


In two articles (Brisson–Ofman1, 2), we have analyzed the so–called ‘mathematical passage’ of Plato’s Theaetetus, the first dialogue of a trilogy including the Sophist and the Statesman. In the present article, we study an important point in more detail, the ‘definition’ of ‘powers’ (‘dynameis’). While in (Brisson–Ofman2), it was shown that the different steps to get the definition are mathematically and philosophically incorrect, we try here to explain why the definition itself is problematic. However, it is the first example, at least in the trilogy, of a definition by division. Our works on Theaetetus’ ‘mathematical passage’, including the present one, challenge the so–called ‘Main Standard Interpretation’. In particular, following (Ofman 2014), we question the claim that Plato praises and glorifies both the mathematician Theodorus and the young Theaetetus. Conversely, our analysis highlights the main cause of some generally overlooked failures in the dialogue: the forgetting of the ‘logos’, first in the ‘mathematical part’, then in the following discussion, and finally the failure of the four successive attempts of its definition at the end of the dialogue. Namely, the passage is closely connected with the problems studied at the end of the dialogue, but also to the two other parts of the trilogy through the method of “definition by division”.Our conclusions are different from the usual ones, because our analysis is conducted simultaneously from the philosophical, historical and mathematical points of view. It had been usually considered either as an excursus by historians of philosophy (Burnyeat 1978), an isolated text by historians of mathematics (Knorr 1975), or a pretext to discuss some astute developments in modern mathematics by mathematicians (Kahane 1985).



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Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus
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Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus
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                                                        trópos • anno X • numero 2 • 2017 – ISBN 978-88-255-1253-3
                                                       DOI 10.4399/97888255125338 – pag. 125-144 (dicembre 2017)




         Powers and Division in the ‘Mathematical Part’
                                    of Plato’s Theaetetus
                                                                                  Luc Brisson*
                                                                              Salomon Ofman**




Abstract: In two articles (Brisson–Ofman1, 2), we have analyzed the so–called
‘mathematical passage’ of Plato’s Theaetetus, the first dialogue of a trilogy
including the Sophist and the Statesman. In the present article, we study an
important point in more detail, the ‘definition’ of ‘powers’ (‘dunàmeic’). While
in (Brisson–Ofman2), it was shown that the different steps to get the definition
are mathematically and philosophically incorrect, we try here to explain why
the definition itself is problematic. However, it is the first example, at least in
the trilogy, of a definition by division. Our works on Theaetetus’ ‘mathematical
passage’, including the present one, challenge the so–called ‘Main Standard
Interpretation’. In particular, following (Ofman 2014), we question the claim
that Plato praises and glorifies both the mathematician Theodorus and the
young Theaetetus. Conversely, our analysis highlights the main cause of some
generally overlooked failures in the dialogue: the forgetting of the ‘logos’, first
in the ‘mathematical part’, then in the following discussion, and finally the
failure of the four successive attempts of its definition at the end of the dialogue.
Namely, the passage is closely connected with the problems studied at the
end of the dialogue, but also to the two other parts of the trilogy through the
method of “definition by division”.
    Our conclusions are diVerent from the usual ones, because our analysis
is conducted simultaneously from the philosophical, historical and mathe-
matical points of view. It had been usually considered either as an excursus
by historians of philosophy (Burnyeat 1978), an isolated text by historians of
mathematics (Knorr 1975), or a pretext to discuss some astute developments
in modern mathematics by mathematicians (Kahane 1985).

Keywords: Theaetetus, Plato, mathematics, powers, divisions.
    ⇤
      Centre Jean Pépin, CNRS–UMR 8230 (lbrisson@agalma.net).
   ⇤⇤
      CNRS – Institut mathématique de Jussieu–Paris Rive Gauche, HSM, Sorbonne Université and
Université Sorbonne Paris Cité (salomon.ofman@imj–prg.fr).


                                                                                                            125
126     Luc Brisson, Salomon Ofman


Introduction

Plato’s three dialogues, the Theaetetus, the Sophist and the Statesman are
explicitly connected at the beginning of the Statesman1 . Many of their
characters are the same: Socrates, the mathematician Theodorus from
Cyrene, a Greek settlement in the north of actual Libya, Theaetetus a
young Athenian, and a friend of him, Socrates, namesake of the first one2 .
This Socrates is silent here, but will take over Theaetetus in the Statesman
(258b). The Theaetetus ends on Socrates’ call for a next day meeting, while
the Sophist opens on Theodorus reminding this announcement and in-
troducing someone coming from Elea3 . Socrates adds he would like to
hear him about the sophist, the statesman and the philosopher (Sophist,
217a). Last, the Statesman begins by Theodorus’ remark that, after the
sophist, the Stranger will search the definition of both the statesman and
the philosopher, allowing Socrates to berate him as misapplied his mathe-
matical knowledge, for he gives the “equal worth (“shc Çx–ac)” to things
which are “beyond any proportion (Çfestêsin ´ katÄ tòn Çnalog–an)”
(257a3–b7), a clear reference to the lesson on incommensurability in the
“mathematical part” of the Theaetetus.
   In this dialogue, the interlocutors try to find a definition of ‘‚pist†mh’
which may be understood either by ‘scientific knowledge’ or, as usually
in English–language texts, simply by ‘knowledge’. It is made clear from
the very beginning that Socrates bears in mind the first meaning, while
Theaetetus the last one. Theaetetus’ answer to the question ‘what is
knowledge’ consists in a list including some hand–crafted techniques4 .
Socrates criticized him not only for giving examples and not a definition,
but because some of them, as skills, do not belong to the definiendum.
In his answer, Socrates replaces them, without more explanation, by
their associated theoretical sciences (146d–147a). Immediately afterwards
begins the so–called ‘mathematical passage’.

     1. Modern translations assume the three dialogues to be a trilogy for instance, Klein 1977 and
Bernadete 2008. This chronology concerns only their literary presentation and entails nothing on
the dating of their composition (Klein 1977: 3).
     2. To distinguish him from the older, its name will be written in italic.
     3. He is simply called ‘xËnoc’ in the dialogue. The usual translation is the “Stanger”, but it is
also rendered by the “Visitor” in some modern translations (for a discussion about this question,
cf. Sayre 2006: 11, n.1). We follow here the usual translation.
     4. ‘te ka» a… t¿n ällwn dhmiourg¿n tËqna…’ (146d1–2); cf. Sophist 221d, in particular
221d1–4, 231b, 267e, contra the Statesman, where they are used sometimes as synonymies, maybe
because of the defects in the language (207d) or the different conceptions of the different in-
terlocutors. For Mary Louise Gill these dialogues are some dialectical exercises for apprentice
dialecticians (2016). However, these questions would need a development beyond the purposes of
the present article.
                    Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus                  127


    The analysis presented here concerns the method used to get the def-
inition of ‘powers’ (‘dunàmeic’). It diVers from the modern doxa or the
so–called Plato’s ‘Main Standard Interpretation’ (Rowe 2015: xiii)5 . As in
our other studies (Brisson–Ofman1, 2; Ofman 2010, 2014), we do not start
from some supposed global interpretation of Plato’s philosophy, but we take
Plato’s words seriously and stick to his words to try to understand the text
the best we can, contrary to the modern ‘hybris’ grounded of some claiming
with Kant to ‘better understand the text than (Plato) himself ’6 . The main
diYculty is the necessity, as in the whole passage, to study it simultaneously
through three diVerent fields: mathematics, philosophy and history.


1. The ‘powers’

According to the ‘Modern Standard Interpretation’ or its shortcut MSI7, the
‘mathematical passage’ of the dialogue (147d–148b) is considered at best as pay-
ing some respect to Plato’s friends, the two (actual or future) mathematicians
Theodorus and Theaetetus, or even as being some distraction and rhetor-
ical nicety with respect to its real philosophical content, the definition of
knowledge. In another article (Brisson–Ofman2), we tried to show that Plato’s
attitude is critical or at least ambivalent towards both Theodorus and Theaete-
tus, an important issue to understand the rest of the dialogue. As a matter of
fact, nothing in the text supports the MSI claims. Namely, once again, it points
to the exact opposite. Theodorus is presented as the kind of teachers filling
the heads of his pupils but unable to develop their own power of thinking8.

      5. The literature about the trilogy is too large to be listed here. A good introduction for the
Theaetetus and the Sophist will be found in Rowe (2015: xli–xlii) and for the Statesman in Brisson
(2003: 289–301).
      6. ‘When we compare the thoughts that an author expresses about a subject, in ordinary speech
as well as in writings, it is not at all unusual to find that we understand him even better than he
understood himself, since he may not have determined his concept suYciently and hence sometimes
spoke, or even thought, contrary to his own intention.’ (Guyer–Wood 1998, Transcendental dialectic, I,
1, B370, p. 396).
      7. Though it is not part of it, since some scholars consider Fowler’s book (1999) as some guide to
understand the history of irrationality or even to understand the method used to get the incommensu-
rability in Theaetetus’ mathematical passage, let us make just two brief remarks. First according to the
author himself, his book is highly speculative and has no historical basis (pp. 18, 42, 369, 371, 373ff.). As a
matter of fact, it is an attempt to show that an important part of early Greek mathematics could have
been obtained through the so–called ‘anthyphairesis’ (for a good analysis of it (cf. Unguru 2002). Now
concerning the Theaetetus, he writes himself that it was too difficult a task to try to connect this method
to the dialogue (p. 381, note 24), and moreover, is not himself convinced that it was used by Theodorus
or Theaetetus (p. 380). For a refutation of the ‘anthyphairesis’, as a supposed method used in Thaetetus
(cf. Knorr 1975; Ofman 2014; Brisson–Ofman1).
      8. These ‘wise and inspired men’ (‘sofoÿc te ka» jespes–oic Çndràsi’) to whom Socrates
sends some of his aspiring disciples for a preliminary training (151b).
128      Luc Brisson, Salomon Ofman


    Another common trend of modern interpretations is the claim that the
exact meaning of the term ‘powers’(‘dunàmeic’)9 , both in mathematics
and in Ancient Greek, is a crucial and obligatory step to understand the
passage10 . In (Brisson–Ofman2), we explained why this hunt for an elusive
unique meaning is based on some misunderstandings. We want to show
here it is an important term indeed, but mainly in order to appreciate
Plato’s critical approach of mathematics ‘à la’ Theodorus as well as the
deep connection of the passage to the rest of the dialogue and to the two
other parts of the trilogy.
    Theaetetus (147d–148b) may be divided into two parts. The first concerns
the account of Theodorus’ mathematical lesson given for his young pupils
(147d3–d6) (Brisson–Ofman1). The second is about Theaetetus and his friend
Socrates working together on the lesson, and trying to go further (147e5–148a4)
(Brisson–Ofman2)11. Let us give a brief summary of the first part.
    Historians and commentators have fought for a long time on the term
‘powers’ (‘dunàmeic’). They can be essentially distributed in two groups: for
some, ‘powers’ would mean the ‘sides’ of squares, for others the ‘squares’
themselves. Contrary to the usual interpretation postulating a unique mean-
ing, a thorough analysis of the passage leads to the conclusion that powers’
(‘dunàmeic’) has multiple meanings according to its diVerent uses12 . As the
Stranger makes clear to Theaetetus in the Sophist, such kinds of ambiguities
are the reason for the search of definitions: to get a mutual understanding
on what the definiendum is (218b–c)13 .

      — Its first instance, the subject of Theodorus’ lesson (‘Per» dunàme∏n
        ti ômÿn JeÏdwroc Ìde Ígrafe’, 147d3), is vague: Theodorus drew
        (probably with a stick on the sand) some ‘figures’ representing the
        squares that involves both meanings, for the drawing of a figure
        shows both its sides and its area.
     9. Some scholars argue against this translation, but it is the usual one in ancient Greek.
    10. Arpád Szabó writes outright: ‘my translation, as well as my interpretation of the text,
is based primarily on the definition of a single word (‘d‘namic’)’ (1978: 36). Myles Burnyeat
speaks about ‘a vexed issue of terminology’ and gives a large part of his analysis to this problem
(1978: 495–502); ‘Among the most debated single terms of ancient Greek mathematics is the word
dynamis’ (Høyrup 1990: 202; cf. also, Caveing 1994: 172–173; Knorr 1975: 94; Vitrac 2008: 73–148).
    11. Here we consider the particular point of the definition by divisions of the ‘powers’ that we did
not detail in this previous article.
    12. This has been already argued, for instance in (Allman1877: 271), but for diVerent reasons,
principally Plato’s weaknesses in mathematics. Paul Tannery claimed the inconsistencies may result
from a not well–defined mathematics vocabulary at Plato’s time (1876, I: 33) and also a series of
mistakes by the copyists.
    13. Cf. the two kinds of ‘love’ in Phaedrus, one ‘which is very justly reviled’ (‘‚loidÏrhsen màl>
‚n d–k˘’) and another, a ‘divine’ one ‘having the same name (Âm∏numon’) as the first’ (266a); or
the two kinds of ‘sophistry’ in the Sophist (231b).
                  Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus          129


    — The second (‘‚peidò äpeiroi t‰ pl®joc a… dunàmeic ‚fa–nonto’,
      147d7–8) is about the infinite quantity of ‘powers’. Once again, it
      is vague but its meaning is either the sides or the squares of areas
      non–perfect integers only, or eventually both, but in any case, diVer-
      ent from the first one.
    — The third (‘Ìt˙ pàsac ta‘tac prosagore‘somen tÄc dunàmeic’,
      147e1) introduces what Theaetetus and his friend Socrates learnt
      from Theodorus’ lesson. It means beyond doubt the sides of the
      squares.
    — The last one (‘Ìsai d‡ t‰n ·terom†kh, dunàmeic’ 148a7–8) concludes
      the boys’ work. A definition of ‘powers’ is eventually given: they are
      the sides of the squares whose areas are of non–perfect integers.

    It is not a sequence of more and more precise meanings ending in the
definition; for instance, the third case is not a consequence of the second. It
is already a hint that the boys’ work is not so methodical14 .
    Let us now consider how, using division, Socrates and Theaetetus come
to the last, namely the only one, explicit definition of ‘powers’.



2. The division

‘Division’15 plays a fundamental role in the Sophist as a tool of the dialectic
method, as well as in the Statesman where it is conceptualized16 . It is also
used in the Phaedrus (266b) and in the Philebus. While there have been
many studies of this method in these dialogues, not much is said about it
in the Theaetetus, and more particularly its ‘mathematical part’, though it
plays a fundamental role in the second part of the passage. Namely, it is
the method used by Theaetetus and his friend Socrates to get a definition
of the ‘powers’.
    First, let us consider Plato’s text itself17 .

     14. The possibility that it is not so much the boys who will be judged but their master is left
open by Socrates’ exclamation at the end of the passage: ‘I think Theodorus will not be found liable
to an action for false witness’. (148b).
     15. Namely, what is usually called ‘division’ in the literature refers rather to ‘dichotomy’
i.e. the particular division in two parts. It is connected to an important tool in arithmetic, the
sequence of successive divisions by 2 of any integer (see Ofman 2010: 104–112).
     16. The term ‘methodos’ appears twice in the Sophist and Aristotle refers to it as the ‘method
of division’ (‘ô diÄ t¿n diairËsewn ÂdÏc’, Post. An. II, 5, 91b12).
     17. All the translations of Theaetetus’ ‘mathematical part’ are from (Brisson–Ofman1, 2).
130       Luc Brisson, Salomon Ofman


      Theaetetus (1) We divided the integer18 in its totality in two parts19 . One which has
      the power to be the product of an equal times an equal20 (148a) we likened to the
      square as a figure, and we called it square and equilateral.21
      Socrates Good, so far.
      Theaetetus (2) Then, the one in between, such as three or five or anyone which has
      not the power to be a product of an equal times an equal, but is the product of a
      greater times a lesser, or a lesser times a greater,22 that is always encompassed by a
      greater side and a smaller one, we likened it this time to a rectangular figure,23 so
      that we called it a rectangular integer.24
      Socrates That’s excellent. But how did you go on?
      Theaetetus (3) All the lines squaring an equilateral and plane integer, we defined as
      ‘length’, while we defined as ‘powers’ the ones squaring the rectangular, because,
      although not commensurable as length with the formers, (148b) they are commen-
      surable as areas that they have the power to produce. And it is the same in the case
      of solids.25

2.1. The question at stake

The boys considered the ‘integer in its totality’ (‘Çrijm‰n pànta’) and they
‘divided it into two’ (‘d–qa dielàbomen’). What does it mean?
    The sentence (1) has to be connected to the long analysis at the end
of the dialogue on the relation between the ‘whole’ and the ‘parts’ or the
‘compound’ and its ‘elements’ (202e–207c). Socrates emphasizes there the
importance of the question, requesting Theaetetus’ full attention26 . First
he asks him if he agrees that the ‘consummate arithmetician knows all
the integers (ti pàntac Çrijmo‘c)’, adding ‘because he has the knowledge

    18. To avoid the confusion attached to the modern very large notion of ‘number’, we will always
translate ‘ÇrijmÏc’ by ‘integer’.
    19. Cf. infra, §2.2.
    20. The same sentence ‘is the product of an equal times an equal’ (‘“son sàkic g–gnesjai’) is
found in Euclid’s Elements (for instance, book VII, def. 18). It is a shortcut for saying that the integer is
the product of the form n ⇥ n, so that its result is a perfect square.
    21. ‘T‰n Çrijm‰n pànta d–qa dielàbomen· t‰n m‡n dunàmenon “son sàkic g–gnesjai
tƒ tetrag∏n˙ t‰ sq®ma Çpeikàsantec tetràgwnÏn te ka» sÏpleuron prose–pomen.’
    22. A shortcut (cf. also note 20, supra) to mean that the integer can only be written as a product
of two diVerent integers i.e. it is any integer except the perfect squares. Nevertheless, the terms
‘equal’ and ‘unequal’ are not names of objects but of relations (x is equal/unequal to y), so that it is a
hint of the utmost importance of relations or ratios (‘logoi’) in mathematics.
    23. A rectangle in a strict sense that is with unequal sides i.e. not a square.
    24. ‘T‰n to–nun metaxÃ to‘tou, ¡n ka» tÄ tr–a ka» tÄ pËnte ka» pêc Ác Çd‘natoc
“soc sàkic genËsjai, Çll’ ´ ple–wn ‚lattonàkic ´ ‚làttwn pleonàkic g–gnetai, me–zwn
d‡ ka» ‚làttwn Çe» pleurÄ aŒt‰n perilambànei, tƒ prom†kei aﬁ sq†mati Çpeikàsantec
prom†kh Çrijm‰n ‚kalËsamen.’
    25. ‘VOsai m‡n gramma» t‰n sÏpleuron ka» ‚p–pedon Çrijm‰n tetragwn–zousi, m®koc
±risàmeja, Ìsai d‡ t‰n ·terom†kh, dunàmeic, ±c m†kei m‡n oŒ summËtrouc ‚ke–naic, toÿc
d’‚pipËdoic É d‘nantai. ka» per» tÄ stereÄ ällo toio‹ton.’
    26. ‘tƒ d‡ dò ‚nte‹jen ¢dh prÏssqec t‰n no‹n’ (198b).
                    Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus               131


of all integers in his soul (pàntwn gÄr Çrijm¿n e sin aŒtƒ ‚n t¨ yuq¨
‚pist®mai)’ (198b). He gets a strongly positive answer from Theaetetus.
    Though generally unnoticed, such a statement is stunning, especially
coming from someone considered as a brilliant student in mathematics.
Since the integers are infinite in number, how would it be possible to know
all the integers, a fortiori to have them in soul?27

2.2. The problems of the division

Then the discussion switches to the diVerences between ‘the all’ (‘t‰ pên’),
‘the totality’ (‘tÄ pànta’) and ‘the whole’ (‘t‰ Ìlon’) (204a–205a), Socrates
asking (‘pÏteron’) Theaetetus if the ‘all’ is identical (‘aŒtÏn’) or ‘diVerent’
(‘Èteron’) of the ‘whole’ (204a–e). This time, a much less assertive Theaete-
tus chooses tentatively the second, changing his mind once again at the
end of a long questioning by Socrates. He finally agrees that all these three
terms (‘the all’, ‘the totality’, ‘the whole’) are the same28 .
    This development starts from the hypothesis that any ‘perfectly scien-
tific knowledge’ (‘tele–wc (. . . ) ‚pist†mhn’) is a compound of elements
that are ‘irrational and unknowable’ (‘äloga ka» ägnwsta’)’ and can
only ‘be perceived through the senses’ (‘a sjhtà’) (202d). Moreover, it
entails forcefully that the ‘integer in its totality’ is, for Theaetetus, nothing
more than all its elements i.e. all the integers. This identification corre-
sponds to Socrates’ example, at the end of the dialogue, of the wagon
which is certainly more than ‘hundred pieces of wood’ (207a–208c). What
is lost is the layout between them i.e. their ‘relations’ (‘logoi’)29 .


    27. Let us emphasize that it does not mean that it is impossible to know some properties on the
whole integers. For instance, we know that all the integers are either odd or even. But it is evidently
impossible to know all the integers: for instance, it is already impossible to know (at least till now) all
the divisors of any integer chosen at random, and it is certainly impossible to have them in a human
mind, not even in the huge memory of modern computers.
    28. The first definitive answer of Theaetetus to Socrates is that the combination is the ‘complete
totality’ (‘tÄ âpanta Ímoige doko‹men.’, 203c). Only when Socrates shows him that this answer
leads to, as the boy says, ‘something monstrous and absurd’ (‘ÇllÄ dein‰n ka» älogon’), he
accepts to change his mind once again (203e). Then, he reluctantly agrees to consider the other
branch of the alternative ‘because perhaps that will be better than the other way.’ (‘ka» tàqa g>
ãn mêllon o’twc ´ ’ke–nwc Íqoi.’, 203e), to conclude: ‘I am not sure; but you tell me to answer
boldly, so I take the risk and say that they are diVerent.’ (‘Íqw m‡n oŒd‡n safËc, Ìti d‡ kele‘eic
proj‘mwc Çpokr–nasjai, parakindune‘wn lËgw Ìti Èteron.’, 204b). For a diVerent analysis
see Harte (2002: 40–41).
    29. As a matter of fact, to get a scientific knowledge of what a wagon is, the enumeration of
its parts is not enough: the ‘logos’ (207c) is needed. And the end of the dialogue is entirely about
the meaning of this term. Conversely, Theodorus proudly says, previously in the dialogue, that
‘rather too soon’, he ‘bent away’ from the ‘logoi’ (‘ômeÿc dË pwc jêtton ‚k t¿n yil¿n lÏgwn
pr‰c tòn gewmetr–an Çpene‘samen’, (165a).
132      Luc Brisson, Salomon Ofman


     However, is it not true that the ‘integer in its totality’ is just the union
of all the integers (a modern would speak of the set of all the integers)?
A positive answer would forget the ordering, so that the integers would
be arbitrarily given. Thus, for instance, Theodorus’ sequence of odd inte-
gers from 3, 5 to 17 could as well contain 10001 (a myriad plus one) than
730 . Such an oversight will impede the boys’ next definition, but it entails
also an ambiguity in the first sentence, the division of ‘the integer in its
totality’. As the compound (‘the integer’) and its elements (‘the integers’)
are homonymous, the sentence may mean:

      — The set of all integers is divided into two subsets.
      — Each and every integer is divided by two.

Nevertheless, the following sentences will make it clear that Theaetetus
means the first: another hint about the non–rigorous character of the boys’
reflections, the very beginning of their ‘definition’ is ambiguous i.e. not
well–defined.


3. Arithmetic and geometry

‘Power’ (‘dynamis’) is used here by the boys to relate arithmetic and geome-
try. It plays also a central role in Plato’s doctrine of ‘being’, namely to link
intellection and sense perception in the Sophist (246c–247e).

3.1. A first correspondence

The boys make an important correspondence, between arithmetic and
geometry. Leaded by the figurative names given to the class or to their
elements, the ‘square and equilateral’ integer, they associate to any of the
perfect square integers a rectilinear figure, the square whose area is this
integer. However, while in Theodorus’ lesson, all the magnitudes, lines
or surfaces, are given as a certain number of feet, here no unit is specified.
Only abstract integers are taken into consideration by the boys whose idea
seems to be that any integer n is represented by a finite line, namely n times
an arbitrary (segment of ) line u considered as the length unit31 . Then, to
    30. The examples used later are mostly about letters and syllables (as for instance ‘S’, ‘O’ and
‘SO’, 203a–e), the same words in Greek than respectively for ‘elements’ and ‘compounds’. Then the
necessity of an ordering (or relation between the letters) is evident, since ‘SO’ is certainly diVerent
from ‘OS’. It is diVerent for the integers: Ancient Greeks did not use a notation by position (cf. for
instance in Plato’s Laws, the integer 5040 is given as ‘tettaràkonta ka» pentakisqil–wn’ (738a)
contra ‘pentakisqil–wn ka» tettaràkonta’ (771c).
    31. As it is usually done in Euclid’s Elements.
                   Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus          133


n ‘having the power to be a product of an equal times an equal’ i.e. n is a
perfect square or in a modern representation n = m2 for some integer m,
they associate the square of side m32 .

3.2. The figures

The definition by divisions into two parts, or dichotomies, is largely used in
the Sophist and the Statesman. It is the privileged way to get a definition (i.e.
what a thing is, Sophist 285d–e). The Stranger in the Statesman claims that
‘by far our first and most important object should be to exalt the method
itself of ability to divide by classes’33 . Thus, its use by the boys34 seems to
show that their method is correct. Let us consider it more in details.

3.2.1. The squares

As the boys acknowledged previously (147d7, cf. supra, §1), the objects of
their study are ‘infinite in quantity’, thus it is not possible to consider the
problem ‘one case in turn’ as Theodorus did in his lesson (147d5). Hence,
their goal here is to treat simultaneously an infinite quantity of cases.
   As expected, the division gives two classes. The first is the ‘square and
equilateral’ integer, defined as the one which has ‘the power to be the
product of an equal times an equal’. Theaetetus presents them rightly as
‘having the power (‘dunàmenon’) to be’ – rather than simply ‘to be’ – ‘the
product of an equal times an equal’, for they are also always the product of
two unequal integers (as 1 and the integer itself ). For example, 4 is equal to
the product of 1 and 4 as well as to the product of 2 and 2.

    32. The length of the side is m times u. However, once the unit u is fixed, it can be forgotten
and we can say the length of the side is m. This is the common representation of integers (or more
generally magnitudes) in Euclid’s Elements. Socrate–Theaetetus’ idea could be represented graphically
in the figure below:




    33. ‘polÃ d‡ màlista ka» pr¿ton tòn mËjodon aŒtòn timên to‹ kat> e“dh dunat‰n
e⁄nai diaireÿn’ (286d). The term ‘e⁄doc’ is not easy to translate.
   34. However, the words are not the same in both cases: in the Sophist, it is ‘diaireÿn’, the same
verb used in Euclid’s Element, to define for instance the odd and the even integers, while Theaetetus
uses the verb ‘dialambànein’.
134     Luc Brisson, Salomon Ofman


3.2.2. The Rectangles35

The definition of the second class, the one of ‘rectangular integers’ in
sentence (2), raises some additional problems. Within ‘the integer in its
totality’, the boys ‘separate’ (‘dialambànein’, 147e) the integer which is a
perfect square36 , from the one that is not. This is problematic since none of
the two rules laid down by the Stranger of the Statesman for a definitional
division is respected.
     The first is to get symmetrical parts. For instance, one cannot divide the
‘mankind’ by ‘separating’ (‘Çfairo‹ntec’) Greeks and Barbarians, or the
‘integer’ by ‘cutting oV ’ (‘ÇpotemnÏmenoc’) ten thousand from all the other
integers (Statesman, 262d). A good division would be ‘to divide ‘the integer
into odd and even’37 , or ‘the mankind into male and female’38 . The mistake
in the definitional division is to ‘take oV one small part on its own, leaving
many large ones behind, and without reference to real classes’39 . Indeed,
‘it’s not safe to make thin cuts’40 , ‘it’s safer to go along cutting through
the middle of things’41 . Now, let us consider the boys’ cutting–up (another
sense of ‘dialambànein’) the ‘integer in its totality’. It is certainly neither
symmetric, nor a division in half. As a matter of fact, between two successive
perfect square integers there are many non–squares (for instance between
two successive squares as 10 000 = (100)2 and 10201 = (101)2 there are 200
diVerent non–squares or rectangular integers)42 .
     The second and most important rule is that each resulting part of the
division has a real unity. They need to form a ‘genus’ (‘gËnoc’, 262d), i.e. to
have some common characters. Barbarians are only defined as non–Greeks,
and all the integers minus ten thousand are only defined as ‘non–ten thou-
sand’ which is not a real unity. On the contrary, the odd integers and the
even integers have each a real unity, allowing a real definition43 and it is the
same with the male humans and the female humans. Now, if we consider
the ‘rectangular’ integers, their only common character is: not being ‘square’
integers. In other words, the division cut oV the ‘integer in its totality’ into
    35. A rectangle here means a figure with unequal sides i.e. excluding the squares (cf. supra,
note 23).
    36. ‘tetràgwnÏn te ka» sÏpleuron’ (148a).
    37. ‘t‰n m‡n Çrijm‰n Çrt–˙ ka» perittƒ’ (262e).
    38. ‘t¿n Çnjr∏pwn gËnoc ärreni ka» j†lei’ (ib.).
    39. ‘mò smikr‰n mÏrion „n pr‰c megàla’ (262b.)
    40. ‘ÇllÄ gàr, ¬ f–le, leptourgeÿn oŒk ÇsfalËc’ (ib.).
    41. ‘diÄ mËswn d‡ ÇsfalËrteron Ënai tËmnontac’ (ib.).
    42. Though for modern elementary logics the asymmetry is meaningless because both sets are
infinite, nevertheless see infra, Appendix.
    43. Definition VII.6 in Euclid Elements: ‘An even number is that which is divisible into two equal
parts.’; definition VII.7: ‘An odd number is that which [. . . ] diVers by a unit from an even number’.
                   Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus          135


square and non–square, as in ten thousand and non–ten thousand, or the
mankind oV into Greek and non–Greek (Barbarian).
    Another generally unnoticed problem is that the boys give two defini-
tions of ‘rectangular’ integers. The first one is purely negative: they are not
‘square’ integers. The other is not really a definition but appears as a truism:
they are the product of two unequal integers. Theaetetus emphasizes this
point, adding they are products of ‘a greater and a smaller’44 .
    The first part is obvious. Since they are not ‘square’ integers, they have
not, by definition, the power to be a product of an ‘equal times an equal’
(i.e. to be a perfect square). But why are they a product of unequal integers?
Namely, according to the popular mantra claiming that 1 was absolutely not
an integer for the Ancient Greeks, this would be flatly false, for no prime
integer45 could be considered as the product of two integers!46 Thus, clearly,
at least here for Plato, 1 is an integer, and any integer n is indeed such a
product i.e. n = n ⇥ 147 .
    Last, they repeat the correspondence done in §3.1, supra. Once an ar-
bitrarily (segment of ) line u set as the unit of length, to any ‘rectangular
integer’ n equal to the product of p and q, they associate a rectangle of sides
p and q48 .

3.3. True results, incorrect reasoning

Using the graphical names given to the two classes (or their elements)
defined previously (the ‘square and equilateral’ and the ‘rectangular’), they
associate a figure to any integer (‘square’ or ‘rectangular’)49 . However, while
the figure associated to each element of the first class (the square integer)
is well–defined (cf. supra, §3.1), the situation is diVerent for the second
class. Generally, there are several possible rectangles associated to the same
rectangular integer. For instance, since the products of 1 and 6, as well as of
2 and 3 gives 6. Thus, for any unit u fixed, both the rectangles of sides (1,6)
and (2,3) may be associated to the rectangular integer 6. The boys do not
consider this problem. However, in sentence (3), they make an important
connection between arithmetic and geometry.
    44. Cf. supra, notes 20 and 22.
    45. ‘A prime number is that which is measured by a unit alone’ (Euclid’s Elements, definition
VII.11).
    46. For instance, there would be no integers such that 3 would be equal to their product.
    47. Cf. also infra, §3.3. Double definitions are also sometimes found in Euclid’s Elements, for
instance in the definition VII.7, often blamed by historians of mathematics, of the odd integer which
either cannot be divided in half or diVers from an even integer by a unit.
    48. For the omission of u, see supra, note 32.
    49. Theodorus’ lesson is based on some drawings, and the boys tried naturally to extend this
method to any integer (Brisson–Ofman1).
136      Luc Brisson, Salomon Ofman


      a) For an integer n which ‘has the power to be the product of an equal
         times an equal’, they associate the square of side m (i.e. for n = m2 , the
         square of side m where m is an integer). They did not say it explicitly,
         but once again we have to suppose an arbitrary unit of length u is
         fixed, and to the integer n = m2 is associated the square of side m.
      b) For a ‘rectangular integer’ n = p ⇥ q, they associate first a rectangle of
         sides p and q, and then a square of same area as this rectangle50. As we
         saw, there is a problem with the choice of the rectangle. Nevertheless,
         the boys did not see it. Since they clearly know that there are in general
         many ways to get the same product51, so they thought, in their soul,
         they have a way to associate to each such integer a unique rectangle.
         Namely, in the same way as Theodorus drew the powers in his lesson,
         to each ‘rectangular integer’ n, they thought to associate the rectangle
         of sides (1,n) and then the square of area n. Since there is only one
         such rectangle, the problem of the choice between different rectangles
         disappears. This follows closely Theodorus’ construction52.

    Nevertheless, the boys did not see the problem of a possible ambiguity
in the construction, that the reader has to fix by himself. It is the first of
several cases where, while the results are true, the boys’ reasoning is not
correct53 . This example shows that, though truth is certainly a property
of knowledge, it does not characterize it, as Socrates makes clear in his
criticism of Theaetetus’ proposal of his last definition of knowledge as ‘true
opinion’54 .

    50. Cf. supra, note 32.
    51. The case of the integer 6 is considered in details in 204b10-c3.
    52. In (Brisson–Ofman1) we showed the simplest way for Theodorus to construct the ‘powers’
associated to 3 feet, 5 feet, etc. till 17 feet, was as follows: first to draw a rectangle of sides (1 foot, n feet)
(where n is equal to 3, 5,. . . , 17), then to use an immediate corollary of Pythagoras’ theorem: in a right
triangle BDO of hypotenuse OB and height DH, the rectangle on OH and HB is equal to the square
on DH (in modern notations, OH ⇥ HB = DH2 ). In particular, if we fix HB equal to the unit and OH to
n times this unit, then DH is the side of the square equals (in area) to the rectangle (1 foot, n feet). For
more details (Brisson–Ofman1). The boys generalized this method to any rectangular integer.




    53. For a much deeper mathematical problem about the same question, we refer to the Appendix
in (Brisson–Ofman2).
    54. ‘True judgement may well be knowledge. So, let that be my answear’ (‘kindune‘ei d‡ ô
Çlhjòc dÏxa ‚pist†mh e⁄nai, ka– moi to‹to Çpokekr–sjw’, 187b5–6).
                    Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus               137


4. An ‘awkward’ definition

It is remarkable that the definitions of both ‘length’ and ‘powers’, given
by Socrates–Theaetetus, disappeared from the field of mathematics, while
the problem of the commensurability/incommensurability was still central
for a very long period, at least till Euclid’s time (cf. book X of the Elements).
This is generally a hint of a problem and indeed, as we will see, the boy’s
definitions are incorrect, according to both Plato and the mathematicians of
his time.

4.1. A too ‘generous’ division

First, the division into ‘square’ and ‘rectangular’ integers is useless. As a
matter of fact, the boys would get the same result at once, if for any integer
n (‘square’ or ‘rectangular’):

    — they had associated the rectangle of sides (1, n)
    — and to this rectangle, they had associated the square of same area as
      the rectangle55 .

Once again, Theaetetus is too ‘generous’56 , giving too a long entangled
exposition57 . But there is more to come.

4.2. A problematic reasoning

At the end of the passage, the boys define the ‘powers’ and the ‘length’
(‘m®koc’). According to their definition, the latter is the side of the squares
associated to the ‘square’ integers, in other words, the sides which are
integers58 . As the boys redefine ‘length’ in an unusual way, they also redefine
‘powers’: the sides of the squares of areas non–perfect square integers. In
modern words, an element of the class ‘length’ is a length equal to an
integer (for instance 3), and a ‘power’pis a length equal to a square root of
any non–square integer (for instance 3). Since the term is here redefined,
its previous meaning must have changed; that is supplementary evidence
that its meaning cannot be unique in the passage.
    55. Cf. supra, note 52.
    56. He is ‘filÏdwroc’ (146d) because, instead of giving ‘a short (. . . ) answer, (he went) an
interminable distance round’ (147c).
    57. Cf. previous paragraph.
    58. In modern notation, a ‘square’ integer is an integer n of the form m2 , where m is some integer.
Thus, the side of the square associated to n is pm2 = m. Conversely, any integer m is the side of the
square associated to the perfect square integer n = m2 . Thus, once a unit of length u is set, the ‘length’
according to the boys’ definition is the class of all the lines of length an integer (with respect to u).
138     Luc Brisson, Salomon Ofman


    Moreover, the redefinition of ‘length’ is surprising because Theodorus’
lesson was precisely about lengths ‘incommensurable to the unit’ (147d5).
However, the boys decided to call these lengths ‘powers’. It seems senseless
as long it is not linked to some parts of Euclid’s Elements, in particular the
definitions in book X59 . In the Elements, the word ‘powers’ (‘dunàmeic’) is
not used to qualify some lines as in Theaetetus’ account but as a property
connected to a dative form (‘dunàmei’, cf. for instance def. 2 and 3, as well as
for the verb ‘d‘nasjai’ under the form ‘dunàmenai’ (‘being able’ to do/be
something) as in definition 4). Definition 3 opposes lines ‘incommensurable
in length only’60 to lines ‘incommensurable in power’61 . Moreover, this
definition contains a result: the existence of an infinite quantity of lines
‘incommensurable in length only’62 . Thus, the boys changed the usual
meaning of ‘in length’ and ‘in power’ for ‘length’ and ‘powers’. In other
words, the lines commensurable to the unit are defined as elements of the
class ‘length’, while the incommensurable lines are defined as ‘powers’.
Plato shows, once again (cf. supra, note 22), the boys replacing a relation
(‘logos’) between objects by the objects themselves.
    Both definitions result from the division of the class of the ‘integer in its
totality’. But to which class do the elements of both classes (‘length’ and
‘powers’) belong together? Though Theaetetus did not name the elements
of the class ‘length’, we know they are, as well as the ‘powers’, sides of
squares, hence lines. Thus, what we get finally are lines that are either an
integer (i.e. the elements of the class ‘length’), or that are not such, but ‘have
the power to produce’ squares that are a whole number (i.e. an integer) of
times the corresponding square–unit (i.e. the ‘powers’)63 .
    It is not easy to define such a mixed class because some of its elements
come from arithmetic, others from geometry. In Plato’s terms, something
is missing: a common unity to form a ‘genos’.


     59. Thus, these definitions came from much earlier works, at least the middle of the 5th century.
On this point, we agree with A. Szabó (1978: 65–66) contra P. Tannery (op. cit., supra, note 12).
    60. As for instance the diagonal and the side of the square.
     61. According to the last line of the passage, an example would be the side of the cube of volume
                                                                                                 p
3: the square built on his side is still incommensurable to the unit (in modern term the side is 3 3 and
              p
its square is 3 9 which is incommensurable to 1).
    62. ‘ka» Çs‘mmetroi a… m‡n m†kei mÏnon, a… d‡ ka» dunàmei’ Along with Theodorus’ lesson,
and contrary to Euclid, the boys did not consider the case of incommensurability in ‘power’ i.e. lines
whose squares are still incommensurable (in modern terms a and b such that the ratio a2 /b2 is still
irrational) except briefly and indirectly at the very end of the passage, when Theaetetus speaks of an
analogous definition for the side of a cube (‘ka» per» tÄ stereÄ ãllo toio‹ton’, 148b2; in modern
terms the cubic roots of integers). Nevertheless, it shows that they were aware of the existence of
such lines, and that such magnitudes were well–known at the end of the 5th century.
     63. In modern terms, lines whose lengths are either integers or square roots of integers, with
respect to an arbitrary unit of length.
                  Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus         139


4.3. Back to Euclid

Let us return to the definition X.4 of the Elements. Two classes are defined.
On the one hand, the class of the straight lines commensurable to an as-
signed straight line, the ‘unit’, on the other hand the lines incommensurable
‘in length’ to the same assigned line but commensurable ‘in power’. Thus,
integers have vanished and the statement is entirely set in geometrical
terms. It is a homogeneous geometrical ‘definition’ and the opposition
is between commensurable and incommensurable magnitudes. On the
contrary, according to the boys’ definition, in line with Theodorus’ lesson,
the opposition is between integers and incommensurable lines that entails
an amalgamate of two mutually exclusive elements, the discrete and the
continuous. Moreover, while the boys’ definition of ‘powers’ results from
its opposition to ‘length’, each class in the Elements is defined by a common
property of its elements64 , not by exclusion of each other65 .
    Worse still, Theaetetus remarks as in passing that ‘powers’ are incom-
mensurable to ‘length’, and in particular to the unit–length. This makes
their definition to be mathematically a total failure. Not that the statement
is false, it is on the contrary perfectly exact, but it needs a very long demon-
stration and the proof of many intermediate propositions. In other terms,
this alleged definition is definitively not a definition but a deep theorem,
using a large part of books VII and VIII of Euclid Elements66 .



Conclusion

We have tried to show the importance of the problem of the ‘powers’ in the
second part of Theaetetus’ account and its close connection to the whole
dialogue, and even to the whole trilogy. However, contrary to the ‘Main
Standard Interpretation’, the fundamental question is not the exact meaning
of ‘powers’, that is ‘square’ or ‘side of square’. Plato’s text shows the flaws in

    64. ‘Those magnitudes are said to be commensurable (‘s‘mmetra’) which are measured by
the same measure, and those incommensurable (‘Çs‘mmetra’) which cannot have any common
measure’ (Definition 1, Book X).
    65. Definition X.1 considers ‘incommensurable’ (‘asymmetra’) simply as non–commensurable.
The prefix ‘a’ in ‘asymmetra’ means indeed that the absence of ‘symmetra’, thus both terms are
well–defined according to Plato’s rules in the Sophist and the Statesman. However, we certainly do
not suggest that Euclid’s definitions respected necessarily Plato’ criteria of division.
    66. According to the tradition, it will be proved later by Theaetetus (excluding by the way his
friend Socrates), when he will have become a bright mathematician emancipated from Theodorus’
teaching. For details, (Brisson–Ofman2).
140      Luc Brisson, Salomon Ofman


the reasoning of Theodorus’ brightest pupils67 . They failed because they
considered mainly objects like ‘lengths’, ‘powers’ and not relations like
‘commensurability/incommensurability in length or in power’, though
Theodorus’ lesson was precisely about the relations of commensurabil-
ity/incommensurability. The same mistake is at the origin of the failures of
all Theaetetus’ attempts to define knowledge: from a Protagorean point of
view68 , only sensible objects are worth of interest, not their relations. It is a
self–defeating point of view, for the only possibility to study the former are
through our senses i.e. through some relations between us and them. That
is why, at the end of the dialogue, the ‘logos’ (i.e. the ‘relation’) appears under
the guise of two dreams (201c–202c), and the attempts to get its definition
ends, once again, in a failure (206c–210b).
    Mathematics is definitively not about sensible objects; it is about mea-
sure (‘mËtron’) and commensurability, the foundations of any science and
technique (Statesman, 284a–b). This is certainly not restricted to mathemat-
ics. To the objects ‘in power’ (‘dunàmei’) that occupy a central part of the
passage, correspond the techniques and sciences as powers, because they
are the ‘powers’ to do something (‘to persuade’, ‘pe–jein’, 304c; ‘to go to
war’, ‘polemhtËon’, 304e; to judge rightly, ‘Êrj¿c dikàzein’, 305b,c; . . . ),
but also at the end of the Theaetetus itself, what Socrates’ technique ‘has the
power to do’ (cf. infra, end of note 71). Back to the Theaetetus, we are able
now to understand the huge paradox of Socrates claiming that to have the
knowledge of the integers is to have all of them in one’s soul69 . Faithful to
the midwifery, he brings the young boy to clarify the premises put forth by
him70 . which results in the failures of the search of the inquiry. The main ob-
stacle is not, as usually assumed, the complete absence of intelligible forms,
but the dreamed hunt of some objects of knowledge, let it be imprints
(194c) or birds (197c), while knowledge is an elusive ‘power’ of thinking
ratios71 .

    67. As a matter of fact, we have emphasized here the shortcomings in the boys reasoning.
But they certainly improved Theodorus’ presentation about incommensurability, at least on two
fundamental points. First, they consider the general problem, involving infinite quantities, instead of
taking ‘each case in turn’. And above all, instead of using sensible measures such as the foot, and
drawing graphical proofs, they referred to abstract units and abstract demonstrations where figures
are simply used as helper for words. This alone would justify Socrates’ favorable comment at the
end of Theaetetus’ account (148b) (cf. also infra, note 71).
    68. Protagoras is said to be Theodorus’ master (‘didàskaloc’, 179a), and in turn, he is Theaete-
tus’ master; cf. also, supra, note 20.
    69. Cf. supra, §2.1
    70. You forget, my friend, that I (. . . ) claim none of them as mine (. . . ) and am merely acting as a
midwife to you’ (157c) (cf. also Brisson 2008).
    71. The same may be said about Socrates’ remark at the end of the passage: ‘Excellent my boys’
(‘^Aristà g’ Çnjr∏pwn, ¬ paÿdec’, 148b). One has to understand it more or less as: ‘Well, now
let us see how your master has helped you to think by yourselves’, the fundamental purpose of
                   Powers and Division in the ‘Mathematical Part’ of Plato’s Theaetetus              141


Appendix

For a modern pure logicist72 , the set of all the integers and the set of the
perfect squares (or the set of all even integers) are equivalent for they are all
numerically infinite. However, a smarter mathematician would consider the
limit of the ratio of the number of their elements less than a given number
when this number increases. When this ratio is near to 1 (respectively to 0,
respectively is very large), the size of the second set would be more or less
the same (respectively much smaller, respectively much bigger) than the
first. Let us consider some examples. For any integer n

     a) There are n/2 even integers between 1 and n, so that the ‘ratio’ in
        the above sense of all integers to the even ones is 2. Thus, there are
        twice ‘more’ integers than even ones, which is consistent with the
        intuition.
     b) The ‘ratio’ of all the integers to the perfect squares is greater than
        the ratio of n over pn (since there are at most pn perfect squares less
        than n) and this ratio is extremely large (since n/pn = pn). Thus,
        the second set (of perfect squares) is much ‘smaller’ than the first
        one (of all integers).
     c) Conversely, if we consider the ratio of all the integers to the ‘rect-
        angulars’ (i.e. the non–perfect square integers) less than n, it is the
        same as the ratio of n to (n – pn), which is very close to 1 when n is
        very large73 . Thus, there are almost as ‘many’ ‘rectangular’ integers
        as integers, that is consistent with both the result in point b) above
        and, once again, the intuition.

    Though it is a modern presentation, this is probably what Plato, and
the mathematicians of the 5th –4th centuries had in mind, since in ancient
Greek, one of the meanings of ‘äpeiron’ was something ‘very large’ (Deti-
enne–Vernant 1974). Indeed, the modern logicist will distinguish between
finite and enumerable infinite, and all the enumerable infinites are the

Socrates’ maieutic, as asserted at the end of the dialogue. Indeed, the result of the long discussion
with Theaetetus is for the latter to ‘be pregnant with better thoughts’, to ‘be less harsh and gentler to
your companions, and first and foremost, to ‘have the wisdom not to think you know that which you
do not know’ (‘swfrÏnwc oŒk o Ïmenoc e dËnai É mò o⁄sja’, 210c); then, Socrates to conclude:
‘This is all my art has the power to do’ (‘toso‹ton gÄr mÏnon ô ‚mò tËqnh d‘nata’). See however
also, supra, note 67.
    72. By this term, we do not mean the modern logician, but someone, more often than not
a philosopher, considering any work, ancient or modern, from the point of view of (elementary)
logics.
    73. Since n/(n pn) = 1/(1 (pn/n) and pn/n = 1/pn is very small when n is very large. For
instance, for n = 10000, we get: n/(n pn) = 10000/9900 = 1, 01.
142       Luc Brisson, Salomon Ofman


same for him. For the smarter mathematician, there is a measure for the
diVerent enumerable infinites. If our analysis is correct, Plato gives here,
inside a mathematical background and in the extreme case of infinite classes,
a first introduction on the two kinds of comparison detailed in the States-
man. One considers on the one hand, in agreement with the logicist, the
pure relation of the more and less (283c–d), on the other hand, with the
smarter mathematician, the existence of a measure. The latter opens the
way to the possibility to decide what is correct, and what is not, in diVerent
situations i.e. to the ‘just measure’ (283e–284c). Anyway, even in Plato’s time,
such considerations were only within the scope of someone well–skilled in
mathematics. This may explain the diYculties for the understanding of this
passage even in the (late) Antiquity74 rather than some alleged mathematical
weakness of his author75 contradicted by all Plato’s writings.


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