sternfeld - zyskind - platos meno

8/12/2019 Sternfeld - Zyskind - Platos Meno

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Plato's Meno: 86E-87A: The Geometricul

Illustration of The Argument by Hypothesis'

ROBERT STERNFELD &H. ZYSKIND

he formal structure of Plato's conception and application of the

geometrical illustration to the question of virtue's teachabilityis clear. The objective is to state an hypothesis which will de-

termine whether or not agiven property is ascribable to agiven object;that is, whether the property, inscribable triangularly, is ascribable to

the oint object,a

givenarea and a

given circle,or whether the

property,teachable, is ascribable to the object, virtue. Yet the hypothesisadopted to provide the determinative criterion in the case of thegeometrical illustration is stated sosuccinctly that it has been subjectto diverse interpretations. We use W. H. S. Rouse's translation of87 A 3-6 to serve as a point of beginning for our interpretation."If the space is such that when you apply it to the given line of thecircle, it is deficient by a space of the same sizeas that which has beenapplied, one thing appears to follow, and if this beimpossible, another. "2

Further, R. S. Bluck3 hasgiven a selective review of the literature onthis question which we take as representative and from which thelinguistic issues are drawn as follows: Does in "its"given line(87 A 3) refer to ywptov (space) or to (circle), as Rouse pre-sumes and Butcher argues despite the necessity to twist the Greeka bit at this point? 4 Is Cook Wilson correct inholding that "the givenline" has to be a line of thecircle; for if not, then the line "wouldnotbring it [the space] into relation with the circle.?"5Or can "the given line"be considered as "aline or side of the figure into which, as its equivalentarea, it has been transformed," as A. S. L. Farquharson argues? 6Does ... olov with the addition ofiX't'6in the clause specifyingthe deficient area (87 A 5), state that the area is similar to(Butcher)


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or the same size as (Rouse) the applied area? And arelated questionreferring to the deficient area, is e:(1te:LVto be read in its strictlytechnical sense, as found in a passage of Proclus7 reporting Eudemus

on the Pythagoreans and in Euclid, or can this be read in a loosernon-technical senseas A. Heijboer suggests in his reading "to leavesufficient room? "8 Does &8voc.'t'ovette yfi (87 B) refer to the in-scribability of the area or merely to its applicability, which latterButcher argues - again apparently forcingthe language? Or, are thecriticisms of T. L. Heath9 and Cook Wilsoni correct in theirinsistingthat Butcher's analysis must begeneralized to provide "a real diorismos,or determination of the conditionsor limits of the possibility of a

solution of the problem whether in its original form or in the form towhich it is reduced?"" And doesthe Wilson-Heath solution (whichgeneralizes Butcher's suggestion) depend upongeometrical knowledgebeyond that available when the Meno was written, as Heath himselfsuggests, and is its reduction to aproblem involving two mean pro-portionals, solved by the use of conic sections, and only approximatedby mechanical devices in Plato's day, a satisfactory interpretationof what Plato intended ?12 Oragain, is there any significance in the use

of the masculine accusative participle (87 A) which in-volves a personal use of the accompanying xxcl


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apparently presenting no geometrical difficulties. Butcher thusinterpreted the question as "whether forany given rectangle, can it beinscribed triangularly within the circle?" though he admittedly failed

to give a method which woulddefinitely determine that a given areawas not inscribable if it did not meet his criterion as Platorequires.(We have not found in Butcher's article any flat rejection of the broad-er problem because itwas too simple such as Thompson attributed toButcher).

Our interpretation beginswith the notion that Plato was referringto an illustration readily understandable by any educated Greek andavailable to any ninth grade student today. Historians of Greek

mathematics agree that the geometrical contents of BooksI, II, andVI of Euclid's Elements were knownlong before the end of the fifthcentury B. C.16 And our construction is based upon this material,which must havebeen available to educated Greeks(including Meno)at the time Plato wrote the Meno.And we believe, in contrast toButcher, that the solution must be immediately available, but at thesame time, it must make sense of the detailed cluesPlato offers inpresenting this illustration.

Weidentify

"itsgiven

line"(87

A3)

as one side of theequilateraltriangle which could be inscribed in the circle - this sidebeing also

the baseline ofany area to be triangularly inscribed within the circle.The area is "applied" to this line in the formof a square. Euclid showshow any rectilinear figure can be redrawn as asquare (II, 14 - inour figure1 ?, A or A2to Si or Sz ) .The squared area can then be appliedto this line so as to determine the thirdproportional (VI, 11 - in ourfigure "h") which taken with the sideof the equilateral triangle willconstitute the two lengths containing the same area rectangularlyas originally given in rectilinear and then in square form - but nowstretched out along the given baseline (in our figure BCDE orBCIJ).The proportion here is simply as follows: the side of theequilateraltriangle is to the side of the square as the side of the square is tothe third proportional. Finally, since one wishes to inscribe thearea within the circlein a triangular shape, one must be ableto drawanother area at least the same within the remaining portion of thecircle (87 A). This means that arectangle congruent to the onestretch-ed out along the baseline (the side ofthe equilateral triangle inscribable


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within the circle) is drawn adjoining the first rectangle on the side ofthe first rectangle parallel to the baseline. Both theoriginal rectangleand the second oneadjoining it are constructed along the baseline on

the side of the baseline toward the center of the circle (in our figureDEFG or IJMN). The point of focus is now upon the line of thesecond rectangle which does not adjoin the first rectangle but whichis parallel to the line common to both rectangles and to the baseline(FG or MN). The question is now simply a matter of whether thatline (or extensions ofit) cut the circumference of the circle. There arethree possibilities: (i)the line iscompletely outside the circle. In thiscase, the area is not inscribable triangularly within the circle. (ii) The

line is tangent to the circle. In this case, the area is inscribable and isexactly the equilateral triangle which can be inscribed in the circle.(iii) The line falls within the circleand cuts the circumference in twopoints. In this case, the area is inscribable in two ways upon thebaseline - each point where the line or its extension cuts the circum-ference is the third point of a triangle with the given area inscribedwithin the circle (in our figure, BFC and BGC or BKC and BLC).

Put in algebraic terms, the height, h (the third proportional) of the

rectangle drawn by stretching out the original area along the base linemust itself be re-drawn orextended again "above" its original drawingand yet the duplicated figure remain in the circle. Then one hasdetermined that thetriangle is inscribable within thecircle and ofcourse that the height of the inscribable triangle is equivalent to 2 h,this triangle having the same area as the original area. If however,2 h is greater than the portion of the diameter perpendicular to and"above" the baseline, the area is not inscribable within the circle.

Arithmetically,since the

heightof

any equilateral triangleinscribable

within any circle can be expressed as a function of theradius, what isbeing stated simply is that if and only if h < 3r /4, the area isinscribablewithin the circle.

This interpretation appears to us to take account in a direct andeasy manner of all the clues Plato gives and to get the either/or resultsthe problem demands. Thus, auTOV(87 A 3) refers to xeptovas thebaseline ofthe area inscribed triangularly as well as to the side of theequilateral triangle inscribable within thecircle; thus, providing thelink to the circle as Wilson demands andbeing the line into which thefigure, equivalent in area, is transformed, as A. S. L. Farquharsonsuggests. Second, the applied area falls short by an areathe same as the applied area, ... fi . In context thiscould


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not mean only the same, for then only one area could be inscribable;nor could it mean the same orless, for this would makeno sense; ithas to mean at least the same.Accordingly, Heijboer's reading of

d7te?v as "leave sufficient room for" isequally acceptable. Thisreading is consistent with - virtually calls for - the clause'smeaning that the remaining portion of the diameter (possible heightof the triangle within the circle) is the same as or more than (or again,"leaves sufficient roomfor") another area of the samesize, similarlysituated. And ct


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