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MICHELE CAMEROTA’S NEW BIOGRAPHY OF
GALILEO:
THREE ESSAY REVIEWS
Michele Camerota. Galileo Galilei e la cultura scientifica nell’età della Controriforma (Rome: Salerno Editrice, 2004), pp. 703 ¤ 34.00 ISBN 88 8402 455 2
(softcover).
The appearance of a new biography of Galileo is a major event
in the history of early modern science, one that we wish to mark
with a little symposium of reviews by three scholars with a very
different background. Indeed, the reviews offer a wide spectrum
of opinions, and their authors have highlighted different aspects
of Camerota’s book. Both Palmieri and Ducheyne would have
appreciated a much more extensive treatment of the science of
motion, for example. Without contradicting them, Finocchiaro
points out that Camerota’s contextualization of Galileo’s 1611
letter to Daniello Antonini is an important contribution to this
area, showing that Galileo abandoned the view that speed is
proportional to the distance traversed at the end of his Paduan
years, rather than later as has often been thought. And while
Palmieri wishes for a more thorough analysis of the recent literature on patronage, Ducheyne considers Camerota’s treatment of
the political, socio-economical, familial, cultural, and patronage
aspects of Galileo’s life precisely one of the book’s strengths.
Lastly, whereas Palmieri suggests that Camerota’s book will be of
interest mostly to non-specialist readers, Finocchiaro mentions
depth as one of its strengths and lists a number of instances in
support of his view, from the Copernican implications of the
debate on comets to the special role attributed to the phenomenon of the tides by important ecclesiastical authors.
Not in spite, but because of the divergent judgements they
express, we very much hope that the three reviews published
here will stimulate further debates on Camerota’s book and on
how to write a biography on Galileo and his world.
Domenico Bertoloni Meli (Book Review Editor)
© Koninklijke Brill NV, Leiden, 2005
Also available online – www.brill.nl
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REVIEW ESSAY I
MAURICE A. FINOCCHIARO
University of Nevada, Las Vegas
In the last half a century, there have been several major biographies or comprehensive accounts of Galileo’s works. In 1957
Ludovico Geymonat published one in Italy, which was later translated into English with the subtitle “a biography and inquiry into
his philosophy of science”; Geymonat’s account was useful, but
much has happened since then in Galilean scholarship, as well
as in the world at large, and such developments usually act as
catalysts and motivations for new accounts.1 In 1964, there appeared in Italy a work by Pio Paschini that provided a comprehensive and detailed account of Galileo’s life and works; although
the scope of this work remains unsurpassed, and although it
remains of some use even today, when the book was first published it was already about 20 years out of phase since it had
been researched and written in 1941-44, at which time it was not
published due to circumstances that are both complex and controversial, so much so that they have given rise to one of the
latest episodes of the Galileo affair.2 In 1966, Alexandre Koyré’s
Études galiléennes, which had been first published in 1939, was
reprinted; this work was immensely influential and inspiring,
and nobody can question its brilliance, insight, historical sensitivity, and analytical acuity.3 Nevertheless, its focus was Galileo’s
thought and methodology, and it paid little attention to biographical developments or the Galileo affair; moreover, by the
time Koyré’s work was reprinted, it was being superseded by
more recent developments.4 At any rate, it is important, indeed
1
L. Geymonat, Galileo Galilei (Turin, 1957); idem, Galileo Galilei, trans. Stillman Drake (New York, 1965).
2
P. Paschini, Vita e opere di Galileo Galilei, 2 vols., ed. Edmonde Lamalle, in
Miscellanea Galileiana, (Pontificiae Academiae Scientiarum Scripta Varia 27), vols.
1-2 (Vatican City, 1964); idem, Vita e opere di Galileo Galilei, ed. Michele Maccarrone (Rome, 1965); cf. Maurice A. Finocchiaro, Retrying Galileo, 1633-1992
(Berkeley, 2005), 318-37.
3
A. Koyré, Études galiléennes, 3 vols. (Paris, 1939); idem, Études galiléennes,
collected into a single volume (Paris, 1966); idem, Studi galileiani, trans. Maurizio
Torrini (Turin, 1976); idem, Galileo Studies, trans. John Mepham (Hassocks,
1978).
4
Such as Thomas B. Settle, “An Experiment in the History of Science,” Science
133 (1961), 19-23.
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essential, that the work of influential masters be assimilated in
such a way that one is able to move beyond the letter of their
teachings.5 In 1968, Maurice Clavelin published in France a
comprehensive account of the background, development, and
significance of Galileo’s scientific and epistemological contributions; although the breadth and depth of Clavelin’s book remain
in some ways unsurpassed, it was not and did not claim to be a
biography.6 In 1978, Stillman Drake published a “scientific biography” that represented a synthesis of his own epoch-making
research and of other recent Galileo scholarship; it established
beyond any reasonable doubt that Galileo was a skillful and indefatigable practitioner of the art of experiment. However, although one can appreciate Drake’s book as a milestone, one
need not deny its weaknesses: its portrayal of Galileo as a-philosophical or anti-philosophical was both naïve and untenable, and
its failure to include a significant account of the details of the
trial suggests that Drake’s monograph falls short of what one
would expect even from a “scientific” biography.7 Since 1978
there have been several important works on selected aspects of
the topic, such as William Wallace’s account of the historical
and conceptual connections between Galileo and the progressive wing of Aristotelianism; Mario Biagioli’s account of the
development of Galileo’s career from the point of view of the
historically specific institution of patronage and social psychology of courtly etiquette; and Annibale Fantoli’s account of the
Galileo affair.8 However, since then there have been no attempts
to give an account of Galileo’s life and works aiming to include
5
Cf. M.A. Finocchiaro, History of Science as Explanation (Detroit, 1973), 86-116;
idem, “Logic and Scholarship in Koyré’s Historiography,” Physis 19 (1977), 5-27;
idem, Galileo and the Art of Reasoning (Boston, 1980), 202-23.
6
Maurice Clavelin, La philosophie naturelle de Galilée (Paris, 1968); idem, The
Natural Philosophy of Galileo, trans. A.J. Pomerans (Cambridge, Mass., 1974); idem,
La philosophie naturelle de Galilée, 2nd edn. (Paris, 1996); cf. Finocchiaro, Galileo
and the Art of Reasoning, 246-53.
7
Stillman Drake, Galileo at Work: His Scientific Biography (Chicago, 1978); cf.
Finocchiaro, Galileo and Art of Reasoning, 237-45; idem, “Drake on Galileo,” Annals
of Science 59 (2002), 83-88.
8
William A. Wallace, Prelude to Galileo (Boston, 1981); idem, Galileo and His
Sources (Princeton, 1984); idem, Galileo’s Logical Treatises (Dordrecht, 1992); idem,
Galileo’s Logic of Discovery and Proof (Dordrecht, 1992); Mario Biagioli, Galileo
Courtier (Chicago, 1993); Annibale Fantoli, Galileo per il copernicanesimo e per la
Chiesa (Vatican City, 1st edn. 1993, 2nd edn. 1997, 3rd edn. 2003); idem, Galileo
for Copernicanism and for the Church (Notre Dame, 1st edn. 1994, 2nd edn. 1996).
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not only his scientific work in kinematics and astronomy, but
also his practical and reflective contributions in methodology,
and the vicissitudes of his trial and condemnation.9
Such a task has now been attempted by Michele Camerota,
and consequently his book may be welcome as a work for which
the time was ripe.10 Besides such breadth, the book usually discusses such topics in enough detail that we may add depth as
another merit. Moreover, since the book is, generally speaking,
well documented on primary and secondary sources, it deserves
the attention of every serious scholar of the subject. Of course,
it is a different story to determine how successful Camerota’s attempt is, how insightful the in-depth discussions are, and how
cogent the documentation is; ultimately, readers will have to
make this determination themselves. Some of the impressions of
this reader will now be elaborated. I shall begin with some favorable
impressions.
It is well known that as early as 1604 Galileo was convinced of
the truth of the law of squares, according to which the distance
from rest traversed by a freely falling body increases in proportion to the square of the time elapsed; and it is well known that
at that time he was trying to explain this fact about falling bodies
by trying to derive the law of squares from the principle of space
proportionality, according to which the velocity acquired by a
falling body is directly proportional to the distance traversed.11
This derivation is fallacious and this principle is false, and by the
time he wrote the Two New Sciences Galileo had rejected them;
instead he gave an argument to refute the principle, and he derived the law of squares from the principle of time proportionality,
9
I discount such successful popularizations as Dava Sobel, Galileo’s Daughter
(New York, 1999) and such amateurish efforts as James Reston, Jr., Galileo: A Life
(New York, 1994); that is, I am not taking them into account in the present discussions of scholarly contributions, although it would be a serious mistake to ignore them in a discussion of the popular-cultural significance of Galileo and the
impact of historical scholarship; cf. Finocchiaro, Retrying Galileo, 1633-1992, esp.
359-65.
10
M. Camerota, Galileo Galilei e la cultura scientifica nell’età della Controriforma
(Rome, 2004). References to this work will be given in parenthesis in the text,
unless the references are too many, or mixed with others, or accompanied by
comments.
11
Galileo to Sarpi, 16 October 1604, in Galileo Galilei, Opere, 20 vols., National Edition, ed. Antonio Favaro (Florence, 1890-1909, rpt. 1929-1939 and
1968), 10: 115; Galilei, Opere, 8: 373-74; cf. Finocchiaro, History of Science as Explanation, 88-96.
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which he called the definition of uniform acceleration, according to which the velocity acquired by a freely falling body is
directly proportional to the time elapsed.12
Thus, sometime between 1604 and 1638 Galileo rejected space
proportionality in favor of time proportionality. When exactly
did this happen? Did it happen in 1629-31 as Drake suggested,
or in 1609 as Koyré implied?13 Camerota argues cogently that it
must have happened before 9 April 1611 because this is the date
of a letter to Galileo by his disciple Daniello Antonini in which
the writer refers to an argument attributed to his teacher to the
effect that a body whose velocity increases in proportion to its
distance from rest would move instantaneously, and this is the
key point of Galileo’s refutation of space proportionality in Two
New Sciences. Although Camerota himself credits Mario Helbing
and Ottavio Besomi for having recently made the point, Camerota
deserves credit for having stressed and contextualized it.14
Next, it is also well known that a few months after the publication of Galileo’s Sidereal Messenger, Kepler published a Conversation with the Sidereal Messenger in which he essentially endorsed
Galileo’s telescopic discoveries. Galileo’s booklet had been published in March 1610, and Kepler’s endorsement came in May,
before having access to a telescope and observing the new phenomena first-hand. Eventually, in August, Kepler was able to use
12
Galilei, Opere, 8: 197-210, esp. 203-4; idem, Two New Sciences, trans. and ed.
S. Drake (Madison, 1974), 153-67, esp. 159-60; cf. M.A. Finocchiaro, “Vires Acquirit Eundo: The Passage Where Galileo Renounces Space-Acceleration and
Causal Investigation,” Physis 14 (1972), 125-45; idem, “Galileo’s Space-Proportionality Argument: A Role for Logic in Historiography,” Physis 15 (1973), 65-72;
idem, “Cause, Explanation, and Understanding in Science: Galileo’s Case,” The
Review of Metaphysics 29 (1975-76), 117-28.
13
Drake, Galileo at Work, 314-19; Koyré, Études galiléennes, 138 n. 2; idem, Galileo Studies, 124 n. 137; cf. Camerota, Galileo Galilei, 146.
14
Antonini to Galileo, in Galilei, Opere 11: 84-86, at p. 85; Camerota, Galileo,
146-47, 595 n. 268; cf. Ottavio Besomi and Mario Helbing, “Commento,” in Galileo Galilei, Dialogo sopra i due massimi sistemi del mondo, tolemaico e copernicano, 2 vols.
Critical edn. by O. Besomi and M. Helbing (Padua, 1998), 2: 546-47. It should be
noted that here Camerota ends up agreeing with Koyré and disagreeing with
Drake; that this is not an isolated case; that indeed in both text and notes Camerota is constantly criticizing Drake’s claims and favorably elaborating Koyré’s
views; so much so that one could describe the book as an attempt to debunk
Drake and to update and revive Koyré by writing the kind of biography that Koyré
might have written if he had ever taken up such a project. As someone who has
shown both appreciation and criticism of both Drake and Koyré, I would not be
bothered by such a situation if that were the position to which the documents,
evidence, and arguments lead us; but that is a crucial and questionable “if.”
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an instrument sent by Galileo to the Elector of Cologne and to
make the confirming observations; and in the fall, Kepler published a report more explicitly favorable to Galileo.15
An important aspect of this part of Camerota’s account is the
following. He stresses that although Kepler’s initial Conversation
is in part a defense of Galileo’s discoveries, it is also a defense
of his own ideas and the ideas of other precursors. Thus for a
while many people interpreted Kepler’s Conversation as an antiGalilean document exposing Galileo’s pretensions and undercutting his originality. These people included Giovanni A. Magini
(professor of mathematics at the University of Bologna), Martin
Horky (who was about to publish the Brevissima peregrinatio contra
Nuncium Sidereum), Michael Maestlin (former teacher of Kepler),
the Neapolitan mathematician Giovanni Camillo Gloriosi, Martin Hasdale in Prague, Georg Fugger (the imperial ambassador
to Venice), and even the Lincean Academician Francesco Stelluti.
Again, as some of Camerota’s own notes suggest, here he is
adopting and utilizing the work of other scholars, in particular
Massimo Bucciantini’s work on Galileo and Kepler.16 Nevertheless Camerota is right to stress the relatively “mixed” character
of Kepler’s Conversation with Galileo and how it was exploited by
more hostile critics.
One of the most insightful and best argued parts of Camerota’s
book is a chapter-length section entitled “Between Ptolemy and
Copernicus” (333-54) in a longer chapter dealing with the period between the 1616 condemnation of Copernicanism and the
1623 election of pope Urban VIII. Camerota documents the
relatively well-known fact that that period witnessed a wide diffusion and acceptance (especially among Catholics) of Tycho
Brahe’s world system, according to which the earth stands still at
the center of the universe, but the planets revolve around the
sun while the sun revolves with diurnal and annual motion around
the earth. After giving this historical background, Camerota takes
up the more evaluative and controversial questions whether the
reason why Galileo was not impressed by the Tychonic system
was Copernican bias and zealotry; whether his low opinion of
the Tychonic system amounted to a failure to consider it at all;
15
Camerota, Galileo, 160, 174-75, 184-85.
Camerota, Galileo, 601 nn. 129, 131; cf. M. Bucciantini, Galileo e Keplero:
Filosofia, cosmologia e teologia nell’età della Controriforma (Turin, 2003), 180-84.
16
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and whether his neglect or dismissal of that system invalidated
or weakened his case in favor of Copernicanism. There is a
common view advanced in both scholarly and lay-popular circles
that answers all such questions in the affirmative; and this view
then leads to a common apologetic line of defense of the Inquisition to the effect that its condemnation of Galileo was essentially
correct, at least scientifically, methodologically, epistemologically,
and logically, whatever its shortcomings may have been from a
judicial, moral, or Christian-charity point of view. Camerota is at
pains to refute this apologia and the historical interpretations
and philosophical evaluations on which it is based. And I believe
he does a very good job.
Some of his remarks are worth quoting:
It is wrong, therefore, to accuse Galileo of having operated in a prejudicial
manner in favor of the Copernican view, masking the alleged weakness of
his own arguments … by obscuring the reasons of opponents, especially
Tycho’s followers. In reality, Galileo’s conviction that the Copernican and
Tychonic hypotheses were not explanatorily equivalent originated from the
consideration that the heliocentric view possessed greater coherence and
simplicity for the reduction of the complexities of planetary motions, and
also was in accordance with the formulations of the new science of motion
(15-16).
Similarly,
it has been often stressed that despite his ‘realistic’ claims, Galileo never
provided a definitive ‘proof’ in support of the Copernican system. This is
true as long as by ‘proof’ one means incontrovertible evidence such as the
measurement of stellar parallax, that is, an accomplishment that was absolutely unattainable in light of the development of the instrumentation of
that time. That said, one must admit that Galileo’s battle for a new cosmology displayed such a quantity of ‘sensory observations … confirming agreements and very strong arguments’ as to exclude any suspicion that one is
facing an ineffectual attempt by a fanatic Copernican ‘activist’ (352).
On the contrary, the situation should be described by saying that
“the demonstrations for the earth’s motion are much stronger
that those for the other side” (352); this description has the
advantage of stressing that in regard to the question of the earth’s
motion the Tychonic and Ptolemaic systems are identical and
both contrast to the Copernican theory. To understand why that
description is correct,
one should not forget what was the strength of the ‘proofs’ and reasons
mentioned to support the alternative doctrines. Above all, one must remember the extra-scientific ‘worries’ (involving the defense of the world view
contained in Sacred Scripture and/or in the texts of the philosophical tradition) that fed in large measure the anti-Copernican hostility and motivated
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the acceptance of the other (Ptolemaic and Tychonic) cosmological views
(354).
In this discussion, Camerota does not quote, cite, or acknowledge other scholars who have advanced similar views.17 But we
may agree that the point is important enough to deserve repetition; moreover, this apologetic line has a very long history, and
it keeps on being revived, and so it is useful for Camerota to
focus on its most recent advocates. 18
The controversy over comets is another well-known episode.
Three comets became visible for a few months in the latter part
of 1618. In early 1619, an anonymous Disputatio astronomica was
published in Rome, critical of the Aristotelian view that comets
have a terrestrial origin and sublunary location, and favorable to
the Tychonic view according to which the 1618 comets were
located beyond the moon and followed circular orbits. It was
soon learned that the author was Jesuit Orazio Grassi, professor
of mathematics at the Roman College. In June 1619, there appeared in Florence a Discourse on the Comets that was a joint work
of Galileo and his disciple Mario Guiducci, but displayed only
Guiducci as the author. The Discourse was critical of Grassi’s view
and advanced a theory according to which comets had a terrestrial origin, but a superlunary location, and followed a rectilinear path away from and orthogonal to the earth. Grassi replied
immediately, defending his views and going to the counterattack
in the Libra astronomica ac philosophica, published the same year
under the pseudonym of Lotario Sarsi. Four years later, Galileo
responded with a detailed and wide-ranging rebuttal in The Assayer
(1623), publishing it under his own name. However, Grassi did
not acquiesce and three years later published a rebuttal in the
Ratio ponderum Librae et Simbellae (1626).
17
See, for example, M.A. Finocchiaro, “Aspetti metodologici della condanna
galileiana,” Intersezioni 4 (1984), 503-32; idem, “The Methodological Background
to Galileo’s Trial.,” in Reinterpreting Galileo, ed. William A. Wallace (Washington,
DC, 1986), 241-72; idem, trans. and ed., The Galileo Affair: A Documentary History
(Berkeley, 1989), 8-10; idem, trans. and ed., Galileo on the World Systems: A New
Abridged Translation and Guide (Berkeley, 1997), 5, 53-55.
18
As for the longevity of this apologetic line, see Finocchiaro, Retrying Galileo,
1633-1992, 79-85, 138-53, 218-21, 280-94, 306-26. Among the recent advocates of
this line, Camerota mentions in passing Arthur Koestler, The Sleepwalkers (New
York, 1959); Enrico Zoffoli, Galileo: Fede nella ragione: Ragioni della fede (Bologna,
1990), 106; Mario D’Addio, Il caso Galilei: Processo, scienza, verità (Rome, 1993), 19,
21; and Remigio Presenti, Galileo e Bellarmino: Leggenda e verità: Lettura moderna di
una disputa antica (Montepulciano, 2001), 70.
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This was a very complex dispute that became extremely bitter
and polarized. The intellectual issues often appear trivial and
the respective positions confused and confusing. Camerota’s
account contains an insightful discussion of a detail (370-72)
which I think goes a long way toward explaining how and why
such apparent non-issues could have generated so much animosity. He stresses that Grassi’s Libra contained an argument to the
effect that the Galilean Discourse was committed to Copernicanism
and thus violated the anti-Copernican decree of 1616; for Galileo claimed that the 1618 comets followed a rectilinear path
away from and orthogonal to the earth, and he knew that they
exhibited an observable northward deviation, but he was also
aware that this apparent deviation could be explained only in
terms of the earth’s own annual motion. I believe that the contortions of the Galilean view of comets were partly defensive
moves and partly overreactions to this venomous charge. In short,
the controversy about comets was not just about comets, but was
connected with both the cosmological debate over Copernicanism
and the theological controversy over its religious propriety.
Another one of Camerota’s insights involves the related topics
of the divine-omnipotence argument favored by pope Urban VIII
and the ending of Galileo’s Dialogue on the Two Chief World Systems (1632). Again, much is relatively well known about this episode:
the Dialogue ends with a statement of pope Urban VIII’s favorite
objection to Copernicanism, and Galileo was required by ecclesiastic authorities to include this argument in his book. This was
the argument that regardless of how much evidence there is for
the geokinetic explanation of tides (and more generally, for the
earth’s motion), we can never be absolutely certain that this is
so (that is, we can never say that this must be so), because God
is omnipotent, and so He could have created a world in which
the tides are caused not by the earth’s motion but by something
else (or more generally, a world in which the earth does not
move), but to say that the tides must be caused by the earth’s
motion (or more generally, that the earth must move) limits
God’s power to do otherwise. On the other hand, there is much
that is controversial about this argument: whether or how strongly
the argument in endorsed in the text of the book; whether or not
the argument is valid, which in turn depends on whether one
states the argument in a stronger or weaker version.
Now, an important and novel point here elaborated by Camerota
is that
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according to a centuries old tradition, the phenomenon of the tides represented the most impenetrable among the secrets of nature, so much so that
not infrequently the problem was regarded as unsolvable in principle, and
that the roots of this unsolvability lay deep in the divine will to mortify the
vainglory of human reason and to remind it or make it aware of its limitations (455).
He documents this thesis with evidence from Diogenes Laërtius,
Seneca, the pseudo-Augustinian De mirabilibus Sacrae Scripturae,
Isidore of Seville, Julius Caesar Scaliger, Leonard Lessius, Eustachius of Saint Paul, and (importantly) from the ending of
Galileo’s “Discourse on the Tides.”19 Although much more needs
to be taken into account, this thesis helps to explain why Church
authorities did not want the tides mentioned in the title of Galileo’s
Dialogue and why this book angered Urban: the point is not (as
commonly claimed) that the Church did not want to put its
stamp of approval on the conclusiveness of the tidal argument,
or that Urban resented the fact that in the dialogue his favorite
argument is advanced by the relatively unintelligent character
Simplicio; rather they felt it was irreverent (toward divine omnipotence) to seriously consider any explanation of the tides (let
alone a geokinetic explanation), and Urban resented the fact
that Simplicio had been made to advance an insufficiently strong
version of the argument (457-59).
There is at least one other discussion that impressed me favorably, namely Camerota’s adaptation of Giorgio de Santillana’s thesis
that a crucial document in the Inquisition proceedings of Galileo’s
trial, namely the executive summary presented to the cardinalinquisitors and the pope, is a biased account of the proceedings.20
But I should not be carried away by favorable impressions, and
must give some space to more critical remarks. For despite the
just-mentioned and other insights and merits, the book contains
a larger number of discussions with which I would want to take
issue, and many of them involve more significant or general
questions. For example, I find the book ambivalent on the question of Galileo’s attitude and degree of pursuit and belief toward
Copernicanism: sometime Camerota claims (as we saw above)
that Galileo’s interest was gradual, nuanced, and moderate, rather
19
For this last reference, see Galilei, Opere 5: 395; Finocchiaro, Galileo Affair,
133.
20
Camerota, Galileo, 504-6; cf. G. de Santillana, The Crime of Galileo (Chicago,
1955), 277-83; Galilei, Opere, 19: 293-97; Finocchiaro, Galileo Affair, 281-86.
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than total and fanatical, but most of the time he speaks as if the
reverse were true. Thus, he claims (98) that as early as the late
1590’s Galileo explicitly accepted Copernicanism, basing this claim
on Galileo’s letters to Kepler and to Jacopo Mazzoni, which however
need a more critical reading. Camerota claims that in the sunspot letters Galileo held the Copernican view “with complete
clarity” (253) and “without hesitation” (259), as if the heliocentrity
of Venus’s orbit and the changeability of the heavenly region
were the only two elements of the Copernican system. And he
claims that by 1615 Galileo “believed that Copernicus’s doctrine
had been fully ascertained” (283), as if Galileo had forgotten
about such unanswered problems as the extruding power of diurnal
whirling or annual stellar parallax.21
Another over-arching and crucial issue is that of Galileo’s attitude toward mathematization and the application of mathematics
to nature. Here Camerota’s account is not ambivalent, but rather
attributes to Galileo a position that might be labeled apriorist
mathematicism, reminiscent of Koyré.22 Camerota’s argument
makes much of the Galilean statement from The Assayer that the
book of nature is written in the mathematical language of triangles, circles, etc. The difficulty is that, like Koyré, Camerota’s
analysis of such remarks is excessively abstract insofar as he takes
them out of context, neglecting both the context of the passages
where they occur and the context of Galileo’s scientific practice.
In any case, to portray Galileo as mathematicist and apriorist,
Camerota seems to ignore what could be regarded as the epistemological climax of the Two New Sciences, where Galileo discusses the crucial issue of whether the particular mathematical
analysis of uniform acceleration which he has just elaborated
corresponds to the way heavy bodies really fall.23 Similarly in one
of the methodological highpoints of the Dialogue, Galileo discusses the question of whether his mathematical analysis of the
problem of centrifugal extrusion on a rotation earth corresponds
21
For a fuller documentation and elaboration of these necessarily condensed
remarks, see M.A. Finocchiaro, “Galileo’s Copernicanism and the Acceptability of
Guiding Assumptions,” in Scrutinizing Science: Empirical Studies of Scientific Change,
ed. Arthur Donovan, Larry Laudan, and Rachel Laudan (Dordrecht, 1988; Baltimore, 1992), 49-67.
22
The most emblematic passage occurs in Camerota, Galileo, 559-60; but see
also, for example, pp. 20-23, 386-87.
23
Galilei, Opere, 8: 198-205; idem, Two New Sciences, 154-61.
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to physical reality.24 The key point is that even if we are assured
that nature is written in the mathematical language of triangles,
circles, etc., which particular type of geometrical figure corresponds to which particular phenomenon is a question that can
only be resolved by observation and experiment.25
Yet another kind of difficulty besets Camerota’s discussion of
a third important and general issue. It relates to the Galileo
affair. One of the most highly debated issues has been whether
in 1633 he was condemned for disobedience or for heresy, and
the discussions are usually carried out as if these two alternatives
were mutually exclusive. The disobedience would consist in his
alleged violation of the restrictions developed in 1616, either the
special injunction that forbade him to discuss the earth’s motion
in any way whatever, or cardinal Bellarmine’s warning prohibiting him from defending the idea. The alleged heresy would be
the Copernican doctrine, which had been condemned by the
Index’s decree of 5 March 1616. The disobedience interpretation makes the trial and condemnation of Galileo an instance of
merely disciplinary proceedings, thus diminishing its general
significance and in turn the seriousness of the presumed error
by the Inquisition, and consequently offering a common apologetic line of defense. The heresy interpretation makes the issue
a doctrinal one, thus aggravating Galileo’s alleged crime, but
also the Church’s own misdeed, and perhaps even its claims to
(doctrinal) infallibility. Camerota argues that “the sentence of
June 1633 struck Galileo not for having disobeyed the ‘injunction’ of 1616, but for having entertained an opinion declared
heretical by reason of its contrast with the scriptural text” (518).
And his argument is very good, indeed perhaps the best I have
seen in support of such a thesis, and it should give pause to
anyone who holds the reverse thesis (that Galileo was condemned
for disobedience only). Nevertheless, the cogency of Camerota’s
argument applies only to the positive part of his thesis; the negative
part that denies the alternative interpretation is unjustified. The
crux of the matter is that on this particular issue, both sides are
wrong insofar as they deny the alternative, although they are
24
Galilei, Opere, 7: 229-37; Finocchiaro, Galileo on the World Systems, 193-202.
Again, for a fuller documentation and elaboration of these necessarily condensed remarks, see M.A. Finocchiaro, “Physical-Mathematical Reasoning: Galileo on the Extruding Power of Terrestrial Rotation,” Synthese 134 (2003): 217-44;
idem, Galileo on the World Systems, 348-52; idem, Galileo and the Art of Reasoning, 62102.
25
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both partly right for what they choose to affirm; in short, they
both presuppose the false dilemma that Galileo was condemned
for only one of the two alternatives (disobedience or heresy),
but not both. In fact, I believe the most tenable position is that
Galileo was condemned for both disobedience and heresy. A key
part of my argument would be that the disobedience in question
involves intellectual matters of what to hold or defend, and so it
is not merely a disciplinary matter but becomes a doctrinal one;
moreover, heresy ultimately is simply to believe what the Church
commands not to believe, or not to believe what it commands us
to believe, in short disobedience in matters of belief. So there is
no contradiction between the two reasons, but they are rather
different ways of looking at the situation.26
There is no space here to do anything more than mentioning,
without elaboration, other, relatively minor difficulties. For example, I found Camerota’s account of Galileo’s tidal theory (35463) superficial as a whole, despite his sound sensitivity to those
aspects of it that may be scientifically correct; in particular,
Camerota’s explanation (358) of Galileo’s basic diagram seems
to interchange point L with B and C with D. I also believe Camerota
does not document satisfactorily the claim (424) that in the spring
of 1630 (while in Rome to have his manuscript of the Dialogue
approved for publication) Galileo attended a meeting (something of a party) organized by the Vallombrosan monk Orazio
Morandi; in fact, although we have Morandi’s letter of invitation,27 there is no documentation that Galileo accepted the invitation. And Camerota seems too quick to dismiss (440-41) the
Platonic cosmogony discussed speculatively by Galileo in the
Dialogue and mentioned again in the Two New Sciences; for although there is no question that the Galilean speculation commits the scientific error of assuming that the law of squares is
universally valid at interplanetary distances, it is not at all obvious that the consequences drawn from this false assumption are
themselves false or invalidly derived.28
26
Such an interpretation, of course, needs further elaboration; I have made
a start in M.A. Finocchiaro, “Science, Religion, and the Historiography of the
Galileo Affair: On the Undesirability of Oversimplification,” Osiris, second series,
16 (2001), 114-32; idem, Retrying Galileo, 1633-1992, 273-74.
27
Morandi to Galileo, 24 May 1630, in Galilei, Opere, 14: 107.
28
For the references to the Platonic cosmogony, see Galilei, Opere, 7: 53-54,
8: 284. For an argument that the consequences Galileo draws are not invalidly
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In summary, Camerota’s biography of Galileo is welcome for
its timeliness, useful for its breadth, valuable for its depth, usually well-documented, and occasionally insightful; but on some
key issues it is best taken as a good occasion to reaffirm or
develop more tenable positions.29
* * *
REVIEW ESSAY II
PAOLO PALMIERI
University of Pittsburgh
Michele Camerota’s Galileo Galilei e la cultura scientifica nell età
della Controriforma is a biography of Galileo Galilei (1564-1642).
The author declares at the beginning that he aims at furnishing
an account of the development of Galileo’s ideas. The book
consists of an introduction, ten chapters, a bibliography, and a
name index.
In the Introduction Camerota claims that he disagrees with
some historiographic reconstructions which, in his view, have
dominated Galileo scholarship over the past decades. These
reconstructions, however, are only vaguely hinted at, so that it is
difficult to understand what exactly the author is rejecting. Further, in the Introduction, Camerota summarizes a few of his
opinions and methodological considerations, which he develops
derived, see Finocchiaro, Galileo and the Art of Reasoning, 82-84; for an argument
on the other side, see I. Bernard Cohen, “Galileo, Newton, and the Divine Order
of the Solar System,” in Galileo, Man of Science, ed. Ernan McMullin (New York,
1967), 207-31.
29
It may be useful to also note a few points bordering on typographical and
editorial matters. Among quasi-typographical errors, the most important could be
corrected by replacing ‘sfera’ by ‘regione’ (439); ‘oriente’ by ‘occidente’ (443);
‘11 gennaio’ by ‘11 febbraio’ (483); ‘Pasquale Zaccaligo’ by ‘Zaccaria Pasqualigo’
(495); and ‘precedute’ by ‘seguite’ (531). And among semi-editorial flaws, one
should mention that of the works cited in the notes, some are, but some are not,
listed in the bibliography, with no discernible criterion of inclusion; the userfriendliness of the copious and informative endnotes (569-671) is marred by the
lack of running heads with page or even chapter numbers; that although the book
has a useful name index, it has no subject index; and the potentially useful information about monetary matters is limited insofar as the exchange rate between
scudi, florins, and lire is only implicitly given once (79) and that between them and
ducats (116) is not given at all.
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in the rest of the book. Given that they are emphasized in the
introductory pages, I assume that they represent at least some of
the original contributions that the author believes to be the
result of his research. Let us consider the gist of them briefly.
They boil down to the following two points.
First, according to Camerota, there exists a scholarly consensus that Galileo was crucial for the development of modern science and that his astronomical discoveries and his battle for the
Copernican system were instrumental in that development. To
this, Camerota adds that the astronomical discoveries and the
campaign in favour of Copernicus constituted the “connective
element” (tessuto connettivo) between Galileo’s physics and his
astronomy. It is this connective element that explains the main
lines of the development of Galileo’s ideas.
Second, Camerota claims that he will place his narrative in the
context of the intellectual debates of Galileo’s own time. The
author believes that the dialogue form of Galileo’s two masterpieces, the Dialogo and the Discorsi indicates his “combative” and
“pedagogical” stance with respect to his surroundings. Hence
the need for historical contextualization.
As for the contents of the ten chapters, Camerota proceeds
chronologically, following the main events of Galileo’s life. We
are therefore given an account of the young Galileo as a student
and later professor at the University of Pisa in the late 1580s, the
subsequent Paduan period until 1610, and finally the Florentine
years, which culminated in the vicissitudes of the trial at the
hands of the Inquisition and the condemnation of 1633. Rather
than furnishing a chapter-by-chapter summary, I propose to focus on a few questions that, in my view, are representative of
Camerota’s style of work and may provide the reader with a
general sense of the strengths and weaknesses of the book.
Let us consider, as an example, the issue of the isochronism
of the pendulum. Galileo repeatedly claims that a pendulum’s
vibrations occur in equal times regardless of the amplitudes of
the arcs of oscillation. He first announces this conviction, as far
as we can tell, in a letter in 1602. He also argues that he has
ascertained that this is the case by observing two oscillating
pendulums and numbering their vibrations with an assistant. The
so-called question of the “isochronism” of the pendulum has
long been debated by Galileo scholars. It is an interesting example of Galileo’s patterns of investigation and the difficulties of
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interpretation that they have given rise to. In 1976 James MacLachlan repeated the experiments that apparently Galileo himself
had performed. He found that pendulums are in fact not isochronous and tried to explain away the discrepancy on the basis
of a difficult textual passage in Galileo’s Two New Sciences.30 He
also quite convincingly showed that one of Galileo’s experiments
concerning two pendulums made of different materials must
have been an imaginary experiment. More recently, David Hill
has argued that Galileo was in fact aware that pendulums are not
isochronous. Hill studied manuscript folios where it seems that
Galileo recorded notes of experiments showing time inequalities
in the vibrations of a pendulum.31 From this, intriguing questions arise: did Galileo publish claims concerning pendulums
that he knew to be false? And if this is the case, what might have
been Galileo’s motivations? Did he really perform the experiments that are recounted in his writings? Somewhat disappointingly, Camerota does not mention any of these important questions,
relegating them to a comforting endnote. He observes that “today we know that there are minimal differences (practically
negligible) between the times of oscillations of large and small
vibrations in a pendulum. Galileo’s rule is considered valid only
in the case of small vibrations…” (593). This may well be true,
but I am still preoccupied with MacLachlan’s and Hill’s conclusions.
Camerota has a tendency, as it were, to raise expectations that
are subsequently not met. At one point Camerota mentions Mario
Biagioli’s book on Galileo and patronage at the Medici court,
announcing that although the book is “stimulating,” Biagioli’s
thesis according to which “…both Galileo and the Aristotelian
philosophers competed for the same niche of patronage… appears misleading and risky (azzardata)” (228); unfortunately, I
found nothing in Camerota’s book that explains why Biagioli’s
thesis is risky and misleading.32 Throughout the book, one often
ends up wanting to know more about the author’s own views!
At times one also finds sweeping generalizations that mar otherwise balanced analyses. Camerota claims, for example, that
30
J. MacLachlan, “Galileo’s Experiments with Pendulums: Real and Imaginary,” Annals of Science 33 (1976), 173-185.
31
David Hill, “Pendulums and Planes: What Galileo Didn’t Publish”, Nuncius
9 (1994), 499-515.
32
Biagioli, Galileo, Courtier.
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Galileo’s emphatic distinction between the domains of natural
science and theology had important consequences for the development of an independent science in our modern culture (17).
But was that distinction really as clearly drawn by Galileo as
Camerota suggests? And are modern scientific investigations really
that independent of our value systems? Camerota himself concludes his Introduction with a splendid statement by Lucien Febvre,
warning the reader of the dangers of lifting the idea of a past
author from its original context only because it suits our own
ways of thinking. One wonders whether Camerota has paid tribute to this methodological admonition when stressing Galileo’s
modernity so emphatically.
Finally, the book’s prose is often convoluted and the arguments hard to follow because of the length of the sentences. The
book makes for demanding reading. Nonetheless, Camerota’s
biography will be of interest mostly to the non-specialist reader,
but possibly also stimulate debate among Galileo scholars.
* * *
REVIEW ESSAY III
STEFFEN DUCHEYNE
Ghent University
Why Another Biography of Galileo?
It is not every year that a scholarly biography of Galileo appears.
More than a quarter of a century after Stillman Drake’s seminal
Galileo at Work, Michele Camerota offers us his Galileo Galilei e la
cultura scientifica nell’età della controriforma. Camerota has undertaken a giant’s labour in writing this almost six-hundred page
volume. Since Drake, Galileo scholarship has evolved enormously,
so that two questions spring to mind. First, how do these two
biographies differ? And secondly, what may have motivated an
author to undertake the arduous work of writing the biography
of no lesser figure than Galileo, given the availability of Drake’s
Galileo at Work? Both questions are connected, for, as I will try to
show in the following, the differences in their approach to Galileo’s biography is due to what can be described as Drake’s inter-
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nalist and Camerota’s externalist conception of the history of
science. In fact, the two biographers have set themselves different goals. Both conceptions are highly significant to the understanding of Galileo, in particular, and the intellectual development
of scientists and science in general. Moreover, a comparison of
the two biographies at hand suggests that these conceptions should
always operate in tandem: we need both to grasp the development of science.
What is meant here by ‘externalist’ and ‘internalist’? On the
externalist account, the study of history of science ought to focus on the socio-political, economical, and cultural factors that
influenced, or were relevant for, the scientific practice and theory
of a particular individual or group.33 Externalist scholars focus
on the context of scientific practice. By contrast, on the internalist
account, the focal point of the historian of science should be the
inter-theoretical and conceptual problems with which a scientist
or group of scientists was confronted.34 In the words of William
R. Shea, “history of science must not only account for present
theories in the light of past developments, it must also assess old
theories in terms of the conceptual framework of the scientists who held
them, and judge them against the background of the world picture of their age.”35 As for the two Galileo biographies discussed
here, Drake may more aptly be labelled an internalist and Camerota
an externalist. In turn, Drake’s internalist approach helps to
explain Camerota’s motivation to write his own account, which
may thus be conceived as complementary. This becomes evident
as we look at what Camerota and Drake say about the goals of
their respective biographies.
33
A clear example of an externalist approach is Biagioli, Galileo Courtier.
An example of this approach is Paolo Palmieri, “Re-examining Galileo’s
Theory of the Tides,” Archives for the History of the Exact Sciences 53 (1998), 223-375;
and id., “Mental Models in Galileo’s Early Mathematization of Nature,” Studies in
History and Philosophy of Science 34 (2004), 229-264. Palmieri writes, for example,
in his “Re-examining,” 226: “All in all, Galileo’s claim to having furnished a physical proof of the Copernican astronomy based on a causal link between tides and
the motions of the Earth (and of the Moon, insofar as the tide [sic] monthly
period is concerned) has turned out to be a fascinating intellectual problem to our
understanding of motion in the universe. (…) Although mine might be regarded as
an ‘a-historical’ treatment, I believe it is a proper way to answer the following
historiographical question: did Galileo succeed in attributing to tide phenomena
the status of physical proof of the Copernican motions of the Earth?” (emphasis
added).
35
William R. Shea, Galileo’s Intellectual Revolution (London, 1972), xii (emphasis added).
34
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Externalist versus Internalist Conceptions of the History of Science
In the preface to his biography of Galileo, Stillman Drake declares that the purpose of his biography is to “show Galileo in his
working clothes, tending his scientific garden and watching buds
develop.”36 Critical of the externalist approach to Galileo’s thought,
as he feels that biographies should focus on the internal development of a scientist’s thought:
Galileo was at once a lively participant in late Renaissance Italian culture and
an anachronism in its halls of academic learning. If it was the fault of his
nineteenth-century biographers to overestimate the anachronism, it is perhaps a fault of modern biographers to overemphasize cultural traditions that
surrounded him. Those have great statistical value, but they tell us nothing
about a particular individual human being. Just as there might be no man
of average height, so there may have been no exactly typical scientist in
Galileo’s time. From the thoughts of all his predecessors and contemporaries we cannot confidently describe exactly his own thought; for that, we need
to know just how he spent his time.37
In other words, only a careful scrutiny of the particular problems with which Galileo was confronted and the solutions he
proposed will offer an understanding of his natural philosophy.
Drake’s internalist creed is clearly expressed in this fragment.
Obviously, the discipline par excellence which very often puts scientists before theoretical problems is mechanics. In mechanics,
one typically attempts to lay down the theoretical framework
best suited to interpreting and describing the motions we observe. A rough estimate suggests that Drake devotes about 14%
of his text to analyses of Galileo’s mechanics. Along with Winifred
L. Wisan, Drake attempted to trace and analyze Galileo’s mechanical works chronologically, with an eye to unravelling the
internal development of Galileo’s conceptual account of the
phenomena.38 For Drake, this internal account is crucial, though
he also devotes some time and effort to the context in which
Galileo’s scientific practice developed.
Michele Camerota, by contrast, devotes much less time to Galileo’s mechanics. Only about 6,5% of his biography is dedicated
to mechanics. Galileo’s early works are dealt with in notable
detail: De motu (63-74), Le mecaniche (87-95), and the material
36
Drake, Galileo at Work, xiii, see also xxii.
Ibid., xiv.
38
See Winifred L. Wisan, “The New Science of Motion. A Study of Galileo’s
De Motu Locali,” Archive for History of Exact Sciences 13 (1974), 103-306.
37
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relating to Galileo’s famous letter to Guidobaldo del Monte (13549). However, his later work in the Discorsi e dimonstrazioni
matematiche, intorno a due nuove scienze (1638) receives little attention. Certainly, no biography of Galileo could dispense with his
mature mechanics entirely. Most attention is given to the First
Day of the Discorsi, which contains a discussion of the rota Aristotelis
(549-55). The Second Day is dealt with more succinctly (555-57),
while Galileo’s most innovative scientific contributions from the
Third and Fourth Day are handled in merely two pages (557-58).
This brevity constitutes a notable weakness in an otherwise
monumental book.
On the other hand, Camerota has done an excellent job of
canvassing the political, socio-economical, familial, cultural, and
patronage aspects of Galileo’s life. Camerota clearly explains why
he feels that Galileo requires such a contextualization:
In fact, in keeping with the didactic form of the two masterpieces of his
mature years … Galileo’s oeuvre generally displays the open and problemconscious nature of a dialogue, rather than the indifferent closure of a
monologue. In this constant dialectical (and often polemical) opening up
to the culture of his time clearly emerges the pedagogical and also combative dimension of Galileo’s cultural impulse, which is so full of radical proposals of innovation.39
It is, for this reason, Camerota’s conviction that a proper biography of Galileo should first and foremost focus on the intellectual climate in which he was working, lest one wishes to commit
the peccato dei peccati, anachronism (23). A lot of place is therefore given to Galileo’s milieus and to the opinions of his adversaries. To give a few examples of this procedure: The discussion
of the socio-economical context at the university of Padua is rich
and will certainly help to appreciate Galileo’s feeling of Patavina
libertas (75-149), and his academic and everyday activities with
their financial, social, and familial elements really do come to
life. A long chapter is devoted to Galileo’s view of the relation
between the book of nature and the book of scripture and to his
often tense relation with the Church. According to Galileo, scien39
Camerota, Galileo Galileo, 22: “In effetti, analogamente alla forma espositiva
dei suoi due grandi capolavori della maturità (…), l’opera di Galileo manifesta il
carattere aperto e problematico di un dialogo, piuttosto che la indifferente
“chiusura” di un monologo. In questa costante apertura dialettica (e piú spesso
polemica) nei confronti della cultura del suo tempo, emerge, ben netta, la dimensione “pedagogica” e “combattiva” dell’iniziativa culturale galileiana, tutta
animata da radicali istanze di rinovamento.”
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tific inquiry must be independent of religious doctrines (296,
519). Camerota synthesizes, but adds nothing new to, the discussions surrounding the 1616 decree and Galileo’s 1633 clash with
the Sant’Uffizio. The author vividly portrays the various instances
of inquisitorial control exerted over Galileo’s work. The complex route by which the Dialogo eventually reached the printers
is given in a separate chapter (399-459). Camerota’s presentation abounds everywhere in references to primary and secondary
material. In sum, with respect to portraying Galileo’s milieu,
Camerota has certainly surpassed Drake by a great margin.
In accordance with his negligent attitude towards Galileo’s
mature mechanics, the main goal of Camerota’s biography is
synthetic, as he carefully attempts to bring together and balance
all the relevant contextual factors of Galileo’s career, the underlying belief being that only a full account of the context can
provide us with a full understanding of his scientific practice.
How very different indeed from Drake’s individualistic approach
to Galileo, which was expressed so clearly in the above quote.
Drake’s biography is rather analytic in orientation, as it tries to
attain a close grip on the central theoretical concepts (e.g., momento,
force, velocity, … etc.) in the evolution of Galileo’s scientific
theory.40 It is in this respect that Camerota sometimes lacks depth.
For example, Camerota frequently observes that the stress on
mathematics and mathematization of natural phenomena is central to Galileo’s epistemology (21; Galileo’s epistemology is also
very briefly discussed on 414, and again on 559), but he nowhere
engages in an analysis of Galileo’s mathematical method and its
evolution. Correspondingly, the issue of Galileo’s indebtedness
to Archimedes and Euclid—which is still a matter of discussion—
is not dealt with.41 Because of his lack of interest in these more
analytical matters, Camerota has passed over some interesting
and fundamental questions. However, he should not be accused
of negligence, given that it was his explicit goal to write an
externalist biography, just as Drake may not be accused of being
an internalist. Both authors have done an excellent job within
40
On momento, see also Paolo Galluzzi, Momento. Studi Galileiani (Rome, 1979).
On velocity and force, see also Clavelin, The Natural Philosophy of Galileo.
41
For the possible influence of Archimedes and Euclid on Galileo’s thought,
see Palmieri, “Mental Models,” and id. “The Cognitive Development of Galileo’s
Theory of Buoyancy,” Archive for History of Exact Sciences 59 (2004), 189-222.
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their respective frameworks, all the more as both approaches are
legitimate, which should in fact be combined whenever possible.
Externalist-cum-Internalist Conceptions of the History of Science
Norwood W. Hanson has once declared that purely descriptive
historical sketches and purely analytical interpretations of the
history of science are both inadequate to study the history of
science, stating that “for the historian formal philosophical analyses
are often empty. For the philosopher the historian’s factual
compendia seem blind.”42 Indeed, a combination of both is
necessary. This means that the ultimate biography of Galileo—
if it can possibly exist—should do justice to both approaches.
Hanson’s dictum seems correct, although a full integration of an
analytic with a synthetic perspective may only be possible in the
long run. Camerota’s reaction to the preceding biographies
certainly constitutes an immensely important step toward that
goal. For this reason, it is desirable that his book be translated
soon—but publishers are probably already busy preparing an
English edition.
42
Norwood R. Hanson, “The Irrelevance of History of Science to Philosophy
of Science,” The Journal of Philosophy 59 (1962), 574-586, at 582.
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REVIEW ESSAY
THE POINT OF ARCHIMEDES
KARIN TYBJERG
University of Cambridge
Reviel Netz. The Transformation of Mathematics in the Early Mediterranean
World: From Problems to Equations (Cambridge: University of Cambridge Press,
2004), pp. x+375 £45.00 ISBN 0 521 66160 9.
Reviel Netz. The Works of Archimedes Translated into English, Vol. 1: The Two
Books on the Sphere and the Cylinder (Cambridge: University of Cambridge
Press, 2004), pp. x+198 £75.00 ISBN 0 521 82996 8.
New Texts for a New Historiography of Ancient Mathematics
Two questions are central to the two new books by Reviel Netz:
How should we study mathematical works of the past? And does
mathematics have a history? The first question is implicit in The
Works of Archimedes, a new English translation of Archimedes’ collected works. Both in the translation itself and in a detailed
commentary, Netz demonstrates how the special character of
Greek mathematics can only be decoded through close attention
to ancient Greek writing practices. The first volume (of three)
covers Archimedes’ two treatises Sphere and Cylinder I and II as
well as the important commentary by Eutocius of Ascalon (sixth
century AD).
The second question—whether mathematics has a history—
forms the starting point for the other work, The Transformation of
Mathematics in the Early Mediterranean World, and Netz answers in
the affirmative. He argues for the historicity of mathematics
through a detailed study of the transformations of a single Archimedean problem as it was rewritten by Archimedes’ followers,
the commentator Eutocius, and medieval mathematicians writing in Arabic.
Netz’ works represent the consolidation of a movement in the
history of ancient mathematics that has criticized the presentation of ancient mathematics as part of a history-independent
© Koninklijke Brill NV, Leiden, 2005
Also available online – www.brill.nl
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accumulation of mathematical facts. The focus of the attack has
been classical treatments such as those by Thomas Heath, Otto
Neugebauer and Hieronymus Zeuthen.1 These works were primarily concerned with mathematical content, and because content was thought to be independent of presentation, the Greek
geometrical texts were not just translated into a modern language, but also rewritten in the modern mathematical idiom of
symbolic algebra. Criticism against this approach was first launched
by Sabetai Unguru in his 1975 article “On the Need to Re-Write
the History of Greek Mathematics.”2 Unguru showed that ancient mathematics has a particular, historically determined character, which cannot be understood once transformed into symbolic
algebra. The Greeks, he argued, performed their operations on
concrete geometrical figures using diagrams, while algebra refers to abstract symbols and relations.
Since then most historians of mathematics have taken Unguru’s
insights to heart, but the profession has remained dependent on
the classical accounts of history of mathematics that were so
vociferously criticized. Netz’ new translation of Archimedes thus
serves a dual purpose: it provides the first complete translation
into English of Archimedes’ Sphere and Cylinder and Eutocius’
commentaries, and offers a basic tool for approaching Greek
mathematics the way it was written.
When in Rome…
The principles behind the translation and commentary in The
Works of Archimedes are laid out in Netz’ earlier, startlingly original work, The Shaping of Deduction.3 Here he demonstrates the
scope of his approach to ancient mathematics, which shifts the
focus of attention from mathematical content to style and practices. More precisely he shows that it is impossible to understand
the content of Greek mathematics separately from the way it is
1
Thomas Heath, Greek Mathematics (Oxford, 1921); Otto Neugebauer, A History of Greek Mathematical Astronomy (Berlin, 1975); Hieronymus Zeuthen, Die Lehre
von den Kegelschnitten im Altertum (Copenhagen, 1886/Hildesheim, 1966).
2
Sabetai Unguru, “On the Need to Re-Write the History of Greek Mathematics,” Archive for the History of Exact Sciences, 15 (1975), 67-114. See also Michael N.
Fried and Sabetai Unguru, Apollonius of Perga’s Conica: Text, Context, Subtext (Leiden, 2001).
3
Reviel Netz, The Shaping of Deduction in Greek Mathematics (Cambridge, 1999).
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written. Netz’ painstaking analyses of Greek geometrical idiom,
of the way reference is established to diagrams and of the use of
letters and linguistic formulae, yield some remarkable insights
into the nature of Greek mathematical thinking.
The diagram and the proof are shown to form a unit, where
neither can be understood without the other. Greek mathematics does not deal with abstract mathematical entities, but rather
with real geometrical objects in the real geometrical space of the
diagram. Netz also draws attention to the highly elliptical style of
Greek mathematical writing, where the standard objects of Greek
mathematics, such as “line” and “circle,” are implied rather than
written out. Fixed formulae secure a highly precise and unambiguous mode of expression.
In his Works of Archimedes, Netz abides by the insights of The
Shaping of Deduction, staying close to the Greek text and thereby
giving a direct feel of the mathematical language of the original.
The translation makes it clear which words are being supplied
and which are there in the Greek, and given that Netz follows
Heiberg’s edition, it is easy to use his translation alongside the
original.4 Perhaps this kind of translation is a step towards integrating the history of Greek mathematics into classical studies
and towards treating mathematical works as texts. Throughout
the translation each section of text is accompanied by a discussion of Heiberg’s editorial decisions. This part of the commentary is a form of critical apparatus, but it is easily readable and
discursive, and it draws the reader into the process of establishing the text.
A remarkable contribution of this work is that it includes a
critical edition of the diagrams. Where previous editions have
redrawn the diagrams simply to make the mathematical situation
more readily understood, Netz bases his diagrams on those found
in the manuscripts. Variations are recorded in intriguing thumbnail diagrams placed in the margin together with a discussion of
the variations. As there is close agreement between different
lines of transmission there is a good chance that the preserved
diagrams resemble the originals. These reveal surprising features
of ancient diagrams, for instance that the metrical relationship
between different geometrical objects stipulated in the text do
not necessarily hold in the diagrams. This access to the skewed
4
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Johannes L. Heiberg, Archimedis Opera Omnia (Leipzig, 1910-15).
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and un-metric world of the Greek geometrical diagram is extremely valuable, because it shows how visual conventions, like
mathematical ones, change.
In the interpretive commentary, Netz unfolds the techniques
of interpretation developed in The Shaping of Deduction by using
small, almost indiscernible features of ancient mathematical writing
to understand and characterize it. We are told how a seemingly
coincidental feature such as repetition reveals the Greeks’ discomfort about generalizing between geometrical objects, while a
reference to “the polygon” of a previous proposition introduces
a hesitant generality. The style of the commentary is clear, informative and imaginative and contains discussions of a wealth
of concepts that are of interest for general studies of Greek
mathematics as well as for particular problems raised by the text.
The commentary gives the reader a feel for recurrent themes—
for instance the important tension between generality and particularity—and for the way in which details come together to
create an understanding of the thinking behind Archimedean
geometry. Throughout the commentary Netz demonstrates convincingly that the devil is in the detail, but also that he has no
fear of the devil.
Netz’ approach requires a committed reader. “There are,” he
writes, “many possible barriers to the reading of a text in a foreign language, and the purpose of a scholarly translation is as I
understand it to remove all barriers having to do with the foreign language itself, leaving all other barriers intact.”5 I believe
this to be the right approach to translation, but there can be no
doubt that the barriers are still intact. Both of Netz’ works require not only tenacity, but also background knowledge of Greek
geometry. While the author guides the reader through the proofs
and problems with depth and lucidity, he does not provide enough
help to understand the reasoning of the propositions for readers
not at home with Greek geometrical techniques from Euclid’s
Elements onwards. These skills, however, are not easily learnt,
because most works offering introductions to Greek geometry
are based on exactly the much-maligned geometrical algebra
that this account seeks to replace.
Netz admonishes that “…the way to understand Greek mathematics is not by transforming it into our mathematical language
5
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Netz, The Works, 3.
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but, on the contrary, by becoming ourselves proficient in the
mathematical language of the ancients.”6 But doing Greek mathematics Greek-style is not for the faint-hearted. Until there is a
more basic introduction to this approach to Greek mathematics,
mathematically inclined readers may need to seek assistance in
Eduard J. Dijksterhuis’ Archimedes before they “go Greek.”7
Transformations in Practice—Transformations in History
This focus on practice and original readings is more than a
modern mind’s craving for authenticity or a philologist’s search
for an elusive original hand. Netz’ analysis and attention to practice also form a compelling programme for understanding how
Greek mathematics developed. His claim is that practices do not
just afford a way to understand mathematical content, but that
they shape mathematical content. The questions asked, the answers sought, and the tools employed form the mathematical object,
and different practices thus produce different types of mathematics. As Netz writes, “[a] change of practice, then, will inevitably tend to change the mathematical object itself… .”8 Netz
argues that we can write history only if we focus on practice.
Thus the minutiae of language, letters and diagrams that Netz
studies to understand Greek mathematics also yield the key to
understanding historical change.
In The Transformation of Mathematics in the Early Mediterranean
World Netz maps a global story of the transformation of ancient
mathematics onto a case study of the transmission of a particular
problem mentioned in Sphere and Cylinder II. The problem, which
concerns the division of a sphere according to a given ratio, is
first solved by Archimedes and the two Greek mathematicians
Dionysodorus and Diocles; it is then included in Eutocius’ sixthcentury commentary; and finally solved and discussed in Arabic
accounts by Khwarizmi and most prominently Khayyam.
Netz’ basic thesis derives from Jacob Klein’s Greek Mathematical
Thought and the Origins of Algebra.9 Klein describes ancient and
6
Netz, The Transformation, 3.
Eduard J. Dijksterhuis, Archimedes (Copenhagen, 1956 / Princeton, 1987).
8
Netz, The Transformation, 190.
9
Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (Cambridge
MA, 1968); first published in German in 1934-36.
7
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modern mathematics as two separate phenomena with radically
different approaches to the objects of mathematics. Where Greek
mathematics is steeped in “geometrical thinking” and refers directly to concrete objects, modern mathematics is characterized
by “algebraic thinking” and refers to symbols that only indirectly
refer to objects—modern mathematics is “second-order” in nature. Klein locates the reason behind this change in terms of
contrasting philosophical beliefs, but does not offer an historical
account of how the shift took place.
Netz’ brilliant idea is to take on board the division between
ancient and modern mathematics described by Klein, but to argue
for a continuous transformation forged by changes in mathematical
practice. The main thesis is that ancient Greek mathematicians
worked directly with particular geometrical objects, precisely because
they tried to produce original work in isolation from other
mathematicians’ solutions and therefore faced ‘geometrical reality’ unmediated. In contrast, mathematicians of Late Antiquity
categorized solutions and established functional relationships
because they compared and systematized works of the past, and
thus saw the second-order relationships that form the foundation of the development of algebra. The argument is attractive
in its simplicity. An immediate approach to geometrical objects
leads to direct manipulation with particular geometrical objects,
singular solutions and divergence in methods. A text-critical and
comparative approach leads mathematicians to see connections
between solutions and mathematical objects and to generalize
and functionalize the objects of mathematics. This big picture
combined with the detailed discussion of changes at the microlevel of propositions is one of the undisputed strengths of the
book.
Kuhn’s The Structure of Scientific Revolutions is not mentioned in
this work, but seems to be an invisible presence in the argument.10 Netz of course rejects the idea of revolution in favour of
a continuous development, but appears to adopt the idea that
scientists adhering to different practices (or paradigms) live in
different worlds: he refers to “the world of geometric problems,”
“the world of Eutocius,” and “the world of Khayyam” in his chap-
10
Thomas Kuhn, The Structure of Scientific Revolutions, 2nd edition (Chicago,
1970) is cited by Netz as a source of inspiration in Shaping, 1-8.
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ter headings. The question is whether those worlds are made to
look too incommensurable.
Calculations, Commentators and Contexts: Muddying the Waters
Maybe the changes in practices were not so abrupt? Perhaps it
is possible to gain an understanding of the historical factors that
turned Greek geometry towards the commentator practices of
Late Antiquity and the Middle Ages? Netz’ story provides a tight
argument, which with awe-inspiring lucidity creates a global history out of a single strand in the history of mathematics. My
suggestion here is that the transformation of mathematics from
ancient Greek geometry to the beginnings of Arabic algebraic
mathematics is a richer story, and that Netz perhaps pushes parts
of the story too far out to the periphery. There are two problematic aspects in particular:
1) Netz’ story is purely diachronic and follows a single strand
of history only. I would argue that other traditions played
significant roles in the transformation from a geometrical
to an algebraic approach.
2) The analysis is purely textual and does not consider extratextual features such as changes in audience and in the
social situation of the authors.
I shall first consider the role of the traditions where calculation
and geometrical work are combined, and secondly aspects of the
historical background of Late Antique and Medieval writers.
Netz links the development of algebra to approaches that generalize the notion of magnitude from particular geometrical objects.
In his analyses of Archimedes, Dionysodorus, and Diocles Netz
finds an interesting dynamic between the geometrical style favoured by Greek geometers and the introduction of certain abstract
considerations. The complexity of certain problems pushes the
envelope of geometrical practice to allow for generalizations that
depart from the particular geometrical situation. As the problem
is assimilated by Eutocius and solved by Khwarizmi and Khayyam,
mathematical objects are further generalized to be like the abstract magnitudes of modern mathematics.
In Netz’s account it is the internal tension in Archimedes’
battle with the problem combined with the ‘second-order’ approach of Late Antiquity that engenders this change. Other—in
my opinion more important—traditions are pushed to the mar-
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gins of the story. There are for instance mathematical texts where
some of the same practices that characterize the transformation
to algebra are found. Netz himself points to the traditions of
professional calculators and to Hero of Alexandria’s mix of calculation and geometry. Hero’s work shows that there was an
overlap between traditions of calculators and of Archimedean
geometry, as he draws on both.11 This overlap is important to
Netz’ story, because the change from geometrical object to generalized magnitude begins with an unusual expression in Archimedes. He considers a square “on” (epi) a line—an expression
that comes very close to multiplying an area with a line to get a
volume and thereby to abstracting magnitudes from geometrical
figures. This expression is frequent in Hero’s work, but Netz
writes off Hero’s use of calculation as an incidental variation of
Euclidean mathematics, while he argues that Archimedes’ adoption of the same phrase marks a different register and therefore
creates the possibility of mathematical change.12
Similarly Netz deems the calculations, which are found in
Khwarizmi’s work and are related to Hero’s work,13 irrelevant to
the transformation claiming that geometry and calculation puzzles belong to separate traditions: “In the Ancient Greek World,
the two forms—literary geometry and oral calculation puzzles—
subsisted separately, with occasional contacts, sometimes exploited
for deliberate effect, as we have seen in the case of Archimedes’
solution to the problem of areas and lines.”14 Netz thus wants to
keep these two worlds separate and make the development of
algebra solely a product of Archimedes’ introduction of a different register.
It seems unlikely that when authors such as Khwarizmi and
Hero deliberately combine traditions, that it is the use of a
Heronian phrase deep in an Archimedean problem that drives
the whole transformation. It is fascinating that the crossing of
11
Karin Tybjerg, “Hero of Alexandria’s Geometry of Mechanics,” Apeiron, 37
(2004), 29-56. See also Jens Høyrup, “Hero, Ps-Hero, and Near-Eastern Practical
Geometry,” Antike Naturwissenschaft und ihre Rezeption, 7 (1997), 67-93; and Bernard Vitrac, “Euclide et Héron: Deux approches de l’enseignement des mathématiques dans l’antiquité?” in Science et vie intellectuelle à Alexandrie (Ier - IIIe siècle
après J.-C.), ed. Gilbert Argoud (Saint-Étienne, 1994), 121-145.
12
Netz, The Transformation, 113-114.
13
Otto Neugebauer, The Exact Sciences of Antiquity, 2nd edition (Providence,
1957).
14
Netz, The Transformation, 141.
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registers can be found at this micro-level in Archimedes, but to
locate the transformation of mathematics in one Archimedean
problem and exclude works that combine calculation and Archimedean practices is to attach too much weight to the Archimedean line of the story.
Netz’ analysis of the practices of the Late Antique commentator tradition and Arabic mathematics hinges on the notion of
“deuteronomic.” Deuteronomic texts are defined broadly as “texts
depending fundamentally on earlier texts.”15 This practice spans,
according to Netz, the period from the third to the fifteenth
century AD and covers a wealth of genres such as commentary,
translation, epitomies, editions, and encyclopaedic collections.
Deuteronomic authors work on mathematical texts rather directly
with mathematical objects, and their work is focused on systematizing and completing these texts rather than creating their own
solutions. Their practices include filling gaps in arguments, adding further cases, and standardizing style and content—the overall aim being that of systematizing works of the past.
There is, however, the danger that the delineation of such a
deuteronomic tradition may result in including both too little
and too much. First, it is not obvious that such a tradition can
indeed be isolated, as all mathematical authors depend on previous work to some extent.16 The division may therefore overemphasize the dichotomy between classical and deuteronomic authors.
Second, the category may include too much and jumble together
works that differ in approach and belong to different cultural
contexts. This makes it difficult to situate historical change simply in the deuteronomic practices.
The sharp distinction drawn by Netz can also come to inform
the way sources are evaluated and established. The fact that Diocles
and Dionysodorus produced their own solutions to the Archimedean problem is explained by their looking for original and
different solutions, but it could also be presented as them filling
out the gap left by Archimedes in an almost deuteronomic manner.
Eutocius just quoted the solution, but then his starting point was
15
This definition and characterization of Late Antiquity was previously set out
in Reviel Netz, “Deuteronomic Texts: Late Antiquity and the History of Mathematics,” Revue d’histoire des mathématiques, 4 (1998), 261-88.
16
A point made in Karine Chemla, “Commentaires, éditions et autre texts
seconds: quel enjeu pour l’histoire des mathématiques? Reflexions inspirées par
la note de Reviel Netz,” Revue d’histoire des mathématiques, 5 (1999), 127-148.
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different, because he believed he had found Archimedes’ own
solution. Turning to textual matters, Archimedes’ own solution
is only preserved in Eutocius’ commentary, and Archimedes’ work
has to be extracted from Eutocius’. Netz does so with utmost
scholarly caution, but to a certain extent the decision depends
on an evaluation of the merits of the two authors, and on that
point Netz’ judgement is very clear.
Eutocius’ work is not treated kindly. It is repeatedly characterized as pedantic, unoriginal and mechanical, and where Archimedes is assumed to be conscious of small changes in the linguistic
register, Eutocius is presented as driving historical change in a
wholly unconscious manner—“he stumbles across functions and
equations without ever thinking about it.”17 Any goals that Eutocius
may have had—apart from those that characterize the typical
deuteronomic author—are erased from the account.
Alain Bernard has recently criticized Netz’ approach to Late
Antiquity for not taking the specific background of the practices
into account.18 He argues that the Roman invasion was a primary
influence on literary culture of Late Antiquity, and for the Greek
speakers living in a world dominated by Roman rule the study of
Greek works became a way of asserting their Greekness. An
education in Greek literary culture, paideia, was central and a
rhetoric of bookish education the order of the day. This overcondensed account already adds something to our view of Eutocius:
although his programme may be conservative, his activities can
no longer be pejoratively characterized as “merely pedantic,”
and we get a sense of some of the cultural factors that may given
importance to the commentary as a genre.
Style is also strongly dependent on historical context, and locating the difference between works purely in the textual practices means that texts written for entirely different audiences are
compared directly. Most of Archimedes’ work appears to have
been written for a small elite community of experts. In Sphere
and Cylinder I he invites only those “friendly with” or “engaged
in mathematics” to approach the work. This must be contrasted
with the work of Khwarizmi who as a court-scholar would—at
17
Netz, The Transformation, 64 (Netz’ italics).
Alain Bernard, “Comment définir la nature des textes mathématiques de
l’antiquité grecque tardive? Proposition de réforme de la notion de ‘textes deutéronomiques’,” Revue d’histoire des mathématiques, 9 (2003), 131-173.
18
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least partly—be addressing the caliph, whose mathematical abilities were not comparable with those of Archimedes’ correspondents.19 A fairer comparison would be with Archimedes’ Sand-reckoner,
which was dedicated to King Gelon. Eutocius’ audience is different again, given that his text may have been a part of a teaching
programme in Greek paideia.
Different historical settings thus separate Eutocius, Khwarizmi,
and Khayyam whose practices and goals are otherwise in danger
of being set apart simply by their relative intellect and “ambition.” Anchoring the story of the transformation of mathematics
more in specific historical practices may help to see the historical continuity between the Hellenistic period and Late Antiquity
and to differentiate Eutocius from Arabic treatments on a richer
basis than that of perceived mathematical talent.
Netz is, of course, aware of the issue concerning the historical
setting. In a recent survey of new approaches, he includes, for
instance, sensitivity to historical context among the important
features of recent work in the history of mathematics.20 Adding
multiple layers and contexts to his account might have muddied
the waters and obscured the line of argument, which would
certainly have been a great loss. However, the title of the historical work—The Transformation of Mathematics in the Early Mediterranean World—is perhaps a little optimistic and somewhat inaccurate
(Baghdad, for instance, is far from the Mediterranean). In order
to do justice to the title, such a study would have required consideration of more cultures, more traditions, and a firmer historical anchoring of the texts. But as a textual study and a case
study of the trajectory and development of a mathematical problem, the book is outstanding.
History and Beauty
Netz’ view exists in a dynamic tension between two stances that
do not often co-exist in the history of mathematics: Netz is uncompromisingly anti-relativist and uncompromisingly historical.
In his account mathematics is deeply embedded in practices,
19
Khwarizmi’s On the Art of Al-Jabr wa l-Mukabala is dedicated to Caliph AlMamun (reigned AD 813-33).
20
Reviel Netz, “Introduction: The History of Early Mathematics—Ways of ReWriting,” Science in Context, 16 (2003), 275-86.
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values, and questions that change with history. At the same time
he confidently divides the actors of his story into geniuses and
water-carriers, allowing some characteristics to transcend the
historical context—the aesthetic accomplishment and brilliance
of mathematical work.
I shall not question Archimedes’ genius here—no one would.
But perhaps it is easy to overrate the historical significance of
genius. This is not to say that Archimedes’ influence has not
been profound, but simply that the history of mathematics does
not coincide with the history of Archimedes’ work and cannot be
told exclusively from this vantage point. Archimedes alone cannot provide Netz with the solid point from which to move the
world.
But Netz’ interest is perhaps not primarily in Archimedes’
historical value. It is clear from his treatment that he ascribes a
value to Archimedes that goes beyond his consequences for the
history of mathematics. At the end of the first book of Sphere and
Cylinder a scribe added: “The hidden rhetoric is that nothing can
be more beautiful than this, unadorned mathematical text.”21
This is the not-so-hidden rhetoric of Netz’ work—that the beauty
of the unadorned mathematical text is the point of Archimedes.
21
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Netz, The Works, 183.
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