Why rob Archimedes of his Lemma?

Luigi Borzacchini (Dipartimento di Matematica, Università di Bari)

Abstract. The method of exhaustion is one of the greatest achievements of Greek mathematics, but the history of its development is not clear. First and foremost Archimedes’ role has been keenly debated, by and large undermined, so that even his name seems condemned to disappear in the name of the Eudoxus-Archimedes Lemma. In this paper we try to revaluate his role by a new interpretation (or, more precisely, by the refinement of an old one) of the historical development of the theory, underlining the theoretical relevance of the problem of addition/subtraction and comparison between curves.

Mathematics Subject Classification. 01A20

Keywords. Eudoxus-Archimedes Lemma, Method of exhaustion, ratiobetween magnitudes

Greek mathematics did not employ the idea of continuity developed by Aristotle in his Physica [2]. For example in Theorem I.1 of Euclid’s Elements [11], which is the construction of an equilateral triangle from its side, nothing requires that the circles built using the side as radius and the extremes of the side as centers, will meet in one (actually two) point(s). Moreover Euclid never employs the idea of continuity with a technical meaning and was often blamed for this fault (Heath [11],I, 234-237).

Actually the Aristotelian theory of continuity remained, in antiquity and in the Middle Ages, a purely philosophical (and physical) theme.

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There is however in Euclid a weak form of continuity, i.e. the system of lemmas, definitions, theorems, developed by Eudoxus, Euclid, Archimedes, which includes the so called Archimedes Lemma or Eudoxus-Archimedes Lemma (EA-Lemma) and which we consider one of the greatest achievements of Greek Mathematics: the method of exhaustion.

The relationship between continuity and this lemma has been enlightened only in modern mathematics, for example by Hilbert, who gave his fifth group of axioms (continuity axioms) with two axioms: the above-mentioned EA-Lemma and the axiom of linear continuity. Other modern authors also analysed the problem, for example Dedekind and Cantor.

Without introducing here the modern theory we can simply say that for these authors the EA-Lemma is not sufficient to yield thenotion of continuity. Furthermore, the fact that the Euclidean axiomatic system is insufficient to lead cathegorically to continuity was made clear by Dedekind [6] who described non-continuous models for that axiomatic system.

The topic actually was the end of a long evolution that concerned many of the most celebrated figures of Greek philosophy and mathematics.

The cluster of geometric propositions that give us the background of the EA-Lemma and that we will give in the following section, are however not well connected to each other, and this caused a lot of discussion on the real theoretical structure of the topic and its historical development, including also the evaluation of the role played by the great Greek mathematicians, Hippocrates, Euclid, Eudoxus and Archimedes, whereby a general undermining of Archimedes’ role steadily appeared.

In this paper we begin by describing this evolution, and thus presenting the evidence of the historical questions concerning thetheory. In the second section we resume the most relevant interpretations of this evidence addressed by modern historians ofmathematics. In the third section we give a new interpretation (or, more precisely, we propose the refinement of an old one), whereby a role is played by the opposition between straight and curved lines, and according to which it seems correct to restore what is due to Archimedes.

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1. Evidence

There are two quite sharply different kinds of evidence: philosophical and mathematical. Let us begin with the former.

1.1 Philosophical Evidence. Anaxagoras flourished in the middle of the fifth century B.C.,

when the philosophical stage was dominated by Eleatic learning, precluding not-being, void and somehow also infinite. His reactionclaimed the recognition of both the infinitely small and the infinitely great: “Neither is there a smallest part of what is small, but there is always a smaller (for it is impossible that what is should cease to be). Likewise there is always something larger than what is large. And it is equal in respect of multitude(plēthos) to what is small, each thing, in relation to itself, beingboth large and small” ([8], 59 B3).

Democritus lived at the end of the fifth century and it is hardto know his real mathematical stature. However in Plutarch we findhis reflections [8, 68 B 155] about the plane parallel sections ofa cone: if they were equal the cone would be a cylinder, if they were unequal the figure would look like a stepped cone, i.e. wouldshow indentations on its lateral surface.

To Aristotle (second half of the fourth century) we owe the theory of infinite and continuity which lasted until modern mathematics and, in a different mathematical cloth and with sharper characters, for continuity even until today. For the infinite Aristotle distinguishes between the actual (“there is nothing more”) and the potential infinite (“there is always something more”). Curiously they are exact logic opposites to one another: the opposite of “infinite” is not “finite” but another “infinite”. For continuity we find in Aristotle two definitions: acontinuous magnitude is (infinitely) divisible and is such that every dichotomy has just one separating point that belongs to bothparts. In our mathematics we can roughly recognize in the potential infinite the enumerable infinite, in the first definition of continuity the density in itself and in the second definition of continuity Dedekind’s continuity [6]. However, the difference between them could not be appreciated before modern Mathematics.

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Common character of these last three authors is that both infinite and continuity appear as physical questions. But what happens in mathematics?

1.2 Mathematical EvidenceBefore Eudoxus we have no trace of issues concerning

continuity. Also the Pythagoreans do not show any idea about, for example the difference they address between point and unity is just in that the first “has position” whereas the second does not,and the same definition can sometimes be found in Aristotle too. For Plato also a sequence of consecutive integers was continuous.

At the end of the fifth century questions concerning incommensurability arose (Borzacchini [3]), and questions concerning the proportion theory for continuous (and hence also possibly incommensurable) magnitudes became more and more urgent. According to tradition, Eudoxus’ theory of proportions that we read in the V book of Euclid’s Elements was aimed at answering thesequestions. Probably Euclid did something more in that book than a simple restyling of Eudoxus’theory, but we will ignore this question here (see Knorr, Archimedes and the pre-euclidean proportion theory [13]).

In that book, among the definitions, we read that:

“Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another” (def. V.4)

“Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equal multiples whatever can be taken of the first and third and any equal multiples whatever of the second and the fourth, the former equal multiples alike exceed, are alike equal to, or alike fall short of, the latter equal multiples respectively taken in correspondingorder” (def. V.5)

The proportion theory developed in the book V is then applied to the theory of incommensurable magnitudes in Book X. The first theorem of that book is:

“Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this

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process be repeated continually, there will be left some magnitudewhich will be less than the lesser magnitude set out” (Prop. X.1).

Actually the condition that the left magnitude be “greater thanits half” is not necessary, but the generalized proof had to be out of the reach of Euclidean geometry. On this theorem, Eudoxus/Euclid built the so called method of exhaustion, whose main results in the book XI were the propositions: 2 (“circles areto one another as the square on the diameters”), 6 (“pyramids which are of the same height and have polygonal bases are to one another as the bases”), 10 (“any cone is a third part of the cylinder which has the same base with it and equal height”), 12 (“similar cones and cylinders are to one another in the triplicateratio of the diameters in their bases”), 18 (“spheres are to one another in the triplicate ratio of their respective diameters”). Observe that: i) the propositions deal with surfaces and solids, ii) the method is employed either to compare solids (cones and cylinders) or to reduce ratios between homogeneous figures/solids to the ratio of their rectilinear dimensions.

Further evolution of the method can be found in Archimedes [1],who based the theory on the celebrated EA-Lemma. It appears in a similar form in three works of Archimedes, in Quadratura Parabolae (QP), De sphaera et cylindro (DSC) and De lineis spiralibus (DLS) (Latin translation by Heiberg [1], English translation by Heath [10]).

tōn anisōn chōriōn tan yperochan, a yperechei to meizon tou elassonos, dynaton eimen au tan eauta syntithemena pantos yperechein tou protethentos peperasmenon chōriou

“spatiorum inaequalium excessum, quo majus excedat minus, sibi ipsum additum quodvis spatium datum terminatum excedere posse”

“the excess by which the greater of unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area”QP

tōn anisōn grammōn kai tōn anisōn epifaneiōn kai tōn anisōn stereōn to meizon tou elassonos yperechein toioutō, o syntithemenon auto eautō dynaton estin yperechein pantos tou protethentos tōn pros allēla legomenōn

“inter lineas inaequales et inequales superficies et inequalia solida maius excedere minus ejusmodi magnitudine, quae ipsa sibi

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addita quamvis magnitudinem datam earum, quae cum ea comparari possint, excedere possit”

“of unequal lines, unequal surfaces, or unequal solids the greater exceeds the less by such a magnitude as is capable, if added [continuously] to itself, of exceeding any magnitude of those which are comparable with one another”, DSC

tan anisan gramman aki tōn anisōn chōriōn tan yperochan, a yperechei to meizon tou elassonos,autan eauta syntithemenan dynaton eimen pantos yperichein tou protethentos tōn pot allala legomenōn

“ut excessus linearum vel spatiorum inaequalium, quo majusexcedat minus, sibi ipsi adiectus quamvis magnitudinem datam excedat earum, quae inter se comparari possint”

“if there be two unequal lines or two unequal areas, the excessby which the greater exceeds the less can, by being continually added to itself, be made to exceed any given magnitude among thosewhich are comparable with it and with one another”(DLS

and usually stated today as: “given m and M, m greater than zero and M greater than m, there is a positive integer n such that the addition n times of m is greater than M,” something similar was already in Aristotle: “by continual addition to a finite magnitudeI must arrive at a magnitude that exceeds any assigned limit” (Physica 266 b2 [2]).

In addition Archimedes ascribed (Introduction to QP [1]) the use of the lemma to earlier geometers (credibly Eudoxus).

2. Historical Interpretations

The first question is: Why does Archimedes mention the difference betweentwo unequal magnitudes instead of a single magnitude in the lemma? This simpler version would also be similar to the fourth definition of the bookV of the Elements, which many authors believe coincident with the lemma.

This problem is strictly interwoven with the historically deeper question of the relationship between Archimedes’ Lemma and Euclid-Eudoxus’ Theory (Def. V.4, V.5, Bisection Principle X.1, and Book XII of the Elements).

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The problem does not appear in Eutocius’ commentary to Archimedes [1].

The first works about the history of Greek Mathematics envisaged the issue as concerning the history of the method of exhaustion, and set out the question of its ascription between Hippocrates of Chio, Eudoxus, Euclid and Archimedes.

In 1883 Stolz [23] introduced the term “Archimedes’ Postulate”,but ascribed to Archimedes making an explicit the implicit axiom, which had already been used by Eudoxus, and probably was known even earlier. Since then, Archimedes has ‘lost’ his lemma.

Gino Loria ([18], 134) split the authorship between Hippocrates(for the actual beginning of the method) and Eudoxus (its actual founder).

Moritz Cantor chose Eudoxus as the real father of the method, even though he considered Hippocrates the first to employ the lemma, for his theorem about the ratio between two circles ([4], 243). In addition he pointed out that the Theorem X.1 should be ascribed to Euclid ([4], 269)

Heath in his The works of Archimedes [10] judged the lemma a sort of appendix to Def. V.4.

Paul-Henri Michel [19] underlined the connection between Def. V.4, VI.3 and the EA-Lemma, and recognized Eudoxus as its author.

F. Lasserre judged the lemma “an intermediate stage between anthyphairesis… and Eudoxus’ general definition” ([17], 93) which “dates from a time before any general theory of proportion … in order to extend anthyphairesis the application of the theory to volumes and probably to surfaces also” ([17], 109). “Anthyphairesis” was substantially the procedure of “taking away” the same number of parts to reveal the “same ratio” between two commensurable magnitudes.

Mueller in his Philosophy of Mathematics and deductive structure in the Elements [20] claimed the formal equivalence between X.1 and Def.V.4, and maintained that they imply the lemma, with its contrapositive being true if we exclude infinitesimals.

Dijksterhius faced the question in his Archimedes [9], pointing out correctly that the right translation had to deal with the two<compared> (and not the generic <comparable>) magnitudes, so thatwe can distinguish three different interpretations of the words “tōn pros allēla legomenōn”:

i) Of those which are comparable with one anotherii) Of those which are compared with one another, i.e.

the magnitudes homogeneous to the two actually given.

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iii) Of the (two aforesaid magnitudes) which are compared with one another.

If (i) is not absurd, the difference between (i) and (ii) is unnoticeable, and hence (ii) is the most credible interpretation for better agreement with the actual applications of the lemma. (iii) is the least supported interpretation.

In addition, Dijksterhuis pointed out that the axiom should be distinguished from the Euclidean Definition V.4, and hence posed the question, “What motive may have induced him [Archimedes] to include this assumption.” ([9], 147) As an answer, Dijksterhuis remarked that “in Greek mathematics there existed, side by side with the strict and official method of the indirect passage to a limit, also the less strict, but heuristically more fertile methodof indivisibles, and that Archimedes himself diligently used it asa method of investigation,” as in the so-called “Archimedes method”, where a curve is regarded as a sum of points, a surface as a sum of lines and a solid as a sum of surfaces. Archimedes’ Lemma simply amounted to postulating that “if two magnitudes satisfy Definition V.4 their difference also satisfies the same definition with respect to any homogeneous magnitude.” ([9], 148) If we observe that in the above method of indivisibles, a point could be the difference of two lines, a line the difference of twosurfaces and a surface the difference of two solids, the lemma “excludes the existence of actual infinitesimals” ([9],149). Dijksterhuis also reminded us that Archimedes did not believe his method a sound proof technique, and claimed that its ‘mathematicaldeficiency’ was exactly the method of indivisibles. More preciselyDijksterhuis maintained that “the invisibles had only been banished from the published treatises, but that in the workshop ofthe producing mathematician they held undiminished sway” ([9], 320)

Credibly Dijksterhuis was right in the translation and in remarking the difference between the lemma and Definition V.4, which characterizes homogeneous magnitudes in terms of their additivity and comparability, and says nothing about the occurrence of the difference of magnitudes in the lemma.

Dijksterhuis’ approach tried to connect the EA-Lemma and Definition V.4 with the exclusion of indivisibles (infinitesimals). However, this point does not convince me for tworeasons (plus one more). First, where can we find a 4th - 3rd century Greek mathematicians who believed that a point could be the difference of two lines? Earlier Pythagoreans or disciples of

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the sophist Antiphon could maintain this thesis, but their ‘points’ were not without dimensions and their techniques were by that time outside the Greek theoretical geometry, and Plato did not believe in the existence of points either. Second, if such a mathematician existed, the lemma would not have kept him from anything, because he could also believe that a sum of lines can exceed a surface. The third reason will be shown in the following.

Infinite division and infinitesimals were indeed not mathematical themes. Dijksterhuis remarked that the ‘weak’ point of Archimedes’ heuristic method (based on static concepts and an actually infinite division of the magnitudes) was the employment of indivisibles because statics was also employed in his rigorous mathematical works. Actually, infinite divisibility and continuityin the 4th and 3rd century were more physical than mathematical themes. This is quite evident: continuity and infinite divisibility appear in pre-Socratics, and they play a great role in Aristotle’s physics, whereas they do not play any role in the works of Euclid and other mathematicians, and in Archimedes they are present in his mechanical method but never in his mathematicalworks. Last but not least, from Aristotle’s De Caelo 306 a 25-30 [2]we realize that infinite divisibility in the 4th century could be accepted more by physicists than by mathematicians. In other words, the infinite divisibility was not more ‘mathematical’ than statics, and Knorr in his Archimedes and the Elements [14] underlines that the access to the use of indivisibles was opened for Archimedes by his statics applications. The strictly physical character of the indivisibles also smooths the opposition between Dijksterhuis and Knorr about the core of the (supposed by Archimedes) ‘weakness’ of his ‘method’ (statics or indivisibles?).

Knorr in his appendix to the English translation of Dijksterhuis’ Archimedes [9] shares his opinion “that Archimedes perceived subtle difficulties not covered by the Euclidean formulation and so introduced his axiom, moreover , that the two axioms are different, and that Archimedes intended his axiom to besupplementary of Euclid’s” (431-2). In his Archimedes and the pre-Euclidean proportion theory [13] he rightly underlines the comparison between the lemma and the theorems of Book XII (XII.2 and XII.10) based on X.1 (the Bisection Procedure) more than with Book V of the Elements, but tries to restate the equivalence between Eudoxus’sdefinition and the Bisection Principle, and the EA-Lemma, underlining that the lemma was necessary for Archimedes’ not having access to the Elements as we know them in his youth,

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postulating Archimedes’ earlier access only to an Eudoxean and pre-Euclidean proportion theory (leading directly to Book XII). InInfinity and Continuity. The interaction of mathematics and philosophy [15], he claims that the Bisection Principle X.1 was not due to Eudoxus, who employed it as a unproved lemma, but to Euclid. However, even if we accept Knorr’s analysis [14] of the evolution of Eudoxus’s method and Archimedes’ ideas, the lemma seems substantially unaffected by this evolution, for it appears almost as a constant in Archimedes’ works, from the earlier QP to the later DLS and DSC.

Another interpretation was set out by Hjelmslev (Eudoxus’ axiom and Archimedes’ lemma [16]), that employed the same translation advocated by Dijksterhuis, and followed a similar path but stressed that Archimedes introduced it to deal with new magnitudes, curves and curvilinear figures, and not for the sake of excluding the method of indivisibles. It was criticized by Dijksterhuis and Knorr because the employment of the EA-Lemma in Archimedes occurs in problems analogous to those of Book XII, and because Eudoxus (in Elements’ V book) in his theory did not view curvilinear magnitudes as in principle distinct from rectilinear magnitudes, and would surely accept that if two finite equal magnitudes have the property of Def. V.4, their difference will too (Dijkstrehuis [9], 149). These objections do not seem thoroughly convincing to me, but underline some weaknesses in Hjelmslev’s interpretation.

In my opinion, however, it grasped a real point. The following section aims to defend Hjelmslev’s interpretation from its criticsand support the claim of Archimedes’ depth and originality in the EA-Lemma.

3. Archimedes’ Problem

3.1 Chronological order.First another look at the three ‘releases’ of the EA-Lemma

given by Archimedes. They are substantially equal, but there are little differences that are useful to underline.

We observe that the chronological order of the three texts is, according to Heiberg [1], Heath [10] and Knorr [14], the following:

QP DSC DLS.

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A first difference concerns the geometrical objects dealt with in the lemma: chōriōnin QP, grammōn,epifaneiōn, stereōnin DSC, gramman chōriōnin DLS. Grammēis the ‘line’, epifaneia is the ‘surface’, stereōsis the ‘solid’. So far so good. A little problem arises, however, with chōriōn: chōra chōros chōrionmean ‘place, space’, but in mathematics it is always 2-dimensional.

A second difference concerns the last words of the lemma. DSC and DSL have tōn pot allala [pros allēla] legomenōn, whereas QP has peperasmenon chōriou. The first means “of those which are compared with one another”, whereas the second means “finite area”.

These two remarks seem to show that Archimedes’ meant that the exceeded objects were finite and homogeneous according to their dimensionality, i.e. lines with lines, surfaces with surfaces, solids with solids, so that the QP release, concerning only finitesurfaces, needed no homogeneity specification.

The mathematical concepts involved in the lemma are: addition, subtraction, comparison between magnitudes, hence the starting point for our proposed interpretation could be: what did it mean to add/subtract or to compare in Euclidean geometry?

3.2 Comparison and addition/subtraction of magnitudes.Surely it did not mean to add/subtract or to compare numbers

that measure magnitudes, i.e. lengths or areas or volumes, becauseEuclidean geometry was strictly non-metric. The answer was quite easy for rectilinear segments. It was less easy for other magnitudes.

The base of the answer was surely the idea of equality. Ever since the seminal paper of Kurt von Fritz about Gleichheit, Congruenz und Ähnlichkeit [25] we know that such an idea was clear and crucial in Euclid. We could say that it was based on an earlier ‘equality as coincidence’ (common notion 4 [11]), and then developed as ‘equality up to transformation,’ meaning that figures which could coincide after a series of specific operations were equal. These operations include, ever since Hippocrates of Chios, cut-and-pasteand uses of common notions concerning equality. Such operations were not called ‘addition’ and ‘subtraction’ because of the Greeks’ keen awareness of the abyss between geometry and arithmetic, magnitudes and numbers, but nonetheless we will anachronistically employ those terms in order to avoid useless unwieldiness.

For rectilinear figures addition/subtraction meant basically assembling and disassembling, and it implicitly also solved the problem of comparison. The basic insight was arguably the easy

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connection between comparison (in terms of lesser, equal, greater)and additivity between figures: a figure was greater of another just if the former could be built by assembling the latter with another figure. Additivity plays a role also in the Eudoxean theory in the Elements, from Book V to Theorems X.1 and X.2, where the commensurability/incommensurability is decided according to the behaviour with respect to a difference procedure, and hence presupposes additivity.

Comparability and additivity were the background of the theory of ratios (to recognize this, it is sufficient to take seriously Definitions V.4 and V.5 of the Elements), but we must observe that in Book V and in X.1, the magnitudes seem always be only straight segments: this is explicit in the figures and also the text is barely understandable if referenced to ‘general’ magnitudes. Book V is based on the idea of a ‘multiple’ of magnitudes, which in turn is based on the idea of ‘measure’ between magnitudes. It is clear what it does mean to measure a segment by another segment, but what does it mean to measure a circle by another circle or (worse) by a parabolic segment?

Nevertheless the Eudoxean/Euclidean proofs seem to us quite general because we unconsciously translate their magnitudes in real numbers, we identify segments and their measures, according to the Cartesian equivalence. The gap had instead to have been blatant in Archimedes’ eyes.

More precisely: with regard to figures, we must remark that the non-metric character of theoretical Greek geometry implied that a sum or difference of two of them was not a value, but had always to be a figure with a specific ‘shape’, and comparison, if not based on inclusion, was possible between figures of the same prescribed ‘shape’, and by a given assembling/disassembling procedure: this was also the real meaning of the squaring of the circle problem, which actually meant “to find a series of suitable geometric transformations to reduce a circle to a square”

By the way, for rectilinear figures the problem was completely and generally solved by Theorem VI.25 which allowed the transformation of any figure in an equal figure with prescribed shape. Also multiplication of figure and solids was well known: the doubling of a square can be obtained by the diagonal of the same square and we have notice of Hippocrates’ discovery that the doubling of a cube can be obtained by the construction of two middle proportionals between 1 and 2. In fact, if 1:x = x:y = y:2,then x is the cubic root of 2.

But what about curves or curvilinear figures?

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3.3 Comparison and addition/subtraction of curves.What does it mean to add two circles or two hyperbolas or

(worse) a square and a circle or a rectangle and a segment of hyperbole? What does it mean to compare two circles or two parabolas or (worse) a circle and a rectangle or a parabola and a square?

In Greek mathematics there was a keen interest in the reductionof curved figures to rectilinear ones. The earliest problem was the above mentioned squaring of the circle: Hippocrates of Chio is one ofthe first mathematicians we have notice of. He flourished in the middle of the fifth century B.C. and we have a fragment (known through and credibly reworked by Eudemus and Simplicius) concerning the squaring of lunes [24].

After Hippocrates, in the fourth century there were many attempted solutions, but none of them was judged thoroughly satisfying, and the problem was far from being solved. Probably the case for curvilinear figures was held true, because Def V.4 could be applied generally between finite figures (for heterospecific but reasonable figures it is always possible to imagine a multiple of the first including the second: there is always a multiple of a given square including a given circle), butthe case for curves was harder to solve (how could we multiply a straight segment to include a circumference?). We must also remarkthat the earliest version of the EA-Lemma in QP dealt only with surfaces, perhaps because the bidimensional case was more common and less questionable. Indeed we can remark that also Aristotle, even though so sceptical about the comparability of circular and rectilinear lines, thought that the squaring of the circle could be the object of scientific investigation (Categoriae 7b, 31-33 [2]).

We can say in fact that there is relevant evidence in the 4th

century of a negative answer to the idea of comparability between circular and rectilinear segments: in Aristotle (Physica, 248 a10-20, b 5-8, 249 a11-22 [2]) rectilinear and curvilinear belonged tothe same genus but to different species and hence were not comparable. Aristotle employed the term ‘comparable’ (symblēta) for ‘synonymous’ (i.e. having equal name and equal definition, Categoriae 1 [2]) things. We can be sure that rectilinear segments belonged to the same species. Can we be sure that the same held true for curvilinear segments? Can we think of circles and conic sections as having the same definition? I would say perhaps yes, because the circle is a conic section. But what about a straight

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segment, a quadratrix and a circle? If we remember the complex ancient classifications of lines we read in Proclus, that quotes Geminus as his authority ([22], 111), it is likely that they were seen as belonging to different species.

In Aristotle, curvilinear and rectilinear segments surely belonged to different species, because the former were ‘affections’ (pathē), the latter were ‘lengths’: every comparison was absurd.

Aristotle faced this problem after the analogous problem of comparison between circular and rectilinear motion, an astronomical problem that could well have been present in Eudoxus’astronomic theory and to Archimedes, son of an astronomer. Howeversynonymity was basically a ‘linguistic’ characterization. A ‘mathematical’ characterization was necessary and arguably it had to befound in additivity.

The problem of the comparison between continuous magnitudes was also addressed by Proclus. The example was: “that the horn angle is always unequal, never equal, to an acute angle… and that the transition from the greater to the less does not always proceed through equality” (Commentary [22], 234): a right angle is greater than any horn angle (the angle between a circumference anda tangent) and if we continuously decreases the right angle till the coincidence between the two sides (what we call the null angle) it eventually becomes less than the horn angle, but in no intermediate position the horn and the decreasing linear angle areequal.

Heron’s Metrika [12] dealt just with surfaces’ and corps’ measures. The rejection of the heterogeneous segments’ comparison was general. There is only Archimedes’ approximation of the ratio between circumference and diameter in his Dimensio circuli [1] as an exception. How difficult the comparison between curved lines was (and even much harder than the analogous problem for surfaces) canbe realized observing that Pappus in Proposition VIII.22 of his Mathematical Collection [21] employs the theorem saying that the ratio between two circles is equal to the ratio between the square of their diameters to prove that the ratio between the circumferencesis equal to the ratio between the diameters!

Obviously common sense gave a positive answer to the question:a medieval commentator on Archimedes’ Dimensio circuli [1] remarked that“if a hair or silk thread be bent around circumference-wise in a plane surface, who unless demented would doubt that the hair or the thread would be the same whether bent around or extended in a straight line, and would be as much afterwards as before?”

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(Clagett, [5], I, 170), but the theoretical tradition of Greek mathematics was less prone to forget its rigour.

To realize how long it took for mathematicians to deal with these questions it is worthwhile to stress the presence of the question even in modern times. In fact Descartes also gave a negative answer: “The ratio which is between straight lines and curves is not known, and even, I believe, cannot be known by man” (Geometrie [7], 90).

3.4 Archimedes’ solution.The keen interest of Archimedes in the squaring problems are

witnessed by his book on the measurement of the circle, and by hisinterest in the squaring of the spiral and its employment to rectify circular arcs (DLS 18-20 [1]).

His awareness of the problems arising from the application of Eudoxus’ approach to non linear magnitudes is witnessed by the proof of Theorem DSC I.2: “Given two unequal magnitudes, it is possible to find two unequal straight lines such that the greater straight line has to the lesser a ratio less than the greater magnitude has to the lesser,” where the key issue in the proof isto deal with generic magnitudes which have only the Eudoxean property of “having a ratio.” This theorem and DLS.4, “Given two unequal lines, a straight line and the circumference of a circle, it is possible to find a straight line less than the greater of the two lines and greater than the less”, are the basic results inwhich the lemma is employed in the two books De sphaera et cylindro and De lineis spiralibus. And both the theorems deal with strictly ‘foundational’ questions with respect to the problem of comparability among heterospecific magnitudes.

We must remark that there are no similar theorems in Archimedes’ Quadratura Parabolae. The rationale is clear: in that book the lemma is limited to surfaces, for which the lemma is not actually something new with respect to Eudoxus/Euclid’s theory, and this is confirmed by the fact that the main result of the book, i.e. the theorem concerning the squaring of the parabolic segment, is proved by the EA-Lemma (proposition 17) and by the classic Eudoxean method (proposition 24) as well.

This is a point to make clear. Knorr [14] pointed out that “thebisection-principle was perfectly adequate for Archimedes’ purposein QP”. Why then the double proof?

I think that there are many differences between the two proofs and that these make clear the rationale of this oddity. In order to prove the theorem by the Bisection Principle Archimedes had to

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prove three special propositions concerning the relationship between a parabolic segment and the inscribed triangles, and the partial sum of a geometric series of ratio 1/4: the procedure was not general and therefore required ad-hoc auxiliary propositions. On the contrary the proof based on the EA-Lemma was quite general and clearly linked to the mechanical discovery procedure describedin his Ad Eratosthenem Methodus [1].

In addition we can remind that the lemma is addressed as already employed by earlier geometers in the introduction to QP, but not in DSC and DLS.

Hence we can come to this conclusion: QP was the first book in which Archimedes faced his complex problems concerning squaring ofsurfaces and solved them by his mechanical approach. The EA-Lemma in the parabolic segment squaring problem appeared then as a variation of the Bisection Principle, homogeneous to the mechanical approach, but better suited and more straightforward for the specific problem. However the traditional procedure too was perfectly adequate for surface problems, even though required auxiliary propositions to be applied. The double proof was a quitenatural solution for a mathematician who was able to interweave perfectly Euclidean rigour and mechanical engineering. In the later DSC and DLS Archimedes faced problems concerning curved lines where he recognized that the Eudoxean approach was ill suited and his lemma well suited. He eventually also realized thatthe lemma was actually something more than a form of the BisectionPrinciple and deserved a more autonomous establishment.

Moreover, Archimedes set up a whole series of analytical instruments to face the analysis of curved lines. In this light wecan also read his first two postulates in DSC: “that of the lines which have the same extremities the straight line is the least,” “that of the other lines, if, lying in one plane, they have the same extremities, two are unequal whenever both are concave in thesame direction and moreover one of them is either wholly included between the other and the straight line which has the same extremities with it, or is partly included by and partly coincideswith the other; and that the line which is included is the lesser.” (Diksterhuis [9], 145)

Which relationship exists between comparable and commensurable?In the introduction to DSC Archimedes says that the properties

he was analysing “were all along naturally inherent already in thefigures referred to” (i.e. sphere and cylinder), but that earlier

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geometers did not realize “that there exists symmetry between these figures.” This is Dijksterhuis’ translation [9], where symmetry translates symmetria, which seems hard to understand according to the context. If we translate it instead with “commensurability” (Heiberg translates “commensurationem”), we getnew evidence that the real issue at stake was comparability/commensurability between straight and curves surfaces.

If Archimedes says explicitly that previous geometers did not believe circular figures and solids (spheres, circles, cylinders) commensurable, obviously for a mixed case (curvilinear and rectilinear lines/figures) the problem had to be even worse. On the other side incommensurable rectilinear lines were comparable (because synonymous and having a ratio in Eudoxus’ theory). Hence we could say that comparability was a weaker property than commensurability. Curvilinear with curvilinear (for curves belonging to the same species!) and rectilinear with rectilinear were comparable, and among them some pairs of lines could be commensurable and some pairs incommensurable. Curvilinear with rectilinear were deemed as probably not comparable.

A less relevant question could also be: what did it mean to compare in a limited geometry?

We must remember that the infinite appears in Euclid in the postulate and definition of parallel, i.e. just for straight segments, whereas his surfaces and solids are always finite.

In Archimedes the simpler version of the lemma on Q.P. talks about a ‘finite’ (peperasmenou) figure, thus confirming this kind of constraint.

What about the definition and the lemma?With respect to X.1, we must also remark that its employment

(in Book XII of the Elements or in Archimedes’ QP, DLS, DSC) required that the excess between credibly comparable magnitudes becomparable with them. On the other hand the proof of X.1 regards only straight segments explicitly given as magnitudes. This gap can be filled by an implicit lemma saying that “it is possible to build the difference between comparable magnitudes as a still comparable (with the above) magnitude” which could be the Eudoxeanlemma ascribed by Archimedes to ‘earlier geometers,’ and is the core of the EA-Lemma, as also observed by Dijksterhuis.

This gap should be not very problematic in Book XII of the Elements (comparison between circle and rectilinear figures, or

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between cones and cylinders), but Archimedes discovered that it was far from obvious in more complex applications, when the magnitudes were sharply heterospecific, as segments of parabola and triangles, most of all if linear (as for the spirals). Archimedes employs X.1 when possible, but he had often to deal with sharply heterospecific magnitudes and unconstructible differences so that his lemma had the aim of dealing explicitly with them.

Here we come to our third remark against Dijksterhuis’ indivisible-based interpretation: who would believe that the excess of a segment of parabola on a triangle or (worse) that of acircumference on a straight line could be an indivisible? Isn’t itmuch more credible that in a non-metric geometry such an excess was instead a rather fuzzy concept?

This interpretation gives the rationale for the strange form ofArchimedes’ Lemma, does not require hypotheses on Archimedes’ knowledge of the Elements, gives no role to the removing of the indivisibles, and enlightens the complex questions concerning comparability and additivity/subtractivity in Greek mathematics.

4. Conclusion

After Eudoxus, Euclid and Archimedes had to face a more complex problem than incommensurability. They had also to take in account comparability.

The aim of Definition V.4 of the Elements probably was then to characterize ‘comparability’ as ‘having a ratio’ in terms of possible additivity, but without any attention to curves. The method of exhaustion was applied to cases in which the excess could intuitively be believed comparable, so that the lemma about the excess could remain implicit. A complete solution of Eudoxus’ problem would require measure theory, but this was beyond the reach of Greek geometry.

The aim of Archimedes’ Lemma was somewhat more complex, becauseit had to deal also with more complex excesses (Archimedes had to square spirals and parabolic segments), but its basis was the same: we can only compare curves and figures (and their excesses) which are ‘comparable’ (having a ratio). Hence the Archimedean version was aimed at making explicit the closure of the ‘comparability’ relationship between heterospecific magnitudes

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under addition and subtraction, and this was done by asking that “the greater exceeds the less by such a magnitude as is capable, if added [continuously] to itself, of exceeding any magnitude of those (two) which are compared with one another.”

To this extent, it is not correct to claim an equivalence between the Eudoxean Definition V.4 or X.1 and the EA-Lemma: the lemma has to make explicit an implicit lemma in X.1. And for the same reason, maybe it is not completely right to say that “the bisection principle was perfectly adequate for Archimedes’ purposes in QP” (Knorr, Archimedes and the Elements [14]): actually, according to Archimedes, without the EA-Lemma and in a non-metric geometry, it was not perfectly adequate in Book XII either!

To summarize, we can say that in a non-metric and limited geometry comparing was a very complex affair. In the fourth century, Greek geometry succeeded in developing a sort of almost-metric and potentially-infinite geometry on the straight line, from Architas’s paradox about the finite universe to Aristotle’s potential infinite, Eudoxus’s proportion theory and parallels theory, that allowed an method of exhaustion more or less extensible to applications of the bisection procedure between credibly comparable magnitudes. But its extension to other sharplyheterospecific lines was far from being at hand, and Archimedes’ theory and lemma were an attempt to make explicit the implicit Eudoxean hypothesis about the closure of the ‘comparability’ relationship under addition/subtraction.

References

[1] Archimedes. Opera Omnia. I.L. Heiberg and E.S. Stamatis edts.Stuttgart, Teubner, 1972

[2] Aristoteles. Aristoteles graece ex recensione Immanuelis Bekkeri edidit Academia regia Borussica, Berlin, ap. G. Reimerum, 1831

[3] L. Borzacchini. Music, incommesurability and continuum: a cognitive approach, Arch. for History of Exact Sciences, to appear.

[4] M. Cantor. Vorlesungen über Geschichte der Mathematik. Leipzig, B.G.Teubner, 1907.

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[5] M. Clagett. Archimedes in the Middle Ages. Madison, University of Wisconsin Press, 1964

[6] J.W.R. Dedekind. Stetigkeit und irrationale Zahlen. Braunschweig, Vieweg und Sohn. 1872.

[7] R. Descartes. Geometry/Geometrie. New York, Dover

[8] H. Diels and W. Krantz, edts. Die fragmente der Vorsokratiker. Zürich, Weidmannsche Verlag. 1964.

[9] E.J. Dijksterhuis. Archimedes. Princeton Univ. Press, 1987 (dutch original edition, 1938)

[10] T.L. Heath. The works of Archimedes. New York, Dover, 1922.

[11] T.L. Heath. Euclid’s Elements. New York, Dover, 1956

[12] Heron. Heronis alexandrini opera quae supersunt omnia. W.Schmidt edt. Leipzig, Teubner, 1899-1914

[13] W.R. Knorr. Archimedes and the pre-euclidean proportion theory Archives Internat. Hist. Sciences 28 (1978), 183-244

[14] W.R. Knorr. Archimedes and the Elements: proposal for a revised chronological ordering of the Archimedean Corpus. Archives for history of Exact Sciences 19 (1978), 211-290

[15] W.R. Knorr. Infinity and Continuity. The interaction of mathematics and philosophy in Kretzmann, N. edt. Infinity and continuity in ancient and medieval thought. Ithaca, Cornell Univ. Press, 1982.

[16] J. Hjelmslev. Eudoxus’ Axiom and Archimedes’ lemma Centaurus 1 (1950), 2-11

[17] F. Lasserre. The birth of mathematics in the age of Plato. Larchmont NY, American Research Council, 1964.

[18] G. Loria. Le scienze esatte nell'antica Grecia. Memorie della Regia Acc. delle Scienze, Lettere ed Arti di Modena, Sez. Scienze, X,2 (1893).

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[19] P.H. Michel. De Pythagore a Euclide. Paris, Les Belles Lettres, 1950

[20] I. Mueller. Philosophy of Mathematics and deductive structure in Euclid’s Elements. MIT Press, 1981

[21] Pappus. La collection mathematiques. Paul ver Eecke edt. Paris,Libr. Scient. et Techn. Albert Blanchard, 1982.

[22] Proclus. A commentary on the first book of Euclid's Elements. G.R. Morrow edt.. Princeton, Princeton University Press, 1970.

[23] O. Stolz. Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes Math. Annalen 22 (1883), 504-519

[24] I. Thomas, edt. Greek Mathematics. Cambridge Ma., Harvard Univ. Press, 1993.

[25] K. von Fritz. Gleichheit, Kongruenz und Ähnlichkeit in der antiken Mathematik bis auf Euklid. Archiv für Begriffgeschichte. 4 (1959), 7-81

Luigi BorzacchiniDipartimento di Matematica,Università di Bari.Via E.Orabona70125 BariItalye-mail: gibi@dm.uniba.it

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