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Digital Object Identifier (DOI) 10.1007/s00407-006-0114-8Arch. Hist. Exact Sci. 61 (2007) 337

Two Traces of Two-Step Eudoxan Proportion Theoryin Aristotle: a Tale of Definitions in Aristotle,

with a Moral

Henry Mendell

Communicated by J.L. Berggren

In memoriam David Fowler

In a classic paper, Wilbur Knorr [1978] presents an account of a pre-Euclidean

proportion theory which he attributes to Eudoxus. This theory is characterized by the

fact that in a proof that two magnitudes a and b have the same ratio as magnitudes

c and d, the case where a and b are commensurable is treated first (for generality, a

determinate case), and the case where they are incommensurable is then derived from the

commensurable case by a reductio ad absurdum (for generality, an indeterminate case).

It remains a significant question whether there are compelling contemporary traces of

the theory.1

Arguments using proportions or mentioning proportion theory are ubiquitous in

Aristotle and can furnish us with important evidence for early proportion theory. I shall

argue that two traces, each nigh incoherent on its own, when taken together constitute

such evidence in Aristotles corpus, as well as evidence for a more general technique

of proof by determinate and indeterminate cases. These examples both afford us with

evidence for Knorrs interpretation and reveal that Aristotle sometimes treats definitions

operationally, where the definition is an effective procedure for constructing a problem.This procedure may be a non-necessary method, i.e., where, at least for some cases,

the constuction may be achieved by other means. It may even be demonstrated and

restricted, i.e., where the definition applies directly only to a subclass of the definiendumand where the other cases are then reduced to this subclass. For Aristotle, definitions

that reveal essence are connected to demonstration. However, the fact that for Aristotle

such real definitions could be operational and restricted and that such demonstrated

real definitions occur in his own use of mathematics, I suspect, will come as news. In

fact, Aristotle had the conceptual apparatus to conceive of the definition of same ratio

as an undemonstrated definition of the restricted case, or as a basic theorem proving

the restricted case, or as a fundamental theorem which reveals how to work from the

restricted definition to all other cases, or as a generalization of all cases taken individually

to the kind taken as a whole based on the fundamental theorem.

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4 H. Mendell

In this paper, then, I shall be presenting two distinct but related theses. One is the

argument that there is evidence in Aristotle for Knorrs Eudoxan proportion theory.

However, the vehicle of this argument will be the claim that operational, restricted, and

demonstrated definitions play an important role in Aristotles scientific theory. Some

may even regard this as the more significant thesis. In fact, the realization that Aristotle

can speak of these as definitions leads directly to the moral, that a central pillar of modern

reconstructions of the early history of proportion theory is very unstable.

TYPES OF DEFINITION

Operational definition: the definition of A provides an effective method for

finding As, where the construction need not be necessary, i.e., where, at least

for some cases, the finding of As may be achieved by other means.

Restricted definition: the definition of A only applies to some As, but by

various methods, such as appeals to continuity and various intuitions, we

may determine for any x whether x is A or x is not A.

Demonstrated definition: a proposition such as AllAs are definition-of-Aor a

proposition from which the definition-of-A can be straight-forwardly derived

(e.g., in Aristotle, by transposing the terms in the demonstrative syllogism so

as to compose them into a new term) is a fundamental theorem about A.

The first trace of Knorrs pre-Euclidean proportion theory occurs in Aristotles anal-

ysis of faster and equally fast in Physics Z 2. It provides an example of demonstrateddefinitions that are restricted and operational in the above senses. The second is in a

argument that infinite bodies cannot have finite weight in De caelo A 6, where Aris-totle does distinguish between a commensurable and incommensurable case, as Knorr

observed [1976, 1989].2 Yet against Knorrs use of this passage it might be argued that

the incoherence of Aristotles discussion counts against its reflecting any method at all.

Quite the contrary, I shall argue that the best way, perhaps even the only way, to under-

stand Aristotles argument is against the background of two types of proof reflected inKnorrs account.

These passages contribute, I believe, to a more general picture of early Greek math-

ematics, namely, that it was customary for mid-4th cent. BCE mathematicians to distin-

guish between a simple determinate case and an indeterminate case. They also show that

we have to look more carefully at some of Aristotles claims about definitions in math-

ematics. Indeed, it may even lead to a modicum of scepticism about a most important

use of Aristotelian discussions to reconstruct early proportion theory: his mention of the

2 Knorr also uses the argument as evidence for the bisection principle, which is ubiquitous

in Aristotles arguments on proportion. Indeed, one formulation of Aristotles definition of the

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Two-Step Eudoxan Proportion Theory in Aristotle 5

definition of same ratio as magnitudes having the same,

or alternating

subtraction (Topics 3.158b2935).3

1. Two-step proportion theory

According to Knorr, the Eudoxan theory of proportion was characterized by a family

of techniques for determining one ratio to be the same as or larger than another. We

can discern from these techniques a conception of ratio and proportion, but we need not

think that they were gathered as a unified theory under a single definition in the sense

of Elements v, i.e. definitions of same ratio and larger ratio. One begins with a casewhere only commensurable magnitudes are involved and then reduces the case where

the magnitudes are incommensurable to the case of commensurables. I shall call this

family of proof techniques the two-step method.

In presenting the theory, Knorr suggests two definitions of same ratios, where A,

B are magnitudes and m, n are integers:4

Def. 1. A : B = m : n if and only if n A = m B.

Def. 2a. A : B < m : n if and only if n A < m B.

Def. 2b. A : B > m : n if and only if n A > m B.

As Knorr points out, it is unimportant whether these are thought of as theorems or as

proper definitions. We may then take an abstract example to see how a general proof

would be reduced to these.

We need to prove that two pairs of magnitudes A, B, and a, b, are such that A : B =

a : b. This may be treated as two theorems, one where A, B are commensurable and onewhere they are not. We first suppose that A and B are commensurable and show that A :

B = a : b, e.g., by finding the common measure M of A and B. Since A = n M and B =

m M, then A : B = n : m. We also prove that n : m = a : b, so that A : B = a : b. We next

prove that A : B = a : b where A, B are incommensurable. Suppose that A and B are

incommensurable and that A : B > a : b. Suppose also that A : B = a : b, so that A A : B > A : B = a : b. However, by the first theorem, for the

commensurable case, we showed that A : B = a : b, an instance of the commensurable

case. Hence, a

: b > a : b, so that a

> a. We also prove separately that a

< a. Hence,A : B = a: b < a : b, in contradiction with the previous argument. The case where we

assume A : B < a : b is reduced to contradiction in a similar way. The two-step method

is, in fact, more general and can also be used to show, given magnitudes A, B and a, b

(arcs in every extant example), that a > b or that A : B = a : x & x < b (in effect A : B >

a : b). Thus, the technique provides one avenue for comparing magnitudes. It is used by

Archimedes, Theodosius, and Pappus.5

3 The passage formed the basis for O. Beckers reconstruction of early proportion theory

[Becker, 1933]. All modern reconstructions start with Becker.4 Knorr [1978, 233].5 Archimedes, Plane Equilibria i 67 (the fundamental theorem on the balance) and Pappus,

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6 H. Mendell

The lemma, as proved in a scholion to Theodosius, Sphaerica iii 9, is particularlyimportant to my discussion (Scholia (ed. Heiberg), 148.1819):

Diagram 1

If three magnitudes of the same kind are supposed, AB,

G, DE, with AB larger than G and DE of any size, let

it be required to find some magnitude smaller than AB,larger than G, and commensurable with DE.

Let BZ be equal to G. Bisecting DE and bisecting the

half of that and repeatedly doing this, we are left with

some magnitude less than AZ [cf. Elements x 1]. Letthere be left DH and let it be less than AZ, while it is

a measure of DE. DH either measures BZ or it doesnt

measure it.

First let it measure it and let ZQ be equal to DH. Since

DH measures ZB, and DH is equal to ZQ, therefore DHmeasures BQ. But it also measured DE. Therefore BQ is

commensurable with DE, while being smaller than AB

and larger than G.

Let DH not measure ZB, and let DH in measuring out ZB

exceed it by ZQ which is smaller than DH [and therefore

ZQ < AZ]. Therefore, BQ is commensurable with DE,

while being smaller than AB and larger than G.

assuming also the existence of the fourth proportional), while the previous theorem, iii 9 (to show

that one arc on a great circle is larger than another), unambiguously concerns one magnitude being

larger than another. After supplying in vi 5 an alternative proof of Theodosius,De sphaera iii 5, thatone of adjacent arcs on an oblique great circle in a given configuration is larger, Pappus provides

in vi 69 two proofs of the case where the arcs are not adjacent. The second proof of Pappus is

two-step (vi 7 for the commensurable case and vi 89 for the incommensurable case, vi 8 that

one arc isnt smaller than the other and vi 9 that the arc isnt equal to the other). Pappus is clearly

using the techniques taken from De sphaera iii 9, 10 to prove this theorem. As to the theorems onspherics, Theod., De sphaera iii 9 and Pappus vi 79, the incommensurable case is divided into acase where a < b is proved impossible by reduction to the commensurable case via the lemma,and the case where a = b is shown to entail what was refuted in the first incommensurable case.

Similarly for Theod., De sphaera iii 10, the incommensurable case is broken up into two caseswhere the hypothesis that A : B = a : x and x > b leads to a contradiction by reduction to the

commensurable case via the lemma, and then the hypothesis that x = b is shown to entail what

was refuted in the first incommensurable case. Since Pappus, Collectio v 12 is a lemma in thediscussion of isoperimetric theorems, presumably based on Zenodorus (2nd cent. BCE), and may

be distinguished from Archimedes use in mechanics as well as from the family of arguments from

spherical geometry, there are really three extant traditions of the method, with one, namely that

from spherical geometry, merely tangentially concerned with proportions (however, in presenting

v 12, Pappus refers to the lemma from the Sphaerica). This accords with my overall thesis that thetechnique concerns the comparison of magnitudes, for which proportionality constitutes a crucial

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Two-Step Eudoxan Proportion Theory in Aristotle 7

Here, the two cases are: one where one magnitude measures another (the determinate

case), and another where it does not (the indeterminate case). The second case is direct

and is not here reduced to the first; it is genuinely distinct. We shall see an argument of this

sort in Aristotle mis-identified as an argument of the commensurable/incommensurable

sort.

It is convenient to distinguish types of determinate/indeterminate cases. We can

call the case where we examine the consequences of a measuring b, where a is a

part of b, the measure case. We can call the case where a exceeds b in measuring

it, i.e. where a doesnt measure b and some multiple m of a exceeds b, the supra-

measure case. Note that it is unimportant whether m is the least multiple of a to exceed

b.

Determinate // Indeterminate Cases

Measure // Suprameasure Cases: A measures B // A does not measure B &

m A m > B

Commensurate // Incommensurate Cases: A and B are commensurate // A and

B are incommensurate and the commensurate case is reduced to the commen-

surate by a reductio.

2. Restricted definitions, operational definitions, demonstrated definitions,

and Aristotelian definitions

In what sense is the two-step method a definition of proportion? Obviously, it is not

a definition in the sense of the definitions in Euclid, Elements v 5, 7, a statement whichprovides necessary and sufficient conditions for determining whether the term defined

applies. Def. 1 can at best be thought of as a definition of same ratio for commensurable

magnitudes, while Def. 2 at best tells us when ratios of magnitudes are larger or smaller

than ratios of commensurable magnitudes. Neither tells us what to say when A, B are

incommensurable. Given some a, b, we need a separate theorem that gives us a method

for determining whether A : B = a : b. We could even imagine Def. 1 as emerging froma yet simpler notion: A : B = a : b if A measures B as many times as a measures b (m

A m = B & a m = b).

This definition of proportion is restricted and operational, in that the strict def-

inition applies only a simple determinate case. The extension to the other cases is

operational, in the sense that the mathematician may discover and prove families of

procedures which extend the determinate case to all indeterminate cases and, in partic-

ular, by reductio, to the incommensurable cases, where a version of the required lemma

is used. Restricted operational definitions and such demonstrated definitions may seem

alien to some modern students of Aristotle, who may object that they could not have been

4th-century notions; yet we shall see that a reasonable argument can be made that even

Aristotle considered such definitions legitimate. And if they are legitimate to Aristotle,

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8 H. Mendell

speaking about definitions in mathematics we should not always assume that he means

a statement providing necessary and sufficient conditions.

Crucial to Aristotles treatment of definitions and demonstrations is the intuitive

operation of term construction from propositions and proposition construction from

terms. So if Some As are B is a true claim, then BA is a legitimate term to use in

ones science. If BA is a legitimate term to use in ones science, this is because one

minimally has a proof or assumption, Some As are B.6

This feature of Aristotles philosophy of science explains how Aristotles core notion

of a real definition can be an explanatory syllogism that reveals the essence (cf. An.Post B 10). We start with nominal definitions.7 Suppose that A belongs to all B. We

6 Cf. Mendell [1998, 18694, 203210, 2225].7 Aristotles account of definitions has three crucial ingredients, the stipulative definition of

the term (normally called namelike or nominal definition), the investigators non-accidental

knowledge of the instantiation of the term, and the demonstration of the essence of the definien-

dum which is transformed in two ways into definitions. Charles [2000] argues that these constitute

successive stages of learning the definiendum. Since I am only here concerned with the arrange-

ment of a scientific treatise, the issue of the order of investigation is unimportant. In an actual

ancient scientific treatise as read by an Aristotelian, it is possible for the stipulative definition to

occur first with the existence proof later, or for the existence proof to occur before the stipula-

tive definition, or for one or both to be omitted altogether, or for the existence proof to come

only with the demonstration of the essence and real definition (in fact, an Aristotelian reader of

Euclids Elements might well regard all of these possibilities as occurring there). There is also muchcontroversy whether Aristotle classifies three or four kinds of definitions in An. Post. B 10. Thesecandidates are definitions which indicate what the name signifies and do not involve an assumption

of the existence of the definiendum (at least included in the class of nominal definitions), in-

demonstrable definitions, definitions that are the transformed conclusions of demonstrations of the

essence, and real definitions that are the demonstration itself transformed. For a brief survey of the

problem, cf. Barnes [1993, 2223]. I am not here concerned with the question whether nominal

definitions are the same as or different from the definitions that are conclusions transformed. If

these are two classes, for the purposes of mathematics, this difference would probably be merely

functional. That is, the definition of triangle in the list of definitions of a science would be non-

existential, while the conclusion of the demonstration, now having existential import, might be

verbally the same but belong to the other class and with a different semantic content. Charles

[2000, esp. 437], produces a general argument along these lines, where he builds on the basic

point that a real definition is of the thing, while a nominal definition is of the word. Bolton [1976]

argues that the nominal definition and the definition as conclusion are the same class and that both

assume existence. We can agree on the phenomena without worrying about the classifications.

Hence, it is consistent with Boltons view that Aristotle insist that only words that are instantiated

can have nominal definitions and yet also insist that a nominal definition at the beginning of a

treatise not include existence as part of its sense. Secondly, the difference between a nominal

definition in the list of definitions at the beginning of a treatise and the transformed conclusion is

that the transformed conclusion does not actually occur in the treatise as a definition separate from

the nominal definition but might well occur as a term in a demonstration. No extant mathematical

text has both of these two claims: rectilinear figures with three sides are called triangles and

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Two-Step Eudoxan Proportion Theory in Aristotle 9

can imagine a stipulative definition: let A be definition-of-A, e.g. let thunder be

noise-in-the-cloud. One can substitute one expression for the other, noise-in-the-cloud

for thunder. Secondly, definition-of-A may be thought of as a nominalization of a

statement, here, Noise belongs to cloud. Next one seeks an explanatory account of

why A belongs to B, e.g. why does it thunder, i.e. why does noise (of the sort that is

in-the-cloud) belong to cloud, i.e. extinction-of-fire:

Noise belongs to extinction-of-fire.

Extinction-of-fire belongs to cloud.

Thus, noise belongs to cloud.

The real definition of thunder is a transformation of this explanatory account, thunder is

noise-resulting-from-extinction-of-fire-in-clouds, where the terms noise, extinction-

of-fire, and cloud are transposedto construct a single term (cf.An. Post. B 10.94a12,

1213). In other words, for Aristotle, the real definition can be the result of a transposition

of a demonstration, while the nominal definition becomes its conclusion in the sense

that the terms of the conclusion, noise and cloud are likewise transposed to form a

single term: thunder is noise-in-the-clouds. The example here is an efficient cause, but in

mathematics we would expect the demonstration that can be transposed into a definition

to provide an account, for example, of why a triangle exists. Aristotle would regard a

mathematical technique for proving that A belongs to B as a causal explanation of why

A belongs to B. Here is a basis for operational and demonstrated definitions in Aristotle.

When I speak of demonstrated definitions, I shall mean loosely any definition which

is built out of a demonstration, whether in Aristotles strict sense of a transformation of

a demonstration or any fundamental theorem which reveals a term.

the definition as conclusion transformed actually occurs, perhaps earlier, with the existence proof,

at Charles second stage, before it is a conclusion; otherwise, there will be three verbally nigh

identical definitions, the nominal definition with the definiendum term mentioned, the definition

as applied to an existing entity with the definiendum term used, and the conclusion transformed. In

other words, the proof of existence legitimates the definiendum expression, e.g., triangle, and its

nominal definiens as terms in anAristotelian demonstration. It would have been very reasonable of

Aristotle to have argued, as he so often does, that in one sense the two definitions are the same and

in another different. Hence, I am indifferently calling the nominal, i.e., stipulative definition, thenominal definition as legitimated by an existence proof, and the transformed conclusion of the

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10 H. Mendell

There is also a foundation for restricted operational definitions.8 In an exposition of

this same account of definition at De anima B 2.413a1120 (cf. Met. B 2.996b1822),Aristotle provides a restricted operational definition for squaring:9

Since what is clear and more grasped according to the account comes from things that

are unclear but more evident, we should attempt to explain this (the soul) again in thisway. For it is not only necessary that the defining account make clear the fact that, as

most definitions do, but also that the cause occur in it and be displayed. The statements

of definitions are now like conclusions. E.g. what is squaring? there being an equilateral

rectangle equal to an oblong. But such a definition is an account of the conclusion. The

one which says that squaring is the discovery of a mean states the cause of the matter.

The nominaldefinition, there-being-an-equilateral-rectangle-equal-to-an-oblong, is not

exactly identical to the conclusion; it is a nominal form of the conclusion and is

constructed out of it. The real definition of squaring then is a transposition of a syl-

logism, there-being-an-equilateral-rectangle-equal-to-an-oblong-rectangle (the conclu-sion of the syllogism) because-of-the-discovery-of-a-mean-proportional-of-the-thing

(the middle term in the syllogism). We may reconstruct the syllogism in this way:10

There-being-an-equal-equilateral-rectangle belongs to (i.e. is constructed via) discovery-

of-a-mean-proportional.

Discovery-of-a-mean-proportional belongs to oblong-rectangle.

Thus, there-being-an-equilateral-rectangle belongs (i.e. there is an equilateral rectangle

equal) to oblong-rectangle.

This definition is restricted since it only applies to a rectangle. All other cases of squaringmust be discovered by a reduction to the case of squaring a rectangle and are squarings

because of that reduction. For example, in the Prior Analytics (B 25.69a3033),Aristotleproposes the following reduction of the problem of circle squaring:

Squaring belongs to rectilinear-figure.

Rectilinear-figure belongs to (i.e., is equal to) circle-with-lunules-becoming-equal-to-a-

rectilinear-figure.

8 In a very useful part of his discussion, Bolton [1976, 53840] argues that there are three

types of nominal definitions as conclusions of demonstrations, those that provide a part of the

essence and so provide necessary conditions, those that provide sufficent conditions by picking

out a characteristic subclass of the definiendum, and those that are members of the class via

accidents. However, Bolton does not regard this classification as pertaining to real definitions. On

demonstration of the essence, also cf. McKirahan [1992, inter alia 198208].9 Normally, operations are not definienda in Greek mathematics, although verbs are commonly

defined. Cf. Netz [1999, 9194, 9699].10 Part of the interpretation of this syllogism involves filling in the appropriate sense of be-

longs. For a discussion of this problem, cf. Mendell [1998, 1708]. The sense of belongcan vary

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Two-Step Eudoxan Proportion Theory in Aristotle 11

Circle-with-lunules-becoming-equal-to-a-rectilinear-figure belongs (i.e., is equal)

to circle.

Thus, squaring belongs to circle.

We can readily imagine a general nominaldefinition of squaring as finding a square

equal to a given figure, but Aristotle does not treat the squaring in this way.It is also clear that this definition is operational in the sense that the real definition

provides the foundation for squaring by an operation that is effective (finding the mean

proportional), but not always necessary. For one can square some figures without finding

a mean proportional (e.g., by cutting and pasting a 1 4 rectangle or an isosceles triangle

with base twice the height). Furthermore, the operation is proved, namely one proves

that any rectangle can be squared in this way and so concludes that any figure that can be

found equal to a rectangle can be squared, e.g. by a theorem along the lines of Elementsvi.13 and 17 or ii.14.11 In other words, squaring involves a cluster of methods of reducing

a problem to squaring a rectangle.In a similar way, we can imagine a definition of same ratiowhich takes into account

only the determinate or commensurable case, but allows us to use principles of continuity

to extend the application of same-ratio to incommensurable cases.

3. The definition of Faster in Aristotle, Physics Z 2

The first passage providing traces of arguments with determinate and indeterminate

cases is Aristotle, Physics Z 2.232a23-b20. Aristotle provides three definitions of fasterand then proceeds to prove all three from a yet more fundamental definition. Even

so, all three definitions are restricted; yet each constitutes an expansion on the previous

definition. To make this account at all plausible, the reader needs to bring in rich kinematic

postulates that will allow one to infer from a principle about initial segments to smaller

segments or that will allow one to extend the motion of one of the moving objects as

needed. Once this is done, general principles of continuity will allow one to expand the

restricted and demonstrated definitions to other required cases. Hence, the definitions

are also operational. Finally, we shall see that the arguments involve a delicate interplay

between determinate and indeterminate cases, where one determinate case provides

the foundation for an indeterminate case, which then is used to establish the second

11 It is common for scholars to try to identify Elements vi 13 (the finding of a mean proportional)and its application to the problem at Elements vi 17 (three lines are in continuous proportion iffthe square on the middle is equal to the rectangle of the extremes) with the proof mentioned

by Aristotle and not Elements ii 14 (the construction of a square equal to a given rectangle).However, it seems to me unlikely that either corresponds precisely to Aristotles theorem, while

the distinction between the two theorems may be more an artifact of the structure of the Elementsthan of the state of mathematics in Aristotles time, where proofs involving ratios of figures are

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12 H. Mendell

determinate case. However, it is also evident that the determinate cases are considered

primary.

In my discussion, I shall allow myself two functionsor more properly abbreviations.

DX(t ) is the distance that X travels during a time interval t. TX(d) is the time interval of

Xs travel over distance d. Furthermore, d1

d2

shall mean that d1

is an initial segment

of distance d2, and similarly for times, while d1 < d2 will have its normal meaning

that d1 is smaller than d2. Obviously, if x y, it follows that x < y, but the converse

will not always hold. Finally, since we shall be comparing changes, X, t, d will be

the Xs travel over distance d in time t. In general, it is adequate to specify either the

distance traveled or the time of travel, while the other may be omitted. Hence, X,t,

will abbreviate X, t, DX(t ), and X,, d will abbreviate X, TX(d),d.

Aristotle sets up an argument for the continuity and finitude of change with a dis-

cussion of faster and equally fast. We expect the definition of faster to be:

1. A, tA, dA is faster than B, tB, dBiffdA : dB > tA : tB.

Although it may be implicit in some of Aristotles argumentation,12 this is not, however,

the definition we find in Aristotles works. Instead we find some combination of the

following three claims concerning A is faster than B, with the precise logical relations

left open for the moment:

2a. A and B travel in time tAB and DA(tAB) > DB(tAB).13

2b. A and B travel a distance dAB and TA(dAB) < TB(dAB).14

2c. A travels a distance dA and B travels a distance dB and dA > dB and TA(dA) dB and tA > tB oreven where dA < dB.

For the Lyceum, however, they constituted all the components adequate for defini-

tions of faster and equally fast. So ps.-Arist., Mechanica 1.848b58:

For the faster is spoken of in two ways, since we say (2b) that something is faster if it

traverses an equal place in less time, and (2a) if it traverses more place in equal time.

This nigh sets out the notions (2a) and (2b) as separate notions of faster, which need to

be proved equivalent. Of greater interest is the definition of faster that Aristotle gives at

the beginning ofPhysics Z 2 (232a257):

. . . then it is necessary that (2a) the faster traverse a greater distance in the equal time

and (2b) an equal distance in the lesser time and (2c) more distance in the lesser time, just

as some define the faster.18

Here, Aristotle maintains (2a), (2b), and (2c) as necessary conditions for faster and

tells us that the three cases constitute someones definition of faster. The fact that he

attributes the definition to some people is not on the radar screens of commentators. The

ancient commentators Themistius, Simplicius, and presumably Alexander completely

ignore it, as do modern commentators such as Heath [1949, 12830] and Ross [1934,p. 641, ad loc.],19 and for an apparently good reason. Although it is as close as Aristotle

gets to an official definition, they do not understand it as a definition, because they work

with a notion of definition that is, in fact, more restricted than Aristotles.

Aristotle lends some force to their view by proceeding to give proofs of all three

and thereby implies that he has a yet more fundamental notion of faster. This more

fundamental notion seems to be:

2d. What is changing earlier is faster, or What changes earlier is faster. (232 a289).20

17 Cf. Arist. Phys. H 5.249b27250a4.18 Physics Z 2.232a237: Epe d pn mgeoj ej megh diairetn (ddeitai gr

ti dnaton x tmwn eina ti-sunecj, mgeoj dstn pan sunecj) (Since

every magnitude is divisible into magnitudes, since it has been proved that it is impossible for

something composed of atoms to be continuous, while every magnitude is continuous), nght tton n t sJ crnJ mezon a n t lttoni son a n t lttonipleon inesai, aper rzonta tinej t tton.

19 Ross explains, i.e. some people actually use these three attributes as forming the definition

of the faster. I do not see this innuendo in Aristotles aper rizontai tinej t tton(just as some people define the faster). Aristotles implication, with or without approval, is that

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14 H. Mendell

Furthermore, it is unclear whether this expresses a necessary condition or a necessary

and sufficient condition. Hence, it may be (constantly faster):

2d.1. If A and B change over comparable changes (and start at the same time and move

over the same path?), then A is faster than B [if and] only if for any point C in As

change, A arrives at C before B arrives at the corresponding point in its change.

or (faster for the entire change):

2d.2. If A and B change over comparable changes (e.g. both in weight or in distance

or in color, etc.) and point C is some point at the terminus of Bs motion (i.e. Bs

motion that is under consideration, since B might continue), then A is faster than

B [if and] only if A arrives C before B arrives at C.

In fact, for the arguments explicitly provided by Aristotle, defining faster only for an

entire change (2d.2) is fully adequate. The biconditional version of (2d.2) is not required

for the first three proofs, while its role in the fourth proof is at best unstated.

A problem also arises in the assumption that A is faster than B. Although Aristotle

merely says that A is faster than B, without specifying the changes compared, the set-ups

for the first three theorems assume changes where we have A,, dAB being faster than

B,, dAB while the distance of Bs motion in each theorem is always less than dAB, it

will appear that the faster motion assumed in the theorem is distinct from the times and

distances compared in the theorem and its proof. I shall keep this fact explicit in my

discussion of the proofs. I will only turn to the difficulties that this assumption involves

after my discussion of Aristotles four proofs on the definitions of faster.

It is also unclear what the difference is between (2d.2) and (2b), while (2d) seems to

entail (2b) trivially. Hence, the fact that Aristotle gives two proofs of (2b) might suggest

that he conceives of (2d) as providing a simple basic case. It is not clear, however, why

Aristotle needs a circuitous route to (2b).

It is significant to my argument that Aristotle does not in fact argue for the bicondi-

tionals implicit in (2ac) but instead for the following variants:

2a. If A and B travel in equal times tAB, then A is faster than B (i.e, A, tAB, is faster

than B, tAB, ) only if DB(tAB) < DA(tAB)).

2b

. If A and B travel a distance dAB, then A is faster than B (i.e, A, TA(dAB), dAB isfaster than B, TB(dAB), dAB) only if TA(dAB) < TB(dAB).

2c If A and B travel in equal times tAB (implicitly in the proof), then A is faster than

B (in fact, unexpectedly, A, tAB, is faster than B, tAB, , as in (2a)) only if for

any distance d where DB(tAB) < d < DA(tAB), it is the case that TA(d) < tAB.

This is surprising as well, since, at the very least, we would expect converse claims. It

is especially awkward in the matter of (2c), where the converse sufficient condition,

2c. If A travels dA and B travels dB and dA > dB and TA(dA) < TB(dB), then A,, dA

is faster than B,, dB.

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Two-Step Eudoxan Proportion Theory in Aristotle 15

is both more straightforwardly true and does not require the implicit reference to 2a

and the related assumption that A travels DA(TB(dB)) further than dA. This is because

the antecedent of 2c and the condition of the period of travel in parentheses are merely

implicit from its proof. I shall return to this matter in my discussion of the argument for

2c.

The arguments all assume the following basic true principle:

5. If A travels over d1, then d2 d1 if and only if TA(d2) TA(d1).21

(5) follows from an intuitive principle which Aristotle belabors in Physics Z, espe-cially in his attacks on atomism:

Basic Kinetic Assumption: If A has a motion, then A moves a distance and in a time.

or more precisely:

Basic Kinetic Assumption: If A travels over d, then d > 0, so that if A travels overd, TA(d) > 0.

(5) follows since, if d2 d1, then d1 d2 > 0, and so too for the times. Hence, if A

travels over d1 d2, then d1 d2 > 0, so that ifA travels over d1 d2, TA(d1)TA(d2) =

TA(d1 d2) > 0. Let us turn to Aristotles arguments in the order he presents them.

I. Argument for (2a) at 232a2731: Let ZH be the time it takes for A to travel GD.

Since A,, GD is faster than B,,GD, A gets to D before B so that B is at some earlier

position in this time (2d). Hence, in an equal time TA(GD), A travels more.22

Diagram 2.

21 Aristotle, Physics Z 7.237b234; cf. 9.266a18. Note the importance of aspect in the firstpassage, since everything that is moving is moving in time, and [over] a larger magnitude in more

time .... What Aristotle says is strictly true. However, it is also obvious that he would do better

with an account that allows for interrupted changes, a fairly simple extension of his account.22 Physics Z 2.232a2731: stw gr t f A to f' B tton. pe tonun

ttn stin t prteron metabllon, n crnJ t A metabblhen p to G

ej t D, oon t ZH, n totJ t B opw stai prj t D, ll'poleyei, sten t sJ crnJ pleon deisin t tton. (Let A be faster than B. Since then what is

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16 H. Mendell

Comment: The argument assumes that A and B travel the same path. We can sum-

marize the argument:

Assume TA(GD) is ZH and A,,GD is faster than B,, GD (this is required to apply

(2d.2)).

Hence, ZH = TA(GD) TB(GD), by (2d.2).Hence, DB(ZH) GD, by (5).

Hence, DB(ZH) < GD = DA(ZH).

Hence, without the principle that leads to the last claim, the theorem actually proved

is:2a.1. IfA and B travel a distance dAB,then A,, dAB is faster than B,, dAB onlyifDB(TA(dAB))

dAB.

From this Aristotle may legitimately conclude:

2a.2. If A and B travel in a time tAB and A and B travel a distance dAB such that tA = TA(dAB),

then A, tA, is faster than B, tA, only if DB(tA) < DA(tA).

To get (2a.2), however, we need three intuitive assumptions about motions. I shall

not formalize them, because we should think of them as informal notions that fit our

conceptions of continuity, space, and time. Among these are:

i) it makes no difference whether the times are the same or merely equal;

ii) it makes no difference whether the distance traveled is an initial segment or merely

equal to an initial segment;

iii) it makes no difference what B does after the time of travel being compared, B doesnt

have to travel GD for A to be faster than B in the interval of travel in time ZH, but

we can extend Bs travel to GD.

This is comparable to ignoring an auxiliary construction in the conclusion of a geomet-

rical theorem. We readily accept that having internal angles equal to two right angles

is true of triangles, and not merely of triangles with one side extended and a parallel

to the opposite side, etc. Given (2d), which covers merely the case where the faster

changes in less time, these are not trivial assumptions. Why should we be permitted to

extend its motion? Allowing auxiliary constructions of motions turns out to be essentialto Aristotles proof technique.

However, we reasonably suppose that Aristotle really wants to prove (2a). To do this,

we need to postulate or establish that if there are motions A, tAB, faster than B, tAB, ,

we can construct motions over dAB = MAX[DA(tAB), DB(tAB)], where A,,dAB isfaster than B,, dAB. So the part of the proof that needs to be brought in would start

with As being faster than B in time tAB, but would then extend the motion to the same

distances. From this Aristotles proof would follow. I will consider this construction

postulate or problem later as an extension of (2d.2). For now, lets just call it the Kinetic

Construction Postulate. All these assumptions and others become more important in

what follows.

II Argument for (2c) at 232a31b5: If we incorporate the contents of its proof from

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Two-Step Eudoxan Proportion Theory in Aristotle 17

2c.1 If A and B travel over GD and A,, GD is faster than B,,GD and tAB = TA(GD), then,

for any distance d where DB(tAB) < d < GD (d < GD is also implicit) only if TA(d) < tAB.

Let ZH be the time it takes for A to travel GD. Since B is slower (since A is faster),

in time ZH it only gets to a point earlier than D, e.g. E (2a). Take any point Q between E

and D. Since GQ > GE,A reaches any point Q between E and D in less time than it takesB to reach E. Here, TA(GQ) = ZK. Hence, for GE < GQ < GD, TA(GQ) < TB(GE).

23

Diagram 3.

Comment: The set-up is exactly the same as for the argument for (2a). This makes

the structure of the argument completely dependent on Case (2a) or at least its set-up

conditions (2d). It also assumes (5). Thus, we can summarize the argument:

Assume B is slower than A (and A and B travel the same path from G), while TA(GD) =

ZH.Hence, A is faster than B, i.e., A,, GD is faster than B,, GD.

Hence, DB(ZH) DA(ZH) = GD, by (2a) with the fact that the paths traveled are

the same.

Hence (by 5), for any distance d where DB(ZH) d GD, it is the case that

TA(d) TA(GD) (lets call this minimal 2c).

Suppose for an arbitrary d, DB(ZH) < d < GD.

Then it is possible to construct d = d, such that DB(ZH) = DB(TA(GD)) d GD

(by the intuitive construction rules).

Hence, TA(d) ZH = TB(DB(ZH)) (by minimal 2c).Hence, TA(d

) < TA(GD) = ZH = TB(DB(TA(GD))) (by the intuitive construction

rule used in the proof of 2a).

23 ll mn a n t lttoni pleon. n gr t A gegnhtai prj t D,t B stw prj t E t bradteron n. oon pe t A prj t D gegnhtai npanti t ZH crnJ, prj t Q stai n lttoni totou . a stw n t ZK. tmn on GQ, diellue t A, mezn sti to GE, d crnoj ZK lttwn to

pantj to ZH, ste n lttoni mezon deisin. (In fact it will also move more in less[time]. For in the [time] in which A has come to be at , let B be at E, since it is the slower. And

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18 H. Mendell

Hence, for any distance d where DB(TA(GD)) < d < GD, it happens that TA(d) PR, and we supposed that DB(C) = LX. Then

A travels LX in PS and B also travels LX in C, and PS < PR < C.24

Diagram 4.

Comment: We can perhaps summarize the argument in this way:

Suppose A and B move LD and A,,LD is faster than B,, LD.

Let C = TA(LD), LX = DB(C).

LX < LD, by (2a).

Let LX < LM < LD, and let PR = TA(LM).

Hence, TA(LM) < C = TA(LD), implicit from (2c).

Hence, TA

(LM) < TB

(LX) = C, by (2c).

Hence, TA(LX) < TA(LM), by (5).

24 In the diagram C is Greek , and X is Greek X. fanern d totwn tit tton n lttoni crnJ deisin t son. pe gr tn mezw n lttonidircetai to bradutrou, at d at lambanmenon n pleoni crnJtn mezw tj lttonoj, oon tn LM tj LX, plewn n eh crnoj P P, n tn LM dircetai, PS, n tn LX. ste e PP crnoj lttwn stn

to X, n t bradteron dircetai tn LX, a PS lttwn stai to f/ X to gr PP lttwn, t d to lttonoj latton a at latton. ste nlttoni insetai t son. (It is obvious from these things also that the faster traversesan equal [distance] in less time. For, since it traverses a larger [distance] in less time than the

slower, but when taken itself by itself will traverse a [distance] larger than the smaller [distance]

in more time, e.g. M [as larger than] , then the time P, in which it traverses M, would

then be more than , in which [it traverses] . Thus if time P is less than X, in which the

slower traverses , then will be smaller than X. For it is smaller than P, while what is

smaller than the smaller is also itself smaller. Thus it will move an equal [distance] in less [time].)

Some manuscripts have X for X. Given my argument, clearly X is the lectio difficilior, while

not impossible. For further arguments on the reading, cf. Ross [1934, 641 ad b514]. Which ofthe two, X or X, Aristotle wrote is indeterminate, but not crucial. The only change would be in

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20 H. Mendell

Hence, TA(LX) < TB(LX), by the previous two claims.

As I have emphasized, the proof presupposes that both A and B travel over LD and that

As motion is faster in this travel. Again, we must suppose the three intuitive principles.

However, this proof provokes many questions. Since we are told to take C = TB(LX),

when we know that this is presupposed by the proof, does this mean that its author hasforgotten that C is presupposed in the initial conditions of the proof? More important,

why does the author want to use (2c) instead of (2a). The fact that LX < LD from

(2a) ensures, by (5), that TA(LX) < TA(LD) = TB(LX). Finally and most disturbing,

Aristotles set-up for the proof of (2b) already presupposes that A and B move LD and

that TA(LX) < TA(LD). This is consonant with the observation made earlier that the

difference between (2b) and (2d) is not clear. It would be harsh, but not entirely unjust,

to accuse the argument of being redundant.

More important, however, is the problem how we would reconstruct the argument

so that the proof is of (2b) and not merely (2b

.1). To do this, we would need to pos-

tulate or establish: if A and B travel dAB and A,, dAB is faster than B,, dAB, then

if dAB = Max[DA(TB(dAB)), DB(TA(dAB))], we can construct motions of A, B such

that A,, dAB is faster than B,, dAB. From this construction it will follow that, in

fact, TB(dAB) > TA(d

AB) (the claim that makes the theorem look redundant), that

DB(TA(dAB)) < d

AB and hence that d

AB = DA(TB(dAB)).

In all these proofs, we have seen that we assume that the change where A is faster

than B is larger than any of the motions actually involved in the proof. The alternative

to this is to take the claim that A is faster than B as vague and as providing constraints

on any discourse about faster. If so, Aristotles arguments, especially for (2c

) wouldbecome incoherent. In any case, it is enough for my purposes here to show that we can

make good sense of these three arguments so that we can treat them seriously.

Now Aristotle concludes (232b1420) his discussion of faster with a second argu-

ment for (2b):

Furthermore, if everything must move in an equal [time] or in less [time] or in more [time],

and that which moves in more [time] is slower while that which moves in equal [time] is

equally-fast, but the faster is neither equally-fast nor slower, then the faster would neither

move in equal [time] nor in more [time]. And so it remains that it moves in less [time], so

that the faster must also travel the equal magnitude in less time.25

The premises seem to be the following, where A and B move the same distance dAB:

6a. TA(dAB) < TB(dAB) or TA(dAB) = TB(dAB) or TA(dAB) > TB(dAB).

6b. If TA(dAB) > TB(dAB), then A is slower than B.

6c. If TA(dAB) = TB(dAB), then A and B are equally fast.

6d. If A is faster than B, then (A and B are equally fast) & (A is slower than B).

From this it follows that:

25 ti d' e pn ngh n sJ n lttoni n pleoni inesai, a t mn b ' ` `

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Two-Step Eudoxan Proportion Theory in Aristotle 21

6e. If A and B travel dAB, then A is faster than B only if TA(dAB) < TB(dAB).

As we saw assumed in the proof of (2c), A is slower than B if and only if B is faster

than A. So, (6b) is equivalent to (with A and B switched):

6b

. If TA(dAB) < TB(dAB), then A,, dAB is faster than B,, dAB.

(6b) and (6e) then entail:

6e. If A and B travel dAB, then A,, dAB is faster than B,, dAB if and only if

TA(dAB) < TB(dAB).

This is basically the same as (2b). Hence, it is unimportant whether the text supports

(6e) or, as less likely, (6e). Furthermore, since (6e) is more explicitly general than (2b),

one might wonder why Aristotle does not give this proof instead or whether the less

geometrical qualities of the proof indicate that it represents a separate part of Aristotles

oeuvre. There is a different more pressing issue. What is the foundation for (6b) and

(6c). We can imagine that (3) should be taken as a biconditional and so entail (6c). So

the problem is really (6b). One could derive it from (2d) understood as a biconditional.

But this is pure speculation. I do not need here to explain further the argument for (6e),

except to make one small observation. Since this argument is independent of Cases (2a)

and (2c), whatever one may think of its worth, it would allow, for example, the author

of the Mechanica to think of (2a) and (2b) as independent conceptions of faster.It is noteworthy that all the arguments discussed so far do not depend on the move-

ments being uniform and that, with the possible exception of the problem of reduncancy

in the first proof of (2b), they are rigorous, at least when appropriately understood.26

However, as a traditional definition of faster, this composite definition seems blatantly

inadequate.

There are then four issues that are central to our present concerns. Foremost is the

fact that without a demonstration of the proportionality principle (1), the definitions

do not consider all cases. However, they are also otiose because one can derive Cases

(2b) and (2c) from Case (2a), as Aristotle shows. Why not just define faster as (2a)

or even (2d)? Indeed, if one accepts a modicum of proportion theory and the ability to

find distances traveled from times and vice versa, then one can derive a general principle

for faster from (2a). Hence, the definition either gives too much or too little. On theother hand, even if we accept his assumption of the converse of (2b), why does Aristotle

not prove the converses of (2a) and (2c). Finally, we need to account for the required

intuitive principles, especially the construction postulates, if that is what they are.

It is easiest to begin with the construction postulate. There is little reason to think that

the construction postulates of Euclid, Elements i, were explicit in the time of Aristotle.Yet, we can give an analogue of the second postulate that allows a geometer to extend a

given line:

Construction Postulate for faster: Suppose A,, dA is faster than B,, dB. Then ifd

A >dA, it is postulated to extend the motion ofA to dA such that A,, d

A is faster than B,, dB,

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22 H. Mendell

and ifdB > dB, it is postulated to extend the motion of B to dB such that A,, dA is faster

than B,, dB.

We can imagine a similar postulate for equally fast:

Construction Postulate for equally fast: SupposeA,, dA

and

B,, dB

are equally fast.Then ifdA > dA, it is postulated to extend the motion of A to d

A such that A d

A and

B,, dB are equally fast.

We can also imagine similar postulates for extending times of motions. One might

object, perhaps even with good justification, that the postulates presuppose that we

already know what the faster and the equally fast are. However, against this one might

note that there is a good intuition that ifA is moving faster than B, it can continue to move

faster than B was moving, and similarly with the slower B. So too if A and B are equally

fast. This is just the sort of intuition that might not be explicit in an early treatise. I would

not claim that the analysis I have given that requires the Construction Postulate for fasteris the only possible coherent analysis of Aristotles argument. However, it shows that

we can make sense of the argument within the confines of his text and his contemporary

mathematical practice. These kinematic versions of geometrical constructions form a

part of general, implicit conceptions of the continuum, constructability, and so forth.

They are not at all unique in Greek applied mathematics. For example, Archimedes has

constructions of extra bodies in On Floating Bodies i 6 and 7, the latter requiring theconstruction of a body with a given volume and a given weight.

I think that the three remaining puzzles drift away if we think of the definition

of faster in Physics Z 2 as restricted and operational in the same way that Knorrsreconstruction of the Eudoxan theory of proportions is. The definition lays out threenecessary conditions for showing that A is faster than B under different conditions. If

the times of the change are the same, look at the distances; if the distances are the same,

look at the times, and if neither is the case, use what mathematics you know to expand

the definitions. A key to this is in the expansion from Case (2a) to (2c), where Aristotle

assumes that A and B move in the same time. As I have been emphasizing, this looks

like a kinematic version of a geometrical construction.

First, the converses of (2a) and (2c) are fairly easy to prove by means of the very

typical form of the second proof of (2b

). I also assume that one change is either fasterthan, as fast as, or slower than any comparable change. Suppose motions A, tAB, dA

and B, tAB, dB with dA > dB. If B, tAB, dB is faster than A, tAB, dA, then, by

(2a), dB > dA. If they are equally fast then dB = dA. But A, tAB, dA is faster than or

equally fast as or slower than B, tAB, dB. Hence, it is faster. The proof of the converse

of (2c) is slightly less trivial since one wants it to be a general claim that if there are

motions A, tA, dA and B, tB, dB, where dA > dB and tA < tB, then A, tA, dA is

faster than B, tB, dB. I leave to the reader its proof, which has the same structure but

requires both construction postulates.

These considerations suggest to me that although Aristotles source may have in-

cluded Aristotles argument much as he presents it, the purpose of the definition cannot

be merely to give necessary and sufficient conditions for saying that one thing is faster

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Two-Step Eudoxan Proportion Theory in Aristotle 23

anomaly that although Case (2c) is used in the argument for Case (2b), its original

form Case (2c) is listed after Case (2b) in the definition, while Cases (2a) and (2b) are

privileged in the tradition. It may even be that Aristotle regards Case (2c) as a lemma

for proving Case (2b). Aristotle uses Cases (2a) and (2b) in the very next proof, but not

(2c).

27

After all, these are the two basic cases in the tradition, but treated as fundamentaltheorems.

We can also readily see how the basic definitions could be expanded so as to prove

even the proportionality principle (1). As an illustration, suppose that the total travel of

B is dB and takes place in time tB less than the time tA that A travels dA with dB < dA.

Suppose that dB measures dA so that n n dB = dA. Then, take n dB of Bs motion

(treating it as repeating), such that we now compare the constructed TB(dA) withTA(dA),

where TB(dA) = TB(n dB) = n TB(dB) = n tB. We are allowed to do this because

we have an intuition that the motion of B may be incremented in this way, namely that

if each segment is as fast as every other, then the whole is as fast as any segment. If A is

faster than B, n tB > tA (case 2b). Of course, we would have to expand the story where

dB is merely commensurable with dA and to where they are incommensurable. But that

would take us into proportion theory and our general understanding of continua. Again,

uniform motion is irrelevant as we are only comparing periods of motion.

In other words, our analysis of Aristotle on faster almost forces us to distinguish

cases which are trivially analyzable into Cases (2a) and (2b), which we could call

the determinate cases, and indeterminate cases such as (2c). (2c) also illustrates how

indeterminate cases can be built directly out of determinate cases. We can even see why

(2a) and (2b) would be taken as the fundamental notions of faster. One starts from the

most basic case of determinate times; the other from the most basic case of determinatedistances traversed.

This distinction between determinate and indeterminate cases mirrors the Eudoxan

method as reconstructed by Knorr. That Aristotle refers to the criteria as someones

definition of faster encourages us to look to a mathematician associated with this

approach. I think that it is very reasonable to consider this definition as deriving from

Eudoxus and his school. Furthermore, it may even constitute a fragment of his lost, but

highly influential treatise, Per tacn (On speeds), but may even go back toArchytas.28

For these are nigh the only candidates we have.

27 Cf. Physics Z 2.232b267, where he refers to (2b) as demonstrated (ddeitai). He alsouses (2a) as well as (3) in the last argument of the chapter, against atomism (2.233 b1532, cf.

1920, 267).28 For Eudoxus, cf. Simplicius, In Arist. De caelo 494.1112. For Archytas, we know little

more than that Diogenes Laertius describes him as the first to apply mathematical principles tomechanics (Vitae viii 83 =DK47A1.16, cf. also Vitruv. vii 14 =DK47B7, which ascribes research

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24 H. Mendell

4. Proof by measure/suprameasure cases and by commensurable/

incommensurable cases: A conflation of techniques in De caelo A 6

InDe caelo A 6.273a21b29,Aristotle produces a somewhat notorious argument thatinfinite bodies cannot have finite weight. I am concerned here with the more innocu-

ous, but perplexing part of the argument, De caelo A 6.273a27b24, and not with theproblematic case where weight is non-uniformly distributed. Earlier I distinguished two

kinds of determinate/indeterminate proofs by cases. In the first the theorem is proved for

the case where one magnitude measures another and then for the case where the magni-

tude is a suprameasure of it, i.e. measures out a magnitude that is larger than the second

magnitude. The other is Knorrs proof for commensurable magnitudes followed by a

proof for incommensurable magnitudes based on the commensurable case. Aristotles

argument is best understood as a proof by cases where one magnitude measures another

and then where it suprameasures it. Yet Aristotle mysteriously treats the indeterminate

case as though it were a commensurable case, and the suprameasure case as an incom-

mensurable case. I shall argue that we can best understand this oddity if we suppose

that Aristotle has conflated the two methods. Thus, the text constitutes evidence for both

types of proof by determinate and indeterminate cases.

In what follows, P(v) is the weight of a body with size v, and V(p) is the size of a

body with weight p. This is convenient because in the part of the argument in question,

Aristotle assumes that weight is uniformly distributed over a given body, a general

problem in Aristotles arguments on the infinite. As in the previous section, v1 v2shall mean that v1 is an initial volume ofv2, while v1 < v2 will have its normal meaning

that v1 is smaller than v2.

[Case 1] [That an infinite body cannot have finite weight] is clear from the following

arguments. For let the weight be finite, and let the infinite body be AB, and the weight

of it be G. Subtract from the infinite some finite magnitude, BD. Let the weight of it be

E. Then E will be less than G. For the weight of the smaller is smaller. Let the smaller

measure it out as many times as you like, and let it come about that as the smaller is to the

larger, so BD is to BZ. For it is possible to subtract as many times as you like from the

infinite.Yet if the magnitudes are proportional to the weights, and if the lesser weight is of

the lesser magnitude, then the larger will be of the larger magnitude. (a27b5) Therefore

the weight of the finite and the weight of the infinite will be equal. (b

56)

Moreover, if the weight of the larger body is larger, then the weight of HB will be larger

than the weight of ZB, with the result that the weight of the finite will be larger than the

weight of the infinite. (b68) And unequal magnitudes will have the same weight. For the

infinite is unequal to the finite. (b810)

[Case 2] It makes no difference whether the weights are commensurable or incommen-

surable. For the same argument will apply [also or even= a] when they are incommen-surable, e.g., if the third E exceeds in measuring G. For if three BD magnitudes are taken

together, their weight will be greater than G. Thus the same impossibility will arise.

[Case 3] Moreover, it is [also or even=a] possible to take commensurable weights.For it makes no difference whether we begin with the weight or with the magnitude For

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Two-Step Eudoxan Proportion Theory in Aristotle 25

some other magnitude, e.g. to BZ. For it is possible to subtract as much as you like from

a magnitude which is infinite. For when these are taken both their magnitudes and their

weights will be commensurable with one another.

Diagram 5.

The basic interpretive problem with the text is that Aristotle provides us with severalcases, but does not make at all clear why these cases are genuinely distinct. Part of the

difficulty, as is clear from Aristotles treatment of the infinite elsewhere, is that for these

types of arguments about the infinite, one can get away with measures and multiples of

measures. Case (1) and Case (2) each seems adequate on its own. In fact, in a similar

situation where he argues that there cannot be an infinite change in a finite time or a finite

change in an infinite time, a little subsequent to his definition of faster, Aristotle says

(Physics Z 2.233b24) that it makes no difference to the argument whether the finiteportion of the finite magnitude measures it, exceeds it, or is deficient.29 Again, I am

only considering three of the four cases, where the weights are uniformly distributed.Aristotle argues for each case that if there is an infinite body then it is of infinite heaviness

since otherwise a finite part of the infinite will weigh at least as much as the whole

infinite. Suppose thatAB is an infinite body and G of finite heaviness, i.e. G = P(AB) and

V(G) = AB.

In Case (1), Aristotle derives three absurd consequences, which we must distinguish

from the main argument. I hence divide Case (1) into four parts, the main argument

(a27b5), and the three consequences (b56, b68, and b810):

Case 1. Aristotle starts with the body and cuts off from AB a finite magnitude DB

AB, whose weight is E. He then assumes that E measures out the whole weight G

and claims that there is a BZ AB such that E : G = DB : BZ, where BZ will be

finite. Aristotle hence appears unnecessarily to assume the existence of the fourth

proportional, a characteristic of the Eudoxan proofs of Euclid, Elements xii. Thefact that E measures G and hence that DB measures BZ guarantees that BZ just is a

multiple of DB and so makes that assumption uninteresting. Behind the language of

proportions is merely the claim that G = n E, so that finite n DB will weigh G.

29 toto d atametrsei t f' AB, lleyei, perbale diafrei gron (And this [finite part BE of AB] either measures out [finite] AB or will be deficient or will

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26 H. Mendell

Case 1 (conclusion a). Hence, the weight of finite part BZ of AB will be equal to

the weight of AB.

Case 1 (conclusion b). If we take HB > BZ, then its weight will be greater than G ,

the weight of the infinite AB, the weight of the part being greater than the weight

of the whole.

Case 1 (conclusion c). Unequal magnitudes will also have the same weight.30

The mathematically salient point for us is that E is taken as measuring G , i.e. a

measure, determinate case. Aristotle next says that it makes no difference whether we

take E commensurable with G or incommensurable with it. Already, we should expect

a division by cases. The result, however, is both striking and somewhat disappointing,

particularly in that Aristotle starts with the incommensurable case.

Case 2 (incommensurable). Suppose E exceeds in measuring out G , i.e. m m E

= G . Then let n E > G , and take n DB from AB, and proceed as in Case (1),where in the example n = 3. Observe that the incommensurable case actually involves

the more general situation where E does not measure G . Thus, it may actually be

unclear whether Aristotle really intends E to be incommensurable with G in the

standard sense. In effect, this is really a supra-measure case.

If we take Case (1) and Case (2) together, we have the method of the scholion to

Theodosius, where our measure of one magnitude (here every portion of the infinite is

a measure of it!) was first treated as a measure of a second magnitude and then treated

as exceeding it in measuring it.

Even the way of expressing the supra-measure case in Aristotle and the scholion

are very similar: Arist. 273b1213: oon e [t E] trton perbllei metron tbroj (e.g., if the third E exceeds in measuring G) and the scholion: a t DH tZB atametron perbaltw lssoni auto t ZQ (and let DH in measuringout ZB exceed it by ZQ which is smaller than DH). The expression, in measuring X

(out) to exceed X is very rare in Greek literature and only known otherwise in extant

Greek mathematics in Platonic commentaries on number theory and commentaries on

Aristotle.31

30 This conclusion is problematic only because the weight is assumed to be uniformly dis-

tributed.31 The scholia to Theodosius are not in the latest ThesaurusLinguaeGraecae CD, TLG E.

The only uses of the expression I have found are these. One author is not in the commentary

tradition: Diodorus Siculus, Bibliotheca historica i.47.3.5 = Hecataeus fr. 25.431 (ca. 300 B.C.E.)tn pda metromenon perbllein toj pt pceij (the foot [of the largest statue inEgypt] in measuring exceeds seven cubits); two are neo-Platonic commentaries on number theory:

Iamblichus, In Nicomachi arithmeticam intro. 53.45: toi plhrontwj atoj metrsei perballntwj llipj (either (the difference of two numbers) will measure them byfilling them up, or by exceeding or by being deficient) and 53.2754.5 for a similar expression:

n d ge perbllonsa mtrhsij. . . (if the measuring is exceding. . . ) and Theon

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Two-Step Eudoxan Proportion Theory in Aristotle 27

Furthermore, contrast this expression with the corresponding expression in Elementsv def. 4 Lgon cein prj llhla megh lgetai, dnatai pollaplasiazmealllwn percein (magnitudes are said to have a ratio to one another which areable by being multiplied to exceed one another), and its use in Elements x 1: T Ggr pollaplasiazmenon stai pot to AB mezon. pepollaplasisw (For Gbeing multiplied will sometime be larger than AB. Let it be multiplied ...). In expressing

variants on the bisection principle, Aristotle rarely, if ever, uses these forms.32 Note,

however, that if Aristotle had used this expression, the distinction between Case (1) and

Case (2) would have been effaced so that Aristotle would only have needed Case (2). In

fact, in the Physics, Aristotle uses either Case (1) or Case (2).33

Case 3 (commensurable). Aristotle points out that here we can begin with the weights

or with the magnitudes. This time he starts with weight E commensurable with G and

marks off from AB the finite magnitude BD of weight E. Next he finds magnitude

BZ AB such that E : G = BD : BZ. Hence, BZ will be commensurable with BD

and is constructable from E, G, and BD.

stin ato to lssonoj mroj (And this happens whenever two numbers are proposedand the smaller in measuring the larger is not able to measure the whole, but leaves out a part

of the larger which is a part of the smaller itself); and the rest are commentaries on Aristotle.

Alexander,In de sensu 115.2024, explains the usage: Every magnitude is measured out by someportion of it. For even if the last measuring it out did not come together with it but exceeded it

(t teleutaon atametron at m sunapartzoito at, ll' perblloi), noless would it have measured it out. Simplicius quotes the passage, In Arist. de caelo libr. comm.220.1819. Three commentators, Themistius, In Arist. physica paraphr. 187.2427, Philoponus,

In Arist. phys, 803.17, Simplicius, In Arist. phys, 949.910, uses a variant of the expression inexplaining Aristotle, Z 2.233b24. See note 29. In any case, the commentators understand the

language of Aristotles argument in the same way as the usage in the scholion to Theodosius.32 Cf. Physics Z 7.237b2833: for when a part [of the motion] is taken which will measure out

the whole, in so many equal times as the parts are, it moves the whole, so that since these are finite,

by each being so-much and by all being so-many-times, then the time would be finite. For [the

time] will be so-much so-many-times, as much as the time of the part multiplied by the number of

parts (lhfntoj gr morou atametrsei tn lhn, n soij crnoij tosotoijsa t mri stin, tn lhn enhtai, st' pe tata peprantai a t psonaston-a t posij panta, a crnoj n eh peperasmnoj. tosautijgr stai tosotoj, soj to morou crnoj pollaplasiasej t plei tnmorwn). I suspect, however, that this passage, which asserts a correspondance between the partsof motion and the parts of the time, only shares in common with Euclids formulation the use of

multiply, which may not be enough for common heritage.33 For Case (1), cf. Physics 10.266a1920: otw d t D prostiej atanalsw

t A a t E t B (in this way by adding to (a measure of finite umph A) I shall exhaustA and by adding to E (the same measure of distance traveled B) I shall exhaust B). For case (2)

cf. Physics .10 266b24: prj peperasmnon gr e prostiej perbal pantjrismnou, a fairn lleyw satwj (by repeatedly adding to a finite I shall exceed

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28 H. Mendell

Cases (1) and (2) start with a given finite magnitude DB and proceed to look at

the weight of DB to get E. Thus, one does not know when one picks DB whether its

weight E is commensurable with G (or rather measures G) or not. In Case (3) Aristotles

point is that we may begin with the weight instead. Aristotle could have constructed the

commensurable case from the magnitude. However, the result would not have added

anything to Cases (1) and (2). Given the set-up, however, this is the only way we can

ensure that we have chosen a weight E commensurable with the whole weight G. For

if we start with the magnitude instead, we can only construct E commensurable with G

if we can pick a finite body BZ, where P(BZ) = G, which itself needs to be constructed

from E. Hence, the three cases are genuinely mathematically distinct, but not for the

reasons Aristotle gives:

Given: n P(x) = P(n x), AB infinite and G finite and P(AB) = G:

Case 1a:Assume, BD AB and BD is finite and n P(BD) = G.

Hence, n BD AB

Hence, n BD is finite and P(n BD) = P(AB).

Case 2a:

Assume BD AB and BD is finite and n P(BD) > G > (n1) P(BD).

Hence, n BD AB and P(n BD) > P(AB).

Case 3a:Assume E is commensurable with G and we can construct V(E) as finite.

Hence, we can construct V(G) such that E : G = V(E) : V(G).

Hence, V(G) is finite.

Aristotles argument, taken out of the context of contemporary mathematics, is in-

coherent. We have seen that Case (2), as argued, really belongs, as the supra-measure

case, with Case (1) as the corresponding measure case. Furthermore, Case (3) actually

considers a different situation, although Aristotle presents it as indifferent between the

two set-ups, with Cases (1) and (2) starting from the magnitude and Case (3) starting

from the weight. However, Aristotle connects (2) and (3) as providing an incommensu-rable and commensurable case, respectively, while it is completely inessential to (2) that

the weights be incommensurable. The whole matter would have been easier if Aristotle

had just told us to multiply E so as to exceed G, in other words Case (2) or even Case

(1) with conclusion (b).

How are we to make sense of this argument? We might imagine a dialogue where

Aristotle provides Case 1, and someone objects that E might not measure G. He then

says that it doesnt matter if the weights do not measure, i.e. are commensurable or

incommensurable. He takes the harder case, even (= a) if they are incommensurable.

Then the objector turns to the issue of the fact that Aristotle begins with the bodies andnot their weights. The difficulty is that we still do not know why Aristotle turns to the

h h i h bl f i f f h

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Two-Step Eudoxan Proportion Theory in Aristotle 29

has been handled. Hence, in turning to the issue of starting with the weights instead of

the body, it seems irrelevant whether or not they are commensurable. Aristotle does not

mention the case where we start with incommensurable weights.

The best (only?) way to make sense of this, I suspect, is if we consider Aristotle

as having two argument forms in front of him. One is the method of the scholion.

This involves taking a measure case, where a measures b, and a supra-measure case,

where a exceeds b in measuring b. The other is the case where the commensurable and

incommensurable cases are distinguished. Such sophisticated argumentation as one finds

in applications of this method in Archimedes, Theodosius, and Pappus, is completely

unnecessary for Aristotles argument, as is clear from its validity. If needed, he could

have added:

Case 4a:

Assume, E is not commensurable with G and n E > G and we can construct V(E)

as finite.Hence, n V(E) is finite and P(n V(E)) = n P(V(E)) > G. Note that P(V(E)) = E.

But this is just (2a) redone. But how else are we to take Aristotles claim that there

are two cases, one commensurable and one incommensurable? In other words, the

measure/supra-measure method is adequate; in fact, either is adequate on its own just as

they need not be distinguished. The addition of the second method, or any method that

would make sense of the distinction, is not necessary.

It is a psychological question whether Aristotle has a penchant for displaying arcane

knowledge in his arguments. However, if we understand him as conflating two arguments

in his presentation in order to make use of the latest in proportion arguments, thisconflation makes perfect sense. Even the order of putting the commensurable case after

the incommensurable case can be explained by the fact that the incommensurable case

is really the supra-measure case that goes with the measure Case (1). I do not see how

else to explain them.

This explanation of the oddities of De caelo A 6, namely that Aristotle conflatestwo proof techniques, requires one more thing. If Aristotle conflates both methods, they

must have been current in the time of Aristotle. Note that this does not imply that these

methods constituted the mid-fourth century concept of same ratio.

5. The two traces summarized

By themselves, De caelo A 6 and Physics Z 2 seem to tell us little about Greekmathematics in the fourth century that we could not learn from other discussions. Taken

together with Knorrs reconstruction, however, they provide interesting evidence for

mathematical techniques in mid-century and probably of the school of Eudoxus. If we

emphasize Aristotles claim that the three claims about faster in Physics Z 2 are a

definition which some people, possibly Eudoxus, held, then in addition to Aristotlesarguments for the three claims we must take very seriously both the order in which

Aristotle presents the three but also the role that they might play in a mathematical kine

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30 H. Mendell

nate case. The conjunction of the three cases as a definition of Physics Z 2 is not at allinadequate. On this reading it represents a important conception of defining a relational

concept such as faster by determinate and special cases which are then expanded to

all cases by means of principles of consistency and continuity, and using analogues of

geometrical constructions.

De caelo A 6 further contributes to this picture. It provides strong, albeit indirectevidence for two related techniques treating determinate and indeterminate cases, where

the measure/supra-measure method serves as a lemma for working with the commen-

surable/incommensurable method. De caelo A 6 distinguishes four cases, the measurecase, an incommensurable case (really the supra-measure case), and a commensurable

case, and the case not here discussed of non-uniform distribution of weight ( b236).

Three of these pertain to these two methods.

Together these passages cannot show that Eudoxus and his school gave a formal

definition of same ratio for the commensurable case, proved theorems about it and

then used reductios and other techniques to extend the theorems to indeterminate cases.

They tell us little directly about 4th cent. BCE manipulations of magnitudes in normal

mathematical contexts. However, given the mass of other, later evidence for a Eudoxan

theory involving a simple commensurable case and an extension to the incommensurable

case, as compiled by Knorr, they constitute contemporary evidence for those procedures.

In fact, they may show us more, that the methods could be treated as definitions, that

the methods as definitions did not need to cover all possibilities except as mathematical

consequences of the cases defined via principles of continuity, and that, as one may

expect, the measure case is distinct from and prior to the commensurable case.

They also tell us something else. We should be cautious about what we take asa definition in 4th cent B.E.C. mathematics and how we should read a report of a

definition in Aristotle. We commonly understand such reports in combination with the

general view that Aristotle has a much narrower notion of definition. My goal was to

show the reverse, that Aristotle is perfectly capable of treating fundamental theorems as

foundations for definitions. Further, I do not claim that my story of a restricted definition

of proportion completed by a real definition was a theory of proportion advocated by

Eudoxus. We know nothing of his conception of definition.34

34 In his review of H.J. Waschkies, Knorr [1980, 507] says, In a half dozen passages Aristotle

himself applies a technique of proportions in dealing with the problem of infinite magnitudes,motions, and powers (e.g., Physics IV 8, VI 2 and 7, VIII 10; De caelo I 6 and 7). In these heinvokes not the theorems but the basic notions of a pre-Euclidean technique. For Physics IV () 8

see below note 39. I suspect that Knorr cites Physics Z 2 and Z 7 primarily because they illustratewhat I have labled determinate and indeterminate cases. Physics 10.266a14b24 illustrates

b b

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Two-Step Eudoxan Proportion Theory in Aristotle 31

6. The moral: Antanairesis, early proportion theory, and Topics 335

This story also has a disturbing moral. Very similar points may be made about the

much more entrenched view that in Aristotles time, people defined magnitudes having

the same ratio as having the same anthyphairesis or alternate subtraction. The basis forthis is a comment by Aristotle, Topics 3. 158b2935, and its explanation by Alexan-

der of Aphrodisias.36 Aristotle says that the definition (or a definition) of same ratio

is having the same antanairesis (ntanaresij). Alexander says that there was an an-cient definition of same ratio: ratios are the same which have the same anthyphairesis

(nufaresij), and adds that antanairesis in Aristotle just is anthyphairesis. Alexan-ders source is very likely Aristotles junior colleague in the Lyceum, Eudemus.37 There

can be little doubt that Alexander and Aristotle refer to the same method as that used by

Euclid in four fundamental propositions (Elements vii 12, x 23) and called by the verb

n

ufaires

ai. The question we need to raise is: what would Aristotle and Eudemusmean by a definition in this case?For convenience, here is a quick, anachronistic description of anthyphairesis.We tend

to define the method recursively, but with some lip service to the difference between it

and, say, continued fractions (hence the anachronism lies both in its algebraic features

and its explicit use of induction). Let there be two magnitudes, x, y, with x > y. We form

a sequence of numbers n1, . . . , and a sequence of magnitudes a1, . . . . Generally, the

sequence of magnitudes is merely a catalyst for determining the sequence of numbers,

whether there is a last term of the sequence of magnitudes, and if there is one what the

last term is.

1. Let a1 = x, a2 = y, and n1 a number such that n1 a2 a1 and (n1 + 1) a2 > a1.

Ifa1 = n1 a2, the sequence of numbers is (n1) and the sequence of magnitudes is

(a1, a2). Otherwise, a3 = a1 n1 a2.

2. Let ni+1 be a number such that ni+1 ai+2 ai+1 and (ni+1 + 1) ai+2 > ai+1. If

ni+1 ai+2 = ai+1, then the sequence of numbers is (n1, . . . , ni+1) and the sequence

of magnitudes is (a1, . . . , ai+2). Otherwise, ai+3 = ai+1 ni+1 ai+2.

We can then reconstruct a ratio x : y, from a series (n1, . . . , nm):

1. Let bm = nm and bm+1 = 1 (alternatively, if we want the unit to be a magnitude b,let bm = nm b and bm+1 = b)

2. Let bi1 = ni1 bi + bi+13. x : y = b1 : b2

35 Saito [2003] has developed a similar thesis to the one here presented, although along different

and interesting lines.36 Alexander, In topic. libr. comm. 545.1519.37 So Knorr [1975, 258]. One cannot draw much from the specific wording of Alexander, In

topic. libr. comm. 545.16: nlogon cei megh prj llhla n at nufaresij(magnitudes hold proportionally to one another which have the same anthuphairesis), even though

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32 H. Mendell

In Euclid, we are only interested in whether there is a last term am+1 (which will be a

common measure) of x and y. To establish by this method that ratios are the same, one

needs to be interested only in the sequence, n1, . . . , ni, . . . . This procedure leads to a

fundamental use of anthyphairesis from at least the Hellenistic age on and probably in

the fourth century B.C.E. as well. Let (n1

, . . . , nk

, . . . , am

) be the anthyphairesis of the

ratio a : b, and let c : d be the ratio reconstructed from (n1, . . . , nk). If k < m, then

c : d > a : b if k is even, and c : d < a : b if k is odd. The limited, but important,

evidence outside the Aristotelian corpus for interest in this sequence in the fourth century

B.C.E. is outside our present concerns.38

The question may be put as follows: when Aristotle and Alexander say that there was

a definition of same ratio as having the same antanairesis, (1) do they mean that there

was a text which stated as much under a list of definitions, or (2) do they mean that there

was a restricted definition that ifa, b and c, dhave respective greatest common measures

by the method of anthyphairesis, then they have the same ratio, or (3) do they mean that

there was a fundamental theorem which proved that a : b = c : d iffa, b have the same

anthyphairesis as c, d? In the last case, we could even imagine a nominal restricted

definition of same ratio along the lines of 1 def. 1 above, or even along the lines of

Elements vii def. 21: Numbers are proportional when the first is equally a multiple ofthe second as the third of the fourth, whether the same part or same parts. One then

extends the definition through the method of anthyphairesis.

I dont here wish to argue about the important internal evidence concerning Ele-ments xiii and x for the role of anthyphairesis, nor do I wish to deny anthyphairesis as afundamental method reflected in Aristotles text.39 I am just concerned with the claim in

Aristotles Topics, a book concerned with strategies in dialectical games, usually betweentwo players, where one player attempts to defend or reject a thesis by answering ques-

tions from a player who wishes to destroy or defend the thesis, respectively. Sometimes

Aristotle seems to suggest that philosophy, the actual activity in developing definitions,

uses a kind of dialectical solitaire.40 However, Topics is concerned with dialecticalstrategies in the arrangement of arguments, where deception and ploy might be central.

In Topics 3, Aristotle offers some warnings about positions that are difficult toattack and easy to defend. These include those that are primary by nature, i.e. first

principles which require only having their terms defined, or are very far from primary

hypotheses and so may use derived principles or very complex proofs.41 The reason isthat in the case of primary hypotheses, the questioner needs to get the interlocutor to

agree to a definition in order to pin down her hypothesis, which she will resist doing,

38 See Fowler [1987].39 A trace of anthyphairesis may also be found at Physics 8.215b1318, and Marrachia [1980,

2134] and Pritchard [1997] suggest that there is also a trace at Met. 15.1020b261a9, but thesea