Apeiron Volume 46 Issue 4 2013 [Doi 10.1515%2Fapeiron-2013-0015] Ranger, Jean-Philipe -- Aristotle on Political Communities- Lessons From Outside the Politics

Elements of Euclid's DataChristian Marinus Taisbak

1. Do we know intuitively what it means to be given?2. Supplements, notations, and conventions

2a. Objects and ratios2b. Predicates and limitations

3. An example: theorem 41 of the Data4. Given in position5. Given in magnitude6. Parallels and angles7. Given in form8. Triangles9. Do we know the use of 'givens'?10. Conclusion11. Epilogue

1. Do we know intuitively what it means tobe given?

If no one had written Euclid's Data, I would never have missed it.Olaf Schmidt

Thus, the appropriate measure of the geometric researches conductedby Euclid and his contemporaries is to be sought not in the Elements,but in the Data

Wilbur Richard Knorr ((1986a), 102)

The confession by my teacher of Greek mathematics, Professor OlafSchmidt of Copenhagen, one of the connoisseurs of Euclid's Elementsof our century, conflicts conspicuously with the statement of WilburKnorr as Schmidt must realize (should Knorr be right) that he hasBrought to you by | St. Petersburg State University

Authenticated | 93.180.53.211Download Date | 12/21/13 1:34 PM

136 Christian Marinus Taisbak

not properly measured Euclid's achievement from his knowledge ofthe Elements alone. To do so, he would have to have read the Data, whichhe admits he has not done at any rate not so thoroughly as Knorr,who gives the following short description of the work ((1986a), 109):

The Data is a complement to the Elements, recast in a form more service-able for the analysis of problems. ... Each of its theorems demonstratesthat a stated term will be given on the assumption that certain otherterms are given. The subject matter overlaps that of the Elements,dealing with ratios and with configurations of lines and of plane fig-ures, both rectilinear and circular. Indeed, only in rare instances doesthe Data present a result without a parallel in the Elements.

I suppose that Knorr could bring this description into harmony withhis statement, quoted above, by giving a suitable definition of 'parallelresults' (see the end of section 3 below). But whereas Schmidt didn'tmiss the Data, in the sense of 'feeling a need for' it, Knorr probablymisses its main point. Common to the mathematician's view and thehistorian's is the belief that he knows what it means to TJC given'.Everyone knows that, of course, although anyone who has tried tofathom the 'common notions' of book I of the Elements ought to havesuspicions about intuitively intelligible words in Greek mathematics.

Many years ago, when first reading Ptolemy's Almagest, I wonderedabout the very strange use of the predicate 'is given', which is appliednot only to the input of a problem, but also to its output (Alm I.38, seefigure 1):

B

Figure 1

AB and AC are both given in magnitude, measured in those units ofwhich the diameter has 120; and let BC be joined. I say that BC is alsogiven. Brought to you by | St. Petersburg State University


Elements of Euclid's Data 137

This last assertion is translated by Manitius as '...dass auch... BC sichbestimmen lsst', probably because he was convinced that hardly anycontemporary mathematician would use the predicate 'is given' ofwhat has been found or proved to be true. But the Greeks did so, andthat is one unfamiliar fact which must be understood if Euclid's Data isto be properly assessed.

In this paper I will try to determine what the predicate 'is given'means in the Data; my investigation will be based on the followinghypothesis: that Euclid is trying to axiomatize its meaning. I cannotcover all of the 94 theorems in the Data,1 but I will point out some'elements' of the work, which were used for geometric analysis in theData and elsewhere.

2. Supplements, notations, and conventions

'By their fruits ye shall know them.' (Matthew 7.20) I am not suggestingthat Euclid (or whoever else wrote the Data) was one of the falseprophets, but now and again he proves assertions with axioms anddefinitions that nowhere are stated, whereas he fails to use an explicitdefinition (3) where it is needed. Theorem 31 is left in No Man's Land:if it is true, it is hard to see why; and if it is false, why was it everpronounced and preserved? One way to clear things up is to elicit thedefinitions and axioms from the results obtained: to know by the fruitswhat the grain was. I shall adopt this method in describing the Data,inserting my own supplements whenever ideas are used without beingdefined in the text. I am sure that I have not invented anything extra,but I may have overlooked some necessary definition. I believe thatEuclid would appreciate my approach, since most of the definitions inhis works were probably written after the fact, as it were, when he (orhis editor) felt a need for them.

1 The text of the Data edited by Heinrich Menge is printed with a Latin translation involume 6 of Euclid's Opera Omnia (Leipzig: Teubner). A translation of the Data withcommentary will be forthcoming in 1992, if I live. Should anything in this paper beless rigorous than it ought to be, I will have the opportunity to amend it then. I wouldappreciate any comments, private or public, that may contribute to a better under-standing of the Data. Brought to you by | St. Petersburg State University


138 Chnstian Mannus Taisbak

There is some variation in the terminology of the Data, probablyreflecting different historical strata. In order not to raise problems thatmay be irrelevant to the present investigation, I have standardized thevocabulary when quoting propositions. For example, I use one term'rectilinear figure' for chrion, euthugrammon, and eidos in theorems49-55. A theorem of the Data is identified by 'thm' followed by itsnumber; a definition is identified by 'def followed by its number, e.g.,thm 25, def 4. My own supplements are signalled by three digit num-bers and upper case letters. THM 125 and THM 225 would be mysupplements to thm 25 of the Data. AXM 103 is one of my axioms.Propositions from Euclid's Elements are identified as, e.g., VI.4, mean-ing the fourth proposition of the sixth book.

2a. Objects and ratiosThe Data deals with objects in the plane. Of course the theory can beextended to space, but that was not done by Euclid. The objects are thewell-known ones from Euclid's Elements. Their status is neither morenor less problematic in the Data than in the Elements. They are:

Points.

Lines (grammai), which comprise three kinds: infinite straightlines, line segments, (arcs of) circles, i.e., such lines as can beproduced by permissible constructions with ruler and com-pass. Normally Euclid does not distinguish between infinitestraight lines and line segments, but the context never leavesany doubt which he means. I allow myself to diverge fromEuclid and speak of straights, segments, and arcs.

Angles.

Polygons, mostly triangles and parallelograms. Sometimes Iwill refer to them as areas, though they should not be associ-ated with any number, rational or real, but only with planefigures; more often than not these figures cannot and will notbe measured in any sense of the word. This is a crucial featureof Greek geometry, on which one must insist.

(Segments of) circles. None will occur in this presentation.Lines, angles, polygons and (segments of) circles are called by thecommon name 'magnitudes' (megethe). I cannot now give any but anBrought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211

Download Date | 12/21/13 1:34 PM


ostensive definition of 'magnitude', but I shall have a little more to sayin section 5.

Besides these objects, the Data involves ratios, logoi between objectsof the same kind, most often line segments or polygons. The ratio of Ato B will be written 'A:B'. A ratio is not an object, but some kind ofconnection or relation between objects. The concept itself is veryvaguely defined in the Elements V, def.3: 'Logos is a sort of state orrelation (poia schesis) in respect of size between two magnitudes of thesame kind.'2 With many reservations one might compare Euclid'sconcept of ratio with the concept of field in modern physics. A ratio isalways there whenever two magnitudes of the same kind or twonumbers are present, but a ratio is not a magnitude, nor a number.Rather, one might say that a ratio is a pair of magnitudes or numbers.Two ratios (say A:B and C:D) can be 'the same', or one can be greaterthan the other, as defined in Elements V, defs. 5 and 7. Euclid neverspeaks of 'equal' ratios.

2b. Predicates and limitations

Any object and any ratio may serve as an argument for the predicate'is given'. To suppress unwarranted connotations of the term 'given' Ishall (sometimes) adopt a shorthand of putting the name of the givenobject in brackets:

[P] means T is given';if ABC is a triangle, then [ABC] means 'the triangle ABC isgiven';

[A:B] means 'the ratio of A to B is given'.Concerning a rectilinear figure in the plane, three questions suggestthemselves: Where is it? How great is it? What is its shape? The Datadefines three corresponding aspects of givenness: the figure may begiven 'in position', or given 'in magnitude', or given 'in form'. We mightsay that its position, its magnitude, or its form is given, but I prefer tofollow the Greek syntax, which uses the dative of respect (thesei, megethei,eidei), to emphasize that these are three different aspects of givenness.

2 On this definition see Euclid-Heath (1926), vol. 2,116.Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211


140 Chnstian Marinus Taisbak

Clearly dimensions are relevant here. A point can be given in posi-tion only; a straight can be given in position and in magnitude; apolygon can be given in position, in magnitude, and in form. Thebracket shorthand will be modified accordingly:

[AB]P means The segment AB is given in position'.[AB]m means 'The segment AB is given in magnitude'.[ABC]f means 'The triangle ABC is given in form'.

Whenever in the Data the predicate 'given' is applied to an objectwithout one of these modifications, it means 'given in magnitude'. Insuch cases I too leave out the index. Seeing that a point can only begiven in position, I shall always use the shorthand [A] rather than [A]pto mean 'the point A is given'.

3. An example: theorem 41 of the Data

In order to acquaint the reader with the way one speaks about givens,I present a typical theorem:

thm 41 If a triangle has one angle given, and the sides contain-ing the given angle have a given ratio to one another, thetriangle is given in form.

B C E

Figure 41

Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211


Elements of Euclid''s Data 141

Let the triangle ABC (figure 41) have one given angle ZBAC;and let the sides BA, AC containing the given angle have agiven ratio to each other. I say that the triangle ABC is given inform.

Let a segment DE be set out, given in position and in magni-tude. And let an angle ZEDF, equal to ZBAC, be constructedon DE and at the point D on it.

The angle ZBAC is given; therefore ZEDF is also given. Now,since at the straight DE given in position, and at the given pointD on it, a straight DF is drawn, making the given angle ZEDF,DF is given in position. And as the ratio AB:AC is given, letDE:DF be the same, and let EF be joined; thus the ratio DE:DFis given. But DE is given, and therefore DF is also given; butalso in position. The point D is given; therefore also the pointF. And both E and D are given; thus each of the lines DE, DF,and EF is given in position and magnitude; therefore the trian-gle DEF is given in form.

And since two triangles ABC and DEF have one angle equal toone angle, ZA = ZD, and the sides containing the anglesproportional, the triangle ABC is similar to DEF, which is givenin form. Therefore also ABC is given in form.

Although much in this theorem needs definition and explication, thetheorem itself is easy to follow and intuitively acceptable. One questionraises itself immediately, however: what does it mean to be given in form?We are supposed to be able to prove this predicate of the triangle whichis set before us, though not given. It is just some triangle with a fewattributes which are given: an angle (in whatever way such a thing canbe given to Euclid), and a ratio of two of its sides (whatever ratio maymean, and however it is possible to give it). Some things are given inposition, some in magnitude, others are just given.

What status does the triangle ABC have? Since we can prove that itis given in form, it must be given in form. But we do not know that fromthe outset. It is not given in position. For if it were, we would know eoipso that it is given in form, as we shall see. It is definitely not given inmagnitude. But it is known to be a triangle, with the properties of anytriangle. The sum of its angles equals two right angles; any two sidestaken together are greater than the third, etc. Thus, there is somegivenness masked in the 'setting ouf of a triangle.Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



Which tools, which auxiliaries do we have at our disposal to provethm 41? We have the Elements, more explicitly the first six books,dealing with plane geometry. In this specific case, we have used 1.23 toconstruct an angle equal to another, and VI. 12 to get two segments inthe same ratio as the given ones. Obviously, thm 41 presupposes a fewtruths, which (you would think) must be procured in the precedingtheorems. At the end of the proof we use Elements VI.6:

If two triangles have one angle equal to one angle and the sides aboutthe equal angles proportional, the triangles will be equiangular andwill have those angles equal which the corresponding sides subtend.

I suppose that something like theorem 41 was in Knorr's mind when hewrote of 'parallel results'. In Elements VI.6 two triangles TI and T2 havingone angle equal to one angle and the sides about the angle in the sameratio are compared. A triangleT3 is constructed equiangular with T] andcongruent with T2. Therefore, T, is equiangular with T2. In Data 41, onetriangle T, is set out, having one angle 'given' and the sides about theangle in a 'given' ratio. A triangle T2 is constructed 'in position' with thesame properties as Tlrand therefore equiangular with it. Since T2 is 'givenin position', it is also 'given in form', and therefore T] is 'given in form'.The Elements compare triangles. The Data deals with individuals, andwith the 'knowledge' we may have of such individual triangles withinthe language of givens. A superfluous discipline? Rather, a useful one ingeometric analysis, where individual objects prevail.

Why are objects said to be 'given' and not Tcnown'? I shall leave thatquestion till the end of this paper; but I am sure that the word waschosen deliberately at some time by some person as a new term toprevent wrong or unwarranted ideas about knowledge.

4. Given in position

In the first theorems of the Data it is not evident that to be given inposition is the most basic of the three kinds of givenness. But thoroughanalysis shows that it is so, and so it must be, since one important groupof objects, namely points, can only be given in position. Points arethought of as separate entities, each with its own identity; each point isdifferent from all other points. Now, the only question one can mean-ingfully ask of a point's identity is: Where is it? To describe thisindividuality, Euclid chose the phrase 'given in position', and defined:Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



def 4 'Given in position' is said of points and lines and angleswhich always occupy the same place.

Heath (Euclid-Heath (1926), vol. 1, 132) says that 'given in position'really needs no definition, and that we are not really the wiser afterreading def 4. But I think we may argue from opposites, and askourselves: If a point is not given (in position), what can it be? Moving,if I am not mistaken. It might change its place, metapiptein, exactly theterm introduced in thm 25 and known from presocratic philosophers,particularly Democritus,3 who use it to mean 'to change' (intransitive).Therefore, I supply two biconditional definitions:

DEF 101 A point or a straight or an arc is given in position ifand only if it does not move.

DEF 102 An angle is given in position if and only if its vertexand legs are given in position.

In the Elements, points are not moving, nor are lines, but elsewhere inGreek mathematics they accomplish useful movements. To mentionone contemporary of Euclid, Autolycus4 has moving points; and Ar-chimedes,5 when producing his helix, sends a point moving on amoving straight.

We cannot help thinking of fixed coordinates in some coordinatesystem when we are told that some point is given in position and alwaysoccupies the same place (relatively to all the others). But the time was notripe for Euclid to refer a point to two straights ('which', to quote Petersent(1866),4], 'generally have nothing to do with the problem considered'),if for no other reason, then because the distance of the point from thoselines could not always be measured by one and the same unit. After all,real numbers were some two millenia from being invented. So Euclidmeant just what he said by 'given in position'. If a point is explicitlyclaimed to be given in position, it must remain where it is, and keepdistinct from all other points. It may surprise a modern reader that so'little' information is of any consequence, but it is.

3 See Dieted 956), vol. 3,278 ff.4 On the Moving Sphere, proposition 1.

5 On Spirals, dei. 1. Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211


144 Christian Marimis Taisbak

How do we get started?AXM 103 Any point or straight in the plane may be taken forgiven in position.

Whenever we need an item it is possible to 'let it be given'. (See, forexample, thm 5). Apparently, we do not have to wait for the Giver toget the idea but may demand it ourselves. After all, the unprovedstatements of the Elements (and any mathematical edifice) are just'demanded' (postulata in Latin, axiomata or aitemata6 in Greek). Note thatthey are demanded, not taken.

It might be appropriate at this stage to introduce that effectiveconstructor, The Helping Hand, the well-known factotum in Greekgeometry, who sees that lines be drawn, points be taken, perpendicu-lars dropped, etc. The perfect imperative passive is its verbal mask. Noone who has read Euclid's Elements in Greek will have missed it. Neveris there any of the commands or exhortations so familiar from our ownclassrooms: 'Draw the median from vertex A', or 'Cut the circle by thatsecant', or 'Let us add those squares together'. The Helping Hand wasalways there to see that these things were done. I wonder how Europeever inherited Greek mathematics without the perfect imperative pas-sive.7

Thus, a point is given if it is taken for given, 'appointed' as it were,or if it is proved to be given by thm 25, i.e., if it is the point of intersectionof two lines that are given in position, either two straights, or a straightand an arc, or two arcs:

thm 25 If two lines that are given in position cut one another,their point of section is also given in position.

6 Another Pythagorean term?

7 Cf. Euclid-Heath (1926), vol. 1,242, note to line 8 of I.I.Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



Figure 25

I reproduce the proof in my notation (figure 25):objects AB, CD: lines (straights or arcs);

E: point;

hypotheses [AB]p/ [CD]p;E is point at which AB and CD intersect;

assertion [E]

argument if not [E], then E would be moving (DEF101);ergo either AB or CD would move; but they donot move (hyp).

What is proved in thm 25 is not that a point of section exists, and it isirrelevant that two arcs or an arc and a straight may have two points ofsection. What is proved is that if they do have a point of section, thepoint is not a moving point, but a standing point given in position. Theproof is a perfectly sound reductio ad absurdum, if DEF 101 is supplied.But the application of thm 25 to prove later propositions may causesome uneasiness.

Thm 25 presupposes that two lines are given. How can a line begiven? According to AXM 103, it can just be taken for given; but if not,thm 26 shows how, while at the same time opening new horizons:8

8 Even without the supplements, which I have added because they are used in someof the reasonings. The concept of direction is as absent from the Data as it is from theElements. Brought to you by | St. Petersburg State University



thm 26 If two points are given, the segment whoseendpoints are the two points is given in position and in mag-nitude;


class of areas that are equal in the sense of the Elements 1.35. In the Data,if a representative of such a class is given in position, any member ofthe class is said to be given in magnitude or to be a given magnitude.Let us define (or, perhaps, lay down as axioms):

DBF 104 A segment or an area or an angle is given in magnitudeif it is given in position.DEF105 A segment or an area or an angle is given in magnitudeif it is equal to one that is given in magnitude.

The latter definition is saved from circularity by the former. They aremodelled on def 1 in the Data:

def 1 'Given in magnitude' is said of areas and lines and anglesto which we can get equals.

Def 1 is always used as a combination of DEF 104 and 105. The Greekterm for 'gef is porisasthai, which means 'procure', 'gef, 'provide', andindicates any kind of purchase without specification of method. But itis evident from the use of def 1 that one cannot procure 'impossible'things, but only objects which can be constructed with ruler and com-pass. I take it that Euclid defines an equivalence class of geometricobjects to be given if and only if one of its representatives is given inposition. This turns out to be one of the most effective definitions in theData.

6. Parallels and angles

Three (non-collinear) given points entail three straights, three angles,and one triangle, all given in position. But also one given straight andone given point not on the straight can ensure another straight givenin position, namely the parallel through the point:

thm 28 If through a given point a straight be drawn parallel toone that is given in position, the straight drawn will be givenin position.




D

Figure 28

The proof is instructive (figure 28):objects A: point;

BC, DAE: straights;hypotheses DAE parallel to BC, [A];assertion [DAE]p.argument if not [DAE]p

then, with A standing, the straight DAE willmove and change its position to, say, ZAH. But,by hypothesis, ZAH isparallel to BC, which is also parallel to DAE. Therefore ZAH is parallel to DAE;but they do meet each other in A, which isabsurd. Therefore DAE does not move, and isgiven in position.

This proof makes free use of the theorems on parallels from the Ele-ments, in casu, 1.30, from which it is deduced that there is not more thanone parallel to a straight line through a point not on the line. Thm 28proves that this one parallel does not move: while being parallel it isgiven in position.

Theorems 31-38 deal with parallels. Their proofs depend on twotheorems about angles, 29 and 30:

thm 29 If, at a straight given in position and at a given point onit, a straight be drawn making a given angle, the straight drawnis given in position.

thm 30 If, to a straight given in position from a given point , a straight be drawn making a given angle with it, thestraight drawn is given in position.Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



Since the only difference between these theorems is the position of thegiven point, the proofs run parallel and are perfectly sound reductionesad absurdum: if the straight is not given in position, it will move whilekeeping its properties, thereby changing the angle; which immediatelyconflicts with 1.16.

We shall need a partial converse to thm 29 and 30 (in accordance withDEF 102):

THM 125 If two lines that are given in position meet each other,the angle which is contained by them is also given in positionand in magnitude.9

The next theorem is a puzzle, resting totally on thm 25, and neithermore nor less dubious than that one:

thm 31 If, from a given point, a segment given in magnitude bedrawn to meet10 a straight given in position, the segment isgiven in position.

B_

Figure 31

I give a very close paraphrase of the proof (figure 31). Let A be thegiven point, and D the point (on the given line BC) where the straightfrom A meets BC, AD being of given magnitude. Assertion: AD is givenin position. With A as center and AD as radius let the circleEDF have been drawn. The circle is given in position (def 6); and BC is

9 The ancient definitions of angle are discussed thoroughly in Euclid-Heath (1926), vol.1,176-181.1 know of nobody who has reviewed the problem since Heath.

10 This is a rare instance of the verb prosballein, meaning 'to draw towards so as to meetor reach'. There is no room for discussing whether the line is reached or not. It is.Brought to you by | St. Petersburg State University



given in position. Therefore their point of section D is given in position(thm 25). And since A is given in position, AD is given in position (thm26).

If we accept that this is a true mathematical theorem," a few obviousinterpretations are ruled out:

1) It is not a problem of constructing a segment given inmagnitude, from a point to a straight. The line is simply as-sumed to have been drawn, the method being irrelevant to theissue: that it is possible to have such a line drawn from A to BC.Why, then, does Euclid have the circle drawn with A as centerand AD as radius? Because analysis has shown that it is thelocus of points at the given distance from A, so D must be onit. He will need the circle to prove that D does not move, but isgiven in position since it is also on BC.Constructions in the Data are always secondary, i.e, auxiliaryto the proof, just commanded (via The Helping Hand). Theymust be ensured by some problem in the Elements, in casu,something like 1.12. The method of construction is and must beirrelevant for the existence of the object. Therefore a construc-tion in the Data may sometimes look superfluous or misplaced,as in the present theorem. We would need the circle EDF to beable to draw the segment AD. But then we should not need todraw it again, in order to use it as the locus. Some otherexamples of this kind of thing will be seen below.2) It is not a uniqueness theorem, since, once the segment isdrawn, there may exist another segment with the same prop-erties. Why didn't Euclid prove that the segment drawn iseither the perpendicular on BC, or one of two segments, sym-metrically situated on either side of the perpendicular? Becausehe knew that already.12 Euclid does not concern himself withuniqueness unless it has some specific significance.13 Unique-

What else can we do? The text might have been better transmitted. See Menge'sapparatus criticus at 52.20.

12 I shall not revive the discussion of 1.12. See Euclid-Heath (1926), vol. 1,271 ff.13 As it does in X.42 ft.: a binomi(n)al line (ek duo onomafon) can be split in only one way

into its names. Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



ness has been discussed ever since the first proposition of theElements. Should the equilateral triangle be above or below thegiven line segment?3) What, then, is thm 31 about? As far as I can see, it provesdeductively that if such a segment has been drawn, it does notmove. Can such an insight be of any use? Unfortunately, theData does not use thm 31 to prove any other theorems.

Thm 32-33 concerns the correlation between angles and transversallines in parallels:

thm 32 If a straight be drawn into parallels given in positionmaking given angles, the straight drawn is given in magnitude.thm 33 If a straight given in magnitude be drawn into parallelsgiven in position, it will make given angles.

The proofs are straightforward applications of the theory of parallelo-grams from book I of the Elements, combined with thm 25,26 and 29. Inthm 32, a segment is procured, given in position, parallel and equal to thestraight drawn, which is therefore given in magnitude. I shall quote thm33 in extenso with comments in curly brackets (figure 33).

A H E B

Figure 33 c ^- p D

L

Let the straight EF given in magnitude have been drawn into theparallels AB, CD, given in position. I say that it will produce given anglesBEF, EFD. (One of them will suffice, since the straights are parallel. Itmust be proved equal to one that is given in position (DEF104,105).}

On AB let a given point H have been taken, and through H let HG havebeen drawn parallel to EF. (If I were to draw it, I might mark off FG = EH,using 1.33. However, the Helping Hand can draw it, since it exists,without bothering about the method of drawing.} Ergo FE = HG (I.34);and EFis given ; therefore HG is given (defl). Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



And H is given; thus the circle with center H and radius HG will,when drawn, be given in position (def 6). Let it be drawn as KGL; thenKGL is given in position. And CD is given in position; thus G is givenin position (thm 25). (I would say, 'G is (a) point of section between CDand the circle with center H and radius HG. Therefore G is given inposition.' The ambiguity of thm 25 is not reflected in this theorem, sincethe straight EF is supposed to have been drawn beforehand, and thechoice between the two possibilities made.)

And H is given: thus HG is given in position (thm 26). CD is givenin position; ergo the angle HGD is given (DEF 102). And it is equal toZEFD, which is therefore given. And the complement ZFEB is alsogiven.

Theorems 34-38 are about parallels and the division of a transversalline through a given point:

thm 34 If a straight be drawn from a given point into parallelsgiven in position, it will be cut in a given ratio.

thm 35 If, from a given point to a straight given in position, astraight be drawn and cut in a given ratio, and if a straight bedrawn through the point of section parallel to the given one,the straight drawn will be given in position.

thm 36 If, from a given point to a straight given in position, astraight be drawn and a segment be added to it having a givenratio to it, and if a straight be drawn through the endpoint ofthe segment added parallel to the given, the straight drawn willbe given in position.

thm 37 If, into parallels given in position, a straight be drawnand cut in a given ratio, and through the point of section astraight be drawn parallel to the given ones, the straight drawnwill be given in position.

thm 38 If, into parallels given in position, a straight be drawnand a segment be added to it having a given ratio to it, and if astraight be drawn through the endpoint parallel to the givenones, the straight drawn will be given in position.




Figure 34

The five propositions may be summarized with small divergencesfrom the text as follows (see figure 34). 37 and 38 are substantially thesame as 36 and 35. In the latter is a given point, in the former is anarbitrary point on a line given in position. The four of them are partialconverse theorems to thm 34:

Common conditions for thm 34-38:

objects h, j, k: straights;t: transversal line;, , : points;

hypotheses [h]p;P on j and f, A on h and t, B on k and t.

Special conditions:

thm 3414hypotheses [P], k \ \ h, [k]p;assertion [PA:PB].

thm 35 (P external) and 36 (P internal)hypotheses [P], k II h, [PA:PB];assertion [k]p.

14 A counterpart to 34 with internal P (between A and B) is transmitted at the end ofthe proof, but relegated by Menge to the appendix. It may well be genuine.Brought to you by | St. Petersburg State University



thm 37 (analogue to 36) and 38 (analogue to 35)hypotheses lj]p, k \ I h, [PA:PBJ;assertion [k]p.

7. Given in form

The main theorem in the Data, that is, the most general assertion aboutgivenness in form and in magnitude is:

thm 55 If a rectilineal figure is given in form and in magnitude,its sides will also be given in magnitude.

Intuitively, most people know what it means to say that a plane figurehas a shape of its own, different from some and, perhaps, similar toothers: it has something to do with angles and proportion. So the Data'sdefinition of being 'given in form' is hardly surprising, though perhapsa little too sophisticated to be invented by Everyman:

def 3 A rectilineal figure is given in form if and only if15 each ofits angles is given and the ratios of its sides to each other aregiven.

The statement looks like, and in fact is, a theorem. Although Euclidapplies def 3 in many propositions, he does not use it where he needsa definition, in thm 39, the starting point of the theory. Instead he hasrecourse to what must be the more basic concept, being given inposition. I offer some additional definitions, elicited from thm 40 (seebelow), with which, by the way, def 3 can be proved:

DEF 106 A rectilineal figure is given in position if its verticesare given in position.

DEF 107 A rectilineal figure is given in form if it is given inposition.DEF 108 A rectilineal figure is given in form if it is similar toone which is given in form.16

15 The 'if and only if' clause is a defining relative clause in Greek. From the use of def3 it is evident that it must be interpreted as a biconditional.

16 Similar rectilineal figures are defined in VI, def 1. See Euclid-Heath (1926), vol. 2,188.Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



Figure 55

To understand thm 55, we may remark heuristically (see figure 55)that if a rectilineal figure P is given in form, its angles are severallygiven, and its sides have given ratios to each other (def 3). This suggestsintuitively that (in modern idiom) the length of one side (and thereforeof all sides) is a function of P's area. Or, as the Data has it in thm 55, ifP is given in magnitude, its sides will also be given in magnitude.

I now give an analysis for thm 55. P and Q are rectilineal figuresgiven in form (in accordance with def 3), and a and b are arbitrary sidesof P and Q respectively. It suffices to prove that one side is given inmagnitude, since, if one is given, they all are (def 3). Thus we want toprove that if [P]m then [a]m. Now, suppose the conclusion to be true: anarbitrary side a of P is given in magnitude. Let b be set out as a linesegment given in position and therefore also in magnitude. Then a:b isa given ratio (thm 1). Let Q be a rectilineal figure, constructed on theline segment b. If we see to it that Q is similar and similarly situated toP, with b and a homologous sides, Q will be given in form (def 3), andthe ratio P:Q will be 'double' (diplasin; we should say 'square') of a:b,by the corollary to VI.20. Therefore, since the 'double' of a given ratiois itself given (thm 8), we have [P:Q]. And then, since P was given inmagnitude, so is Q.

The analysis leaves us with an implication from one conjunction toanother:

[P]m & [a]m & [&L -> [a:b] & [P:Q] & [Q]m.In the proof of thm 55 this implication is partly reversed:

[PL & [&L - [QL & [P:Q1 & fob] & [a]m.Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211Download Date | 12/21/13 1:34 PM


1st step [b]m > [Q]m (the converse of 55, proved in 52).2nd step [P]m & [Q]m -> [P:Q] (thm 1).3rd step [P:Q] -> [a:b] (thm 54, using thm 8 and the

important thm 24, which will be proved below)4th step [a:b] -[fl]m (thm 2), and therefore (def 3) all the

sides of P are given in magnitude.

Here is a survey in reverse order of theorems 55-49 formulated inmy notation:

thm 55 [P]m -> [A]m;thm 54 [P:Q] -> [a:b];thm 53 [a:fc] for one pair a, b * [a:b] for any pair a, b;thm 52 [e]m -> [P]m (converse of 55);thm 51 [a:b] -[P:Q] (converse of 54);thm 50 [a:b] & P sim Q -[P:Q];17thm 49 a is the same as b [P:Q].

We shall need thm 1 and 2, but in this paper I shall not discuss howsuch statements can be proved. Evidently the answer is either trivial orvery subtle.

thm 1 The ratio of given magnitudes to each other isgiven.thm 2 If a given magnitude has a given ratio to another one,that magnitude will be given.

A theorem of transitivity for given ratios is also needed. We find it inthm 8, which is expressed in the well-known idiom of the 'commonnotions' of book I of the Elements: the word 'magnitude' is absent fromthe text, which merely uses the definite article in the plural: 'thosewhich have...':

thm 8 Magnitudes which have a given ratio to one and the samemagnitude, also have a given ratio to one another.

17 In thm 50, an auxiliary to 51, P is similar and similarly situated to Q; the sides a andb are, consequently, homologous sides (in the sense of VI.4); but P and Q are not givenin form: the theorem is about ratios between magnitudes (and might be put beforethm 25), not about given forms. Knorr could legitimately speak of a 'parallel resulfto the corollary to VI.20. Brought to you by | St. Petersburg State University



[a:b] & lc:b] -> [a:c]The proof of thm 8, which I shall not give, rests upon the 'di'isou'theorem, V.22, an important one for the theory of proportions.18

I shall also not discuss the first part of the Data since it does not dealwith positions. It ends with a solitary theorem proving, to put it inmodern terms, 'the uniqueness of the square root', though one wouldnot easily recognize this fact from the way it is stated:

thm 24 If three line segments are proportional,and the first has a given ratio to the third, it will also have agiven ratio to the second.

objects a, b, c: line segmentshypotheses a:b = b:c, [a:c]assertion [a:b]

To represent the proof of thm 24 I introduce the following notation,which follows the tradition initiated by Dijksterhuis (1929-1930), but ismore like that used by Herz-Fischler ((1987), XIV):

S(e) for 'the geometric square on the line segment ;R(d.f) for 'the rectangle contained by (cf. II, def 1) the segmentsdandf.

Let d be a line segment given in position.19 Get/such that d:f=a:c (VI.12).Then ld:b}, and, since Id}, [f\ (thm 2). Get e such that d:e = e:f(VI.13); thenR(d.f) = S(e) (VI.17). Since [d] and [/], [R(d.f)] (true, but unproved) andtherefore [S(e)]r and therefore [e] (true, but unproved). Since [d], [d:e].The argument continues:

ax = d:f,S(e):Rfo.c) = S(d):R(d.f) (VI.l, V.ll),S(a):S(b) = S(d): S(e) (since a:b = b:c and d:e = e:f)a:b = d:e (VI.22)).

Hence, since [d:e], [a:b].

18 In Euclid's theory of numbers, book VII of the Elements, that theorem is equivalentto the associative law for multiplication, as I show in chapter 8 of Taisbak (1971).

19 . Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



As can be seen, the theorem amounts to proving thatif S(a):S(b) = S(c):S(d) then a:b = c:d.

Normally, one is referred in this connection to VI.22, which is, however,defective as transmitted in the Elements. To prove the case for squares,and not for arbitrary rectilineal figures described upon a and b, asimpler proof can be given, based on V.9.20

The proof of thm 24 makes use of statements about squares andrectangles which have not been proved, namely,

If e and / are given, R(e.f) will be given,and its special case,

If e is given, S(e) will be given.In the Data, these truths are taken for granted (e.g., in thm 52, below).They can be proved using thm 27, thm 29,1.46, and (perhaps) a supple-mentary theorem stating that a rectilinear figure given in position isgiven in magnitude. Thm 39 amounts to such a theorem, but, surpris-ingly, nothing is said about magnitude in that proposition or in thetheorems that follow it.

I noted that in the 3rd step of thm 55 a special case of the followingproposition was used:

thm 54 If two rectilineal figures given in form have a given ratioto each other, their sides will also have a given ratio to eachother.

Euclid divides the proof into two cases in order to be able to use thecorollary to VI.20:

Case 1. The form P is similar to the form Q. Get the thirdproportional c to the similarly situated sides a and b. Then a:c= P:Q (VI.19 cor.),and P:Q is a given ratio. Then (thm 24) theratio a:b is also given. But since P and Q are given in form, eachside in P has a given ratio to each side in Q (thm 53, below, andthm 8).Case 2. The form P is not similar to the form Q. On one side bof Q a rectilineal figure R similar and similarly situated to P isconstructed. Then R is given in form, and so was Q. Therefore

20 See, for example, Taisbak (1982), 70-71. Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



(thm 49) Q:R is a given ratio, and P:Q is given. Therefore P:R isgiven (thm 8). The rest follows from case 1 of the theorem.

In fact, case 2 is not necessary for thm 55, and, as it stands, thm 54 is amuch more general proposition than needed.

Thm 53 shows that the sides compared need not be homologous:thm 53 If two rectilinear figures are given in form, and one sideof one figure has a given ratio to one side of the other, the othersides will also have a given ratio to the other sides.

The assertion follows immediately from def 3 and thm 8.thm 52 If on a segment given in magnitude a rectilinear figuregiven in form is described, the figure is given in magnitude.21

This theorem is proved by drawing the square on the given segmentand using thm 49 and the unproved special case of this theorem: if asegment is given in magnitude, its square (which is a given form) is alsogiven in magnitude. The converse of thm 54 is of interest, though notnecessary for thm 55, since the figures in 55 are not arbitrary but similar:

thm 51 If two segments have a given ratio to each other andarbitrary rectilineal figures given in form are described uponthem, the figures will have a given ratio to each other.

With thm 50 as an intermediary the statement is a generalization of thm49, in which the figures are drawn on the same segment:

thm 49 If on one and the same segment two arbitrary rectilinealfigures given in form are described, they will have a given ratioto each other.

49 is itself a generalization of thm 48, which deals with triangles. Thuswe have reached the 'lower' level which is the basis of the series ofassertions about polygons given in form, namely, assertions abouttriangles. They are made to work through:

thm 47 Rectilinear figures that are given in form can be dividedinto triangles that are given in form.

21 This is one of very few theorems where Euclid's figure has more than four sides. Healways uses just enough to make the reasoning general. In my diagrams for this paperI follow a modern practice, so that the reader is not distracted by doubts as to thegenerality of the theorems. Brought to you by | St. Petersburg State University



8. Triangles

The basic theorems concerning triangles given in form can be surveyedin the following diagram (g in a column means that the item above theline is given). Some of the theorems (40-44) are obvious, while two ofthem (45-46) are surprising and may be considered as applications ofthe obvious ones:

ZA ZB

g ggg(right)g

g

a:b b:c a+b : c b+c : a

gggg g

gg

thm

4041434442

4546

In all the theorems, except 42, one angle is given and something more.In 40 it is one of the other angles. It then follows immediately, from thesum of angles of a triangle, that the third angle is also given. Thus it isnot necessary to have the third angle given, though the statement ofthm 40 takes all three to be given. In 41 the ratio of the sides containingthe given angle is taken to be given, in 43 and 44 the ratio of the sidescontaining another angle, the latter being dubious (under the usualinterpretation). Finally in 45 and 46 the ratio of the sum of two sides tothe third side is taken as given.

It may be instructive to begin with thm 42, which does not involvea given angle; it is based on thm 39, which stands somewhat apart fromthe others, having sides with given magnitudes, not only given ratios:

thm 42 If a triangle's sides have given ratios to each other, thetriangle is given in form.

thm 39 If a triangle has each of its sides given in magnitude, thetriangle is given in form. Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



Note that thm 39 says nothing about the triangle being given in mag-nitude, although this is actually proved. Only the form is relevant forthm 42 and the following propositions.

In the proof of 39 a triangle ABC is set out with sides given inmagnitude. Another triangle is constructed (not by Euclid, nor you, norme, but by The Helping Hand) after the fashion of 1.22 (and with,mutatis mutandis, the same words) on a segment given in position andwith sides equal to the sides AB, BC, AC. The new triangle will havevertices given in position and is therefore given in form. And it is congruent with ABC, which is thereforegiven in form (DEF106,107,108).

In thm 42 a triangle ABC is set out, with sides having given ratios.Using 1.22 another triangle is constructed from segments having thegiven ratios on a segment given in magnitude. The new triangle will begiven in form (thm 39), since its sides are given in magnitude. And it isproved to be similar to ABC, which is therefore given in form.

For a long time while studying the Data I had the impression that, inorder to be constructed, triangles (and figures in general) had to havesome part given in position. Now I shall have to recant, since thistriangle is not in any sense given in position, but only in magnitude(which is irrelevant), and in form. But I am still pondering over themeaning of ekkeisth, 'let it be laid out', from the verb ektithemi, whichgives immediate associations to thesis, 'position'. If ekkeisth is inter-preted as 'let it be set out in position', then the reference to thm 39 willbe unnecessary, and that proposition will be superfluous in this con-text. The history of the two theorems needs thorough investigation. Ishall leave the question open, and go on to:

thm 40 If, in a triangle, each of its angles is given in magnitude,the triangle is given in form.

Some might want a diorism to ensure that the sum of angles be lessthan two right angles, but none is needed, since the object is assumedto be a triangle. The proof also uses the triangle inequality (1.20) as amatter of fact, as a 'masked given'. If def 3 were the basic definition ofbeing given in form, we might expect Euclid to prove that the ratios ofthe sides are given. But what happens? A line segment is taken to begiven in position and therefore in magnitude. On the segment a triangleis constructed with the given angles (1.23,1.32), and thus having verticesgiven in position (thm 25). Euclid then says,




Therefore each side is given in position and in magnitude.Therefore the triangle is given in form. And it is similar to the triangle (VI.4); which, therefore, is given in form.

I feel justified in adding the definitions DEF 106, 107, and 108, andmaintaining that def 3 should be a theorem.

Thm 41 was used as an example in section 3. In its proof anothertriangle is constructed with the given angle, and with the sides contain-ing it having the given ratio, and with vertices given in position, so that(DEF 106,107) the new triangle is given in form. It is proved (VI.6) tobe similar to the one that is set out, which, therefore, is given in form,by DEF 108.

Thm 44 says,thm 44 If a triangle has one angle given and the sides aboutanother angle have a given ratio to each other, the triangle isgiven in form.

Roughly speaking, a figure ought (?) to be given in form, if it is similarto figures having the same properties. On this assumption, one caneasily verify that thm 44 is equivocal. If the given angle is A, and thegiven ratio is ax, the theorem is true only if a>c, or if ZC is a right angle(i.e., if a:c - sin A). Otherwise, two very different triangles will bothhave the properties in question (figure 44). But perhaps there are ideasabout being given in form that I do not yet understand.

a c

Figure 44




In the Data, thm 44 is used only to prove thm 45, in which the two'different7 triangles happen to be congruent. The 'proof of thm 44 restson:

thm 43 If in a right-angled triangle the sides about one of theacute angles have a given ratio to each other, the triangle isgiven in form.

D

objects ABC: triangle;hypotheses ZA right angle,

la:b];assertion [ABC]f.construction DHE right-angled triangle,

ZH right,[DE]P/[DE:DH] = [a:bl

To prove thm 43 Euclid constructs (figure 43) a right-angled triangleDHE with hypotenuse given in position and with the ratio betweenhypotenuse and one of the legs given. Its vertices are proved to be givenin position, whence the triangle is given in form (DEF106,107). And itis similar to ABC (VI.17). Therefore the latter is also given in form.

The proof of this proposition includes another example of a 'super-fluous' construction (see the discussion of thm 31 in section 6). Afterfitting in the segment DH, Euclid goes on to draw the arc of circle KHG,Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



which is exactly the one he had to use to fit in DH, according to IV.l.But then, the fitting of DH is just commanded (being admitted fromIV.l), whereas the arc KHG is part of a new construction to establishthat H is not a moving point (thm 25).

Since the sine of ZC is a:b, some moderns will want to interpret thistheorem in trigonometric terms as indicating that the sine of an acuteangle determines the angle uniquely. I would prefer to call the wholetheory of triangles given in form Trigonosophy', Wisdom of the Trian-gle. At a time when Ptolemy's trigonometry was still far off, the use ofsuch wisdom was very limited. But wisdom it is.

For convenience I shall introduce some very useful auxiliary trian-gles, the lesser and the greater isosceles adjuncts. The lesser adjunct isknown from 1.20 (triangle inequality) and VI.3 (bisector of an angle ina triangle):

In a triangle ABC (figure 67) let the side BA be produced to D,AD = AC, and let CD be joined. Then CAD is a lesser isoscelesadjunct with respect to ZA. From the vertex B let BE be drawnparallel to AC, to meet DC produced in E. Then EBD is thegreater isosceles adjunct. The adjuncts are similar, and their baseangles are equal to half of ZA. (By interchanging the letters Band C, you get two others; while the greater adjuncts arecongruent, the lesser are so only if AB = AC).

As I mentioned above, the following two propositions are not elemen-tary. But they are applications of some of the elements of the Data, andEuclid obviously regarded them as belonging among the elementarytruths about triangles given in form. I therefore include them in mysurvey: Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



thm 45 If a triangle has one angle given, and the sum of thesides about the given angle has a given ratio to the third side,the triangle is given in form.thm 46 If a triangle has one angle given, and the sum of thesides about another angle has a given ratio to the third side, thetriangle is given in form.

Both of these propositions can be (and in the appendix to Menge'sedition are) proved by bringing in the lesser isosceles adjunct; then 45follows directly from 44,46 from 41. Since 44 is equivocal if applied tothis situation, 45 ought to be so, too; but the two triangles which bothmeet the conditions are similar (see figure 45). As this is the onlyinstance (that I know of) where thm 44 is used, the author may wellhave chosen to ignore its ambiguity.

Figure 45 B

In the proofs which Menge edited as original (because they come firstin the manuscripts) the given angle is halved, and VI.3 is applied,followed by some appropriate manipulations with ratios from book Vof the Elements. In the end the propositions follow from thm 44 and 41respectively; and the lesser adjunct has already done its work in VI.3.




9. Do we know the use of 'givens'?

The following theorem illustrates the tricks that can be performedwith isosceles adjuncts and one sort of proposition that is found amongthe 94 in the Data:

thm 67 If a triangle has a given angle, then an area which is thedifference between two squares, namely the square on the sumof the sides containing the given angle and the square on thethird side, will have a given ratio to the triangle.

Let the given angle be ZA. After constructing (figure 67) the lesser andthe greater isosceles adjunct, Euclid asserts (out of the blue, because henever mentioned the proposition in scholion 133) that since BC is anarbitrary transversal line (through one vertex of the isosceles triangleEBD),

S(BD) = R(DC.CE) + S(BC).But BD = BA + AD = BA + AC,so that S(BD) = S(BA + AC) = R(DC.CE) + S(BC).

Now Euclid asserts and proves that R(DC.CE), the difference betweenS(BA + AC) and S(BC),has a given ratio to the triangle ABC. SinceZBAC is given, the lesser isosceles adjunct is given in form (thm 40),whence

[DA:DC] (def3),and so [S(DA):S(DQ] (thm 50).Now, BA:AD = EC:CD (VI.2),

R(BA.AD):S(AD) = R(EC.CD):S(CD) (VI.l),R(BA.AD):R(EC.CD) = S(AD):S(CD) (enallax, V.16).

Since the ratio S(AD):S(CD) is given, so is R(BA.AD):R(EC.CD), and,since AD = AC, so is R(BA.AC):R(EC.CD). But R(BA.AC) has a givenratio to the triangle ABC (thm 66, see below); therefore also R(EC.CD)has a given ratio to the triangle ABC (thm 8).

Scholion 133, which (as far as I know) is met in no other source,obviously belongs to 'geometric algebra':

scholion 133 If, in an isosceles triangle, a straight is drawnarbitrarily (from the apex) to the base, the square on the seg-ment drawn plus the rectangle contained by the parts of thebase is equal to the square on one of the legs.Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



H

Figure 133D

The proof runs easily. Dropping the perpendicular from the apex andapplying II.5 on the parts of the base (figure 133), and finally addingthe square on the perpendicular, one has:

R(EC.DC) + S(CH) = S(HD)R(EC.DC) + S(CH) + S(BH) = S(HD) + S(BH)K(EC.DC) + S(BC) = S(BD)

Thm 67 depends on:

thm 66 If a triangle has a given angle, the rectangle containedby the sides that contain the given angle has a given ratio to thetriangle.

Let the given angle be A, and let BD be the perpendicular on AC. ThenR(BD.AC) is double the triangle; its ratio to R(BA.AC) is BD:BA, whichis given according to thm 40 and def 3.

Thm 67 is enigmatic in several ways. What is its information worth?What kind of analysis preceded it? In what mathematical context? Weobserve that the vocabulary (meizon dunantai) is of the sort that is foundprimarily in book X of the Elements. I feel sure, although I cannot proveit, that this proposition was invented in a context of geometric analysiswhere isosceles adjuncts were frequently used and scholion 133 was auseful auxiliary. But at the moment I fail to see the worth of theinformation that S(AC + AB) - S(BC) has a given ratio to the triangleABC. I can see a vague (?) connection with 11.12 and 13, but for me andyou the main difficulty in appreciating the statement is that we do notget what we are interested in, namely the value of the ratio, which isnot expressed or expressible in any Greek measure, but amounts (intrigonometric idiom) to 4 cot A/2. Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211


168 Christian Mannus Taisbak

This is the sort of frustration that affects us everywhere in the Data.We get very little information, hardly any 'knowledge' of the givens.And why not? Probably because 'knowing' geometrical objects wasproblematic at the time when the concept of given came into being andthe consequences of incommensurability were just being understood.There was next to nothing known about these objects, and very littlethat is worth knowing: length, size, distance, i.e., any of the rheta, theattributes that can be spoken of with numbers. We look for coordinatesto pin down positions, for lengths and areas to measure magnitudes.With forms we are at a loss. Moreover, we do not need them any more.Mathematics lost interest in them when the 'application of areas' wasrendered obsolete by the theory of equations, and perhaps alreadywhen trigonometry took over and codified the alliance between anglesand ratios.

The situation is not much better in the case of those propositionswhich ensure the existence of geometric objects. As an illustration, let uslook at two famous proposition from the Data, 58 and 85 (see figure 58):

B D

Figure 58

thm 58 If a given area be applied to a given segment deficientby a figure given in form, the latitudes of the deficiency aregiven.thm 85 If two straights enclose a given area in a given angle andif their sum is given, each of them will be given.




objects AC, CD: parallelograms;hypotheses [AC]m, [AD]m, [CD]f;assertion [BC]m, [BD]m.

At the beginning of the proof the parallelogram AC has been appliedto the segment AD in such a way that the deficiency CD is given in form.Then AD is bisected at E, and on ED a parallelogram EF is drawn,similar and similarly situated to CD, and therefore about the samediagonal; thus it is possible to draw the schema, the gnomonlike figureEDFGCK.22 The rest follows from adding and subtracting givens:

EF = gnomon EDFGCK + KG, given.AC = gnomon EDFGCK, given.EF = AC + KG, given.

KG is given, and similar to CD. Therefore the sides of KG are given (thm55), whence EB is given, and therefore BD given. Ergo BC is given.

Thm 85 follows immediately from thm 58 (figure 85):

B

Figure 85

Let AC be the given area enclosed by AB and BC, whose sum is given.Let AD be equal to that sum, and let CD be a parallelogram on BD. SinceCD is given in form (def 3), we have the conditions of 58: a parallelo-gram AC is applied to a given segment AD such that a given form CDis deficient. Therefore BD is given, and so BC.

22 Whenever the word schema occurs in the Elements and cognate texts, it means agnomonlike figure. Brought to you by | St. Petersburg State University



Evidently, the construction and proof of thm 58 presuppose ananalysis along the same lines as VI.28. How do they differ? VI.28constructs the parallelogram KG as the difference between EF and thegiven area by means of some propositions in book VI of the Elements.Thm 58 proves that KG is given in magnitude (being equal to the saiddifference) and in form (being similar to the given form). Therefore(thm 55) KC is given. No wonder that Olaf Schmidt never missed theData for this problem. Its contribution is definitely one of Knorr'sparallel results. And the analysis which must have preceded the bisect-ing of the given segment is absent from the Elements, as well as fromthe Data.

10. Conclusion

In conclusion, let me offer a thesis and an image. My thesis is this. Itwill take us nowhere to 'translate' the 'givens' into trigonometry or anykind of analytic geometry. Of course, thm 66 may be said to prove theratio:

2 triangles : R(b.c) = sin A : 1,or even the formula:

2 triangles = be sin A.

But this interpretation is of no consequence as long as numbers arerational only. Ptolemy allowed himself to express given chords withsexagesimals, 'aiming at a continually closer approximation in such amanner that the difference from the correct figure shall be inapprecia-ble and imperceptible',23 but he did so simply because they were givenand therefore ought to be determinable. But Euclid did not think thatway: his 'givens' live in a world of their own which we must masterwithout measure if we want to understand it.

One important feature in a given is its character of being thrust uponus not to be gotten rid of. In I.I we must use exactly the given piece ofstraight line to construct the triangle. We learn a Greek attitude to giftsfrom this, I am sure. And we are facing a piece of Platonic ontology:

23 Almagest 1.10, translated by Thomas (1939), vol. 2,415.Brought to you by | St. Petersburg State UniversityAuthenticated | 93.180.53.211



what we construct is what already exists. We can do nothing by way ofgenerating objects that are not already there. They are given. By whom?I do not know if Euclid answered that question, but he did set himselfto explain axiomatically what it means to be given. The difficulties hereare not new. In his commentary on the Data my namesake Marines,director of Plato's Academy starting in 485 C.E., was no less bewilderedthan I am about the meaning of being given. He pretends at the endthat the theory of givens is very useful, but I am afraid that some willstill not need it if they know the Elements.

My image is this. The Data makes geometry into a play enacted on astage we may call The Geometrical Plane. When the curtain rises to eachact (theorem), some actors (geometrical objects) are on the scene, a fewof which are presented to us in a certain fashion: they are said to begiven. As the play goes on, it reveals that there are more given actorson the scene than we believed at first sight, which revelation turns outto be the point of the play. The important feature of this play is thatthere are more things given than meet the eye. Everything is latent onthe scene when the curtain rises, but only a few objects and attributesare noticed by the audience. As the play runs according to the rules ofthe author, more and more facts are illuminated and emerge, until atlast the scene is full of real entities, while several phantoms (duplicatedcubes, squared circles, trisected angles) have been exiled.

11. Epilogue

It is very much in line with the Data to end abruptly, leaving manypoints unresolved. Consider common notion 2 of the Elements: if equalsbe added to equals, the sums are equals. This is not primarily a state-ment about magnitudes involved in an addition, but about the relationof equality combined with the operation of addition. I repeat: not themagnitudes, but the relation. Compare now Data 3: if given magnitudesbe added together, their sum is given. This is not primarily a statementabout magnitudes involved in an addition, but about the property ofbeing given. That property in its various manifestations is what Euclidis trying to elucidate. Such an investigation can always take into ac-count one more example. There is, then, no natural end to the Data.



Documents

Apeiron Volume 46 Issue 4 2013 [Doi 10.1515%2Fapeiron-2013-0015] Ranger, Jean-Philipe -- Aristotle on Political Communities- Lessons From Outside the Politics