet in his old age he somehow found the strengtbegin writing what would become theTwo New Sciences.And from somewhere in his past he drew the scaling theand placed it first.

Just where the scaling theory comes from is a bit mysrious, because there are few surviving references to it. Hever, it was certainly complete by 1612, the year Galipublished theDiscourse on Bodies in Water.18 The most vex-ing problem that Galileo deals with there is the problem othin lamina of material denser than water which nonethelfloats on the surface, supported by surface tension, aswould say now. At the very end of the Discourse, Galilpoints out that if the lamina were supported at its perimea thin sample~say of square shape and fixed thickne!would be more easily supported if it were cut into masmaller squares, because in reducing the size, the areeach piece, and hence its weight, would go down faster tthe perimeter, which is the source of its support. Galipresents this argument purely hypothetically, as contraryfact, because his own idea of what supports the laminaquite different. Peculiar details aside, it suffices for our ppose that this reasoning is an explicit use of the scatheory as early as 1612.

An even earlier reference, although not as explicit, occin a letter from Galileo to Antonio de’Medici in 1609. Thiremarkable letter is, in effect, a one-page outline of theTwoNew Sciences, almost thirty years before it was printed! Galileo describes his most recent investigations, saying21

‘‘And just lately I have succeeded in finding all theconclusions, with their proofs, pertaining to forcesand resistances of pieces of wood of variouslengths, sizes, and shapes, and by how much theywould be weaker in the middle than at the ends, andhow much more weight they can sustain if the

576Mark A. Peterson

license or copyright; see http://ajp.aapt.org/authors/copyright_permission

osi

tit

sethliea

l-n

ry

hyith

otema

nth

he

neaerha

di-. Hpein

rdd

thinr

thro

fi-tiotha

w

ec

ce.of

t an oflyfy

th-aedm-m.

inndivalbyow

an-moreheettillo,

a

cerntheace,r of

theof

lyrth

vel,thes a’s

weight were distributed over the whole rather thanconcentrated at one place, and what shape woodshould have in order to be equally strong every-where: which science is very necessary in makingmachines and all kinds of buildings, and which hasnever been treated before by anyone.’’

He then goes on to describe certain insights into the mtion of projectiles. This letter makes it clear that the baorganization ofTwo New Scienceswas already in his mind in1609, and even seems to say that the results were at thatquite fresh. On closer inspection, though, one sees thatdetailed results described in the letter belong to the lasttion of the Second Day, building on the basic results offirst eight propositions, which must therefore be even earmaterial. That the projectile theory was still incompletethis point, and also comes later inTwo New Sciences, sug-gests~which is plausible in any case! that the order of treat-ment inTwo New Sciencesis chronological. That is, the scaing theory comes first there because it literally was first, awe will have to look earlier to find it.

The next suggestion for the origin of the scaling theomight come fromTwo New Sciencesitself. The openingscene in the Venetian Arsenal is peculiarly effective—wnot take it at face value? Galileo was obviously familiar wthe Arsenal, and it is known that he was consulted thereship design. What likelier place could there be to encounproblems of scale, in just the way that he says? This assution could place the origin of the scaling theory as early1592, the year he became Professor of MathematicsPadua. On closer reading, though, one sees that eveSagredo is surprised by the failure of scale invariance inshipyard, Salviati is not. He already understands it, andimmediately ready to explain it all to Sagredo. What tscene really says is that Salviati’s mind~read Galileo’s! wasalready prepared to understand what he saw in the Arsebecause he had already done the scaling theory. This mwe must search for the scaling theory still earlier, at the vbeginning of Galileo’s career, or perhaps even earlier tthat.

III. THE LECTURES ON DANTE’S INFERNO

Galileo enrolled at the University of Pisa to study mecine at the age of 17, and dropped out at the age of 21spent the next several years studying mathematics indedently, especially Euclid and Archimedes, and did tutoringmathematics. While living at home he assisted his fatheremarkable experiments on the pitch of plucked strings unknown tension.22 Fifty years later he briefly summarizetheir experimental results inTwo New Sciences, the FirstDay.23 He proved some theorems on centers of gravity instyle of Archimedes, and at the end of his life he alsocluded those inTwo New Sciences, as an appendix. It is cleathat Two New Sciencescontains some very early material.

The young Galileo hoped to make a reputation in maematics with his theorems, and he sent them to a numbeItalian mathematicians. He was fortunate to receive a favable reply from Guidobaldo del Monte, Inspector of Fortications to the Grand Duke of Tuscany, someone in a posito help him.24 They corresponded. When the chair of maematics at Pisa became open, someone, probably Guidobbut perhaps his even more illustrious brother Francesco,had just been made a cardinal,25 arranged for Galileo to beinvited to address the Florentine Academy to give two l

577 Am. J. Phys., Vol. 70, No. 6, June 2002

Downloaded 23 Sep 2012 to 152.3.102.242. Redistribution subject to AAPT

-c

mehec-er

t

d

nrp-satif

eis

al,ns

yn

en-

iner

e-

-ofr-

n-ldoho

-

tures on mathematical topics.4 It was in effect a seminar andjob interview. Galileo’s lectures charmed his audienWithin a few months the young dropout was ProfessorMathematics at Pisa.

Galileo’s audience at the Florentine Academy was nomathematical one. The Florentine Academy was a creatiothe Medici dynasty~which had ascended to the nobility onin the immediately previous generation!, and had as one oits chief functions the glorification of the Medici in everintellectual arena.26 It was far more important for Galileo toplay to this predilection of his audience than to display maematical erudition. In the event he brilliantly combinedclear exposition of mathematics with a topic Florence lovto hear, their great poet Dante, and in particular, the geoetry of Dante’s Inferno, based on evidence from the poeAlthough Galileo did introduce certain original materialhis lecture, he did not call too much attention to it, arepresented himself rather as describing two previous rattempts to determine the plan of Hell. One of these wasAntonio Manetti, who was represented as a member ndeceased of the Florentine Academy itself.27 The other plan,a later attempt, was by Alessandro Vellutello, notFlorentine.28 Galileo begins in a way that seems evehanded, but as the second lecture proceeds, he becomesand more sarcastic about Vellutello’s plan until in the endseems to be defending the virtuous Florentine Managainst the ridicule of the stupid and thoughtless Velluteto the delight of his audience, no doubt. This was howMedici intellectual should defend the honor of Florence!

Only the large scale features of these plans need conus here. Manetti’s Inferno is a cone-shaped region inearth, with its vertex at the center and its base on the surfcentered on Jerusalem. But because Galileo is a masteexposition, let him describe it:

‘‘ . . . imagine a straight line which comes from thecenter of the earth~which is also the center ofheaviness and of the Universe! to Jerusalem, and anarc which extends from Jerusalem over the surfaceof the water and the earth together to a twelfth partof its greatest circumference: such an arc will ter-minate with one of its extremities on Jerusalem;from the other let a second straight line be drawn tothe center of the earth, and we will have a sector ofa circle, contained by the two lines which comefrom the center and the said arc; let us imagine,then, that the line which joins Jerusalem to the cen-ter staying fixed, the other line and the arc should bemoved in a circle, and that in such motion it shouldgo cutting the earth, and move itself until it returnsto where it started. There will be cut from the eartha part like a cone; which, if we imagine it to betaken out of the earth, there will remain, in the placewhere it was, a hole in the form of a conical surface,and this is the Inferno.’’

As an aside to the mathematically adept, Galileo gavevolume of this region, which he knew from his studyArchimedes.

The various levels of Manetti’s Inferno are regularspaced, for the most part, with 1/8 the radius of the eabetween each level and the next. In particular the first leLimbo, is at a depth of 1/8 the radius of the earth belowsurface, and the shell of material down to this depth formcap of this thickness over the whole of Hell. Vellutello

577Mark A. Peterson

license or copyright; see http://ajp.aapt.org/authors/copyright_permission

nt

ly

20is

ngtththr

sual-yheu

oInn

yt,

le

off

iond

t t

togn,

easone-ad

tyw

lec-

whatreseanaloun-tore

sisis.heranntButhad

er-eren iseldno

theent

dl heon

ro-rin-oats.to-thatand

ut

ats,

heyingm-longad

notHehe-

es’d,. He, hede-

Inferno, by contrast, is much smaller, located near the ceof the earth, and only about 1/10 the radius of the earthheight, making it, as Galileo is quick to say, ridiculoussmall, only 1/1000 the volume of Manetti’s.

Near the end of his presentation, Galileo says

‘‘Here one might oppose that the Inferno cannot beso large as Manetti makes it, since, as some havesuspected, it doesn’t seem possible that the vaultthat covers the Inferno could support itself and notfall into the hole, being so thin, as is necessary if theInferno comes up so high. And especially, beyondbeing no thicker than the eighth part of the radius ofthe earth, which is 405 miles more or less, some ofit must be removed for the space of the Grotto ofthe Uncommitted, and even more must be removed@on the top# for the very great depth of the sea. Tothis one answers easily that such a thickness is morethan sufficient; for taking a little vault which willhave an arch of 30 braccias, it will need a thicknessof about 4 braccias, which not only is enough, buteven if you used just 1 braccia to make an arch of30 braccias, and perhaps just 1/2, and not 4, itwould be enough to support itself; and knowing thatthe depth of the sea is a very few miles, or better,even less than one mile, if we believe the most ex-pert sailors, and assigning as many miles as seemnecessary for the Grotto of the Uncommitted, a de-terminate measure not being given by the Poet, ifthis together with the depth of the sea comes to 100miles, the said vault will still be very thick, and farmore than is necessary to hold itself up.’’

Because in Galileo’s units the earth’s radius is about 3miles, and 1/8 of that is 400 miles, it is clear that hedescribing a scale model of the roof of the Inferno, includia certain anteroom hollowed out of it, at a scale of aboubraccia to 100 miles. A normal man is 3 braccias tall, somodel suggests a large domed roof, somewhat smallerthe famous Brunelleschi dome of the Florentine cathedwhich, as Galileo says, is less than 4 braccias thick andports itself beautifully. This is a convincing argument thManetti’s model can support itself—but only until you reaize that the argument assumes scale invariance! Couldreally scale it up by a factor of 100,000? Absolutely not! Tscaled up version is effectively weaker by that enormofactor and would immediately collapse of its own weight.

IV. DISCUSSION

When Galileo realized his mistake, probably just a shtime later, it must have struck him like a lightning bolt.fact, this is just how Sagredo reacts, in strangely emotiolanguage, at the beginning ofTwo New ScienceswhereSalviati asserts that nature isnot scale invariant: ‘‘My brain. . . reels. My mind, like a cloud momentarily illuminated ba lightning-flash, is for an instant filled with an unusual lighwhich now beckons to me and which now suddenly mingand obscures strange, crude ideas . . . .’’29

We need look no further to know why the problemscaling and the strength of materials had urgent meaningGalileo. There is nothing hypothetical about this suggestIt is clear in the record, with no ambiguity at all. He hamade a gigantic blunder in the Inferno lectures, sufficien

578 Am. J. Phys., Vol. 70, No. 6, June 2002

Downloaded 23 Sep 2012 to 152.3.102.242. Redistribution subject to AAPT

erin

0

1eanalp-t

ou

s

rt

al

s

or.

o

turn his whole argument on its head, and with it his claimbe an intellectual champion of his country and his sovereion whom his young career depended.

It may be difficult, from a modern point of view, to takSagredo’s overwrought words seriously. Yet the situation wserious enough, apparently, to force Galileo to concentratethe scaling problem and develop a new theory almost immdiately, to judge from numerous internal clues. Then, insteof publishing the theory, he kept it to himself for almost fifyears! This delay is baffling, if we ask ourselves only howe might have acted. Rather, we must ask how an inteltual of Galileo’s time would have acted and why.

The lectures themselves were evidently a success, sowas the crisis? Here we note that the context of the lectuwas a dispute, a ‘‘duel,’’ between Manetti and Vellutello. Thlectures were actually a savage attack on Vellutello. Toextent that it is difficult to appreciate now, an intellectucareer at a Renaissance court consisted in attacks and cterattacks, which often functioned as a kind of spectasport.30 This kind of ongoing dispute, with reputations on thline, was the very stuff of Italian intellectual life. Galileo’own career offers many examples. And here was the crNothing would be more natural than to anticipate anotround in this dispute, a reply to Galileo from some partisof Vellutello. Galileo should expect to be attacked—it wewithout saying—and he should have a good reply ready.his position on scaling in the lectures was untenable. Heto develop a new position.

In this culture of attack and counterattack, it made a ctain kind of sense to keep results secret until they wneeded. The most famous example of this phenomenoprobably the solution of the cubic equation, which was hsecret for decades until it was finally published by Cardain 1545.31 In the same way, Galileo may have developedscaling theory and then not published it because the momnever actually arrived when he needed it.

It is not pure conjecture that Galileo would have followesuch a strategy, keeping secret an important result untineeded it for self-defense. Indeed, in the controversyfloating bodies we see him doing exactly this. This contversy began when Galileo asserted, by Archimedes’ pciple, that ice must be less dense than water, because it flHis Aristotelian opponents asserted that according to Aristle, ice, being cold, must be more dense than water, andice floats, despite being more dense, because it is broadflat. Galileo quickly disposed of this foolish argument, bthen his opponents found an example of something~a chip ofebony! that really is denser than water and nonetheless floif you carefully put it on the water’s surface~because it isbroad and flat, they said!. At this point neither side in thecontroversy had a satisfactory understanding of what twere seeing. Galileo’s slightly odd response was to brforward a method to prove Archimedes’ principle by geoetry, using some mechanical ideas of balance that hadfascinated him. This theory was clearly something he hkept ready for just such a moment, even though it didreally address the problem of the floating ebony chip.made some new observations of this anomalous floating pnomenon~which we attribute mainly to surface tension! andstretched his geometrical theory to argue that Archimedprinciple could entirely account for it. This strategy workein the sense that he argued his opponents to a standstillnever convinced them, but that was not the point. Ratherhad used his hitherto secret knowledge in a successful

578Mark A. Peterson

license or copyright; see http://ajp.aapt.org/authors/copyright_permission

linsit

hengeegnnugdathtorv

intefohe

tin

arn-

ecarslivliteo

otrehais

heo

as.

in

eo

ahlethn

avtutioab

rno

thro

t

sto,hedgi-s

stoonlystoionthatuldap-m-e.ry

stillin-’s

rac-nc-toan

Ga-utorkim-sionfo-sec-t toti-

ns-arf itsot

oadbleisl

tond-im,m-

d no

ac-and

iffi-noe! It

doita-siblea-

heverare

fense. As we noted earlier, Galileo also used the scatheory defensively in this controversy, right at the end, aclincher, so to speak.32 Thus it is plausible that he had keptfor just such a purpose.

When Galileo finally published the scaling theory at tend of his life, he hints at the story of its discovery, includihis original naivete about scale invariance. We have show Sagredo endorses scale invariance at the beginninTwo New Sciences, but in fact all three participants do so. Othe Second Day, Simplicio, who is often slow to catch osuddenly endorses scale invariance, as if he had just thoof it, and Sagredo has to point out that he had said it thebefore, and this is what they have been talking about. Atpoint Salviati, who is Galileo’s spokesman, says that hehad assumed scale invariance once, until various obsetions showed him the contrary.33 In this way Galileo is com-mendably frank, truthfully representing his former opinionthe Inferno lecture. On the other hand, he never mentionslecture itself, which would still have been awkward for thhonor of the Florentine Academy. No one can blame himthis. Quite possibly by 1638 there was no one else alive weven remembered this detail of the original lecture, and thwas no real need to recount this story. Thus the motivaproblem ofTwo New Sciences, the collapse of Manetti’s In-ferno, is discreetly avoided, and another one, not particulconvincing, a ship that falls apart of its own weight, is ivented in its place.

In fact, Galileo seems to have regarded the Inferno ltures as an embarrassment, and this may be another rewhy he postponed publishing the scaling theory. His fibiographer, Viviani, was unaware of the lectures, despiteing in Galileo’s house during his last years, collecting Galeo’s stories, and devoting himself to Galileo’s memory afhis death. Because Viviani is the source of most stories abGalileo, subsequent biographies have also said little or ning about the Inferno lectures, which were only rediscovein the 19th century. Thus one can plausibly conclude tGalileo never told Viviani about them. This circumstancerather odd when you think that they were very likely tpivotal opportunity that got his career started. A letter1594 from Luigi Alamanni also indicates that Galileo wunhelpful to someone who wanted to get a copy of them34

The letter says, ‘‘About that lecture of Galileo, he isPadua, and I have not been able to get it from him.’’35 Therewere copies with Bacio Valori in Florence, including onwritten out in Galileo’s own hand, but he apparently did nvolunteer this information.

There is another hint that Galileo’s scaling theory woriginally planned for defensive purposes. In working out tscaling theory, Galileo would have noticed a second probin the Inferno lectures, besides the inevitable collapse ofroof. In the first lecture the dimensions of the lowest regioare determined by comparison with certain giants who hbeen placed there by Dante, embedded in ice. In the lecGalileo assumed that these giants have the same proporas normal men, and his only hesitation on this point wwhether giants have the ideal human proportions favoredartists like Albrecht Du¨rer, who wrote on the subject, owhether they have proportions more like ordinary meManetti and Vellutello had differed on this point, and Galilefavored Manetti, of course. But it is a consequence ofscaling theory that giants couldn’t have either of these pportions. If we look inTwo New Sciencesin the section onscaling in animals,36 we find that in fact the animal of mos

579 Am. J. Phys., Vol. 70, No. 6, June 2002

Downloaded 23 Sep 2012 to 152.3.102.242. Redistribution subject to AAPT

ga

nof

,hty

isoa-

he

roreg

ly

-sont-

-ruth-dt

f

t

semeserenssy

.

e-

concern is man: Galileo is concerned aboutgiant men. Heeven quotes a poet describing a giant, although it is Arionot Dante, and he suggests, although it is a rather far-fetcinterpretation of the words, that Ariosto understands thatants would be misshapen.36 This occurrence of poetic giantand their proportions inTwo New Sciencesis entirely to beexpected if the context is the Inferno lectures. The Ariolines make sense as part of a prepared defense: notcould he expound the scaling theory, he could invoke Arioas a poet who understands it. Although this literary allusmight not impress a scientific audience, the audiencemattered was a courtly audience. And although they wonot understand the geometry, they would undoubtedlyplaud this trick, just as they had applauded his skillful cobination of Dante and geometry in the original lecturNearly fifty years later, as he finally wrote the scaling theodown for a very different audience, these associationspersist. This is speculation, of course, but the reader isvited to find a more plausible explanation for what Ariostogiant is doing inTwo New Sciences.

The setting in the Venetian Arsenal seems to promise ptical science, and the scene is so effective that it has futioned as a credential for Galileo ever since, showing himbe a practical man of science, right at home with the foremon the shop floor. There can be no doubt whatever thatlileo really was an ingenious and skillful experimentalist, bthis opening scene fails to suggest how theoretical the won scaling actually was. The Arsenal fades away almostmediately on the First Day, never to reappear. The discuson the Second Day, when the subject finally comes intocus, is about beams with rectangular and circular crosstion because the aim is geometrical proof. And the resulwhich the whole development is tending is entirely theorecal, and is given no practical illustration, namely thatany-thing would break of its own weight if it were sufficientlyscaled up. This is the content of the culminating PropositioVII and VIII on the Second Day. Proposition VII, for example, is ‘‘Among heavy prisms and cylinders of similfigure, there is one and only one which under the stress oown weight lies just on the limit between breaking and nbreaking: so that every larger one is unable to carry the lof its own weight and breaks; while every smaller one is ato withstand some additional force tending to break it.’’ Th17th century version of ‘‘the inevitability of gravitationacollapse’’ is what the Second Day is really about.

It is noteworthy that Galileo describes no experimentsverify the scaling theory, although experiments on the being of a loaded beam would have been very simple for hand entirely characteristic of his Padua years. This circustance suggests that the scaling theory is early, and hapractical motivation.

The Inferno lectures begin by praising the skill and audity of those discoverers who have measured the heavensthe surface of the earth, and point out how much more dcult it is to know the earth’s interior, where it seems thatone can go and return—yet even this our Dante had donis an implied theme of the lectures that the geometer canthis too: by the use of geometry we see beyond the limtions of our senses, reasoning about otherwise inaccesthings. This insight is, in a way, the heart of physics as Glileo came to understand it. Yet in the Inferno lectures tidea occurs almost accidentally, as part of a glib and cleentertainment. The geometry is merely descriptive. Thereno actual discoveries.

579Mark A. Peterson

license or copyright; see http://ajp.aapt.org/authors/copyright_permission

ly

heethsohrfeahtr

b-teemalic

t htnavg

-ch

.’sucroaaric

teienk

fu

o.

m-a‘‘a

th

to

du/

re to

thatdiam-

,

,

theof

thateingd at

ded

dent

nioteden-ther,

f. 1,

It is entirely different with the scaling theory. Suddengeometry reallydid give the key to new discovery,37 a dis-covery that Galileo was still proud to claim as his own at tend of his amazing career. This new insight by way of gometry must have come as a wonderful surprise. For allfaith that Galileo and a few other mathematical enthusiahad in geometry, it was, in practice, the everyday toolartisans and merchants, not the source of new insigArchimedes, to be sure, had used geometry in this wondeway, but no one imagined there could be anothArchimedes. With the scaling theory, however, Galileo hsomething truly new, worthy of comparison witArchimedes, something that validated his faith in geomeand hinted at undreamed of successes to come.

Looked at this way, Galileo’s lifelong reluctance to pulish seems even more inexplicable, but perhaps this patbegan with the experience of the Inferno lectures. He seto have done his best to make people forget the lectures,he kept the scaling theory to himself. What he made pubat least in this case, was a source of trouble, while whakept secret was a source of confidence. The unpleasanof being vulnerable to attack is a lesson that he might htaken to heart then, and it is a view he expresses feelinlater on, on the basis of real experience~although withoutadmitting vulnerability!, in the opening lines ofTheAssayer.17 Galileo frequently claims to have wonderful results that he has not yet revealed, things he has not yetsen to disclose. We know that this was true through muchhis career, and apparently it was true right from the start

Finally, it is an irony that the first success of Galileomathematical physics, which is close to being the first scess of mathematical physics at all, was a response to a plem that was not physical, but rather the collapse ofimaginary structure in a work of literature. This peculistory, while probably not of much use in a standard physclass, has great appeal to an audience of students fromhumanities—students to whom we should pay more attion. Like Galileo, we, as physics teachers, have both sctific and unscientific audiences. Like him, we should thiabout how to reach them.

ACKNOWLEDGMENTS

I would like to thank two anonymous referees for helpsuggestions and criticisms.

a!Electronic mail: mpeterso@mtholyoke.edu1Galileo Galilei,Two New Sciences, translated by Henry Crew and Alfonsde Salvio~Dover, New York, 1954!. Originally published by Elzevir, 1638

2Reference 1, pp. xx–xxi.3Reference 1, p. xii. The full title is ‘‘Discourses and Mathematical Deonstrations concerning Two New Sciences pertaining to MechanicsLocal Motions.’’ Galileo complained that the publishers had substitutedlow and common title for the noble and dignified one carried upontitle-page.’’ His preferred title is unfortunately lost.

4Galileo Galilei, ‘‘Due lezioni all’Accademia Fiorentina circa la figura, sie grandezza dell’Inferno di Dante,’’ inLe Opere di Galileo Galilei, edited

580 Am. J. Phys., Vol. 70, No. 6, June 2002

Downloaded 23 Sep 2012 to 152.3.102.242. Redistribution subject to AAPT

-etsfts.ulrd

y

rns

nd,e

essely

o-of

-b-

n

sthen-n-

l

nd

e

by G. Barbe`ra ~Ristampa della Edizione Nazionale, Florence, 1933!, Vol.9, pp. 31–57. Translated by Mark A. Peterson, http://www.mtholyoke.e

˜ mpeterso/classes/galileo/inferno.html.5Reference 1, p. 90.6Reference 1, p. 2.7Reference 1, p. 112.8Reference 1, p. 3.9Archimedes’ law of the lever, the balance of torques, was second natuGalileo.

10We would now say the stress itself varies linearly across the beam, sothe correct dependence is the second moment of area, going as theeter to the fourth power.

11All propositions in this paragraph are from the Second Day.12Reference 1, p. 6.13Aristotle, in Minor Works, translated by W. S. Hett~Harvard U.P., Cam-

bridge, MA, 1936!, pp. 330–411.14Reference 1, p. 135.15Reference 1, p. 5.16Reference 1, p. 7.17Galileo Galilei, inDiscoveries and Opinions of Galileo, translated by Still-

man Drake~Anchor Books, New York, 1957!.18Galileo Galilei, Discourse on Bodies in Water, translated by Thomas

Salusbury~University of Illinois Press, Urbana, IL, 1960!.19Galileo Galilei,Dialogue Concerning the Two Chief World Systems, trans-

lated by Stillman Drake~University of California Press, Berkeley, CA1967!.

20M. A. Finocchiaro,The Galileo Affair ~University of California Press,Berkeley, CA, 1989!.

21‘‘Le Opere di Galileo Galilei,’’ edited by G. Barbe`ra ~Ristampa della Ediz-ione Nazionale, Florence, 1933!, Vol. 10, No. 207, pp. 228–230.

22Stillman Drake,Galileo at Work~Dover, Mineola, NY, 1978!, pp. 16–17.23Reference 1, p. 100.24Reference 16, p. 13.25Mario Biagioli, Galileo Courtier ~University of Chicago Press, Chicago

1993!, pp. 30–31.26Reference 16, pp. 106–107.27In fact Manetti was a Florentine intellectual of the previous century,

biographer of Filippo Brunelleschi. He lived well before the foundingthe Florentine Academy.

28Vellutello was from Lucca, a notable rival of Florence.29Reference 1, p. 3.30Reference 19.31See, for example, the entertaining account in William Dunham,Journey

Through Genius~Wiley, New York, 1990!.32The way he uses scaling goes as follows. Galileo’s opponents argue

the lamina floats because it is broad and flat, and the water resists bcut. But where the water is cut is along the perimeter, so it is supportethe perimeter, and thus, by scaling, the lamina floats better if it is diviinto smaller pieces, that is, it floats better if it isless broad and flat,contradicting their own position.

33Reference 1, p. 125.34‘‘Le Opere di Galileo Galilei,’’ edited by G. Barbe`ra ~Ristampa della Ediz-

ione Nazionale, Florence, 1933!, Vol. 9, p. 7.35Because he could not obtain the text, Alamanni provided his correspon

a summary of the lectures~five years after they were delivered!!. ‘‘Itconsisted in this, that he reviewed the opinion of the Florentine AntoManetti concerning the site of Dante’s Inferno, published in a book prinby the Giunti, and then he reviewed the opinion of Vellutello, a commtator on Dante, on the same subject, and comparing the one with the ohe showed that of Manetti to be better.’’

36Reference 1, pp. 130–131.37See for example the exchange between Simplicio and Sagredo, Re

bottom of p. 137.

580Mark A. Peterson

license or copyright; see http://ajp.aapt.org/authors/copyright_permission