Phenomenology of diagrams in Book II of the Elements
aa r X i v : . [ m a t h . HO ] O c t Phenomenology of diagramsin Book II of the
Elements
Piotr B laszczyk
Institute of Mathematics, Pedagogical University of Cracow, Podchorazych 2,Cracow, Poland [email protected]
Abstract.
In this paper, we provide an interpretation of Book II of the
Elements from the perspective of figures which are represented and notrepresented on the diagrams. We show that Euclid’s reliance on figuresnot represented on the diagram is a proof technique which enables toturn his diagrams II.11–14 into ideograms of a kind.We also discuss interpretations of Book II developed by J. Baldwin andA. Mueller, L. Corry, D. Fowler, R. Hartshorne, I. Mueller, K. Saito, andthe so-called geometric algebraic interpretation in B. van der Waerden’sversion.
Keywords:
Euclid’s diagram · Visual evidence · Substitution rules · Geometric algebra able of Contents
Rectangle contained by two straight-lines . . . . . . . . . . . . . . . . . . . . . 123.2.1 Current interpretations of rectangle contained by . . . . . . . . . 133.3 Identifying figures through letters and technical terms . . . . . . . . . . 143.3.1 Algebraic view on propositions II.1–3 . . . . . . . . . . . . . . . . . . . 164 Schemes of propositions II.1 to II.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.0.1 David Fowler on proposition II.4 . . . . . . . . . . . . . . . . . . . . . . . 214.0.2 Ian Mueller’s interpretation again . . . . . . . . . . . . . . . . . . . . . . 215 Non-geometrical rules in Book II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.1 Visual evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Overlapping figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.1 Hilbert-style account of visual evidence . . . . . . . . . . . . . . . . . 245.3 Renaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 From visible to invisible and backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.1 Propositions II.5–6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.1.1 Van der Waerden’s and Corry’s interpretations . . . . . . . . . . 306.2 The use of II.5–6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Interpretations of Book II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.1 Historians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.1.1 Ken Saito . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.1.2 Leo Corry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.1.3 Ian Mueller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.2 Mathematicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2.1 Bart van der Waerden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2.2 John T. Baldwin and Andreas Mueller . . . . . . . . . . . . . . . . . . 398 Descartes’ lettered diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 henomenology of diagrams in Book II of the
Elements Proposition II.1 of Euclid’s
Elements states that “the rectangle contained byA, BC is equal to the rectangle contained by A, BD, by A, DE, and, finally, byA, EC”, given BC is cut at D and E. In algebraic stylization, it is conveyed bythe formula A · BC = A · BD + A · DE + A · EC , where BC = BD + DE + EC .In modern theory of rings, this formula is simplified to a ( b + c ) = ab + ac andrepresents distributivity of multiplication over addition. In algebra, however, it isan axiom, therefore, it seems unlikely that Euclid managed to prove it, even in ageometric disguise. Moreover, if we apply algebraic formulea to read propositionII.1, it appears that the equality a ( b + c + c ) = ab + ac + dc is both the starting-point and the conclusion of the proof. Yet, there is some in-between in Euclid’sproof. What is this residuum about? Although an algebraic formalization easilyinterprets the thesis of the proposition, it does not help to reconstruct its proof.The above interpretation is an extreme. Usually, instead of the multiplication A · BC the term A.BC is considered, which stands for a rectangle with sides A and BC . Then, it is assumed that the equality A.BC = A.BD + A.DE + A.EC isapproved by the accompanying diagram. In this way, a mystified role of Euclid’sdiagrams substitute detailed analyses of his proofs.David Fowler, for example, ascribes to Euclid’s diagrams in Book II not onlythe power of proof makers. In his view, they summarize both subjects and proofsof propositions: “The subject of each proposition is best conveyed by its figure(and it is these figures, not what is made of them in their enunciations or proofs,that will enter my proposed reconstruction)” (Fowler 2003, 66).Accordingly, he presents propositions II.1–8 as arguments based on diagrams:“In all of these propositions, almost all of the text of the demonstrations concernsthe construction of the figures, while the substantive content of each enunciationis merely read off from constructed figures, at the end of the proof (Fowler2003, 69). In propositions II.9–10, Euclid studies the use of the Pythagoreantheorem, and the accompanying diagrams represent right-angle triangles ratherthen squares descried on their sides. Yet, to buttress his interpretation, Fowlerprovides alternative proofs, as he believes Euclid basically applies “the techniqueof dissecting squares”. In his view, Euclid’s proof technique is very simple: “Withthe exception of implied uses of I47 and 45, Book II is virtually self-containedin the sense that it only uses straightforward manipulations of lines and squaresof the kind assumed without comment by Socrates in the
Meno ”(Fowler 2003,70).Fowler is so focused on dissection proofs that he cannot spot what actually isand what is not depicted on Euclid’s diagrams. As for proposition II.1, there isclearly no rectangle contained by A and BC , although there is a rectangle with All English translations of the
Elements after (Fitzpatrick 2007). Sometimes weslightly modify Fitzpatrick’s version by skipping interpolations, most importantly,the words related to addition or sum. Still, these amendments are easy to verify,as this edition is available on the Internet, and also provides the Greek text anddiagrams of the classic Heiberg edition. Piotr B laszczyk vertexes
B, C, H, G (see Fig. 7). Indeed, all throughout Book II Euclid deals withfigures which are not represented on diagrams. Finally, in propositions II.11, 14they appear to be a tool to establish results concerning figures represented onthe diagrams.The plan of the paper is as follows. Section § §
3, we analyze basiccomponents of Euclid’s propositions: lettered diagrams, word patterns, and theconcept of parallelogram contained by . In section §
4, we scrutinize propositionsII.1–4 and introduce symbolic schemes of Euclid’s proofs. In section §
5, we in-troduce non-geometrical rules which aim to explain relations between figureswhich are represented (visible) and not represented (invisible) on Euclid’s dia-grams. In section §
6, we analyze the use of propositions II.5–6 in II.11, 14 todemonstrate how the technique of invisible figures enables to establish relationsbetween visible figures.Throughout the paper we confront specific aspects of our interpretation withreadings of Book II by scholars such as J. Baldwin and A. Mueller, L. Corry,D. Fowler, R. Hartshorne, I. Mueller, and B. van der Waerden. In section § §
8, we discussproposition II.1 from the perspective of Descartes’s lettered diagrams. We showthat there is a germ of algebraic style in Book II, however, it has not beendeveloped further in the
Elements . Book II of the
Elements consists of two definitions and fourteen propositions.The first definition introduces the term parallelogram contained by , the second– gnomon . All parallelograms considered are rectangles and squares, and indeedthere are two basic concepts applied throughout Book II, namely, rectangle con-tained by , and square on , while the gnomon is used only in propositions II.5–8.Considering the results, proof techniques, and word and diagrammatic pat-terns, we distinguish three groups of propositions: 1–8, 9–10, 11–14.II.1–8 are lemmas. II.1–3 introduce a specific use of the terms squares on and rectangles contained by . II.4–8 determine the relations between squares andrectangles resulting from dissections of bigger squares or rectangles. Gnomonsplay a crucial role in these results. Yet, from II.9 on, they are of no use.Propositions II.9–10 apply the Pythagorean theorem for combining squares.To this end, Euclid considers right-angle triangles sharing a hypotenuse andequates squares built on their legs. Although these results could be obtained bydissections and the use of gnomons, proofs based on I.47 provide new insights.II.11–14 present substantial geometric results. Their proofs are based on thelemmas II.4–7, and the use of the Pythagorean theorem in the way introducedin II.9–10. henomenology of diagrams in Book II of the
Elements Propositions II.1–3, when viewed from the perspective of deductive struc-ture, seem redundant. Yet, we consider them as introducing the basics of thetechnique developed further in II.4–8. Similarly, II.9–10 also seem redundant –when viewed from this perspective. Yet, they introduce a technique of applyingthe Pythagorean theorem.
In regard to the structure of Book II, Ian Mueller writes: “What unites all ofbook II is the methods employed: the addition and subtraction of rectangles andsquares to prove equalities and the construction of rectilinear areas satisfyinggiven conditions. 1–3 and 8–10 are also applications of these methods; but whyEuclid should choose to prove exactly those propositions does not seem to befully explicable” (Mueller, 2006, 302).Our comment on this remark is simple: the perspective of deductive struc-ture, elevated by Mueller to the title of his book, does not cover propositionsdealing with technique. Mueller’s perspective, as well as his Hilbert-style read-ing of the
Elements , results in a distorted, though comprehensive overview ofthe
Elements . Too many propositions do not find their place in this deductivestructure of the Elements . While interpreting the
Elements , Hilbert applies hisown techniques, and, as a result, skips the propositions which specifically de-velop Euclid’s technique, including the use of the compass. Furthermore, in the
Grundalgen , Hilbert does not provide any proof of the Pythagorean theorem,while in our interpretation it is both a crucial result (of Book I) and a prooftechnique (in Book II). In modern mathematics, there are many important results concerning prooftechnique. The transfer principle relating standard and non-standard analysis,is a model example. Hilbert’s proposition that the equality of polygons builton the concept of dissection and Euclid’s theory of equal figures do not produceequivalent results could be another example. Viewed from that perspective, II.9–10 show how to apply I.47 instead of gnomons to acquire the same results. II.1–3introduce a specific use of the terms square on and rectangle contained by whichMueller ignores in his analysis of Book II (see § In regard to geometric results, II.11 provides the so-called golden ratio construc-tion. It is a crucial step in the cosmological plan of the
Elements , namely – theconstruction of a regular pentagon and finally, a dodecahedron. The respectivejustification builds on II.6.II.12, 13 are what we recognize as the cosine rule for the obtuse and acutetriangle respectively. The former proof begins with a reference to II.4, the later– with a reference to II.7. The Pythagorean theorem plays a role in Hilbert’s models, that is, in his meta-geometry. Piotr B laszczyk
Fig. 1.
Elements , II.14.
In II.14, Euclid shows how to square a polygon. This construction crowns thetheory of equal figures developed in propositions I.35–45; see (B laszczyk 2018).In Book I, it involved showing how to build a parallelogram equal to a givenpolygon. In II.14, it is already assumed that the reader knows how to transforma polygon into an equal rectangle. The justification of the squaring of a polygonbegins with a reference to II.5.As for the proof technique, in II.11–14, Euclid combines the results of II.4–7with the Pythagorean theorem by adding or subtracting squares described onthe sides of right-angle triangles. Thus, II.5, 6 share the same scheme: rectangle + square on A = square on B. When applied, a right-angle triangle with a hypotenuse B and legs A , C isconsidered. Then, by I.47, rectangle + square on A = square on A + square on C. By subtraction from both sides of the square on
A, the equality characterizingII.11 and II.14 is obtained rectangle = square on C. On the other hand, II.4, 7 share another scheme. In II.4, it is as follows square on A = square on B + square on C + 2 rectangles. Addition to both sides another square gives the equality square on A + square on D = square on B + square on D + square on C +2 rectangles. henomenology of diagrams in Book II of the Elements D is a leg of two right-angle triangles: the first with another leg A and ahypotenuse E , the second with leg B and hypotenuse F . Thus, equality obtains square on E = square on F + square on C + 2 rectangles. The use of II.7 starts with the equation square on A + square on B = square on C + 2 rectangles. A square on the height AD of the triangle ABC is added to its both sides;see Fig. 2. The rest of the proof II.13 proceeds in a similar way. Significantly, the accompanying diagram, next to sides of the triangle, fea-tures only the height AD . Thus, the point D represents the way the side BC is cut, namely at random . In this way, it makes a reference to II.7. The line AD represents the side of the square that is added to both sides of the aboveequation. All that illustrates our thesis, that although in II.11–14 Euclid relatessquares and rectangles, the accompanying diagrams depict only the respectivesides of the figures involved. Fig. 2.
Elements , II.13.
References to II.4–7 follow a similar word pattern:
For since the straight-line ...has been cut ... . Then comes a specification how new points are placed on thestraight-line: in half ( equally ), at random ( unequally ), or a new line is addedto the original one. Finally, a relation between resulting squares and rectanglesestablished in the refereed proposition is reformulated due to new names ofpoints. The term can be interpreted as − F C cos( angle between sides F and C ).See section § Using the names of lines in the diagram, BC + DC = BD +2 BC × DC . By adding AD to the both sides of this equation, we obtain BC + AC = BA + 2 BC × DC . Piotr B laszczyk For example, the respective part of II.14 is this: “since the straight-line BFhas been cut equally at G, and unequally at E, the rectangle contained by BE,EF, together with the square on EG, is thus ( ἄρα ) equal to the square on GF”.While II.5 states: “For let any straight-line AB have been cut equally at Cand unequally at D. I say that the rectangle contained by AD, DB, togetherwith the square on CD, is equal to the square on CB”.Clearly, these statements differ only in names of points (for comparison, seeFig. 1 and Fig. 11).In propositions II.1–8, depending on a distribution of cut points, a varietyof squares and rectangles appears on the accompanying diagrams. Next to thoserepresented on the diagrams, Euclid refers also to figures which are not repre-sented on the diagrams; let us call them invisible figures. These are squares on and rectangles contained by some lines. Although the respective lines are rep-resented on the diagrams, the related squares and rectangles are not. Indeed,whereas these invisible figures occur in the statements of propositions, Euclid’sproofs usually start with figures which are represented on the diagrams.We recognized the following pattern in the procedures Euclid adopts in propo-sitions II.1–8. Firstly, contrary to Book I, the diorisomos part of the propositionrefers to figures not represented on the diagram. Secondly, in the kataskeu¯e , ageometric machinery is applied to construct figures represented on the diagram.Thirdly, in the apodeixis , a relation between figures which are represented andnot represented on the diagram is determined. It is achieved by visual evidence,substitution rules, and the renaming of figures. In sections § § square on and rectangle contained by .As we proceed from II.1 to II.8, Euclid’s diagrams get more complicated:they depict more and more squares and rectangles. Then, in propositions II.9–10, they gain a new clarity. Indeed, II.9–10 explore the Pythagorean theorem inequating groups of squares, yet, the accompanying diagrams do not depict thesesquares. For example, II.10 reads: “For let any straight-line AB have been cutin half at C, and let any straight-line BD have been added to it straight-on. Isay that the squares on AD, DB is double the squares on AC, CD” (see Fig. 3).Since the right-angle triangles ADG and
AEG share the hypotenuse AG ,squares on the legs of the first triangle, AE , EG , and on the legs of the second, AD, DG , are equal. The squares on AE are AC , CE ; the squares on EG are F E , F G ; and the squares on the legs of the second triangle are AD , DG . Finally, Euclid determines equalities between squares to get the doubles of thesquares, such as 2
F E which is to be equal to the square on EG . Althoughthe equality of line segments, F G = EF , is derived from I.6, the equality ofrespective squares, F G = F E , is based on an implicit rule: “since FG is equalto EF, the one on FG is equal to the one on EF”. In our interpretation, it is oneof substitution rules discussed in section § In II.9, Euclid does the same trick. The term AC stands for the phrase the square on the straight-line AC .henomenology of diagrams in Book II of the Elements Fig. 3.
Elements , II.10.
Notice, that to justify the equality
F G = F E , Euclid does not refer to thediagram. In fact, the accompanying diagram does not depict any square. It isalso not the case, that this conclusion is taken for granted, since he provides areason. Indeed, throughout Book II, Euclid reiterates the argument: since X isequal to Y, the square on X is equal to the square on Y .Interestingly, the results II.9–10 could be obtained by dissection of the squareon AB , in the case of II.9, and the square on AD , in the case of II.10, and thenby the use of gnomons in a way similar to the proofs of II.5–8. Therefore, weview II.9–10 as introducing a new technique which combined with the results oflemmas II.4–7 is used in the proofs II.11–14.The word pattern of references we identified above finds its diagrammaticcounterparts in II.11–14. Starting from II.11, squares on are represented by lines, rectangles contained by – by lines with cut points. In II.11–14, diagrams looklike ideograms rather than a simple composition of lines introduced throughoutconstruction steps.Let us focus on II.14 (see Fig. 1). The figure A and the rectangle BCDE referto the theory of equal figures developed in Book I. The exposition of the line BF and the cut points G , E , refer to II.5. The semicircle BHF and the radius GH represent a new construction. The line EH represents the conclusion of thesquaring of a polygon process.In Fig. 4, we represent the way how II.5 is applied in II.14. The scheme of thisapplication is as follows: Since the line BF is cut in half at G , and at randomat E , by II.5, the equality obtains rectangle BD + square on GE = square on GF. In section § GF = GH , then GH = GF .0 Piotr B laszczyk Fig. 4.
The scheme of application of II.5 in II.14.
Then, by I.47: square on GF = square on GE + square on HE.
Thus rectangle BD + square on GE = square on GE + square on HE.
By subtraction, the final equality obtains rectangle BD = square on HE.
Yet, instead of all these auxiliary lines that evoke the II.5 construction, Eu-clid’s original diagram is much simpler: it refers to II.5 by the way points G and E are located on the line EF . Note however, that Euclid reached this clar-ity due to the specific use of the term square on . Instead of the square on GH ,he considers square on GF , instead of the square on GE , he considers anothersquare – the gray one in Fig. 4, on the right. And that is why, in II.5, althoughhe mentions the square on CD , he considers the square LEGH – the gray one inFig. 4, on the left.In sum, from the perspective of diagrams, Book II applies figures which arerepresented and not represented on the diagrams. These are squares on and rectangles contained by . II.9–10 apply line segments instead of squares on . II.11–14, besides lines representing squares on , apply also line segments with cut pointsinstead of rectangles contained by . There are two components of Euclid’s proposition: the text and the lettereddiagram. The Greek text is linearly ordered – sentence follows sentence, from henomenology of diagrams in Book II of the
Elements left to right, and from top to bottom. Diagrams consist of straight-lines andcircles. The capital letters on the diagrams are located next to points; theyname the ends of line segments, intersections of lines, or random points.The text of the proposition is a schematic composition made up of sixparts: protasis (stating the relations among geometrical objects by means ofabstract and technical terms), ekthesis (identifying objects of protasis withlettered objects), diorisomos (reformulating protasis in terms of lettered ob-jects), kataskeu¯e (a construction part which introduces auxiliary lines exploitedin the proof that follows), apodeixis (proof, which usually proves the diorisomos ’claim), sumperasma (reiterating diorisomos ). References to axioms, definitions,and previous propositions are made via the technical terms and phrases appliedin prostasis . Fig. 5.
Elements , I.47.
In fact, it is the received account of Euclid’s propositions. For example, inproposition I.47, the protasis , ekthesis , and diorisomos are as follows (numerals insquare brackets added, here they stand for subsequent parts of the proposition):[1] “In a right-angle triangle, the square on the side subtending the right-angle is equal to squares on the sides surrounding the right-angle.[2] Let ABC be a right-angled triangle having the right-angle BAC.[3] I say that the square on BC is equal to the squares on BA and AC.”Indeed, the triangle ABC as well as the squares constructed on lines BA and AC are all depicted on the accompanying diagram (see Fig. 5).Yet, already in the first propositions of Book II, we observe a new phe-nomenon: figures mentioned in the diorisomos are not represented on the dia-grams. Here are ekthesis and diorisomos of proposition II.2: “For let the straight-line AB have been cut, at random, at point C. I say that the rectangle containedby AB, BC together with the rectangle contained by BA, AC, is equal to thesquare on AB.” On the accompanying diagram (see Fig. 6), we can see the lines AB , AC , and BC , however, neither the rectangle contained by AB , BC , nor the one containedby BA , AC is depicted on the diagram: line-segments AB , AC , and BC lay onthe same straight line and do not contain a right-angle. Rectangles which aresupposed to be formed by line segments not containing a right-angle occur inevery proposition of Book II.Furthermore, notice that in proposition II.14, Euclid shows how to “constructa square equal to a given rectilinear figure”. Although the accompanying diagramdepicts a quadrilateral A , it is a symbol of a rectilinear figure rather than theindividual object that is being studied in the proposition: it is not constructed, itsvertices are not denoted by letters, and they are not involved in the constructionscarried out in the proposition.The diagram in proposition VI.31 plays an even more abstract role. The diorisomos part of VI.31 reads: “Let ABC be a right-angled triangle having theangle ABC a right-angle. I say that the figure on BC is equal to similar, andsimilarly described, figures on BA and AC.” Like figure A in proposition II.14,the rectangles on the sides of the triangle ABC are not constructed, they arenot involved in the proof, and the proof does not rely on information read-offthe diagram. Rectangle contained by two straight-lines
Here is the crucial definition of Book II: “Any right-angled parallelogram is saidto be contained by two straight-lines containing a right-angle”. Throughout theentire book, Euclid studies squares, rectangles, and triangles. The term right-angled parallelogram is applied only to rectangles, then it is reshaped to rectanglecontained by two straight-lines . What, then, is the role of the term rectanglecontained by two straight-lines ? How does it differ from a simple rectangle?Let us start with the most general concept, namely that of a figure. In BookI, Euclid defines a figure as follows: “A figure is that which is contained by someboundary or boundaries”. The term boundary applies to a circle only, boundaries apply to polygons. Hence, for example, a triangle is contained by three straight-lines, i.e., its sides. In other words, what we call a polygonal curve today isnot considered to be a single line in the
Elements – according to Euclid, it is acomposition of lines.In Book II, in addition to triangles, Euclid studies squares and rectangles.Definition I.22 clarifies these concepts. It reads: “And of quadrilateral figures: asquare is that which is right-angled, and equilateral, a rectangle that which isright-angled but not equilateral”. The term parallelogram requires the concept ofparallels and is not included in the list of definitions prefacing Book I. It occursin proposition I.34 as parallelogrammic figure . Although not explicitly defined, itis clear what Euclid means: a parallelogram is a quadrilateral with two pairs ofparallel sides (I.33 shows the existence of parallelograms). This term is closely For a detailed analysis of this proposition in terms of individual and abstract com-ponents, see (B laszczyk, Petiurenko 2020).henomenology of diagrams in Book II of the
Elements related to Euclid’s theory of equal figures. Within this theory, in propositionI.44, Euclid shows how to construct a parallelogram when its two sides and anangle between them are given. Still, in Book II, all parallelograms are rectangles.What is, then, the reason for the term rectangle contained by two straight lines ?Firstly, this term is related to the ways figures are referred to in the text ofthe propositions, specifically, it is essential in protasis parts. Secondly, it playsan analogous role to the term square on a side: as the latter enables to identifya square with one side, the former enables to identify a rectangle with two sideswith no reference to a diagram. Thirdly, since the terms square on a straight-line and rectangle contained by straight-lines are applied both to figures which arerepresented (visible) and not represented (invisible) on a diagram, these namesmake it possible to relate the visible and invisible figures. Due to substitutionrules which we detail in section §
5, Euclid can claim that a rectangle containedby X,Y , which is not represented on the diagram, is contained by A, B , wheresegments A , B form a rectangle which is represented on the diagram. rectangle contained by Within the so-called geometric algebra interpretation of Book II, all rectanglesare represented by the formula ab , no matter whether lines a, b contain a right-angle or not. Moreover, the term ab is subject to some processing.To be clear, we do not agree with the claim that this interpretation ignoresthe historical context by implying a multiplication of line segments. One maytreat the terms ab , a ( b + c ) as interpreting the phrases rectangle contained by lines a, b , or a rectangle contained by lines a and b, c . Yet, when that the term ab is applied in the same way to rectangles represented and not represented ona diagram, it blurs the essence of Book II.John Baldwin and Andreas Mueller interpret a rectangle contained by X,Y as a rectangle of length X and height Y . They write: “This definition [II.1]allows us to study the areas of arbitrary rectangles by indexing each rectangle byits semi-perimeter and a cut point in that line” (Baldwin, Mueller 2019, 8). Inproofs, instead of Y , they consider W such that W=Y , and lines X , W containa right-angle. Thus, in fact, they reduce a rectangle contained by to a rectanglerepresented on a diagram. Still, they are the only authors that find Euclid’sdefinition “strange to modern ears”.Ian Mueller tries yet another trick: “ O ( x,y ) is used to designate a rectanglewith arbitrary straight lines equal to x and y as adjacent sides” (Mueller 2006,42). First of all, O ( x, y ) denotes a rectangle and makes no difference betweenvisible and invisible figures. For example, in Mueller’s formalization of II.2, thereis no difference between O ( AD,AC ) and O ( AB,AC ), since AB = AC (see Fig 6).By his notation alone, Mueller ignores the basic problem Euclid seeks to resolvein propositions II.1–8, namely the relation between a rectangle represented onthe diagram, and one not represented on the diagram. As a result, he distorts Euclid’s original proofs, even though he can easily interpret the theses of hispropositions. Jeffrey Oaks provides a similar interpretation, as he writes in a commentaryto proposition VI.16 of the
Elements : “Here ‘the rectangle contained by themeans’ in most cases will not be a particular rectangle given in position becausethe two lines determining it are not attached at one endpoint at a right angle. Infact, the sides determining rectangles cited in Greek works rarely satisfy Euclid’sdefinition at the beginning of Book II [...]. Already in Proposition II.1 Euclidwrites about ‘the rectangle contained by A, BC’ when the two lines may notbe anywhere near each other. And the lines determining the rectangles citedin Proposition II.2 are absolutely not at right angles, since they are colinear.Propositions VI.16 and XI.34, like many propositions in Greek mathematics, areabout the measures or sizes of geometric objects. ‘The rectangle contained bythe means’ does not designate a particular rectangle given in position, but onlythe size of a rectangle whose sides are equal (we would say “congruent”) to thoselines. Location and orientation of this rectangle relative to the other magnitudesin the diagram are undetermined and irrelevant to the argument. It is only therelative ‘measure’ that is intended” (Oaks 2018, 259).At first, Oaks admits that not all rectangles contained by are featured ondiagrams. Going beyond this observation, he links (invisible) rectangles with theconcept of measure.The above interpretations involve terms, figures and measure. We do notinterpret the phrase rectangle contained by , but rather study its role in Euclid’sproofs. Eventually we view it as a proof technique not an object.
In the
Elements , Euclid adopts the following pattern of naming the figures fea-tured on diagrams: squares and rectangles are, first of all, denoted by the letterslocated next to their vertices, they are also denoted by the letters which desig-nate the diagonal. In proposition I.46, Euclid shows how to describe a square ona given straight-line. In the propositions that follow, squares are also identifiedby the phrase square on a straight-line , where the specific name of a line is given.We can illustrate this naming technique by referring to proposition I.47 (Fig. 5represents the accompanying diagram).Thus, in the text of the proposition, the square
BDEC is also called thesquare on BC ; the square on BA is also denoted by the two letters locatedon the diagonal, namely GB . Since the intersection of lines BC and AL is notnamed, rectangles that make up the square BDEC are named with two letters,as parallelogram BL and parallelogram CL .In Book II, Euclid introduces yet another naming scheme for the rectangle:it is identified with its two sides and is called rectangle contained by , and theterm is followed by the names of line-segments containing a right-angle. In fact, Mueller tries to reconstruct only the proof of II.4. For details, see § Elements All these naming rules – that is, by vertices, by a diagonal, square on , and rectangle contained by – apply to figures represented on the diagrams. Here,for example, is the text of proposition II.2 (Fig. 6 represents the accompanyingdiagram).[1] “For let the straight-line AB have been cut, at random, at point C. I saythat the rectangle contained by AB, BC together with the rectangle containedby BA, AC, is equal to the square on AB.[2] For let the square ADEB have been described on AB, and let CF havebeen drawn through C parallel to either of AD or BE.[3] So AE is equal to the AF, CE. And AE is the square on AB. And AF (is)the rectangle contained by BA, AC. For it is contained by DA, AC, and AD (is)equal to AB. And CE (is) by AB, BC. For BE (is) equal to AB. AB. Thus, byBA, AC, together with by AB, BC, is equal to the square on AB.”
Fig. 6.
Elements , II.2.
Thus, the square
ADEB is also named the square on AB . Rectangle AF isalso called the rectangle contained by DA, AC .The phrases square on and rectangle contained by are also applied to figuresnot represented on diagram. In the text of proposition II.4, the term the squareon AC occurs, although there is no such square on the accompanying diagram(see Fig. 8). Also in this proposition, the term rectangle contained by AC, CB occurs, although there is no such rectangle on the accompanying diagram.In fact, rectangles contained by straight-lines lying on the same line and notcontaining a right-angle are common in Book II. Whether represented on dia-grams or not, as a rule, they are contained by individual straight-lines. However,proposition II.1 represents a unique case in this respect. Therein, Euclid consid-ers rectangles contained by A , BD , and A , DE , and A , EC (see Fig. 7). They areto be rectangles contained by BG , BD , by DK , DE , and by EL , EC respectively.Rectangles contained by A , BD , by A , DE , and A , EC are neither representedon the diagram, nor contained by individual line-segments: line A , considered asa side of these rectangles, is not an individual line.To be clear, the relation between rectangles, contained on the one hand by A , BD , and on the other by BG , BD , is not an equality: it is stated that the former is the rectangle contained by BG , BD . The relevant part of the proposition reads:“BK is by A, BD. For it is contained by GB, BD, and BG is equal to A”. Here, BK is represented on the diagram, and Euclid claims that it is contained by BG , BD ,which is simply another name of the rectangle BK . Still, Euclid also claims that BK is contained by A , BD , while the later rectangle is not represented on thediagram. Thus, the relation between figures represented and not represented on adiagram is founded on substitution and renaming rules. We will explicate theserules in section §
5. In general, Euclid engages a bunch of tricks to establishan equation between a rectangles contained by and a figures represented on adiagram.Interestingly, Euclid never refers to proposition II.1. Moreover, its result,when viewed from the modern perspective, is reiterated in propositions II.2 andII.3. Hence, it seems that its role is to demonstrate the substitution rules whichare applied throughout the rest of Book II, rather than to present a specific ge-ometrical statement. All that is required to analyze these rules are propositionsII.1–II.4. From the perspective of substitution rules, proposition II.1 introducesthem, then proposition II.2 applies them to rectangles contained by , and propo-sition II.4 – to squares on . Proposition II.4 involves yet another object, namelythe so-called complement . It shows how to apply the substitution rules to theseobjects.From the perspective of represented vs not represented figures, propositionII.2 equates figures which are represented, on the one side, and not represented,on the other, while proposition II.3 equates figure not represented, on the oneside, and figures represented and not represented, on the other side, propositionII.4 introduces yet another operation on figures which are not represented, as itincludes an object called twice rectangle contained by , where the rectangle is notrepresented on the diagram.
Without paying attention to Euclid’s vocabulary, specifically to the terms squareon and rectangle contained by , one cannot find a reason for propositions II.2 andII.3. Thus, Bartel van der Waerden in (Waerden 1961) considers them as specialcases of II.1. Similarly, Robin Hartshorne, in the
Appendix to (Hartshorne 2000),includes statements of “the most frequently quoted results” of the
Elements .Regarding Book II, he refers to proposition II.1, then skips to II.4.From the modern perspective, especially when the diorismos of Euclid’sproposition is stylized as an algebraic formula, such an interpretation seemsreasonable. For, when II.1 is rendered as a ( b + c + d ) = ab + ac + ad , then II.2is a = ab + ac , given a = b + c , and II.3 is a ( b + a ) = ab + a . Indeed, II.2and II.3 follow from II.1 by suitable substitutions. In fact, however, propositionII.1 is never quoted in the Elements , and due to the role of line A it is a uniqueproposition in the entire Elements . henomenology of diagrams in Book II of the Elements In this section, we provide detailed analysis of propositions II.1 to II.4. Theyaim to reveal non-geometrical rules which enable to relate figures representedand not represented on the diagrams.Here is the text of proposition II.1, starting with the diorismos : Diorismos “Let A, BC be the two straight-lines, and let BC, be cut, atrandom, at points D, E. I say that the rectangle contained by A, BC is equal tothe rectangles contained by A, BD, by A, DE, and, finally, by A, EC.”
Kataskeu¯e “For let BF have been drawn from point B, at right angles toBC, and let BG be made equal to A [...].”
Fig. 7.
Elements , II.1. “[1] So BH is equal to BK, DL, EH. [2] And BH is by A, BC. For it is containedby GB, BC, and BG is equal to A. [3] And BK (is) by A, BD. For it is containedby GB, BD, and BG is equal to A. [4] And DL is by A, DE. For DK, that is tosay BG, is equal to A. [5] Similarly, EH (is) also by A, EC. [6] Thus, by A, BCis equal to by A, BD, by A, DE, and, finally, by A, EC.”Now, we present this proposition in a more schematic form. In what follows,symbol A × BC stands for the phrase “rectangle contained by A, BC”, while BH π GB × BC stands for the phrase “BH is contained by GB, BC”. Diorismos A × BC = A × BD, A × DE, A × EC. Numbering of sentences and names of parts added.8 Piotr B laszczyk
Kataskeu¯e BF ⊥ BC, BG = A ; GH k BC ; DK, EL, EH k BG . Apodeixis BH = BK, DL, EHBH π GB × BC, BG = A −→ BH π A × BCBK π GB × BD, BG = A −→ BK π A × BDDK = BG = A −→ DL π A × DE −→ EH π A × EC −→ A × BC = A × BD, A × DE, A × EC. (1) The formula in red interprets sentence [1]. It is a simple statement withno justification and the starting point of the whole argument. Since the figuresinvolved are represented on the diagram, we interpret it as based on purely visualevidence. All of the rectangles mentioned in the diorismos are not represented on thediagram. Therefore, we are to explain how, starting from the relation betweenthe figures represented on the diagram, Euclid gets the relation between figuresnot represented on the diagram.(2) The next line in the apodeixis scheme interprets sentence [2]. The formulain blue stands for the phrase “BH [...] is contained by GB, BC”. It is one ofthe three possible names for a rectangle represented on a diagram. Thus, it is aresult of renaming figures rather than a geometrical or logical relation. Hereafter,formulas resulting from renaming will be represented in blue.The equality BG = A follows from the construction part of the proposition.Yet, the most puzzling is the phrase “BH is by A, BC”. We interpret it as a resultof substituting A to the formula BH π BG × BC in place of BG . Arguments ofthis kind are applied all throughout Book II. The relation between the rectangle BH , and the one contained by A , BC is by no means an equality; the wordpattern “BH is by A, BC” is systematically used by Euclid. Let us representformulas obtained by this type of substitution in violet.(3) In sentence [3], Euclid reiterates the previous argument.(4) In sentence [4], Euclid skips supposition DL π DK × DE and notes theequalities DK = BG = A . They can be justified by Common Notion 1.(5) In sentence [5], Euclid skips arguments relying on substitution and CN1,and simply states the result. It is a way of shortening repeated arguments, typicalof Euclid.(6) In sentence [6], with ἄρα , Euclid reaches the equality between invisiblerectangles. We interpret this step as a result of another substitution rule: in theequality starting in apodeixis , A × BC is substituted for BH , then A × BD issubstituted for BK , etc.Let us represent the equality obtained by this type of substitution, i.e., sub-stitutions for equality, in magenta.The arrow → in the scheme stands for a conjunction, usually it is γάρ . It isby no means suggested to be a logical implication. (B laszczyk, Petiurenko 2020) develops the idea of pure visual evidence.henomenology of diagrams in Book II of the Elements II.2 (see Fig. 6)In the below scheme, the term AB stands for the phrase “the square onAB”. Thus, for example, AE is AB interprets the phrase “AE is the square onAB”, or “AE is on AB”. Diorismos AB × BC, BA × AC = AB . Apodeixis AE = AF, CEAE is AB AF π DA × AC, AD = AB −→ AF π BA × ACBE = AB −→ CE π AB × BC −→ BA × AC, AB × BC = AB . Lines (3) and (4) of the apodeixis scheme, represent, again, typical of Euclidway of shortening repeated arguments: in line (4), Euclid skips the premise
CEis between CF, BC .In this proposition, the diorismos equates the figure represented on the di-agram, that is, the square on AB , with figures not represented on the diagram,namely rectangles contained by AB , BC , and AB , AC . Fig. 8.
Elements
II.3 (left) and II.4 (right).
II.3 (see Fig. 8)
Diorismos AB × BC = AC × CB, BC . Apodeixis AE = AD, CEAE π AB × BE, BE = BC −→ AE π AB × BCDC = CB −→ AD π AC × CBDB is CB −→ AB × BC = AC × CB, BC . Here, the square CE is also named by the second diagonal, as DB . Apartfrom this fact, this apodeixis is similar to the previous one.In this proposition, Euclid equates the figure not represented on the diagram, AB × AC , with figures both represented and not represented on the diagram, AC × CB, BC .Proposition II.1–3 share the same word patterns clearly represented by thecolors of terms in our schemes. Since there are no explicit references to thesepropositions in the rest of Book II, we treat them as introducing a technique ofdealing with figures which are not represented on diagrams.With II.4, we pass to a proposition which will be referred to in the followingpropositions.II.4 (see Fig. 8) Diorismos AB = AC , CB , AC × CB.
Apodeixis . CGKB is CB HF is HG , HF is AC −→ HF, KC are AC , CB GC = CB −→ AG π AC × CBAG = GE −→ GE = AC × CB −→ AG, GE = 2 AC × CBHF, CK are AC , CB −→ HF, CK, AG, GE == AC , BC , AC × CBHF, CK, AG, CE = ADEB,ADEB is AB −→ AB = AC , CB , AC × CB.
In this proposition, Euclid equates the figure represented on the diagram, AB , with figures represented, CB , and not represented on the diagram, AC ,2 AC × CB . A new component is the square on AC which is not representedon the diagram. Next, “twice the rectangle contained by AC, CB” means thatEuclid handles not represented rectangles the same way as represented ones.Formula HF is HG , HG is AC interprets the following phrase: “HF isalso a square. And it is on HG, that is to say AC”. We formalize it like this: HF is HG , HF = AC → HF is AC . Thus, from the adopted perspective, it is a substitution rule applied to thesquare on , analogous to the one applied to the rectangle contained by . That iswhy it is highlighted in violet.Now, let us focus on the lines (3) and (4) of the apodeixis scheme. Theyinterpret the following part of Euclid’s proof: “AG is equal to GE. And AGis contained by AC, CB. For GC is equal to CB. Thus, GE is also equal tothe one by AC, CB”. The equality AG = GE follows from proposition I.43. AG π AC × CB is the result of the substitution rule we already identified. The henomenology of diagrams in Book II of the Elements conclusion “GE is also equal to the one by AC, CB” is the result of a substitutionto the equality.In regard to the term AG, GE = 2 AC × CB , we cannot provide a clearjustification for this conclusion. The same applies to the conclusion HF, CK, AG, GE = AC , BC , AC × CB.
Although we could justify it by using logical tricks, it is not Euclid’s styleto conceal complicated rules and explicate simple ones. It seems to be a puzzleEuclid could not resolve.The formula in red interprets the following sentence: “HF, CK, AG, GE arethe whole ADEB”. Like in previous cases, we take it to be justified by visualevidence. From a logical point of view, it could be the starting point of thisproof.
In regard to propositions II.1–8, Fowler writes: “In all of these propositions,almost all of the text of the demonstrations concerns the construction of thefigures, while the substantive content of each enunciation is merely read off fromconstructed figures, at the end of the proof, as in lines 49 to 50 of II.4, theproposition just considered” (Fowler 2003, 69)Our schemes evidence that arguments read off the diagrams are staring pointsof the proofs II.1–3. In II.4–8, they are at the end of proofs. However, the enun-ciations of propositions II.1–8 concern figures which are not represented on thediagrams, therefore the essential arguments can not be read off the diagrams.Commenting on the final lines of the proof II.4, Fowler writes:
Therefore the four areas HF, CK, AG, GE are equal to the square on AC, CB,and twice the rectangle by AC, CB ,and the fact that this gives a decomposition of the square, the ostensible pointof the proposition, is merely stated (lines 49 to 50):
But HF, CK, AG, GE are the whole ADEB, which is the square on AB ” (Fowler2003, 69)The point is that HF , CK , AG , GE give the decomposition of the square ADEB , not “AC, CB, and twice the rectangle by AC, CB”.
Interestingly, the equality GE = AC × CB follows from the equality AG = GE rather than from the rule: A π C × D, C = E, D = F, B π E × F → A = B. This alternative argument is obvious within the so-called geometric algebra,as well as Mueller’s interpretation (see § x = A, y = B ⇒ O ( x, y ) ≃ O ( A, B ) . However, we have not identified this rule in Book II. On the contrary, propo-sition II.4 exemplifies different reason, namely
A π C × D, A = B → B = C × D. In words, since we know that figures A and B are equal and one of them is contained by C , D , then the other is also contained by C , D . It means, that theequality established within the theory of equal figures is more fundamental thanthe relation contained by . In a way, Euclid aims to introduce rectangles containedby into equalities of non-congruent figures. Within Mueller’s interpretation theequality between rectangles follows from a relation contained by . Four colors in or schemes of Euclid’s propositions correspond to three groupsof rules: visual evidence (red), renaming (blue) and substitutions (violet andmagenta). In this section we scrutinize these rules.
It is standard to identify two meanings of equality of figures in the
Elements –congruence and the equality of non-congruent figures. The congruence of figuresis usually linked to the idea of coinciding figures involved in
Common Notion
4. Itis also commonly assumed that the idea of coinciding figures plays a crucial rolein proposition I.4. But the superposition of figures presupposes (rigid) motion.The statements in red in our schemes are so simple that they do not engageany other concepts. If any needed justification, CN4 would be a good choice.Although such an interpretation finds no textual corroboration, there are nosignificant differences between the claim that CN4 is founded on visual evidenceand the claim that statements in red are justified by CN4. In fact, Euclid providesno arguments for these statements.
At first, the range of visual evidence is obvious: it justifies the equality of a figureand their dissection parts, which are squares and rectangles. Yet, as we proceedfurther, in propositions II.5–8, Euclid extends its power to another cases. InII.5, the gnomon
NOP is taken together with the square LG and Euclid declaresthat they form the square CEFG : “the gnomon NOP and the square LG arethe whole square CEFB” (see Fig. 11). We interpret it as an equality based onvisual evidence,
N OP, LG = CEF B .In the next proposition, the diagram is the same as regards the gnomonand its complementing square. This time, instead of the square LG , Euclidadds the square on BC to the gnomon NOP (see Fig. 12). His argument isthis: since LG = BC , then N OP, BC = CEF B . However, while the equality henomenology of diagrams in Book II of the
Elements N OP, LG = CEF D is visually obvious, the equality
N OP, BC = CEF B re-quires other kind of justification, as BC is not represented on the diagram. Ifwe add an auxiliary line to represent the square on BC , we will get overlappingfigures. Thus, the status of the equality N OP, BC = CEF B is not as obviousas
N OP, LG = CEF B in II.5.In II.7, Euclid goes a bit further in terms of abstraction. He explicitly con-siders overlapping figures: the rectangle AF , and the square CE . He claims thatthey together form the gnomon KLM and the square CF : “but AF, CE are thegnomon KLM and the square CF” (see Fig. 9). Here is the scheme of the proof. Fig. 9.
Elements
II.7.
Diorismos AB , BC = 2 AB × BC, CA . Apodeixis AG = GE −−−→ CN AG, CF = GE, CFAF = CE −→ AF, CE = 2 AF −→ KLM, CF = 2
AFBF = CB −→ AF = 2 AB × BCDG is AC −→ KLM, DG, GB = 2 AB × BG, AC KLM, DG, GB = ADEB, GB −→ ADEB, GB = 2 AB × BG, AC −→ AB , BC = 2 AB × BG, AC . Here, the red suggest that the respective formulas are based on visual evi-dence. Yet, since rectangles AF and CE overlap, i.e., share the square CF , theydo not represent the same kind of evidence as red formulas in propositions II.1–4.In II.8, Euclid’s considers even more complicated configuration of overlappingfigures. Below we present the scheme of the key step (see Fig. 10). DK = CK = GR = RN −→ DK, CK, GR, RN = 4
CKAG = M Q = QL = RF −→ AG, M Q, QL, RF = 4 AG −→ DK, CK, GR, RN, AG, M Q, QL, RF = ST U −→ ST U = 4
AK.
Fig. 10.
Elements
II.8.
The red formula interprets the following sentence: “the eight, which comprisethe gnomon STU”. Due to an implicit step
DK, CK, GR, RN, AG, M Q, QL, RF = 4
AK,
Euclid managed to bypass an explicit reference to overlapping figures. The miss-ing step could be like this:
AG=MQ=QL , CK=GR , then
AK=MR=GL . How-ever, here the square GR is counted twice. Thus, the square GR could be movedto cover the square DK . Argument of this kind characterize the so-called dissec-tion proofs, for example, the famous Chinese proof the Pythagorean theorem.Significantly, Euclid does not apply dissection combined with a translation.Another option could be a reference to CN2, namely: since AG=QL and
CK=DK , then
AG, CK= QL, DK . However, as a rule, Euclid does not applyCN2 when the resulting figure is not connected, i.e., it does not make a whole .Nevertheless, it is possible when the resulting figures overlap.With the use of modern technology, overlapping figures are handled withshades, or colors. Yet, these are textbooks tricks. Foundational studies seek toeliminate overlapping figures, as we demonstrate in the next section.
Within Hilbert’s tradition of reading the
Elements , the congruence of line seg-ments, angles, and triangles is covered by their respective axioms, Euclid’s propo-sition I.4 specifically is an axiom in the Hilbert system. Euclid’s theory of equalfigures is covered by the idea of the content of a figure. henomenology of diagrams in Book II of the
Elements Robin Hartshorne develops Hilbert’s idea of content further to the modernconcept of measure. Regarding Euclid’s theory of equal figures, he writes: “Look-ing at Euclid’s theory of area in Books I–IV, Hilbert saw how to give it a solidfoundation. We define a notion of equal content by saying that two figures haveequal content if we transform one figure into the other by adding and subtractingcongruent triangles” (Hartshorne 2000, 195).Indeed, figures involved in equalities in red also have equal content in theHilbert-Hartshorne system. However, a justification is far from obvious. Letus take, for example, proposition II.2 and Euclid’s statements
AE=AF, CE .Hartshorne’s definition of a figure reads: “A rectilinear figure [...] is a subsetof the plane that can be expressed as a finite non-overlapping union of trian-gles” (Hartshorne 2000, 196). AE , on the one hand, and AF , CE , on the other,meet the requirements of this definition. The next definition is this: “Two fig-ures P, P ′ are equidecomposable if it possible to write them as non overlappingunions of triangles P = T ∪ .... ∪ T n , P ′ = T ′ ∪ .... ∪ T ′ n , where for each i , the triangle T i is congruent to the triangle T ′ i ” (Hartshorne 2000,197). Then, the definition of equal content follows: “Two figures P, P ′ have equalcontent if there are other figure Q, Q ′ such that: (1) P and Q are not overlapping,(2) P ′ and Q ′ are not overlapping, (3) Q and Q ′ are equidecomposable, (4) P ∪ Q and P ′ ∪ Q ′ are equidecomposable” (Hartshorne 2000, 197).Thus, the proof that AE and AF, CE have equal content would be the sameas the proof of the reflexibility of the relation have equal content , as if AE and AF, CE were the same figures. In fact, from the perspective of set theory, whichmakes the basis of the Hartshorne system, AE = AF ∪ CE . However, it is notenough. To decide that AE and AF, CE are equidecomposable, we not only haveto cover both sides with the same triangles, we also need to add to both sidesanother figure Q . This peculiar step is the price for the solid account of Euclid’svisual evidence.Now, let us take the rectangle contained be BA, AC . Given Hartshorne’sdefinition, it is not a figure at all. Therefore BA × AC cannot be studied withinthis theory.In sum, within the Hartshorne system, one can provide conceptually com-plicated proof of a statement which is obvious in the Elements . However, acomplete reconstruction of Book II is impossible since there is no counterpart ofthe concept rectangle contained by . Hartshorne overestimates his system whenhe claims that “In Book II, all of the results make statements about certainfigures having equal content to certain others, and all of these are valid in ourframework” (Hartshorne 2000, 203) In fact, his system does not enable to iden-tify the real problems of Book II, that is, a relation between the represented andnot represented figures.In modern system of geometry, the measure of a figure, that is a real number,plays the role of figures which are not represented.
Our schemes of Euclid’s propositions clearly expose the role of the names of fig-ures in the analyzed arguments. Rectangles represented on the diagrams arenamed by their vertices, diagonals, and as contained by two line segments.Squares represented on the diagrams, similarly, are named by their vertices,diagonals, and as a square on a side. Figures which are not represented get onlyone name: it could be a rectangle contained by two lines, or a square on a line.Thus, the most important factor is that figures represented on the diagrams canalso be named rectangle contained by , or square on . Then, due to substitutionrules, they can be related to figures not represented on the diagrams.In a model example, in proposition II.2 (see Fig. 6), the rectangle AF isrepresented on the diagram and gets the name contained by DA, AC . Segments DA, AC are represented on the diagram and contain the right-angle. Then, Eu-clid claims that “AF is contained by BA, AC”, for “AD is equal to AB”. How-ever, the rectangle contained by
BA, AC is not represented on the diagram.Moreover, these lines do not contain a right-angle. That is why, in our scheme,
AF π DA × AC is represented in blue – it is simply a new name for a visiblefigure. Nevertheless, to turn AF π DA × AC into AF π BA × AC a substitutionrule is needed, namely rule (3) presented in the next section. First of all, observe that it is not explicit that the relation contained by is com-mutative, therefore we will not apply the following rule X × Y = Y × X . Euclid also does not apply this seemingly obvious rule: if X = U, Y = W , then X × Y = U × W .The first substitution rule, applied all throughout Book II, is the one denotedin violet in our schemes. It is as follows X π Y × V, Y = U ⇒ X π U × V. (1)The point is, while X and Y × V , as well as Y , V , and U , are representedon the diagram, U × V is not. The following line from the scheme of propositionII.2 exemplifies this rule: AF π DA × AC, AD = AB → AF π AB × AC. Mueller writes: “Since Euclid normally takes for granted such geometrically obviousassertions as T ( x ) ≃ O ( x, x ) and O ( x, y ) ≃ O ( y, x ), he could have carried outgeometrical versions of theses arguments” (Mueller 2006, 46). However, we have notidentified such arguments in Book II. It often happens that Euclid permutes letters naming line segments, as here with AB and BA , or AD and DA . This could be a topic for another paper. Generally, itseems that these letters are arranged to follow the drawing of the line, which is toillustrate an argument.henomenology of diagrams in Book II of the Elements A similar rule applies to square on , namely X = Y , Y = U ⇒ X = U . (2)The point is, while the square X and its side Y are represented on a diagram,the square U is not, although the side U is represented.We exemplify it by an argument from proposition II.4: HF is HG , HG = AC → HF is AC . This formula interprets the following phrase: “HF is also a square. And it ison HG, that is to say on AC”. Results based on this rule could be also achievedby a reference to proposition I.36. Yet, it would require introducing anotherpoint and an extra construction. Significantly, Euclid does not refer to I.36 inthis context.Finally, the rule concerning substitution to an equality; in our schemes it isrepresented in magenta. It is as follows: X = Y, X π U × W ⇒ U × W = Y. (3)Since the relation of equality is symmetric, by applying this rule, we can alsoget the following result X = Y, X π U × W, Y π Z × V ⇒ U × W = Z × V. Thus, in proposition II.1, the starting point is this BH = BK, DL, EH.
Then, by rule (1), we get the following results
BH π A × BC, BK π A × BD, DL π A × DE, EH π A × EC.
Finally, by rule (3), we reach the conclusion A × BC = A × BD, A × DE, A × EC.
To be clear, these substitution rules apply to the relation contained by ratherthan an equality of rectangles contained by . In II.11 and II.14, we can find a rectangle contained by equal to a square represented on the diagram. The equalityis achieved by rule (3). Another way to equate rectangle contained by and afigure represented on a diagram is by combining the Pythagorean theorem andCommon Notion 2. This trick is applied in II.11–14. In other words, whenevera rectangle contained by is equal to another figure, it is not a straightforwardrelation.
In this section, we study the use of propositions II.5, 6 in II.11, 14, and show howthrough the technique of rectangles contained by
Euclid has managed to establisha relation between visible figures. From a methodological point of view, he appliesresults obtained in one domain to determine results in another domain. It is likea factorization of real polynomial by its factorization in the domain of complexnumbers, or, finding a solution to a problem in the domain of hyperreals, then,with its standard part, going back to the domain of real numbers.
Propositions II.5, 6 are often discussed in the literature, as scholars seek toprovide a reason for including these seemingly twin propositions into Book II.First, we show how the substitution rules impact the interpretation of thesepropositions.
Fig. 11.
Elements
II.5.
Viewed in terms of construction, they look alike (see Fig. 11 and 12). Line AB is cut in half at C , then point D is placed between C and B , or on theprolongation of AB . Yet, their protasis parts differ in wording: in the first case,Euclid considers equal and unequal lines , in the second case, the whole line and the added line . Still, when we proceed to their diagrams and diorismos , they areagain similar. Moreover, their proofs apply the same trick: at first, Euclid showsthat a rectangle is equal to a gnomom, then he adds a square that complementsthe gnomon to a bigger square.We present a scheme of proposition II.5 starting from when it is establishedthat the rectangle AH is equal to the gnomon NOP .II.5
Diorismos AD × DB, CD = CB . henomenology of diagrams in Book II of the Elements Apodeixis ... −→ AH = N OPDH = DB −→ AH π AD × DB −→ N OP = AD × DBLG = CD −−−→ CN N OP, LG = AD × DB, CD N OL, LG = CEF BCEF B is CB −→ AD × DB, CD = CB . Similarity, we schematize Euclid’s proof of the next proposition starting fromthe conclusion: the rectangle AM is equal to the gnomon NOP . Fig. 12.
Elements
II.6.
II.6
Diorismos AD × DB, CB = CD . Apodeixis ... −→ AM = N OPDM = DB −→ AM π AD × DB −→ N OP = AD × DBLG = BC −→ AD × DB, BC = N OP, BC N OL, BC = CEF BCEF D is CD −→ AD × DB, BC = CD . In both propositions, figures not represented on the diagrams are to be equalto the squares represented on the diagrams. Euclid’s job is to show that these notrepresented are equal to some figures represented on the diagrams. The gnomon
NOP plays the crucial role in that process.When one pays no attention to the distinction between figures in terms ofthe representation on the diagram, these proofs are alike. However, in II.5, when
Euclid takes together the square LG and the gnomon NOP , they make a figurerepresented on the diagram. Thus, equality
N OP, LG = CEF B is based onvisual evidence. In II.6, Euclid adds the square on BC , which is not representedon the diagram, to the gnomon NOP . Thus, equality
N OP, BC = CEF B cannot be based on visual evidence here. In fact, Euclid skips an argument justifyingthis step. Thus, the second proof is more abstract.Let us consider the sequence of propositions II.5–8 from the perspective ofvisible and invisible figures. In II.5, the equality
N OL, LG = CEF B is basedon visual evidence. In II.6, the equality
N OL, BC = CEF B is not so obvious,yet in the company of II.5 it is almost the same. In II.7, Euclid adds to thegnomon
KLM , the complementing square DG and another one placed on thesame diagonal DB . These figures are represented on the diagram, yet the equal-ity KLM, DG, GB = ADEB, GB involves overlapping figures. In II.8, Euclidconsiders overlapping figures but not represented on the diagram. It is typical ofEuclid sequence of micro-steps, similar, e.g. to the first propositions in his the-ory of equal figures, when he considers parallelograms on the same base, then onequal bases (I.35–36), triangles on the same base, then on equal bases (I.37–38).Therefore, when II.5, 6 are considered in isolation, they provide almost the sameresult. When viewed in a bigger picture, they pave a way to a more abstractdiagrams.
Here is van der Waerden’s interpretation: “We see therefore, that, at bottom,II 5 and II 6 are not propositions, but solutions of problems; II 5 calls for theconstruction of two segments x and y of which the sum and product are given,while in II 6 the difference and the product are given. The applications in the Elements themselves are consistent with this view” (Waerden 1961, 121).In his view, II.5 and II.6 are two propositions for one formula, namely( x + y ) = xy + ( x − y ) , which is why they need to be interpreted as solutions ofproblems rather than mere propositions. To illustrate his idea, van der Waerdeninterprets proposition II.11 as the solving of a specific equation. Yet, II.11 alsoallows a standard, say a Hilbert-style interpretation, where II.6 is referred to inorder to get the result CF × F A, AE = EF .We interpret II.6 as lemma which is applied in II.11, while II.11 we viewas the crucial step in Euclid’s construction of dodecahedron – a regular solidforeshadowed in Plato’s Timaeus .Corry’s interpretation is as follows: “if we remain close to the Euclideantext we have to admit that, particularly in the cases of II.5 and II.6, both theproposition and its proof are formulated in purely geometric terms. There areno arithmetic operations involved, and surely there is no algebraic manipulationof symbols representing the magnitudes involved. The entire deduction relies onthe basic properties of the figures that arise in the initial construction or thatwere proved in previous theorems (which in turn were proved in purely geometricterms)” (Corry 2013, 647). henomenology of diagrams in Book II of the
Elements Our schemes clearly show that Euclid’s deduction only partly relies on con-structed figures. Euclid’s results in Book II also concern figures which are notrepresented on diagrams, and not constructed.
We present propositions II.5–6 as lemmas. Indeed, II.5 is applied in II.14, II.6 –in II.11. The word pattern of these references is the same: Euclid simply repeatsthe ekthesis with new names of the respective points. In regard to II.5 it is: “Forlet any straight-line AB have been cut – equally at C, and unequally at D”.As for II.6 it is: “For since the straight-line AC has been cut in half at E, andFA has been added to it”. In this way, the applied propositions are identifiedby the patterns of the cut points: with II.5 it is equally and unequally, withII.6 – at half and a line added. This style of references compels Euclid to provetwo propositions. Nevertheless, we provide a detailed analysis of the use of thesepropositions, as it reveals another relation between visible and invisible figures.In II.1–8, Euclid starts with visible figures to get a relation between invisiblefigures. In II.11, 14, by referencing to II.5, 6, he starts from invisible figures toget a relation between visible ones. Therefore, when one ignores Euclid’s prooftechniques, one can still consider propositions II.11, 14 as a relation betweenvisible figures, and retain a Euclid drawing of individual lines and circles.
Fig. 13.
The use of II.5 in II.14.
In II.14, it is required to construct a square equal to a rectilinear figure A .Due to a triangulation technique, A is turned into a rectangle BCDE . Here iswhere our scheme starts. ... −−→
II. BE × EF, GE = GF GF = GH −→ BE × EF, GE = GH HE , GE = GH −→ BE × EF, GE = HE , GE −−−→ CN BE × EF = HE EF = ED −→ BD π BE × EF −→ BD = HE BD = A −→ A = EH . When we add the diagram of II.5 to the diagram of II.14, the proof supportedby the compound diagram goes smoothly (see Fig. 13). On the one hand, by II.5,rectangle
BCDE and the gray square are equal to the square on GF , which isequal to the square on GH . On the other hand, by I.47, the square on GH isequal to the square on HE and the gray square. By substitution, BCED = EH .The final result concerns figures represented on the diagram, modulo the squareon HE represented by its side.Note, however, that the result was achieved by a reversal of our rule (3).Specifically, by II.5 and CN3, BE × EF = HE . By substitution rule (1), BD π BE × EF . As a result, BD = EH . We can turn this process into thefollowing rule X = Y × Z, W = Y × Z ⇒ X = W. (4)The invisible figure, the rectangle contained by Y , Z , enables the relationbetween visible figures X , W . Fig. 14.
The use of II.6 in II.11.
Here is the scheme of II.11 starting from the reference to II.6. henomenology of diagrams in Book II of the
Elements ... −−→ II. CF × F A, AE = EF EF = EB −→ CF × F A, AE = EB ∠ A = π/ −−→ I. AB , AE = EB −→ CF × F A, AE = AB , AE −−−→ CN CF × F A = AB AF = F G −→ F K π CF × F AAD is AB −→ F K = AD −−−→ CN F K \ AK = AD \ AK −→ F H = HDAB = BD −→ HD π AB × BHF H is HA −→ AB × BH = HA . We marked in red the equality
F H = HD . In fact, equality F K \ AK = F H ,and AD \ AK = HD are based on visual evidence. Yet, Euclid skipped thesesteps.The rectangle F CKG and the gray square are equal to the square on EF ,which is equal to the square on EB .Here, by II.6 and CN3, Euclid gets the equality CF × F A = AB . Then, bysubstitution rules, he aims to turn it into the equality of of figures representedon the diagram, F K = AD . Here, we can notice the pattern of our rule (4) CF × F A = AB , F K π CF × F A ⇒ F K = AB . In the rest of the proof, Euclid handles visible figures, and the next step relieson visual evidence,
F H = HD .Let us have a look at figures Fig. 13, 14. In II.14, when we apply the diagramof II.5 to the line BF , no auxiliary lines are needed to finish the proof (modulothe square on HE). In II.11, when we adopt the same procedure, the square on AB deforms the diagram II.6, in a way.Finally, let us adopt a mechanical perspective known, for example, throughDescartes’ drawing instruments; see e.g. (Descartes 1637, 318, 320, 336). DiagramII.11 is, in fact, a project of a machine squaring a rectangle, where a sliding point E determines its perimeter. As figure A can change in the original diagram II.11,line GE has to change accordingly. In this context, the term at random , appliedalso as a synonym of unequally , may suggest a dynamic interpretation. On theother hand, in II.11, the solution is determined by the right-angle triangle AEB ,and no line can play the role of a variable.
So far, we commented on the recent interpretations of Book II regarding thespecific aspects of our schemes. In this section, we discuss broader analyses.
Saito interprets the propositions of Book II as a relation between visible andinvisible figures. He writes: “The propositions II 1–10 are those concerning in-visible figures, and they must be proved by reducing invisible figures to visibleones, for one can apply to the latter the geometric intuition which is fundamentalin Greek geometric arguments” (Saito 2004, 167).Instead of the description of visible vs invisible, we prefer to address thisduality as represented vs not represented on a diagram. It seems a better choice,since two sides of parallelograms contained by straight-lines containing a rightangle are visible, that is, represented on a diagram. Nevertheless, we should giveSaito the credit for this general observation, especially as Euclid scholars usuallyuphold the dogma that “Greek mathematical proofs are about specific objects inspecific diagrams” (Netz 1999, 241). Since we identify the rules relating visibleand invisible figures, one may view our study as a development of Saito’s basicobservation.
Corry applies Saito’s distinction of visible vs invisible in his analysis of BookII. Accordingly, regarding proposition II.1, he formalizes its diorismos as thefollowing equation R ( A, BC ) = R ( A, BD ) + R ( A, DE ) + R ( A, EC ) , (Eq. 1)and the starting point of Euclid’s proof (the formula in red in our scheme) as R ( BG, BC ) = R ( BG, BD ) + R ( DK, DE ) + R ( EL, EC ) . (Eq. 2)Then, he points out that a relation between these equalities can be explainedin terms of visible and invisible figures. Hence, Corry writes: “what Saito drawsour attention to, in particular, is the fact that the rectangles used in (Eq. 2)are all ‘visible’ in the diagram, whereas those of (Eq. 1) are ‘invisible’. [...] Inother words, situations embodied in (Eq. 2) [...] involve visible figures and hencedo not require further justification other than what the figure itself shows. Thesituation embodied in (Eq. 1), in contrast, does require a proof precisely becausethe rectangles involved are, as indicated by Saito, invisible. In Book II, then,Euclid shows how the properties of invisible figures can be derived from thoseof visible ones” (Corry 2013, 650–651). In (B laszczyk, Petiurenko 2020) we identify a tendency in the
Elements to eliminatevisual aspects in order to achieve a generality founded on theoretical grounds alone.Thus, II.1 to II.4 exemplify a trend rather than atypical arguments. Unfortunately, instead of Euclid’s parallelograms contained by , Corry applies his ownterm, namely “R(CD, DH ) means the rectangle built on CD, DH”. It corresponds toour suggestion that Corry pays no attention to the renaming technique characterizedabove.henomenology of diagrams in Book II of the
Elements Regarding the crucial point, namely “how the properties of invisible figurescan be derived from those of visible ones”, Corry’s explanation is as follows:“The proof itself, on the other hand, is based on (i) taking a segment BG = A,(ii) constructing the parallelograms and proving on purely geometric grounds(using I.34) that DK = A = EL, and (iii) then realizing that, according to thediagram: R ( BG, BC ) = R ( BG, BD ) + R ( DK, DE ) + R ( EL, EC ) . (Eq. 2)So, what is the big difference between (Eq. 1) and (Eq. 2) and in what sensedoes the latter prove the former? Notice, in the first place, that proving DK =A = EL is fundamental since otherwise the three rectangles in the figure cannotbe concatenated into a single one in (Eq. 2). But what Saito draws our attentionto, in particular, is the fact that the rectangles used in (Eq. 2) are all ‘visible’in the diagram, whereas those of (Eq. 1) are ’invisible’” (Corry 2013, 650).Indeed, step (i) is the kataskeu¯e part of Euclid’s proof. As for step (ii), it isCorry’s argument rather than Euclid’s, since the text of the proposition is: “DK,that is to say BG, is equal A”. It means that Euclid does not justify the equalities DK = BG = A . Nevertheless, it is a favorable argument, if needed. Step (iii)is what we consider as visual evidence. However, steps (i)–(iii) do not provide acomplete account of Euclid’s proof. Corry does not explain how Euclid relates thevisible figure R ( BG, BC ) and the invisible R ( A, BC ). The simple observationthat, on the one hand, there are visible figures, on the other hand, invisible ones,does not tell us how Euclid turns equation Eq.1 into equation Eq.2. We believethat our substitution rules enable an adequate explanation.
In most of his review of Book II, Mueller argues against algebraic interpretation;see (Mueller 2006, 41–52, 301–302). In this subsection, we try to separate hisown interpretation from this polemic. In section §
3, we have shown that Muelleradopts a notation which revokes the distinction between visible and invisiblefigures. Let us recall his definitions: “ O (x,y) is used to designate a rectanglewith arbitrary straight lines equal to x and y as adjacent sides”, “I use T ( x ) tostand for the square on a straight line equal to x ” (Mueller 2006, 42, 45)Actually, there is no significant difference between O ( x, y ) and the algebraicterm xy . On the one hand, algebraic interpretation takes it for granted that x ( y + z ) = xy + xz , on the other hand, the rule O ( x, y + z ) ≃ O ( x, y ) + O ( x, z )is self-evident for Mueller. Moreover, like in algebraic interpretation, the term O ( x, y ) is applied to visible and invisible figures in the same way.For example, here is Mueller’s reading of II.1: “It should be clear, oncethe construction is described, II.1 becomes a geometrically trivial proposition”(Mueller 2006, 42). However, according to our scheme of II.1, only the first steprepresented by the formula in red is trivial. The rest of the proof is far fromobvious. As long as Mueller interprets the diorismos parts, his formalism works well.When he seeks to analyze Euclid’s proofs, it leads him astray. Here is his readingof the proof of II.4: T ( BC ) ≃ squareBG , T ( AC ) ≃ square HF , O ( AC, BC ) ≃ rectangle AG ≃ [ by I. rectangle GE , since T ( AB ) ≃ square AE , the theorem follows, that is T ( AB ) ≃ T ( AC ) + T ( CB ) + 2 O ( AC, CB ).The relations between visible and invisible figures, which we explain via sub-stitution rules, are covered by the congruence ≃ alone in Mueller’s interpretation.Thus, the line of arguments O ( AC, BC ) ≃ rectangle AG ≃ [ by I. rectangle GE, aims to interpret Euclid’s two different relations: AG is contained by AC, BC,and AG = GE . Moreover, ≃ is used to explain what we call the renaming offigures. Thus, Euclid’s “CGKB is the square on CB”, Mueller interprets alsoby the congruence: T ( BC ) ≃ squareBG . Since this congruence is supposed tobe transitive – Mueller does not explain why it is so, in the context of BookII – Euclid’s proof seems to go smoothly. However, it breaks as the conclusion T ( AB ) ≃ T ( AC ) + T ( CB ) + 2 O ( AC, CB ) comes out of nothing. Mueller skipsany reference to the relation AE ≃ T ( AC ) + T ( CB ) + 2 O ( AC, CB ). Why?Euclid’s argument is this:
HF, CK, AG, CE = ADEB,ADEB is AB −→ AB = AC , CB , AC × CB.
Since the term O ( x, y ) applies both to visible and invisible figures, withinMueller’s formalization, it would assume such a form:Since T ( AC ) + T ( BC ) + O ( AC, CB ) + O ( AC, CB ) ≃ square AE, T ( AB ) ≃ square AE, then T ( AB ) ≃ T ( AC ) + T ( CB ) + 2 O ( AC, CB ) . It would result in a vicious circle argument. Therefore, Mueller had to skipEuclid’s reference to visual evidence. As a result, he mischaracterized Euclid’sproof. See (Mueller 2006, 45–48).henomenology of diagrams in Book II of the
Elements Van der Waerden is a prominent advocate for the so-called geometric algebrainterpretation of Book II. Recent papers by Victor Bl˚asj¨o and Mikhail Katzrecount this fascinating debate between mathematicians and historians. Fromour perspective, however, it is too abstract, as it does not stick to source textsclosely enough. The analysis of van der Waerden’s arguments below certyfies ourclaim.Van der Waerden writes: “When one opens Book II of the
Elements , onefinds a sequence of propositions which are nothing but geometric formulationsof algebraic rules. So, e.g., II.1: [...] corresponds to the formula a ( b + c + ... ) = ab + ac + ... . II 2 and 3 are special cases of this proposition. II 4 corresponds tothe formula ( a + b ) = a + b + 2 ab . The proof can be read off immediately fromFig 34” (Waerden 1961, 118). Fig. 15.
Van der Waerden’s diagrams for II.1 and II.4.
Our Fig. 15 represents van der Waerden’s Figures 33 and 34. They aim toemulate Euclid’s diagrams accompanying propositions II.1 and II.4. Let us noticethat these diagrams differ from Euclid’s in regard to the names of line segments– it never happens in the
Elements diagrams that different individual lines havegot the same name. Moreover, there is no counterpart of line A in Figure 33.It looks like van der Waerden had to modify Euclid’s diagrams to develop hisinterpretation.Now, let us take the formula a ( b + c + d ) = ab + ac + ad designed to corre-spond to proposition II.1. Which part of the proposition does it formalize: the diorisomos , or the starting point of the proof (the formula in red in our scheme)?In fact, since van der Waerden’s diagram does not represent line A, his account See (Bl˚asj¨o 2016), (Katz 2020)8 Piotr B laszczyk of Euclid’s proof would look like this a ( b + c + d ) = ab + ac + ad −→ ... −→ a ( b + c + d ) = ab + ac + ad. Thus, there is no need for any proof at all.The same applies to his interpretation of proposition II.4. Instead of
HF, CK, AG, CE = ADEB −→ ... −→ AB = AC , CB , AC × CB, van der Waerden-style proof would look like this( a + b ) = a + b + 2 ab −→ ... −→ ( a + b ) = a + b + 2 ab. There is also no need for any proof. Indeed, regarding proposition II.4, hewrites “The proof can be read off immediately from Fig 34”. However, in the
Ele-ments , only the equality in red is read off the diagram, while the final conclusionrequires some arguments.In sum, whatever van der Waerden interprets, these are not Euclid’s propo-sitions.Finally, we find the following speculations: “We were not able to find anyinteresting geometrical problem that would give rise to theorems like II 1–4. Onthe other hand, we found that the explanation of these theorems as arising fromalgebra worked well” (Waerden 1975, 203). There is no need to dispute whetherthe distinction between figures represented and not represented on a diagram isa geometrical problem. Whatever it is, it provides an explanation for Euclid’spropositions II.1 to II.4. As regards strictly geometrical problems , II.4 is appliedin II.12 which is the ancient counterpart for the cosine rule for obtuse triangle.Below diagram illustrates the use of II.4Here is the respective scheme.
Diorismos CB = CA , AB , CA × AD.
Apodeixis −−→
II. DC = AC , AD , CA × AD −→ DC , DB = AC , AD , DB , CA × AD ∠ D = π/ −→ CD , DB = CB −→ AD , DB = AB −→ CB = CA , AB , CA × AD. henomenology of diagrams in Book II of the
Elements Fig. 16.
Euclid,
Elements , II.4, 12.
The reference to II.4 is easily identified by the way the line CD is cut at thepoint A: “since the straight-line CD has been cut, at random, at point A, theone on DC is thus equal to the squares on CA, AD, and twice the rectanglecontained by CA, AD”. Again, we will not dispute whether the cosine rule isan “interesting geometrical problem”.In section § (Baldwin, Mueller 2019) provides a series of arguments for autonomy of geom-etry. It includes historical, conceptual and model theoretical ones. As regardshistory, the paper develops a geometric interpretation of Book II as opposed tovan der Waerden’s ‘geometric algebraic’ interpretation, as they call it. Geometryin this context, implicitly, means for them a study of figures represented on thediagrams.Baldwin and Mueller place Book II within Euclid’s theory of equal figures:“On reflection, there is a natural geometric motivation for the main themesof Book II: Determine a precise method for determining which of two disjointrectilinear figures (polygons) has the greater area” (Baldwin, Mueller 2019, 8).In fact, only two propositions of Book II, namely II.5 and II.14, complete thetheory of equal figures as developed in Book I.Baldwin and Mueller continue: “Thus, Proposition II.2 certainly implies thatif a square is split into two non overlapping rectangles the sum of the areas of therectangles is the area of the square” (Baldwin, Mueller 2019, 8). However, whatthey refer to it is the starting point of II.2, not the conclusion. This starting The sin of the angle DAB, i.e., − cos of the angle BAC, is the fraction AD/BA .That is how we get the modern version of the cosine rule.0 Piotr B laszczyk point (the formula in red, in our scheme), as based on visual evidence needs noproof.Accordingly, they present II.5 as a dissection proof; see (Baldwin, Mueller2019, 9–10). Here is their proof schematized according to the rules we havealready applied to Euclid’s proofs.By construction, DB ∼ = BM , and CD ∼ = M F . The rest is as follows.
ADHK = ACLK, CDHL −→ ACLK ∼ = BF GD −→ CB ( is ) ( CBF E ) −→ CB = BF GD, CDHL, LHGELHGE = CD −→ . Indeed, Baldwin–Mueller’s proof is a series of observations rather than argu-ments. It also does not provide a final conclusion. To get it, one should applythe substitution rule, namely CB = ADHK, CD .In the above scheme, equalities in red interpret the phrase “is composed”,the formula LHGE = CD interprets the phrase “LHGE (which has the samearea as the square on CD)”. Thus, Baldwin and Mueller provide a styling onEuclidean proof rather than an interpretation of the actual Euclid’s proof.Historians often point out that algebraic interpretation ignores the role ofgnomons in Book II. Baldwin and Mueller managed to turn that objection intoa more specific argument, namely: “Much of Book II considers the relation ofthe areas of various rectangles, squares, and gnomons, depending where onecuts a line. While gnomons have a clear role in decomposing parallelograms,the algebraic representation for the area of gnomon, is not a tool in polynomialalgebra. That is, while such equations as ( a + b )( a − b ) = a − b or the formula forproduct of binomials are tools in algebra which have nice geometric explanation,the area of a gnomon has an algebraic expression, 2 ab + a , which does not recurin algebra (e.g., as a method of factorization)” (Baldwin, Mueller 2019, 9).Although Baldwin and Mueller emphasize the role of gnomons, in fact, intheir proof of II.5, Euclid’s gnomon NOP is simply a composition of two rectan-gles:
BFGD , CDHL . As a result, they do not provide a counterpart of Euclid’sdecisive argument, namely
N OP = AD × DB .What is, then, the role of the gnomon in II.5. Starting from the equality N OP = AD × DB , with προσκείθω , Euclid refers to CN2 to get N OP, LG = AD × DB, CD . What Baldwin and Mueller get by visual evidence, Euclid getsby deduction.How about Baldwin–Mueller congruence ACLK ∼ = BF GD ? By no means itis obvious, as we gave up an algebraic mode. Dissection also seems useless.Here is how Euclid gets the result AH = N OP in proposition II.5 (see Fig.11). henomenology of diagrams in Book II of the
Elements −−→ I. CH = HF −−−→ CN CH, DM = HF, DM −→ CM = DFAC = CB −→ CM = ALAL, CH = CM, HF −→ AH = N OP .
While Baldwin and Mueller did not manage to represent Euclid’s reliance ongnomons in II.5, contrary to Euclid, they apply gnomon in their proof of II.14.In II.14, let us remind, it is required to construct a square equal to a recti-linear figure A . Within the theory of equal figures, A is turned into a rectangle BCDE . And here is where our scheme starts. ... −−→
II. BE × EF, GE = GF GF = GH −→ BE × EF, GE = GH HE , GE = GH −→ BE × EF, GE = HE , GE −−−→ CN BE × EF = HE EF = ED −→ BD π BE × EF −→ BD = HE BD = A −→ A = EH . Fig. 17.
Euclid’s proof of II.14 (left), Baldwin-Meuller version (right).
And here is Baldwin–Meuller proof (see Fig. 17). ... −−→
II. IJN O, QJ = QP QP ∼ = QZ −→ QP = QZ −−→ I. QP = QJ , JZ −−−→ CN QP \ QJ = U V P JR = IJN O = JZ . The reference to the gnomon
U V P JR does not add anything to this proof– from the deductive perspective it is useless. Yet, Baldwin and Mueller createda diagram for II.14 in which every argument (every line in the scheme of theirproof) is represented by an individual figure. Yet, they did not manage to finda similar solution for this argument QP ∼ = QZ → QP = QZ . Now, let us compare Baldwin–Mueller’s and Euclid’s diagrams. On the onehand, there is a complex composition of rectangles and squares designed to rep-resent every textual argument (Baldwin and Mueller’s arguments, instead ofEuclid’s). On the other hand, there is a simple composition which represents,actually, abstract arguments. If we take Euclid’s diagram in its original form(see Fig. 1), we can still recognize how II.5 is applied, and the diagram becomesan ideogram. If there are no other reasons for the use of the terms rectangle con-tained by and square on , this diagrammatic representation of Euclid’s abstractarguments is the most solid one.
In this final section, we discuss one of Descartes’ diagrams included in his
Geome-try . By this contrast, we aim to underline the crucial steps in Euclid’s proceduresused in Book II.In the fragment cited below, Descartes illustrates one of his rules for solvingfourth degree equations. The details of his problem are irrelevant for our analysis,therefore they are skipped.Here is Descartes’ account of the diagram represented in Fig. 18:“For, putting a for BD or CD, and c for EF, and x for DF, we have CF = a − x ,and, as CF or a − x is to FE or c , so FD or x is to BF, which as a result gives cxa − x . Now, in the right-angle triangle BDF one side is x , another is a , then theirsquares, which are ( leures carr´es, qui sont ) xx + aa , are equal to the square ofbase, which is ccxxxx − ax + aa ” (Descartes 1637/2007, 388/191). On this diagram, Descartes applies the ancient technique of naming verticesand intersections of lines with capital letters. Nevertheless, when it comes toanalysis, he assigns new letters, specifically one letter to different lines: in this Translation after (Descartes 2007) slightly modified.henomenology of diagrams in Book II of the
Elements Fig. 18.
Descartes,
Geometry , p. 388. case, he puts a for BD and CD . We have already observed that Euclid appliesthe same trick in proposition II.1. Yet, in the following propositions, he does notexplore this idea. As for Descartes, it is his standard procedure. We believe thatis where the algebra begins: giving the same name to different objects. The nextstep in algebraic account of geometry is symbolic computation. As we all know,Descartes developed a technique of symbolic computation. It is the arithmeticof line segments. However, in regard to this initial step, it seems that Descartesdoes not appreciate it. That is how he explains his technique of naming lines:“Finally, so that we may be sure to remember the names of these lines, a separatelist should always be made as often as names are assigned or changed. Forexample, we may write, AB=1, that is AB is equal to 1; GH=a; BD = b, and soon. If, then, we wish to solve any problem, we first suppose the solution alreadyeffected, and give names to all the lines that seem needful for its construction,to those that are unknown as well as to those that are known. Then, making nodistinction between known and unknown lines, we must unravel the difficulty inany way that shows most naturally the relations between these lines, until wefind it possible to express a single quantity in two ways. This will constitute anequation, since the terms of one of these two expressions all together equal tothe terms of the other” (Descartes 2007, 6–9).The same letter, x , y or z , can name unknown , and unequal lines. Yet,Descartes does not mention that the same letter can name different, thoughknown and equal lines. It is his practice, not an explicit method. From his per-spective, the most important notion is that “a single quantity”, i.e. a line seg-ment, can get two names. In his system, these two names make an equation oftwo arithmetic terms. However, within the theory of polynomials developed in Another spectacular example is Descartes’ analysis of Euclid’s proposition III.36; see(Descartes 1637, 302). While showing how to construct roots of the second degreepolynomial, he assigns the name a to lines ON and NL. Yet, that assignment isimplicit.4 Piotr B laszczyk Book III, these names can be transformed into a polynomial equation f ( x ) = 0,and their original meaning, i.e. a reference to a specific line, evaporates.From our perspective, the crux of Descartes’ method consists in giving onename to different objects: that is the starting point of his equations. However,Euclid has a technique of asserting different names to the same object (renaming,in our terms). The transformations of these names allowed him to relate figureswhich are represented and not represented on diagrams. The latter only haveone name: it can be rectangle contained by or square on . Thus, if one looks foran algebraic prelude in Euclid’s Book II, it could be proposition II.1. Yet, thetechnique of assigning one name to different objects has not been developedfurther in the Elements . References
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Annales Univer-sitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia
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