Apeiron Volume 44 Issue 3 2011 [Doi 10.1515%2Fapeiron.2011.014] Raymond, Dwayne -- From a Particular Diagram to a Universal Result- Euclid's Elements, Book I

From a Particular Diagram to a Universal Result:Euclids Elements, Book I1

DWAYNE RAYMONDDepartment of PhilosophyTexas A & M University

College Station, Texas, 77843-4237USA

[email protected]

Abstract

I argue that Euclids highly abstract definitions in Book I of the Elements play a key role,enabling general conclusions from particular diagrams. I argue four points. First, defini-tions limit focus to the most generalized features of elementary geometrical objects. Sec-ond, an explicit reciprocal dependency relation exists between figure and boundary.Third, Euclids derivation style employs a combination of diagrams in tandem with thehighly abstract definitions. Fourth, this combination renders all non-universal featuresof a particular diagram irrelevant. The result limits the role of diagrams in a way that isreminiscent of Aristotles characterization of the geometers use of diagrams.

Keywords: Euclid; Diagrammatic Proofs; Co-relatives.

1 Introduction

It is well-known that Euclid obtains universal results from particular geo-metrical constructions.2 How? In opposition to Ian Muellers view thatEuclids definitions of the most elementary geometrical objects points,lines, straight lines, angles, etc., are mathematically useless and never in-voked in the subsequent development3 of Euclids geometry, I will argue

apeiron, vol. 44, pp. 211218Walter de Gruyter 2011 DOI 10.1515/apeiron.2011.014

1 I should like to acknowledge the helpful comments of anonymous reviewers, RobinSmith and the participants comments from the 2010 Joint Session of the AristotelianSociety and the Mind Association.

2 Euclid The Thirteen Books of the Elements vol I, II, III, trans. Sir T. Heath (Dover,1956).

3 Mueller, I. Philosophy of Mathematics and Deductive Structure in Euclids Elements(MIT Press, 1981), 40.

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that all of his definitions, including those of the most elementary objects,are fundamental to that development. And, in opposition to Reviel Netz,who holds that no use can be made of such definitions in the course offirst-order, demonstrative discourse, 4 I will argue that Euclids definitionsplay a crucial role in establishing a universal result with a particular dia-gram. They are fundamental to Euclids diagrammatic reasoning. That is,the definitions are not simply a kind of second-order discourse aboutmathematics in which [b]efore getting down to work, the mathematiciandescribes what he is doing,5 but rather, Euclids definitions, includingthose of the most elementary objects, are significant to his style of deduc-tion in Book I in at least four ways:

First, the definitions of point and line, etc., limit the focus to the most generalizedfeatures of elementary geometrical objects.Second, an explicit reciprocal dependency relation establishes a non-hierarchical co-dependency between geometrical objects in the form of boundary and figure.Third, Euclids derivation style employs a combination of diagrams in tandem withthe highly abstract definitions. Reciprocal dependency relations contribute to thelogical structure within the diagrammatic reasoning.Fourth, this combination renders all non-universal features of a particular diagramirrelevant and the use of reciprocal dependency relations limits the role of the dia-gram in a way that is reminiscent of Aristotles characterization of the geometersuse of diagrams: the diagrams make the truth of a fact obvious, the geometer doesnot reason on the basis of a drawn line, but on the basis of the facts made clear bythem.6 Highly abstract geometrical objects are linked together via co-relatives.

In short, the particular, non-universal features of a diagram are systemati-cally excluded from having a role in the demonstrations. In this way, thegeometer does not reason on the basis of a drawn line, but on the basis ofthe facts made clear by them.

The combination of definitions and diagrams provides for a style ofdeduction that exploits the heuristic advantages of diagrams, while mana-ging the concern that accidental properties may lead one astray. Thehighly abstract definitions of point and line severely limit ones attentionto highly abstract features in the diagram. The combination of definitionsand diagrams effectively removes accidental features of the diagram fromthe proof. The reciprocal dependency relation (the co-relative pairing) be-tween figure and boundary maximizes our ability to exploit the heuristicadvantages of reasoning with diagrams. The reciprocal relation allows usto move in both directions.7

4 Netz, R. The Shaping of Deduction in Greek Mathematics (CUP, 1999), 955 Netz, Shaping, 95.6 Aristotle Posterior Analytics 77a 13.7 This needs to be studied in the context of the analysis and synthesis.

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2 Discussion

Mueller distinguishes between definitions that are simply part of a tradi-tional list and have no role in the demonstration, and those that are essen-tial to the demonstrations.8 While Mueller acknowledges that there is noevidence that the difference between these two sorts of definitions werefelt by the Greeks,9 there does appear to be a divide between highly ab-stract definitions whose function is not obvious, and more robust defini-tions whose role in a demonstration is more transparent. For example, thedefinitions for a point, a line, and a surface appear to be unlike the defini-tions for figures such as circles.

A point is that which has no parts.10

A line is breadthless length.11

A surface is that which has length and breadth only.12

A circle is a plane figure contained by one line in such a way that all straight linesfalling upon it from one point among those lying within the figure are equal to oneanother.13

It has been suggested that the highly abstract definitions of a point, aline, and a surface merely reflect the way in which the work of prede-cessors enters into the Elements.14 Szabo, for example, relies upon thelack of apparent functionality in proofs when he dates these definitionsearlier than those that have a role in the derivation.15 The complexdefinition of a circle, however, clearly provides a context in which thelengths of lines are equated. Whereas the former appear to play no rolein a demonstration, the latter is designed to contribute to geometricalconstructions. Indeed, the definition of a circle does contribute to Eu-clids constructions, such as in Book I, Proposition 1, as will be shownbelow.

Despite the surface appearances, the highly abstract definitions ofpoint and line are not ornamental. They too contribute to the geometricalconstructions. In fact, their high degree of abstraction is required by thefunction that they have in a demonstration. That function is to excludeall but the most universalized features from the demonstration. How dothey perform this role?

8 Mueller, Philosophy of., 38.9 Mueller, Philosophy of., 40.10 Euclid, Elements vol. I, 153.11 Euclid, Elements vol. I, 153.12 Euclid, Elements vol. I, 153.13 Euclid, Elements vol. I, 153.14 Mueller, Philosophy of., 38; Heaths commentary in Euclid, Elements vol. I, 151.15 Szabo, A. The Beginnings of Greek Mathematics (Reidel, 1978).

From a Particular Diagram to a Universal Result 213


My answer fleshes out how Aristotle characterizes the use of diagramsby a geometer.16 Aristotle holds that diagrams aid in making the facts ob-vious and that the geometer reasons on the basis of facts made clear bythe diagram. I attempt to explain how this is done. The geometer predo-minantly reasons in terms of reciprocal dependency relations that existbetween highly abstractly defined geometrical objects. This combinationlimits our attention to what is paired, which, in turn, is based upon highlyabstract features such as breadthless length.

Reciprocal dependency relations exist between attributes or propertiesthat always exist as a pair, if they are to exist at all.17 According to Aristo-tle, they are tested for by means of what I will call a co-demolition test.18Two entities are reciprocally dependant if the destruction of A demolishesB and the destruction of B co-demolishes A. For example, half of is reci-procally dependant with double of. Something is half of something onlyin relation to something else, namely that of which it is double. Andsomething is double of something else, namely that which half of it. Ifone exists, the other exists. The destruction of half of co-demolishes dou-ble of and the demolition of double of co-demolishes half of. The pair-ings hold between parts in diagrams; the parts are given definitions in theElements and stipulated in a diagram during the construction.

In the Elements Euclid relies upon these relations to create contexts.For example, there is a reciprocal pairing between two intersecting linesand the angle at which they intersect. The angle comes into existencewhen the lines intersect. The angle and the two intersecting lines co-exist.This is not minor feature. It plays a central role in the Elements, as Euclidplaces previously defined material in new contexts. As we will see with thecircle, the context can provide the means by which objects are measured.David Reed has astutely observed19 that Definitions 13 and 14 jointly spe-cify a co-relative relationship between figure and boundary: A boundary isthat which is an extremity of any kind. 20 A figure is that which is con-tained by any boundary or boundaries.21

A figure is contained by a boundary and a boundary contains a figure.To have a boundary is to have a figure and to have a figure is to have a

16 Aristotle Posterior Analytics 77a 13.17 Aristotle presents a more robust account, allowing for different types of co-relatives

in both Categories 7 and Metaphysics V 15.18 Aristotle, Topics VI, 4.19 Reed, D. Figures of Thought (Routledge, 1995), 12. I am indebted to Reed for his

emphasis on context and measurement. Whereas I make several changes to his con-ception of co-relatives, I take over the bulk of his conceptual framework concerningmeasurement.

20 Euclid, Elements vol. I, 153.21 Euclid, Elements vol. I, 153.

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boundary. Thus, the figure and boundary structure provides a frameworkof reciprocal dependency relations. Euclid relies on it to define his figures.For example, every time the figure circle exists, it is contained by a bound-ary and every time there is a boundary, there is a circle. Recall the defini-tion: A circle is a plane figure contained by one line in such a way thatall straight lines falling upon it from one point among those lying withinthe figure are equal to one another.

To be sure, definitions 1 for a point and 2 for a line do not make use offigure and boundary. However, Euclids third definition states the co-relativelink between them: The extremities of a line are points.22 Since an extremityis a boundary, whenever two points exist, there is a breadthless length be-tween them. To remove the possibility that different lines exist between twopoints, Euclid relies upon a construction postulate to limit the range of casesunder consideration. Postulate 1 of Book I allows us to draw a straight linefrom any point to any point. No other shape of line is licensed. Thus, twopoints are co-relative with a straight line. As noted, a pairing is also estab-lished between intersecting lines and angles; lines contain the angle. Thus,lines are the boundary for the figure angle. Unlike lines, no limit is set on thekind of angle that can exist. Rather than ruling out alternative possibilities,the context that creates the angle creates an area to be studied. This area isstudied in the Elements Book I. Reed correctly observed that Euclid createsthe subject matter to be studied and the means to study it as he proceedsthrough his work.23 What inform this, are exactly the test and the depen-dency relations that Aristotle attributes to the geometers in Topics VI, 4.

The reciprocity between figure-boundary provides a non-hierarchicalstructure among the elements. It pairs together such highly abstract com-ponents as two points and breadthless length. The pairings are importantin a demonstration insofar as they restrict relevance. That is, the combina-tion of highly abstract definitions and reciprocal relations informs the dia-gram by limiting our attention to that which is specified in the pairing.The definitions make precise what is specified. The diagrams aid insofar asthey draw our attention to facts.

How does this work? Consider the construction proof for Proposition1.1. In this proof reciprocal pairings create contexts in which comparisonsare made. Postulates specify a range of cases under consideration. In thiscase, straight lines between two points are part of the investigation, as aretwo circles. Common notions enable us to equate items that have alreadybeen measured. For example, Common Notion 1 in Book 1 holds that:Things which are equal to the same thing are equal to one another.24

22 Euclid, Elements vol. I, 153.23 Reed, Figures, 12.24 Euclid, Elements vol. I, 155.



[Protasis (enunciation)] On a given finite straight line to construct an equilateraltriangle. [Ekthesis (setting out)] Let AB be the given finite straight line.

[Diorismos (definition of goal)] Thus it is required to construct an equilateral trian-gle on the straight line AB. [Kataskeu(construction)] With centre A and dis-tance AB let the circle B be de-scribed; again with the centre B andthe distance BA let the circle AE bedescribed; and from the point , inwhich the circles cut one another, tothe points A, B let straight lines A,B be joined. [Apodeixis (proof)] Now,since the point A is the centre of circle

B, A is equal to AB. Again, since B is the centre of circle AE, B is equal toBA. But A was also proved equal to AB; therefore each of the straight lines A,B is equal to AB. And things which are equal to the same thing are also equal toone another; therefore A is also equal to B. Therefore the three straight lines A,AB, B are equal to one another. [Sumperasma (conclusion)] Therefore the triangle,AB is equilateral; and it has been constructed on the given finite straight line AB.(Being) what is required to do.25

In the first stage (protasis), a very general claim is enunciated consisting oftwo parts: a condition (e.g., on a given line) and something that followseither a task (e.g., construct an equilateral triangle) or a result (in the caseof a theorem). The second stage (ekthesis), sets out the very conditionnoted in stage one. Stage three (diorismos) restates the original goal. Thefourth stage (kataskeu) constructs all the geometric objects required forthe proof. The proof that is given in the apodeixis stage makes referenceto the diagrams given in stage four. The proof ends when the desired re-sult stated in the third stage (the diosismos) is obtained. Euclid ends hispresentation indicating that the protasis has been proven.

This procedure works with mathematically relevant features of the dia-gram. Those features are precisely those specified by the following recipro-cal pairings.

Boundary Figure

Two points Line (breadthless length)

A boundary line such that all straightlines falling upon it from one pointamong those lying within the figureare equal to one another

Circle

25 Euclid, Elements vol. I, 241.

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The proof begins with a line whose points are AB. While the drawnline has breadth, the definition and the reciprocal pairing that links twopoints with breadthless length renders all but the points and the breadth-less length irrelevant to the demonstration. Since the object is to constructan equilateral triangle, the demonstration proceeds by constructing twocircles: one with A at the centre and B on the circumference, called AE;a second one with B at the centre and A on the circumference, calledB. Each circle (figure) coexists with its reciprocally dependent bound-ary. In drawing each circle, we simultaneously create a context to comparelines: lines with one point on the center and falling on the circumferenceare equal to one another. Since it is taken for granted that this construc-tion is on the same plane, the two circles intersect at a point, called . Wenow have two additional sets of two points, A and B. Given two pointsa breadthless length co-exists between them automatically. Postulate 1 al-lows us to draw straight lines A and B, completing the respective pair-ings.

Each stage in this construction is governed by reciprocal pairings. Themembers of each pairing are highly abstract, which limited properties.Breadthless length co-exists between two points. All there is to a line islength. Two lengths are measured within a context created by figure-boundary pairing of a circle. The contexts for each of the circles establishthat line AB equals A and that B equals AB. The equivalency betweenA and B is established by means of Common Notion 1: Things whichare equal to the same thing are equal to one another.26 As Reed observes,the common notions do not establish contexts, they relate measuredthings to each other.27 The contexts provide for measuring, and contextsemerge via reciprocal pairings.

The particular diagram aided the study by drawing attention to geo-metrical facts. The highly abstract definitions, in combination with reci-procal relations, make many features of the diagram irrelevant. The scaleof the particular diagram, the thickness of the lines, even to some degreethe accuracy of the sketch, including the straightness of the lines, theshape of the circle, the presumed location of the circles centre, all play norole in the demonstration. The highly abstract definitions and the recipro-cal pairings exclude non-universal items, such as non-essential features andinexact renderings, from having a role in the demonstration. Provided thatthe diagram is reasonably close, minor variations will not alter the result.After all, if there is a circle, it co-exists with its boundary. That boundarycontains the context in which pairs of lines are equated. All lines, even ifthey are not equal when a ruler, are equal by definition.

26 Euclid, Elements vol. I, 155.27 Reed, Figures, 19.



The tight reasoning results because of the reciprocal pairings and thecontexts to which they gave rise. They provide the necessity of the links.And the highly-abstracted pairings contribute to the repeatability of theproof. After all, the role of reciprocal pairs and highly abstract definitionsrenders scale irrelevant to the demonstration. Proposition 1.1 holds forthe construction of an equilateral triangle on any length of line.

3 Conclusion

I have argued against Muellers claim that the definitions of point andline, etc., are mathematically useless and never invoked in the subsequentdevelopment. I have argued instead that Euclids definitions, includingthose of the most elementary objects, are significant to his style of deduc-tion in Book I in at least four ways:

First, the definitions of point and line, etc., limit the focus to the most generalizedfeatures of elementary geometrical objects.Second, an explicit reciprocal dependency relation establishes a non-hierarchical co-dependency between geometrical objects in the form of boundary and figure.Third, Euclids derivation style employs a combination of diagrams in tandem withthe highly abstract definitions. Reciprocal dependency relations contribute to thelogical structure within the diagrammatic reasoning.Fourth, this combination renders all non-universal features of a particular diagramirrelevant, and the use of reciprocal dependency relations limits the role of the dia-gram in a way that is reminiscent of Aristotles characterization of the geometersuse of diagrams: the diagrams make the truth of a fact obvious; the geometer doesnot reason on the basis of a drawn line, but on the basis of the facts.28 Highlyabstract geometrical objects are linked together via co-relatives.Particular, non-universal features of a particular diagram are systematically excludedfrom having a role in the demonstrations.

The combination of definitions and diagrams provides for a style of de-duction that exploits the heuristic advantages of diagrams, while managingthe concern that accidental properties may lead one astray. The highlyabstract definitions of point and line severely limit ones attention tohighly abstract features in the diagram. In large part this role managesconcerns about being led astray by accidental features of the diagram. Thecombination of definitions and diagrams effectively removes these features.

The reciprocal dependency relation between figure and boundary max-imizes our ability to exploit the heuristic advantages of reasoning with dia-grams. The reciprocal relation allows us to move in both directions with avisual demonstration.

28 Aristotle Posterior Analytics 77a 13.

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Apeiron Volume 44 Issue 3 2011 [Doi 10.1515%2Fapeiron.2011.014] Raymond, Dwayne -- From a Particular Diagram to a Universal Result- Euclid's Elements, Book I