ESSAY
10
Diagrammatic Reasoning
in Euclid’s Elements
Danielle Macbeth
Although the science of mathematics has not undergone the sorts of
revolutionary changes that can be found in the course of the history
of the natural sciences, the practice of mathematics has nevertheless
changed considerably over the two and a half millennia of its history.
The paradigm of ancient mathematical practice is the Euclidean demonstration, a practice characterized by the involvement of both text and
diagram. Early modern mathematical practice, begun in the seventeenth century, is instead computational and symbolic; it constitutively
involves the formula language of arithmetic and elementary algebra
that one is taught as a schoolchild even today (see Macbeth [2004]
and Lachterman [1989]). Over the course of the nineteenth century,
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this practice gave way, finally, to a more conceptual approach, to reasoning from concepts, for instance, from the concept of continuity in
analysis or from that of a group in abstract algebra (see Stein [1988]).
This most recent mathematical practice has naturally brought in its
train—at least officially, if not in the everyday practice of the working
mathematician—a demand for rigorous gap-free proofs on the basis of
antecedently specified axioms and definitions. It has also suggested to
many that Euclidean mathematical practice is hopelessly flawed.
But Euclidean geometry is not flawed. Although it has its limitations—not everything one might want to do in mathematics can be
done in the manner of Euclid—this geometry has, over the course of
its two and a half thousand year history, proved to be an extremely
successful, robust, and sound mathematical practice, albeit one that is
quite different from current mathematical practice. My aim is to clarify
the nature of this practice in hopes that it might ultimately teach us
something about the nature of mathematical practice generally. Perhaps
if we better understand the first (and for almost the whole of the long
history of the science of mathematics the only) systematic and fruitful
mathematical practice, we will be better placed to understand later
developments.
Euclid’s Elements is often described as an axiomatic system in
which theorems are proven and problems constructed though a chain
of diagram-based reasoning about an instance of the relevant geometrical figure. It will be argued here that this characterization is mistaken
along three dimensions. First, the Elements is not best thought of as
an axiomatic system but is more like a system of natural deduction; its
Common Notions, Postulates, and Definitions function not as premises
from which to reason but instead as rules or principles according to
which to reason. Secondly, demonstrations in Euclid do not involve reasoning about instances of geometrical figures, particular lines, triangles,
and so on; the demonstration is instead general throughout. The chain
of reasoning, finally, is not merely diagram-based, its moves, at least
some of them, licensed or justified by manifest features of the diagram.
It is instead diagrammatic; one reasons in the diagram in Euclid, or so
it will be argued.
Diagrammatic Reasoning in Euclid’s Elements · 237
1 Axiomatization
or System of Natural Deduction?
In an axiomatic system, a list of axioms is provided (perhaps along with
an explicitly stated rule or rules of inference) on the basis of which to
deduce theorems. Axioms are judgments furnishing premises for inferences. In a natural deduction system one is provided not with axioms
but instead with a variety of rules of inference governing the sorts of inferential moves from premises to conclusions that are legitimate in the
system. In the case of natural deduction, one must furnish the premises
oneself; the rules only tell you how to go on. The question, then, whether
Euclid’s system is an axiomatic system or not is a question about how
the definitions, postulates, and common notions that are laid out in advance of Euclid’s demonstrations actually function, whether as premises
or as rules of construction and inference. Do they function to provide
starting points for reasoning, as has been traditionally assumed?1 Or
do they instead govern one’s passage, in the construction, from one diagram to another, and in one’s reasoning, from one judgment to another?
Inspection of the Elements suggests the latter. In Euclid’s demonstrations, the definitions, common notions, and postulates are not treated
as premises; instead they function, albeit only implicitly, as rules constraining what may be drawn in a diagram and what may be inferred
given that something is true.2 They provide the rules of the game, not
its opening positions.
Consider, for example, the first three postulates. They govern what
can be drawn in the course of constructing a diagram: (i) If you have two
points then a line (and only one) may be produced with the two points
as endpoints; (ii) a finite line may be continued; and (iii) if you have a
point and a line segment or distance, then a circle may be produced with
that center and distance. In each case, one’s starting point, points, and
1 The underlying assumption perhaps is that, as both Plato and Aristotle thought,
any science, including mathematics is, or should strive to be, axiomatic. Insofar as
Euclid’s system is a paradigm of science, then, it must be axiomatic.
2 As we will see, Euclid in fact almost never invokes his definitions, postulates,
and common notions in the course of a demonstration. They are nevertheless readily
identifiable as warranting the moves that are made.
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lines, must be supplied from elsewhere in order for the postulate to be
applied. And nothing can be done, at least at first, that is not allowed by
one of these postulates. But once they have been demonstrated, various
other rules of construction can be used as well. For instance, once it has
been shown, using circles, lines, and points, that an equilateral triangle
can be constructed on a given finite straight line (proposition I.1), one
may in subsequent constructions immediately draw an equilateral triangle, without any intermediate steps or constructions, provided that one
has the appropriate line segment. Propositions such as I.1 that solve
construction problems function in Euclid’s practice as derived rules of
construction. Once they have been demonstrated, they can be used in
the construction of diagrams just as the postulates themselves.
Euclid’s common notions, and again most obviously the first three,
again govern moves one can make in the course of a demonstration, in
this case in the course of reasoning. They govern what may be inferred:
(i) If two things are both equal to a third then it can be inferred that
they are equal to one another; (ii) if equals are added to equals then it
follows that the wholes are equal; and (iii) if equals be subtracted from
equals, then the remainders are equal.3 These common notions manifestly have the form of generalized conditionals, which is just the form
rules of inference must take when they are stated explicitly.4 Furthermore, in this case as well, theorems, once demonstrated, can function
in subsequent demonstrations as derived rules of inference. Once it has
been established that, say, the Pythagorean theorem is true (I.47), one
may henceforth infer directly from something’s being a right triangle
that the square on the hypotenuse is equal to the sum of the squares
on the sides containing the right angle. Indeed, Euclid’s Elements is so
called because the totality of its theorems and constructions provide in
this way the elements, rules, for more advanced mathematical work.
Definitions can also license inferences, though, as we will see, they
have other roles to play as well. If, for example, a diagram in a proposition contains a circle then the definition of a circle licenses the passage
to the claim that its radii are equal. If it contains a trilateral figure,
3 These are, of course, not formally valid rules of inference; they are instead what
Sellars has taught us to call materially valid rules.
4 That rules of inference are inherently conditional in form and essentially general
is argued by Ryle ([1950]).
Diagrammatic Reasoning in Euclid’s Elements · 239
that is, a figure bounded by three straight lines, all of whose sides are
equal then the definition of equilateral triangle licenses one to conclude
that the figure is an equilateral triangle.
But not everything that happens in the course of a demonstration
is governed by an explicitly stated rule, whether primitive or derived.
There are two sorts of cases. First, Euclid draws (explicitly or implicitly) various obviously valid inferences, such as that two things are not
equal given that one is larger than the other, despite the fact that the
rule governing the passage is nowhere explicitly stated. Furthermore, in
order to follow a demonstration in Euclid, one must read various things
off the relevant diagrams, again according to rules that seem nowhere to
be stated. For example, given two lines that cross, or cut one another,
in a diagram, Euclid assumes that there is a point at their intersection.5
The point of intersection seems simply to “pop up” in the diagram as
drawn, and is henceforth available to one in the course of one’s reasoning.6 This happens not once but twice in the very first proposition of
Book I of the Elements, to construct on a given finite straight line an
equilateral triangle.
The demonstration begins with the setting out: Let AB be the given
straight line. A statement of what is to be done follows: to construct
an equilateral triangle on AB. Then the construction is given:
(C1) With center A and distance AB let the circle BCD be described.
(This is licensed by the third postulate, though Euclid does not
mention this.)
(C2) With center B and distance BA let the circle ACE be described.
(Again, the warrant for this, the third postulate, is not mentioned.)
(C3) From point C, in which the circles cut one another, to the points
A, B let the straight lines CA, CB be joined. (This is implicitly
5 Thomas L. Heath ([1956]: vol. 1, 242) takes this to be an objection: “Euclid has
no right to assume, without premising some postulate, that the two circles will meet
in a point C.” The objection is, Heath says, “a commonplace.”
6 I borrow this use of the expression “pop up” from Kenneth Manders, to whom
I am indebted for helping me to appreciate just how important this feature of Euclidean diagrams is. See his [1996] and [2008].
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Figure 1
licensed by the first postulate on the assumption that there is such
a point C.)
Figure 1 contains the resulting diagram.7 And the apodeixis then follows:8
(A1) Given that A is the center of circle CDB, AC is equal to AB. (This
is licensed, without mention, by the definition of a circle.)
(A2) Given that B is the center of circle CAE, BC is equal to BA. (This
again is implicitly licensed by the definition of a circle.)
(A3) Given that AC equals AB and BC equals BA, we can infer that
AC equals BC because what are equal to the same are equal to
each other (that is, Common Notion 1).
(A4) Given that AB, BC, and AC are equal to one another, the triangle
ABC is equilateral. (This is warranted by the definition of equilateral triangle, on the assumption that there is such a triangle
ABC.)
This triangle was constructed on the given finite straight line AB as
required, and so we are done. In the course of this demonstration, first
a point pops up at the intersection of the two drawn circles, and then
later a triangle pops up, formed from the radii of the two circles. This
7 This
image is taken from the Elements: vol. 1, 241.
word ‘apodeixis’ is generally translated as ‘proof.’ For reasons that will
emerge, I leave it untranslated.
8 The
Diagrammatic Reasoning in Euclid’s Elements · 241
sort of thing is, furthermore, ubiquitous in ancient Greek geometrical
practice. One simply reads the relevant geometrical objects off the diagrams, apparently without any explicitly stated warrant for doing so.
Although nothing can be put into a diagram that is not licensed by one
of the given postulates or a previous construction, there seem to be no
stated rules governing what, in the way of pop-up objects, can be taken
out of it.9 We will have occasion to come back to this.
2 Generality in Euclid’s Demonstrations
I have suggested that Euclid’s Elements is not best thought of as an axiomatic system (even one that is incomplete, not fully rigorous) because
it seems instead to function more like a system of natural deduction.
The Common Notions, Postulates, and Definitions function in the Elements not as judgments that will, for the purposes of proof, be taken to
be true but instead as rules of passage governing allowable moves in the
demonstration. We need now to consider whether the diagram provides
an instance about which to reason or instead something more general.
We have seen that Euclid’s demonstration that an equilateral triangle can be drawn on a given finite straight line begins with a “setting
out” (ekthesis): Let AB be the given straight line. And this is generally
true of Euclid’s demonstrations. Although what is to be demonstrated
is something wholly general, the demonstration invariably proceeds by
way of such a setting out. To anyone familiar with proofs in standard
quantificational logic, it is very easy to take this setting out as an analogue of Universal Instantiation. We read, for instance, in Russell’s
[1956]:
Given a statement containing a variable x, say ‘x = x,’ we
may affirm that this holds in all instances, or we may affirm any one of the instances without deciding as to which
instance we are affirming. The distinction is roughly the
same as that between the general and particular enunciation in Euclid. The general enunciation tells us some9 Could Euclid’s definitions serve this purpose? Can it be inferred, for example,
from the fact that a line is a breadless length that at the cut of two lines there is a
point (i.e., something that has no parts)? Perhaps, but more would need to be said.
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thing about (say) all triangles, while the particular enunciation takes one triangle and asserts the same thing of this
one triangle. But the triangle taken is any triangle, not
some one special triangle; and thus although, throughout
the proof, only one triangle is dealt with the proof retains
its generality.
(Russell [1956]: 64)
In quantificational logic, in order to prove that (say) all A is C given
that all A is B and that all B is C, one must first turn to an instantiation
of the premises in order that the rules of the propositional calculus may
be applied. One reasons, in effect, about a particular case, and then at
the end of the proof one takes what has been shown to apply generally
on the grounds that no inference was drawn in the course of the proof
that could not have been drawn were any other instance to have been
considered instead. According to Russell, demonstrations in Euclid work
the same way.10 Indeed, one might be puzzled as to what the alternative
might be.
Manders offers a suggestion following Leibniz, namely, that “[D]iagrams [are] textual components of a traditional geometrical text or argument, rather than semantic counterparts” (Manders [1996]: 391). That
is, the diagram does not provide an instance (a semantic counterpart),
but instead what Leibniz calls a “character”: “For the circle described
on paper is not a true circle and need not be; it is enough that we take
it for a circle.”11 But what exactly might this mean? Grice’s analysis of
the distinction between (as he puts it) natural and non-natural meaning
provides the tools for at least an outline of a plausible answer (Grice
[1957]).12
As Grice points out, there is an intuitively clear distinction between,
for example, the sense in which a certain sort of spot on one’s skin can
mean measles and the sense in which three rings on the bell of a bus can
10 Netz ([1999]: §6) takes essentially the same view: What is to be demonstrated
is general but the demonstration itself (that is, the setting out, construction, and
apodeixis) is particular; its generalizability is “a derivative of [its] repeatability”
(ibid.: 246; see also ibid.: 262).
11 Leibniz [1969]: 84; quoted in Manders [1996]: 291.
12 See also Dipert [1996].
Diagrammatic Reasoning in Euclid’s Elements · 243
mean that the bus is full.13 The task is to provide an analysis of the
difference between the two senses of ‘means,’ natural and non-natural,
respectively; and one of Grice’s examples to that end is particularly
revealing for our purposes. Grice asks us to compare the following two
cases:
(1) I show Mr. X a photograph of Mr. Y displaying undue familiarity
with Mrs. X.
(2) I draw a picture of Mr. Y behaving in this manner and show it to
Mr. X.
And, as he immediately goes on to remark,
I find that I want to deny that in (1) the photograph (or
my showing it to Mr. X) meantNN [that is, non-naturally]
anything at all; while I want to assert that in (2) the picture
(or my drawing and showing it) meantNN something (that
Mr. Y had been unduly familiar), or at least that I had
meantNN by it that Mr. Y had been unduly familiar. What
is the difference between the two cases? Surely that in case
(1) Mr. X’s recognition of my intention to make him believe
that there is something between Mr. Y and Mrs. X is (more
or less) irrelevant to the production of this effect by the
photograph . . . But it will make a difference to the effect
of my picture on Mr. X whether or not he takes me to be
intending to inform him (make him believe something) about
Mrs. X, and not to be just doodling or trying to produce a
work of art.
(Grice [1957]: 282–3)
Although the photograph can serve to convey to someone the fact that
Mr. Y is unduly familiar with Mrs. X independent of anyone’s intending
that it so serve, the drawing cannot. To take the drawing as conveying
a message about Mr. Y’s behavior, rather than as a mere doodle or as
a work of art, essentially involves taking it that someone produced it
with the intention of conveying such a message, and that that person
13 In the opening paragraphs of his essay, Grice lists five different ways the use of
‘means’ differs in the two cases.
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did so with the intention that that intention be recognized (and play
a role in the communicative act). Only in that case does the drawing
mean (non-naturally) that Mr. Y is unduly familiar with Mrs. X.
A photograph has natural meaning in virtue of a (causally induced)
resemblance between the image in the photograph and that of which it
is a photograph. A drawing, Grice argues, can in certain circumstances
have instead non-natural meaning. That is, it can have the meaning
or content that it does in virtue of one’s intending that it have that
meaning or content (and intending that that intention be recognized
and play a certain role in the communicative act). So, we can ask,
does a drawn figure in Euclid have Gricean natural meaning or instead
Gricean non-natural meaning? If it is a drawing of an instance, a particular geometrical figure, then it has natural meaning. It is in that
case a semantic counterpart; it is the thing, say, the particular triangle ABC, that is referred to in the text of a demonstration when one
judges of triangle ABC that it is thus and so. But perhaps the drawing,
like Grice’s drawing, instead has non-natural meaning. Perhaps it is
not (say) a circle but instead, as Leibniz says, is taken for one. If so,
then the Euclidean diagram can mean or signify some particular sort
of geometrical entity only in virtue of someone’s intending that it do
so and intending that that intention be recognized. One’s intention in
making the drawing—an intention that can be seen to be expressed in
the setting out and throughout the course of the construction—is, in
that case, indispensable to the diagram’s playing the role it is to play
in a Euclidean demonstration.14
A Euclidean diagram can be interpreted either as having natural
or as having non-natural meaning in Grice’s sense. If it has natural
meaning then it does so in virtue of being an instance of a geometrical
figure. But if it has non-natural meaning then it can be inherently
general, a drawing of, say, an angle without being a drawing of any
14 Jody Azzouni makes a related suggestion in his [2004]. He writes that “[T]he
proof-relevant properties of diagrammatic figures . . . are not the actual (physical)
properties of diagrammatic figures, but conventionally stipulated ones the recognition
of which is mechanically executable . . . [T]he situation is exactly the same with
games: Pieces are conventionally endowed with powers (e.g., to ‘take’ other pieces
under certain circumstances) . . . [A]ll powers a game piece is stipulated to have are
mechanically recognizable” (ibid.: 125).
Diagrammatic Reasoning in Euclid’s Elements · 245
angle in particular. The geometer draws some lines to form a rectilineal
angle, in order, say, to show that an angle can be bisected. The geometer
does not mean or intend to draw an angle that is right, or obtuse, or
acute. He means or intends merely to draw an angle. Hence he says
(I.9) that the angle he draws is “the given rectilineal angle,” not also
that it is right, or obtuse, or acute. That which he draws, regarded as
something having natural meaning, will necessarily be right, or acute, or
obtuse; but regarded as having non-natural meaning, it will be neither
right nor acute nor obtuse. It will simply be an angle. One can no
more infer on the basis of the drawing so conceived that the angle in
question is (say) obtuse, based on how it looks, than in Grice’s case one
could infer (say) that Mr. Y has put on a little weight recently, based
on how the drawing of him looks. A drawing of Mr. Y could have as its
non-natural meaning that he has put on weight recently, just as in the
context of a Euclidean demonstration a drawing of an angle can have as
its non-natural meaning that it is (say) obtuse, but that would require,
in both cases, that quite different intentions be in play.
If the figures drawn in a Euclidean diagram have non-natural rather
than natural meaning, then they can, by intention, be essentially general. It seems furthermore clear that they would in that case function
as icons in Peirce’s sense, rather than as symbols or indices, because
they would in some way resemble that which they signify.15 But the resemblance is not, at least not merely and not directly, a resemblance in
appearance. As Peirce notes, “[M]any diagrams resemble their objects
not at all in looks; it is only in respect to the relations of their parts
that the likeness consists” (Peirce [1931]: 282). We can, then, think of a
drawn figure in Euclid as an icon that (though it may also resemble its
object in appearance) signifies by way of a resemblance, or similarity,
in the relations of parts, that is, in virtue of a homomorphism. For
example, on such an account a drawn circle serves as an icon of a ge15 Peircean icons can have either natural or non-natural meaning. In particular,
individual instances of geometrical figures can be icons of the relevant sorts of things;
they have in that case natural meaning. A drawn circle regarded as an instance
of a circle is an icon of a circle that has natural meaning because it so functions
independent of anyone’s intention that it do so. But a drawn circle can also function
as an icon with non-natural meaning. In that case it can be essentially general, an
icon of a circle not further specified.
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ometrical circle not in virtue of any similarity in appearance between
the two but because there is a likeness in the relationship of the parts
of the drawing, specifically in the relation of the points on the drawn
circumference to the drawn center, on the one hand, and the relation of
the corresponding parts of the geometrical figure, on the other.
A drawn circle is roughly circular; it looks like a circle just as a dog
looks like a dog. But a dog looks like a dog because it is a dog, that is,
a particular instance of doggy nature (as we can think of it). A drawn
circle, I have suggested, can look like a circle for either of two reasons.
It can look like a circle for the same reason that a dog looks like a dog,
namely, because it is a circle, a particular instance of circle nature. Or
it can look like a circle because it is a icon with non-natural meaning
that is intended to resemble a circle first and foremost in the relation
of its parts. Because what it is an icon of is circle nature, and because
what is essential to a circle’s being a circle is that all points on the
circumference are equidistant from the center, and it is this relationship
of parts that is to be iconically represented, the icon itself comes to look
roughly circular. The appearance of circularity is induced in this case
by the intended higher order resemblance rather than being something
that is there in any case (as circularity is there in any case in a drawing
of a particular instance of a circle).
Now if drawings in Euclid function as icons with non-natural meaning, this would explain why Euclid never mentions either straightedge or
compass—which one might expect him to if it was an instance that was
to be drawn. If one wishes to draw a particular straight line then the
best way to do that is with a straightedge, and similarly in the case of a
circle: If one wants to draw a particular circle then one is best off using a
compass. If, however, one’s aim is to draw something with non-natural
meaning, in particular, an icon, say, of a line or circle, then all that
matters is that one’s drawing is able to produce the desired effect, to
convey one’s intention. In this case there is no reason at all to mention
some particular means of producing the drawing precisely because the
drawing, to serve the purpose it is to serve need not look very much like
that for which it is an icon. This is not true of an instance: An instance
ought as far as possible to look like what it is. It follows that instances
Diagrammatic Reasoning in Euclid’s Elements · 247
are harder to draw than icons, and this is in fact generally true. It is, for
instance, much easier to draw stick human figures and “smiley” faces,
which are of course inherently general, than it is to draw someone in
particular, the way some particular person actually looks. Even very
small children can do the former; most of us even as adults cannot do
the latter very well at all.16 There is no need, then, for mechanical aids
in drawing Euclidean diagrams if those diagrams function as has been
suggested here, as essentially general icons with non-natural meaning.
More interestingly, the idea that diagrams are iconic and general
by intention would also explain why ancient Greek mathematicians almost never make the sorts of mistakes they would be expected to make if
their reasoning were based instead on an instance. If one were reasoning
about an instance, it would be critical carefully to distinguish between
what can and cannot be inferred on the basis of one’s consideration of
that instance. And this ought, in principle, to be quite difficult. It is, for
instance, much harder to learn what a particular person looks like (so
as to be able to re-identify that person) from a chance photograph than
it is to find this out from, say, a caricature because in the caricature the
work of discovering what are the salient and characteristic features of
the person’s appearance has already been done. Similarly, if a Euclidean
drawing were an instance (with natural meaning), it would be hard to
distinguish between what Manders has called co-exact features, ones on
the basis of which an inference can be made, and exact features, which
have no implications for one’s course of reasoning (see Manders [1996]:
392–3). But in fact, all the evidence suggests that this is not hard at all.
As Mueller remarks regarding the familiar diagram-based “proof” that
all triangles are isosceles, “perhaps a ‘pupil of Euclid’ might stumble on
such a proof; but probably he, and certainly interested mathematicians,
would have no trouble in figuring out the fallacy on the basis of intuition
and figures alone. And in the history of Euclidean geometry no such
fallacious arguments are to be found. There are indeed many instances
16 Notably, some very young autistic children lacking essentially all linguistic ability
are able to draw very realistically, that is, things as they actually look, an ability
that deteriorates as they learn language. This suggests that we generally draw not
what things look like but instead something essentially general, something more like
an icon of a concept. See Selfe [1977].
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of tacit assumptions to be made, but these assumptions were always
true. In Euclidean geometry . . . precautions to avoid falsehood are really unnecessary” (Mueller [1981]: 5). This is unsurprising if the figures
in the diagram function not as instances but as icons (with non-natural
meaning) that are inherently general. The features that Manders identifies as exact are no more there in the diagram than information about
the relative size of human limbs is there in a stick figure.
We saw that drawings of geometrical figures conceived as icons (with
non-natural meaning) can resemble particular instances in appearance
in virtue of the fact that there is a resemblance in the relation of the
parts. A drawn circle, intended as an icon of a circle, can look very
much like a particular instance of a circle. It is easy, then, based on
such an appearance, to think that the drawing is such an instance, not
such an icon at all. There is good reason nonetheless for thinking that
in Euclidean demonstrations the diagram functions iconically (i.e., nonnaturally) rather than to provide an instance about which to reason.
First, as Manders has noted, the latter idea “seems incompatible with
the use of diagrams in proof by contradiction” (Manders [1996]: 391).
To demonstrate, for instance, that a circle does not cut a circle at more
than two points, one first sets out that two circles do cut at more than
two points, say, at four. This is not a situation that can obtain; there
is no instance to be drawn. And yet the diagram is drawn (as we will
see). If the diagram is read instead as a Peircean icon with non-natural
meaning there is no difficulty. What the diagram means non-naturally,
the content it exhibits, is exactly what one aims to show is impossible,
that a circle cuts a circle at more than two points.
Nor is this the only sort of case in which it is impossible to draw
an instance. We know, because Euclid tells us in the opening section of
the Elements, that a point is that which has no parts, and that a line
segment is a length that has no breadth (the extremities of which are
points). Such entities are clearly not perceptible; there is nothing that
a thing with no parts, or a length with no breadth, looks like. It follows
directly that there is no way to draw an instance of either a point or a
line. On the other hand, such things, and their relations one to another,
can be iconically represented. A drawn line length, for example, can
Diagrammatic Reasoning in Euclid’s Elements · 249
represent (be intended iconically to signify) a line with endpoints. A
drawn dot can represent (be intended iconically to signify) a point. A
drawn circle is, again, a slightly different case because drawn circles do
look roughly circular; that is, there is a look that geometrical circles can
be said to have. But as in the case of Grice’s drawing, the role of a drawn
circle in the context of a Euclidean demonstration is not, on the account
we are pursuing, that of an instance but instead that of an icon with
non-natural meaning, one that is intended to represent the relation of
the points on the circumference to the center, the fact that those points
are equidistant from the center (whether or not in the figure as drawn
the points on the circumference look equidistant from the center). All
that is represented in a Euclidean straight line (conceived as an icon) is
a breadthless length lying evenly with the points on itself, that is, a certain relationship between the line and the points that may be found on
it; and similarly, all that is represented in a Euclidean circle (similarly
conceived) is the relation of the points on its circumference to the center.
In each case, what is important for the cogency of the demonstration is
not what the figure looks like but instead what is intended by it whether
as set out in Euclid’s definitions and postulates, or as stipulated by the
particular problem or theorem in question. The similarity in appearance
(say, between the icon of a circle and an actual instance of something
circular) does help to convey the intended meaning, just as it does in
the example of Grice’s drawing; nevertheless, what is meant is carried
by the intention, not by the similarity in appearance. It is for just
this reason that the demonstration can be seen to be essentially general
throughout.17 What one draws in Euclidean diagrams are not pictures of
17 This is, furthermore, consistent with the final step in the demonstration. As Ian
Mueller explains, “[I]n ancient logic the sumperasma is the conclusion of an argument.
In the Elements, however, the sumperasma is not so much a result of an inference
as a summing up of what has been established” (Mueller [1974]: 42). As he notes
in his [1981], “[T]he word sumperasma [which marks the last stage of a Euclidean
demonstration] can . . . mean ‘completion’ or ‘finish’ . . . [T]he sumperasma merely
sums up what has taken place in the proof . . . It merely completes the proof by
summarizing what has been established” (ibid.: 13). So conceived, the move from
a claim involving letters to the explicitly general claim at the end of a Euclidean
demonstration is essentially a move from an implicit generality, such as that a horse
is warm-blooded (inferred perhaps from the claim that a horse is mammal together
250 · Danielle Macbeth
Figure 2
geometrical objects but the relations that are constitutive of the various
kinds of geometrical entities involved. As the point might be put, a Euclidean diagram does not instantiate content but instead formulates it.
3 Diagrammatic Reasoning in Euclid’s Elements
We have seen that Euclid’s system is best thought of on analogy with
a system of natural deduction as it contrasts with an axiomatization.
And I have argued that figures in Euclidean diagrams do not provide
instances but instead should be read as having non-natural meaning
and to function, in particular, as icons. Diagrams so conceived are
constitutively general. What remains to be shown is that the reasoning
involved in a Euclidean demonstration is not merely diagram-based but
instead diagrammatic.
It is clearly true that in a Euclidean demonstration at least some
steps in the chain of reasoning are in some way licensed by the diagram.
Consider, for example, proposition I.5, that in isosceles triangles the
angles at the base are equal to one another, and if the equal straight
lines be produced further, the angles under the base will be equal to
one another. The diagram is shown in Figure 2.18 It is then argued
that because AF is equal to AG and AB to AC, the two sides FA,
with the fact that mammals are warm-blooded) to the explicit generality that all
horses are warm-blooded.
18 The image is taken from Euclid: vol. 1, 251.
Diagrammatic Reasoning in Euclid’s Elements · 251
AC are equal to the two sides GA, AB, and they contain a common
angle FAG. This inference, from the two equalities, AF to AG and AB
to AC, to the double conclusion, first that the two sides FA, AC are
equal to the two sides GA, AB, and also that both those same pairs
of sides contain the angle FAG, might be thought to be diagram-based
in the following sense. Having, first, drawn an isosceles triangle with
sides AB and AC equal, and then having produced the straight lines
BD and CE by extending (respectively) AB and AC, and taken F on
BD at random and cut AG off from AE to equal AF, one has thereby
inevitably produced not only FA, AC equal to GA, AB, which is already
more or less given, but also the angle FAG, which, as inspection of the
diagram shows, is identical both to the angle FAC and to the angle
GAB. It is no more possible to construct the required diagram without
realizing this identity of angles than it is to embed one circle in a second
circle and the second in a third without thereby embedding the first in
the third. (This is of course the principle behind Euler diagrams, which
can be used to exhibit the validity of various valid syllogistic forms
of reasoning, here barbara.) So, just as one can infer on the basis of an
Euler diagram that some conclusion follows, so one can infer on the basis
of the above diagram that the relevant claims are true. The diagrambased inference is, in both cases, valid, that is, truth-preserving, and
it is truth-preserving precisely because there exists a homomorphism, a
higher level formal identity, that relates relations among elements in the
diagram with relations among the entities represented by those elements.
Reasoning using Euler diagrams provides a paradigm of what we
might mean by diagram-based inference. In that case, one draws icons
of things of a given sort, in particular, circles to represent collections of
objects, and furthermore draws them in spatial relations that are homomorphic to relations of inclusion and exclusion over collections. If,
for example, all dogs are mammals then one draws the circle representing the collection of dogs inside the circle representing the collection
of mammals. If no fish are mammals then one draws the circle representing the collection of fish wholly separate from that representing the
collection of mammals, without any overlap at all. Then one can just
read off the diagram what the relationship is between the collection of
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dogs and the collection of fish: The relevant icons are wholly disjoint
so one infers that no dogs are fish. The inference, the passage from
the premises about dogs, mammals, and fish, to the conclusion about
dogs and fish is licensed by the diagram one draws. It is diagram-based.
Do diagrams Euclidean demonstrations function in the same way as the
basis for an inference? We will see that they do not.
The first indication that diagrams in Euclid function differently from
Euler diagrams is the fact that although Euler diagrams can make something that is implicit in one’s premises explicit, they cannot in any
way extend one’s knowledge. Euclidean demonstrations, by contrast,
do seem clearly to be fruitful, real extensions of our knowledge. And
the reason they are is surely connected to the fact that objects can pop
up in a Euclidean diagram. We find out that an equilateral triangle
can be constructed on a straight line only because a triangle pops up
in the course of the construction. Before that point there is nothing
whatever about any triangles in anything we have to work with in the
demonstration. Quite simply, there is no sense in which an equilateral
triangle is implicit in a line, even given Euclid’s axioms, postulates, and
definitions. Nevertheless, as I.1 shows, such a triangle is there potentially: given what Euclid provides us, we can construct an equilateral
triangle on a given straight line. The triangle just pops up when we
draw certain lines. And this is true generally in ancient Greek mathematics: One achieves something new largely in virtue of this pop-up
feature of its diagrams. To understand the role of diagrams in Euclidean
demonstrations, then, we need to understand these pop-up objects.
In a Euclidean demonstration, what is at first taken to be, say, a
radius of a circle is later in the demonstration seen as a side of a triangle.
But how could an icon of one thing become an icon of another? How,
for example, could an icon of a radius of a circle turn into an icon of a
side of a triangle? Certainly nothing like this happens in the course of
reasoning about an Euler diagram. The iconic significance of the drawn
circles is fixed and unchanging throughout the course of reasoning. If
the lines one draws in a Euclidean diagram had this same sort of iconic
significance, the reasoning would clearly stall because in that case no
new points or figures could pop up. It is also obvious that nothing could
Diagrammatic Reasoning in Euclid’s Elements · 253
possibly be both an icon of (say) a radius of a circle and an icon of a
side of a triangle at once, because these are incompatible. An icon of
a radius essentially involves reference to a circle; radii are and must be
radii of circles. An icon of a side of a triangle makes no reference to a
circle. So nothing could at once be an icon of both. But one and the
same thing could serve now (at time t) as an icon of a radius, and now
(at time t′ ≠ t) as an icon of a side. The familiar duck–rabbit drawing
is just such a drawing; it is a drawing that is an icon of a duck (though
of course no duck in particular) when viewed in one way and an icon of
a rabbit (no one in particular) when viewed in another.
Some drawings, such as the duck–rabbit, can be seen in more than
one way. It would seem, then, that we can say that in the diagram of I.1
one sees certain lines now as icons of radii, as required to determine that
they are equal in length, and now as icons of sides of a triangle, as required in order to draw the conclusion that one has drawn an equilateral
triangle on a given straight line.19 The cogency of the reasoning clearly
requires both perspectives. But if it does then the Euclidean diagram
is functioning in a way that is very different from the way an Euler diagram functions. Much as in the case of the duck–rabbit drawing, and by
contrast with an Euler diagram, various collections of lines and points in
a Euclidean diagram are icons of, say, circles, or other particular sorts of
geometrical figures, only when viewed a certain way, only when, as Kant
would think of it, the manifold display (or a portion of it) is synthesized
under some particular concept, say, that of a circle, or of a triangle.20
The point is not that the drawn lines underdetermine what is iconically
represented, as if they have to be supplemented in some way. It is that
the drawing as given, as certain marks on the page, has (intrinsically)
19 Euclid’s
definitions would seem to play a crucial role here: Only if one can see
in the diagram that the definition is satisfied can one find the relevant object in it.
20 This idea, that a system of signs may function not by way of combinations of
primitive signs that have designation independent of any context of use, but instead
by appeal to primitive signs that designate only given a context of use, is appealed
to also in Sun-Joo Shin’s ([2002]) reading of Peirce’s notation for his alpha logic. In
my [2005], I argue that Frege’s notation functions in an essentially similar way. The
primitive signs of the language express senses independent of their occurrence in a
sentence but designate only in the context of a sentence and relative to an analysis,
relative, that is, to a way of regarding the whole two-dimensional array of signs.
254 · Danielle Macbeth
the potential to be regarded in radically different ways (each of which is
fully determinate albeit general, again, as in the duck–rabbit case). It
is just this potential that is actualized in the course of reasoning, as one
sees lines now as radii and now as sides of a triangle, and which begins to
explain the enormous power (and beauty) of Euclidean demonstrations
as against the relative sterility (and lumpishness) of Euler diagrams.
In Euclidean diagrams, geometrical entities, points, and figures, pop
up as lines are added to the diagram. Cut a line AB with another
line CD and up pops a point E as the point of intersection, as well as
four new lines AE, BE, CE, and DE; take the diagonal of a square and
up pop two right triangles, and so on. We do not find this surprising
in practice; indeed, one need have no self-conscious awareness that it
is happening as one follows the course of a Euclidean demonstration.
Nevertheless, as we have seen, it clearly is happening, and the cogency
of the demonstration essentially depends on it. Such pop up objects, I
have suggested, depend in turn on our capacity to “re-gestalt” various
collections of lines, to see them now one way and now another. And
what this shows is that it is not the lines themselves that function as
icons (even in light of one’s intention that they be so regarded) but only
the lines when seen from a particular perspective, when viewed in one
way rather than another equally possible way.
Consider again our two crossed lines AB and CD that cut at E,
and suppose that they are functioning as icons independent of any perspective taken on them. One could then argue that the point E can
belong to only one of the four segments AE, BE, CE, or DE, leaving the
other three without an endpoint at one end. This would be a perfectly
reasonable inference if we were dealing with a simple icon, one whose significance was fixed independent of any perspective taken on it, because
it is true that if you divide a (dense) line by a point then that point can
be the endpoint of only one of the two line segments, leaving the other
to approach it indefinitely closely (because the line is dense), without
having it as its endpoint. Suppose now that we offered this argument
to an ancient Greek geometer. How would he respond? He would laugh
us away, much as he laughs away those who would “[attempt] to cut up
the ‘one’” by pointing out that the line drawn to iconically represent
Diagrammatic Reasoning in Euclid’s Elements · 255
the unit can surely be divided.21 The geometer laughs because one is
in that case taking the diagram the wrong way, because one fails to
understand how it works as a diagram.
When two lines AB and CD cut at E, only one point pops up and
it is the endpoint of all four lines AE, BE, CE, and DE, which also just
pop up with the cut. And they can do because E is the endpoint of any
one of these lines only relative to a way of regarding it, much as certain
lines in the duck–rabbit drawing are ears, or a duckbill, only relative
to a way of regarding those lines. Another example makes the same
point. Suppose that I claimed on the basis of the drawing of a straight
line segment ABCD that, contrary to what Euclid holds, two straight
lines can have a common segment: AC and BD have BC in common.
Again the geometer would laugh—and rightly so. Of course one can see
the drawn line as iconically representing AC, or BD, or AD, or BC, and
indeed one may need so to regard it in the course of a demonstration; but
that does not show that the lines themselves have a common segment.
This would be required only if the drawn line functioned iconically to
represent these possibilities independent of any perspective that was
taken on the drawing. Between two points there is only one straight
line, not many; two points can seem to have a common segment (as
above) only if one, perversely, mistakes the way the diagram functions.
I have been arguing that the lines in a Euclidean diagram, like the
lines in a duck–rabbit drawing, function as icons of various sorts of geometrical objects only relative to a perspective that is taken on them.
But those diagrams have a further feature as well, one that is not found
either in the duck–rabbit drawing or in an Euler diagram, a feature that
is essential to the fruitfulness of Euclidean demonstrations. Euclidean
diagrams are distinguished in having three levels of articulation as follows. At the lowest level are the primitive parts, namely, points, lines,
angles, and areas. At the second level are the geometrical objects we
are interested in, those that form the subject matter of geometry, all of
which are wholes of those primitive parts. At this level we find points
as endpoints of lines, as points of intersection of lines, and as centers of
circles; we find angles of various sorts that are limited by lines that are
21 Plato,
Republic Book VII : 525e.
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also parts of those angles; and we find figures of various sorts. A drawn
figure such as (say) a square has as parts: four straight line lengths, four
points connecting them, four angles all of which are right, and the area
that is bounded by those four lines. Of course, in the figure as actually
drawn, the lines will not be truly straight or equal, and they will not
meet at a point; the angles will not be right or all equal to one another.
But this does not matter because the drawn square is not a picture or
instance of a square but instead an icon of a square, one that formulates
certain necessary properties of squares. At the third level, finally, is the
whole diagram, which is not itself a geometrical figure but within which
can be discerned various second-level objects depending on how one
configures various collections of drawn lines within the diagram. What
we will see is that these three levels of articulation, combined with the
possibility of a variety of different analyses or carvings of the various
parts of the diagram, account for the ways in which one can radically
reconfigure parts of intermediate wholes into new (intermediate) wholes
in the course of a Euclidean demonstration, and thereby demonstrate
significant and often surprising geometrical truths.
As already noted, a Euclidean demonstration comprises a diagram
and some text, most importantly for our current purposes, the kataskeue
(construction) and the apodeixis. The kataskeue provides information
about the construction of the diagram and is governed by what can
be formulated in the diagram as legitimated by the postulates and any
previously demonstrated problems. The apodeixis, which is governed by
what can be read off the diagram as legitimated by the definitions, common notions, and previously demonstrated theorems, should be read, on
our account, not as the medium of the proof but instead as providing
instructions regarding how various portions of the constructed diagram
are to be read, construed, or analyzed, that is, how they are to be carved
up in the course of the demonstration. It is the diagram that is the site
of reasoning, on our account, not the accompanying text.
Consider one last time the first demonstration in Euclid, that on
a given straight line an equilateral triangle can be constructed. The
diagram is displayed, again, in Figure 3. According to the reasoning
already rehearsed, we know that AC = AB because A is the center of
Diagrammatic Reasoning in Euclid’s Elements · 257
Figure 3
circle CDB, and that BC = BA because B is the center of circle CAE,
hence that AC = BC = AB, on the basis of which it is to be inferred that
ABC is an equilateral triangle constructed on the given line AB, which
was what we were required to show. Our task concerning this little chain
of reasoning is to understand how, based on a claim about radii of circles,
one might infer, even given the diagram, something about a triangle.
Were the reasoning merely diagram-based, that is, were it reasoning
in (natural) language but, at least in some cases, justified by what is
depicted in the diagram then the problem seems intractable. No mere
diagram of some drawn circles, however iconic, can justify an inference
from a claim about the radii of circles to a claim about a triangle. If, on
the other hand, we take the diagram not merely as a collection of icons of
various geometrical entities but instead as a display with the three levels
of articulation outlined above, and so as a collection of lines various
parts of which can be configured and reconfigured in a series of steps
(as scripted by the apodeixis), the solution of our difficulty is obvious.
One considers now one part of the diagram in some particular way and
now another in some (perhaps incompatible, but perfectly legitimate)
way as one makes one’s passage to the conclusion. In particular, a line
that is at first taken iconically to represent a radius of a circle is later
taken iconically to represent a side of a triangle. It is only because the
drawing can be regarded in these different ways that one can determine
that certain lines are equal, because they can be regarded as radii of
a single circle, and then conclude that a certain triangle is equilateral,
because its sides are equal in length. The demonstration is fruitful, a
258 · Danielle Macbeth
Figure 4
real extension of our knowledge, for just this reason: Because we were
able to take a part of one whole and combine it with a part of another
whole to form an utterly new and hitherto unavailable whole, we were
able to discover something that was simply not there, even implicitly,
in the materials with which we began.
4 More Examples
Consider now the Euclidean demonstration of Proposition II.5: If a
straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square
on the straight line between the points of section is equal to the square
on the half. The diagram is shown in Figure 4.22 We letter it for ease
of reference in our written discourse; were the presentation an oral one
would need only to point to relevant aspects of the drawn diagram. To
understand this diagram, that is, to know how to “read” it, configure
its parts (at least to begin with), we need to know the provenance of its
various parts. We are given that line AB is straight, and that it is cut
into equal segments by C and into unequal segments by D. We know by
the construction that CEFB is a square on CB, which we know from
proposition I.46 is constructible, and that DG is a straight line parallel
to CE and BF, KM parallel to AB and EF, and AK parallel to CL and
BM, shown to be constructible in proposition I.31. That is, already we
see here the various ways the drawn lines are taken to figure in various
iconic representations of geometrical figures. CB, for example, is first
22 The
image is taken from Heath [1956]: vol. 1, 382.
Diagrammatic Reasoning in Euclid’s Elements · 259
taken to be the half of line AB, but then as a side of a square. BM is a
line length equal in length to DB but also a part of the line BF, which
is another side of that same square. It is in virtue of these various relations of parts iconically represented in the drawn diagram, and of these
reconfigurings of parts, that the diagram can show that the theorem is
true. It does so in a series of six steps scripted by the apodeixis:
(1)
These (shaded) areas are
equal, as is shown by Prop.
I.43: The complements of a
parallelogram about the diameter are equal to one another.
(2)
It follows from (1) that these
areas are equal, because the
same has been added to the
same.
(3)
These areas are known to be
equal on the basis of what we
know about the relationships
that obtain among the lines
that form the boundaries of
the two shaded areas.
(4)
It follows from (2) and (3) that
these areas are equal because
things equal to the same are
equal to each other.
260 · Danielle Macbeth
(5)
It follows from (4) that these
areas are the same because
the same has been added to
equals.
(6)
It follows from (5) that these
areas are equal because the
same has been added to
equals.
But that is just what we wanted to show: that the square on the half
is equal to the rectangle contained by the unequal segments plus the
square on the line between the points of section. By construing various
aspects of the diagram in these various ways in the appropriate sequence
one comes to see that the theorem is, indeed, must be, true. But if that
is right, then (again) it is the diagram that is the site of reasoning in
Euclid, not the text. The text, on this account, is merely a script to
guide one’s words, and thereby one’s thoughts, as one walks oneself
through the demonstration.23 The demonstration is not in the words of
the apodeixis; it is in what those words help us to see in the diagram.24
23 Think of doing a calculation in Arabic numeration, say, working out the product
of thirty-seven and forty-two. One first writes the two Arabic numerals in a particular
array, one directly above the other, then, in a series of familiar steps performs the
usual calculation. As one writes down the appropriate numerals in the appropriate
places, one may well talk to oneself (‘Let’s see; two times seven is fourteen, so . . . ’).
Nevertheless, the calculation is not in the words one utters, if one does utter any
words, even silently (and one need not), but in the written notation itself. Just
imagine trying to do the calculation without appeal to Arabic numeration! One
calculates in Arabic numeration. In much the same sense, I am suggesting, one
reasons in a Euclidean diagram.
24 That the words of the demonstration in Euclid are merely a record of the speech
of someone presenting the demonstration to an audience is also indicated by the way
the written text is presented in Euclid: “unspaced, unpunctuated, unparagraphed,
aided by no symbolism related to layout.” “Script,” in Euclid, “must be transformed
into pre-written language, and then be interpreted through the natural capacity for
seeing [better: hearing] form in [spoken] language” (Netz [1999]: 163).
Diagrammatic Reasoning in Euclid’s Elements · 261
(This would perhaps be even more obvious if we considered someone
trying to discover a demonstration of some proposition. The inscribing
would not be of words but of diagrams.)
As this simple demonstration again shows, in order to follow, that
is, to understand, a Euclidean demonstration, one must be able to see
various drawn lines and points now as parts of one iconic figure and
now as parts of another. The drawn lines are assigned very different
significances at different stages in one’s reasoning; they are interpreted
differently depending on the context of lines they are taken, at a given
stage in one’s reasoning, to figure in. It is just this that explains the
fruitfulness of a Euclidean demonstration, the fact that its conclusion is,
as Kant would say, synthetic a priori. In Euclid, the desired conclusion
is contained in the diagram, drawn according to one’s starting point
and the postulates, not merely implicitly, needing only to be made explicit (as in a deductive proof on the standard construal or in an Euler
diagram), but instead only potentially. The potential of the diagram
to demonstrate the conclusion is made actual only through an actual
course of reasoning in the diagram, that is, by a series of successive
refigurings of what it is that is being iconically represented by various
parts of the diagram. Parts of wholes must be taken apart and combined with parts of other wholes to make quite new wholes. And this is
possible, first, in virtue of the three levels of articulation in the diagram,
and also because the various parts of the diagram signify geometrical
objects only relative to ways of regarding those parts. A given line must
actually be construed now as an iconic representation of (say) a part of
another line and now as an iconic representation of a side of a square, if
the demonstration is to succeed. The diagram, more exactly, its proper
parts, must be actualized, now as this iconic representation and now as
that, through one’s construal of them as such representations, if the result is to emerge from what is given. Only a course of thinking through
the diagram can actualize the truth that it potentially contains.25
Another example, this time of a reductio proof, is the demonstration
of III.10, that a circle does not cut a circle at more points than two.
The diagram (again lettered to enable us to explain in the written text
what it involves) is shown in Figure 5.26 The following is encoded, or
25 We
might, then, translate ‘apodeixis’ not as ‘proof’ but as ‘reasoning.’
image is taken from Heath [1956]: vol. 2, 23.
26 The
262 · Danielle Macbeth
Figure 5
formulated, iconically in the diagram. First, we have by hypothesis (for
reductio) that the circle ABC cuts the circle DEF at the points B, G, F,
and H; and here (again) it is especially obvious that we do not picture
the hypothesized situation, which is of course impossible, but instead
formulate in the diagram the content of that hypothesis. BH and BG
are drawn (licensed by the first postulate) and are then bisected at K
and L respectively. (We know that we can do this from I.10.) KC is
then drawn at right angles to BH, and LM at right angles to BG. (We
know from I.11 that we can do this.) Both KC and LM are extended,
KC to A and LM to E, as permitted by the second postulate. What we
have then are, first, two straight lines BH and BG, both of which are
chords to both circles; that is, we can see BH as drawn through ABC or
as drawn through DEF, and similarly for BG. And we also know that
AC bisects HB and is perpendicular to it, and similarly for ME and BG.
Now the reasoning begins:
(1)
The center of the highlighted circle
is on the highlighted bisecting line,
by III.1 porism.
Diagrammatic Reasoning in Euclid’s Elements · 263
(2)
The center of the highlighted circle
is on the highlighted bisecting line,
again by III.1 porism.
(3)
As the diagram shows, the two bisecting lines meet at only one point;
so this point must be the center of
the circle (because it is the only
point that is on both lines).
(4)
The center of the highlighted circle
is on the highlighted bisecting line,
by III.1 porism.
(5)
Again, the center of the highlighted
circle is on the highlighted bisecting
line, by III.1 porism.
264 · Danielle Macbeth
(6)
The two bisecting lines meet at only
one point and so that point must
be the center of the circle (because,
again, that is the only point on both
lines known to contain the center).
(7)
The two circles have the same center
because one and the same point that
was shown first (3) to be the center
of the circle ABC was shown also (6)
to be the center of DEF.
But we know from III.5 that if two circles cut one another then they will
not have the same center. Our hypothesis that a circle cuts a circle at
four points is false. Furthermore, because the demonstration involved
only three of the four points presumed to cut the two circles, it clearly
is cogent no matter how many points one supposes the two circles to
cut at. Any number more than two will lead to contradiction. It follows
that a circle does not cut a circle at more points than two.
In this demonstration we are concerned with lines, circles, more exactly, the circumferences of circles, and their interrelationships. All that
we assume about the various figures is what we are told about them in
the construction, and all that we infer at each stage is what we know
follows from what we know about the figures that are taken at that
stage to be iconically represented in the relevant part of the diagram.
By focusing on the various parts of the diagram, now this part and now
that, and conceptualizing these parts appropriately, then applying what
one knows to the relevant figure thereby iconically represented, one can
easily come to realize that, given what is known and assumed, the two
circles have one and the same center. But this is absurd given that they
cut one another. The assumption that a circle can cut a circle at more
than two points must be rejected. It is in just this way that one reasons
Diagrammatic Reasoning in Euclid’s Elements · 265
by reductio in a Euclidean diagram, by assuming in a diagram what one
aims to show is false and then showing by a chain of strict and rigorous
reasoning in the diagram that that assumption leads to a contradiction.
5 Conclusion
We have seen that diagrams in Euclid are not merely images or instances of geometrical figures but are instead icons with Gricean nonnatural meaning. As such, they are inherently general. But as we also
saw, this alone is not sufficient to explain the workings of a Euclidean
demonstration. In particular, Euclidean reasoning cannot be merely
diagram-based, even where the diagram is conceived iconically, because
then no account can be given of the pop-up objects that are essential
to the cogency of Euclidean demonstrations. Euclidean reasoning is instead properly diagrammatic; one reasons in the diagram in Euclidean
geometry, actualizing at each stage some potential of the diagram. This
is furthermore made possible, we have seen, by the nature and structure
of the signs that make up that system. Because Euclidean diagrams are
collections of primitive signs with three levels of articulation, the parts
can be variously construed in systematic ways to mean, iconically represent, now this geometrical figure and now that. The conclusion that
one draws on the basis of a Euclidean demonstration is, for just this reason, contained in one’s starting points only potentially. The steps of the
demonstration must be taken to actualize that conclusion. A Euclidean
demonstration is not, then, diagram-based, its inferential steps licensed
by various features of the diagram. It is properly diagrammatic. One
reasons in the diagram, in Euclid, that is, through lines, dia grammon,
just as the ancient Greeks claimed (Netz [1999]: 36).
The picture of Euclid’s Elements that has emerged is, then, very
different from that with which we began. First, a Euclidean demonstration, as we have come to understand it, is not a proof in the standard
sense at all; it is not, that is, a sequence of sentences some of which are
premises and the rest of which follow in a sequence of steps that are
deductively valid, or diagram-based. Indeed, the demonstration does
not lie in sentences at all. It lies in a certain activity, in a course of
reasoning focused directly on a diagram. And this activity is needed,
we have seen, because it alone can actualize the potential of the diagram
266 · Danielle Macbeth
to yield something new. The diagram, the actual marks on the page (or
some appropriate analogue), must be seen now this way and now that if
the result that is wanted is to be achieved. But if so, then Euclid’s geometry is not an axiomatic system, that is, a collection of primitive and
derived sentences; it does not involve reasoning about an instance (of
a geometrical figure); and it is not merely diagram-based. It is instead
a mode of mathematical inquiry, a mathematical practice that uses diagrams to explore the myriad discoverable necessary relationships that
obtain among geometrical concepts, from the most obvious to the very
subtle.
Acknowledgments
This paper is a much longer version of the paper I presented at the
PMP 2007 conference on the philosophy of mathematical practice in
Brussels. I have been helped in preparing this version by comments
Ken Manders made at the time.
Bibliography
Azzouni, J. [2004]. “Proof and Ontology in Euclidean Mathematics.”
New Trends in the History and Philosophy of Mathematics. Ed.
by T.H. Kjeldsen; S.A. Pederson; and L.M. Sonne-Hansen. Odense,
Denmark: University Press of Southern Denmark.
Dipert, R.R. [1996]. “Reflections on Iconicity, Representation, and Resemblance: Pierce’s Theory of Signs, Goodman on Resemblance, and
Modern Philosophies of Language and Mind.” Synthese 106. 373–97.
Grice, H.P. [1957]. “Meaning.” Philosophical Review 66. 377–88.
Heath, T.L. [1956]. The Thirteen Books of the Elements. 2nd ed. First
published in 1908. Toronto: Dover.
Lachterman, D.R. [1989]. The Ethics of Geometry: A Genealogy of
Modernity. New York and London: Routledge.
Leibniz, G.W. [1969]. Philosophical Papers and Letters. Trans. by L.E.
Loemker. Dordrecht: Reidel.
Macbeth, D. [2004]. “Viète, Descartes, and the Emergence of Modern
Mathematics.” Graduate Faculty Philosophy Journal 25. 87–117.
Diagrammatic Reasoning in Euclid’s Elements · 267
— [2005]. Frege’s Logic. Cambridge, MA: Harvard University Press.
Manders, K. [1996]. “Diagram Content and Representational Granularity.” Logic, Language, and Computation. Ed. by J. Seligman and D.
Westerståhl. Vol. I. Stanford: CSLI Publications and Stanford University Press.
— [2008]. “The Euclidean Diagram.” The Philosophy of Mathematical
Practice. Ed. by M. Paolo. Oxford: Oxford University Press.
Mueller, I. [1974]. “Greek mathematics and Greek logic.” Ancient Logic
and its Modern Interpretation. Ed. by J. Corcoran. Dordrecht and
Boston: D. Reidel.
— [1981]. Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. Cambridge, MA and London: MIT Press.
Netz, R. [1999]. The Shaping of Deduction in Greek Mathematics. Cambridge: Cambridge University Press.
Peirce, C.S. [1931]. Collected Papers of Charles Sanders Peirce. Ed. by
C. Hartshorne and P. Weiss. Vol. 2. Cambridge, MA: Harvard University Press.
Russell, B. [1956]. “Mathematical Logic Based on the Theory of Types.”
Logic and Knowledge: Essays 1901–50. Ed. by R.C. Marsh. Book
first published in 1956. Article first published in 1908. London and
New York: Routledge.
Ryle, G. [1950]. “‘If,’ ‘So,’ and ‘Because’.” Philosophical Analysis: A Collection of Essays. Ed. by M. Black. Ithaca, NY: Cornell University
Press.
Selfe, L. [1977]. Nadia: A Case of Extraordinary Drawing Ability in Children. London: Academic Press.
Shin, S.-J. [2002]. The Iconic Logic of Peirce’s Graphs. Cambridge, MA
and London: Bradford Books and MIT Press.
Stein, H. [1988]. “Logos, Logic, and Logistiké: Some Philosophical Remarks on the Nineteenth Century Transformation of Mathematics.”
History and Philosophy of Modern Mathematics. Ed. by W. Aspray
and P. Kitcher. Vol. XI. Minnesota Studies in the Philosophy of Science. Minneapolis, MN: University of Minnesota Press.