Symbolism in the Plato Scholia III Robert S. Brumbaugh

7/28/2019 Symbolism in the Plato Scholia III Robert S. Brumbaugh

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SYMBOLISM IN THE PLATO SCHOLIABy Robert S. Brumbaugh

III. A FINAL SUMMARY

ISome years ago, I became interested in the relevance of the geometricaldesigns used to schematize arguments and divisions in the Plato scholia tomore general questions in the history of logic. In particular, at the outset,I was puzzled by Arethas's 'architecture of argument' designs, and impressedby the Gorgiasscholion in which relations of four classes were plotted outsimultaneously. Had the Platonic tradition actually developed a geometricalsymbolic logic? What, if anything, did the 'syllogism' diagrams show aboutthe interaction of Aristotelian syllogistic and Platonic dialectic? And what didthe persistence and diffusion of these designs through space and time showabout the suggestiveness, viability, and vitality of the 'symbolism' itself?In the course of cataloguing microfilms of extant Plato manuscriptsfor thePlato Microfilm Project, ProfessorRulon Wells and I located and collected allthe scholia that involved mathematical or geometrical-logical designs. Forthe first time in a long while, ProfessorW. C. Greene's ScholiaPlatonicahadpaid some attention to these schemata in the oldest Plato manuscripts, repro-ducing them with line drawings. It seemed to us, however, that photographsmight bring out details significant for study of symbolism, not included inGreene's schematizations. Further, there was the question of what hadhappened in the later stages of Platonic copying and annotation.In two preceding articles, I have classified and listed locations and alllater occurrencesof the scholia included in Greene, plus several late or minoritems needed to complete the record.1The present article is concerned withcollection and classification of new uses of logical or mathematical symbolismin later scholia, ranging from the eleventh to the sixteenth centuries. By andlarge, these will be found to confirm the conclusions suggested in the twoearlier articles; and perhaps a summary of these conclusions is in order as aprologue to the present, concluding, set of figures.The chequered careers of these symbolic designs suggest four generalconclusions, as well as several subordinate ones. First, while there seems tohave been a latent idea of true 'symbolic logic'-a mathematical formalizationof argument and explanation-underlying this tradition, the idea failed toachieve clear realization. For this failure, there are both philosophical and

1 Robert S. Brumbaugh, 'Logical andMathematical Symbolism in the PlatoScholia', this Journal, XXIV, 1961, 45-58;'Logical and Mathematical Symbolism inthe Plato Scholia; II: A Thousand Years ofDiffusion and Redesign,' this Journal,XXVIII, 1965, 1-13.As before, I want to thank the BollingenFoundation and the Yale Library PlatoMicrofilm Project for assistance in the workthat has made the collection of symbolic

scholia possible. My colleague, ProfessorRulon Wells, has worked through thismaterial with me, and offered many excellentsuggestions. Mr. Frederick Ludwig, of theYale Photographic Service, has, as for Part I,lent his technical skill to preparing legibleprints enlarged from our microfilm. I alsowant to thank my son, Robert C. Brum-baugh, for his assistance in the task oflocating, printing, classifying, and indexingthe symbolic scholia.I


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1

i143D.of types ofVatican, cod.gr. 270, fol. 135r.4] (P. 7)

b-Meno 82C. Florence, cod. Laur. Plut.85.6, fol. 41v. [SN 13] (p. 8)c-Meno 82B. Rome, cod. Angel. gr. o107,fol. 9v. [SN 13] (p. 8)

C

d

d- Timaeus 53C. (Synthesis of themolecular equilateral triangles).Tfibingen, cod. Tuib. gr. Mb 14fol. 302. [SN I9] (p. 8)

e-Meno 82B. Vienna,B.S.B., cod. phil. gr. 39fol. 4ov. [SN 13] (p. 8)

Theaetetus 147. Vienna, cod. phil. gr.fol. 68v. [SN 2] (p. 8)Bessarion's scholion;Pl.3b ff.) Venice, cod. Marc. gr. 186,338v. (p. 8f.).

f- Timaeus Locris scholion, illustrating Timaeus 53C.Florence, cod. Laur. gr. 103, fol. I168r. [SN 9ig(p.8)


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a- Timaeus 35A. Double dia-pason ratios (8 to 36). Tfibin-gen, cod. Tfib. gr. Mb 14,fol. 274 (P. 9)

b- Timaeus 35A. (Double dia-pason ratios). Munich, cod.Mon. gr. 237, fol. 216r.[SN 17] (p. 9)

c-Astronomical/astrologicalsymbolism. Wheel for calcu-lating date of Easter. Endpapers, Paris, B.N., MS. grec18io [SN 22] (p. Io)


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3

Republic 5 iA. (Arbelus redesign of the 'divided line'.) Munich, cod. Mon. gr. 490, fol. 326(b)r. (p. io)

db-Left half of bottom marginalscholia to Republic 5IIA (dividedline redesigned as tree) and Republic5o4A (wheel design: soul at centre,senses around perimeter. Com-plete texts in footnotes). Munich,

"o cod. Mon. gr. 490, fol. 326(b)r.(p.o)c-Laws 7ooB. Bessarion's scholion.A(Classification of kinds of song).Venice, cod. Marc. gr. 188, fol.30r. (p. io)d-Laws 888E. Bessarion's scholion.(Triple classification of ta pragmata,as in Plato's text). Venice, cod.c Marc. gr. 188, fol. ioov. (p. Io)

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:-*~.~i''''i.:.:-: .'''' iiii : i : i iiiiili'i~-_l~?--ii''ii~iiiiiiiiiii~ iii:-- : -_i:

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95v.-(p.-IO)~ * ',c06

f, g-Astrological/astronomical sym-bolism. Two sequences of charactersfrom end-paper astrological scholionin Milan, cod. Ambros. D 56. (p. Ii)


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2 ROBERT S. BRUMBAUGHtechnical reasons. But it is worth noting that the emergence of formalizedgeometric logic did not fail by much: some features of diagram design werenoted in the first part of this article which, with proper stabilization as con-ventions and new combination, could have been significant components ofsuch a logic.Second, the sub-set of 'syllogism' and 'transitive class-inclusion' figuresreflected an attempt of ancient scholars to show that Plato's dialectic followedthe rules and patternsof Aristotle'ssyllogisticlogic. (It is hard to say how earlythis first was reflected in sign-design, but already by the second century A.D.the Plato scholars seem to have agreed in expecting and imposing syllogisticpatterns on the dialogues of Plato.) The mixture was, as we can see moreclearly in retrospect, a compromise both in idea and technique, and thecompromise was one in which the components neutralized each other. Thiscan be seen even on the technical level of the notation itself. While thepatterns of arbelus, triangle, and arc give a Platonic emphasis to the abstractpatternsof class-relations,the abbreviated quantifiers(oud,pn, t) add a naturallanguage element essential to specifying whatthe precise relations in a set are.Further, the terms are written out as words, centring attention on themeaning and content relations of each particularargument, and thus divertingattention from common form. Further development of geometrical designwas, it seems, blocked by the natural language elements, while rejectionof thegeometric designsas adventitious was equally blocked by a felt significancethefigures had.2A third conclusion, evident on inspection of the list of schemata, is theubiquity of trees. The use of the same design for family-trees and genus-species classificationsshows, I think, that the metaphor of 'genus' as 'family'had been a living one. But this time bothAristotelian and Platonic logiccombined to frustrate development in the most fruitful direction. Trees are'in' again in contemporaryscience and logic. But they are 'in' because theirbranching structure is so powerful for representing alternativeselated o time.(For example, where moves in a game are representedas successivebranchingpossibilities; or where consequents in modal logic are related by branchedpatterns to possible antecedents, or axioms and rules to possible consistentconsequents.) Somehow the Platonic tradition's 'genealogical trees of form'are trees ofknowledge, but not trees of life. Particularlyaftertheir specificationto genera and species by Porphyry, the pattern became atemporal and

2 I suggested, in the second article, thatsyllogistic and dialectic are related ratherlike (I) the logic of types and usage of con-temporary linguistic analysis and (2) thelogic of contemporary mathematical logic,respectively. I think this is essentiallycorrect; and the comparison suggests twofurther comments. First, it seems likely thatthe simplification and streamlining ofAristotle's highly complex Organon to the(eminently unsatisfactory) 'Aristotelian' logicof 19th century textbooks was the result of asimilar Platonic-Aristotelian interaction.

Second, it is worth noting that very fewlogicians have recognized how great thedifference in level of abstraction is betweena modern formal and an original Aristotelianlogic. Where a matrix of two rows canrepresent the relevant possible types ofproposition of modern propositional calculus,a matrix of 512 rows of I and o would berequired to represent the many differenttypes of proposition Aristotle's Organontakesinto account in the De Interpretatione nd theAnalytics.


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PLATO SCHOLIA 3inflexible, so that any genealogical notion weakened to a fruitless taxonomicone.A further conclusion evident from inspection of the space-time career ofthis collection is that there is some kind of magic in symbolic design, evenwhere the symbolism is not adequate. This seems to me the most likelyexplanation of the extraordinarypersistenceof the traditional trees, arcs, andarbeli. For the student of symbolic form, if my explanation of this persistenceis correct, this is probably the most interesting result of the present study.(However, this suggestion must be qualified by noting another feature of thecollection. This is an isolatedradition of symbolism, as shown by the very lowfrequency of occurrence in the total set of scholia studies of symbolic elementsfrom other logical, astronomical, or magical sources.)A subordinate,but interesting,point explored in the firstpart of this articlewas the modification by Arethas of purely geometrical symbolism to archi-tectural re-design. I can now add that this, with the notions of logic andsymbolism which it represented, finds no later imitators in the scholarlytradition.

IINew MathematicalndLogicalScholiaSN I. Phaedo7IE.First-figure syllogism, arbelus design, proving that the soul, sinceit returns cyclically, is immortal.-Tiibingen cod. gr. Mb 14.SN 2. Theaetetus47Two small geometrical figures, illustrating the 'roots and surds'classification.-Vienna phil. gr. 8o.SN 3. Sophist22A-23 C.Six new division diagrams (Bessarion's)representingthe six openingdefinitions of the Sophist by dichotomous divisions.-Venice Marc. gr. 186.SN 4. ParmenidesI43D ff.Complex arbelus, with conventional signs used as variables forterms, illustrating the generation of the number series.-Vatican cod. Barberinusgr. 270.SN 5. Symposium 87A.Syllogism triangle.-Vienna phil. gr. 2I; Venice Marc. gr. 590.SN 6. Phaedrus 44A ff.Two line-divisions, representing kinds of madness.-Vienna phil. gr. I56.SN 7. Phaedrus 45C.A pair of class-inclusion sorites arbeli schematizing the proof forthe immortality of the soul.-Florence Laur. c.s. gr. 78; variant, with order of terms from left

to right reversed, Vienna phil. gr. I56.


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4 ROBERT S. BRUMBAUGHSN 8. Alcibiades IIIa.A two-stage tree, formed by adding geometrical elements to thescholion of T (Greene, p. 91). Occurs in codd:-Paris B.N. grec 18o8; same, 1809; Florence, Laur. pl. 59.1;same, 85.9; Venice Marc gr. 590; same, 186.SN 9. AlcibiadesII 5A.An arbelus syllogism diagram, with slight variant text from that ofGreene's scholion (Greene, p. 44).-Vat. gr. 1030; another variant, Venice Marc. gr. 186.SN Io. Protagoras 3oA ff.A new figure (presumably Bessarion's): a chi with side arcs, illus-trating Socrates'sargument identifying piety and justice.-Venice Marc. gr. I86.SN I1. Gorgias 56A, 498A, 502C.Variations on the scholia vetera.456A: The standardmatrix and arc scholion, but with three addedgeometrical elements.-Vat. gr. 1297.498A: A pair of arbeli representing syllogisms in the first figure, amodified and transformed version of the class-inclusion arcsof the traditional scholion (Greene, Gorgias 98A, schema 2).-Vat. gr. 933.502C: Arbeli: a variation on the SV version in which the argumentis extended to four terms.-Florence Laur. gr. Plut. 85-9.SN 12. Gorgias 5oA.An arbelus representationof a syllogism in the first figure provingthat medicine is an art that uses words.-Tiibingen gr. Mb I4; Munich gr. 514.SN 13. Meno82B ff.A distinctive geometrical figure, unlettered, to demonstrate thedouble area theorem.-Vienna sup. phil. gr. 39; Florence Laur. Plut. 85.6; RomeAngel. gr. o107.SN 14. Republic 04C.A wheel figure representingthe soul and its five peripheral senses.-Munich gr. 490.

SN 15. Republic 14A.Redesign of the 'divided line' as a complex tree and as two arbeli.-Munich gr. 490.SN I6. Same.The divided line (Bessarion'sversion, presumably) illustrating apeculiar compromise between the 'equal segments' and 'unequalsegments' directions for construction.-Venice Marc. gr. 186.SN 17. Timaeus 5A.Complex arbelus with intervals indicated, illustrating scale andconstruction of the world-soul.-Tiibingen gr. Mb 14; Munich gr. 237.


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PLATO SCHOLIA 5SN 18. Timaeus35A.Simpler arbelus, illustrating intervals of the tetrachord.-Tiibingen gr. Mb 14; Munich gr. 237.3SN 19. Timaeus53C ff.

A drawing to show the synthesis of the equilateral plane trianglesout of six elementary triangles.-Tiibingen gr. Mb 14. (Compareno. 85 in the list in Part II of theprevious articles.) A similar scholion to Tim. Locris n FlorenceLaur. gr. 103.SN 20. Lawspassim.Bessarion's seven new figureswith geometrical elements; divisions,trees and chirather than triangles or arbeli.-Venice Marc. gr. 188.SN 2I. Astronomical/astrologicalymbolism.A pair of astrological matrices, with, apparently, a mixture ofcipher and standard numerical keys.End papers of cod. Milan Ambros. gr. D56 sup. (See Notes on thePlates, p. I1; description of this is not based on a microfilm, buton my own notes from 1953 examination).SN 22. Astronomical/astrologicalymbolism.Pair of wheels designed for the calculation of the date of Easter.End papers of Paris B.N. MS gr. 18io.

IIIA further conclusion follows from the examination of this source-material,which I had occasion to emphasize in Part I. That is, that there is some senseof magic in the felt effectiveness of these symbolic designs, despite the fact thatthe symbolism is far from adequate to its ideal function. This, it seems, is theinference to be drawn from the persistence over seven centuries of the trees,chi figures, linking arcs, triangles, and arbeli. On reflection, one can seeseveral points of relevance between the philosophical ideas of this traditionand the persistenceof symbolic form in its annotations.In the first place, the Platonic tradition has always used a metaphor of'intellectual vision', or 'seeing things synoptically'. From the outset, therefore,a geometrical representationof formal relations matches this key metaphor in

epistemology. Second, ever since Republic ii suggested that ten years of puremathematical study would help rulers to solve more concrete problems, therehas been a drive towards 'formalization' in Platonism. By 'formalization', Imean the creation of some ideal, abstract calculus, algebra, language, orschematization which isolates very general patterns of form: those, forexample, common to valid argument, or to musical and astronomicalconcordsand periods, or to indirect proofs in the many specific subdivisions of mathe-matical science. The ideal of a 'universal algebra' or 'universal geometry'3 Professor Rulon Wells suggests that thefact that all linking arcs are functional in themusical-mathematical diagrams using arbelus

designs, and are still retained where they are

not functional in Arethas's sorites diagrams,may reflect an origin of the arbelus in mathe-matics, its transfer to syllogistic logic cominglater.


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6 ROBERT S. BRUMBAUGHwhich reflects this dialectical drive toward form remains operative andattractive from Plato to the twentieth century. A geometrical logic, even avery rudimentaryone, is a step towardsbringing out relationalpatternsin thisformal way. Third, the Platonic scientist has always held to a belief in theisomorphismof the parts of reality: in a sense, this is an analytic necessity ofhis position, since, in a convergent hierarchy, there will be some orms sharedby any sets of phenomena. However, the belief is much strongerthan that: itis a conviction that there is a transferabilityof quantitative relations betweensubject-matters,so that a single set of ratios ought to be equally illuminatingin ethics and astronomy, in music and cosmology. Now, this belief is bothreinforced and beautifully projected by the recurrence, in the geometricalscholia, of the same designs of pattern and relation in all the diverse contextsof the Platonic writings. The arbelus, transferringfrom mathematical ratiotheory to logical transitive class-inclusion, then specialized within this latterrole to the schematization of any first-figuresyllogisms, illustrates the point.So, too, do the identical taxonomic trees as they trace out the genera andgenealogies of things, thoughts, and words alike. In this transferability, thesymbolism has a positive and functional suggestiveness. Fourth, and perhapsas a resultant of the other three characteristics mentioned (though I mustconfess I do not see any logical necessity for such connection) Platonism hasalways had a preferencefor symmetryn its outlines and classifications: when-ever possible, for example, a classification will be set up as a dichotomousdivision, a hierarchy will be represented as an isosceles triangle, a sym-metrical triadic matrix will order types of soul and state. If symmetry andbeauty go together (as Plato, in the Philebus, eems to insist that they do), thensymmetrical geometric forms are exactly what one expects a priorito be the'right' patterns of classification and relational connection. Finally, thistradition has always (or almost always) avoided a 'univocal' vocabulary infavour of an analogical contextual definition of key terms. This means, ofcourse, that the relationsather than any univocal relataare the keys to 'mean-ing'. And the symbolic effectiveness of geometry is exactly the way that itmakes the relational patternsstand out against a logical space as background.All of these ideas are effectively projected by the symbolic scholia. (The'mathematical' scholia have similar symbolic overtones as well, as I tried toindicate in my Plato's Mathematicalmagination,Bloomington (Indiana) 1954,particularly Appendix B.) On the other hand, the dependence on naturallanguage for the logical peculiarities and concrete contours of classificationand argument, natural to Aristotelian logic, became fused with these motifs ofPlatonic dialectic, with the hybrid mutual dependence of verbal and geo-metrical elements that we have already seen embodied in the sign-designsofthese scholia.And so, with the present plates and notes on later designs, a first survey iscompleted of the geometrical logic that has left its record in the margins of themanuscripts of Plato. The survey has found no consistent innovations herethat compare with, say, the anticipation of modern propositional calculus inStoic logic. Even if that were all that one could say, it would still remain truethat any such body of primary source material as these scholia are for thehistory of logic is worth collection, classification, and study.


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PLATO SCHOLIA 7My own conclusion, however, is rather more general and more construc-tive. Ideas, such as the idea of a 'universal formal system', 'universalalgebra',or 'geometry of thought', do not appear in history fully definite and clear.Rather, they occur first as guiding ideals, often latent, flickering into overt

awarenessthen fading again. Sometimes, as with Plato's ideal of a 'systematicsolid geometry', fifteen years is enough for their mature realization. At othertimes, as with the great ideal of a symbolism adequate to express a generalform for all laws of nature and of thought, fifteen hundred years is not enoughfor precise technical implementation and fully conscious appreciation. Andthe idea of a 'symbolic universal logic' is of this latter kind: Eudoxus, Lull,Leibniz, Euler, and Boole are names that mark important moments in itspath. At last, in the twentieth century, the implementation and technicalprecision seemed to have been attained by Whitehead and Russell in PrincipiaMathematica.But the paradoxes discovered by Kurt Goedel, and by Russellhimself, show that there still remains some distance to go. Meanwhile, therehas been the continuing suggestiveness of the geometrical logic of the Platoscholia (transferred,as we have noted, to Aristotelian scholia in about 1500),transmitting and suggesting, if not refining, this latent ideal. The scholia thusplay their part in the history of ideas that constitutes a long-term writing of acrucial footnote to Plato.It is a history of vicissitudes and frustratinglapses. Arethas, for example,with his redesign of arbeli as architecture of argument figures,would, had hebeen followed, completely have changed the suggestive effectiveness of thewhole notation. The thirteenth century, when one might hope for an effectiveinteraction of these designs with other traditions of symbolism-astrologicalor alchemical-and the fourteenth, where the calculi of Lull could haveinteracted, show only two examples of such an interaction. The obviousnotion of using conventional diacritical marks (already on hand in the secondcentury A.D.) as variables in argument pattern was not hit upon until thesixteenth century, and then in a single isolated case. Conventions of shadingand different line-design that could have become standardized to produce atrue geometric logic appeared fitfully, but were never generally adopted. Thefourteenth century, for reasons I do not at present understand, saw a suddenabrupt erosion and decline in these designs. Even Cardinal Bessarion, whoappreciated geometrical patterns, invented thirteen new ones, and had a keeneye for places where they would illuminate an outline or a logical puzzle in thetext, was extremely conservativein the designshe chose for these innovations.But for all its lapses and missed opportunities, this is the historyof a greatidea, latent in the symbolismwe have seen. It is my conviction that this modestsymbolism preserveditself through space and time so durably because what itsymbolized was a philosophically significant intellectual vision.

NOTES ON THE PLATES4Ia. Parmenides43D. This scholion is unique in its use of arbitrary symbols asvariablesn the logicaldesign,withan attachedkeygivingthe particular aluesthey

4 In the present plates, unlike those ofPart I, no definite convention has beenfollowed in using black-on-white or white-on-black reproductions. We have in each casechosen the form which gave best legibility.


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8 ROBERT S. BRUMBAUGHrepresent as terms in the present construction of the number series. The distinction ofsingle and double arcs indicates two successive stages in the construction; first, eventimes even numbers are generated, then odd times odd; then as a second stage, oddtimes even. (How the primes are accounted for is a much discussed question. Probably,however, they are derived by addition as a final step to fill out the series. See my Platoon the One, New Haven 1962.) As early as Thrasyllus, various conventional signs wereused by editors to indicate characteristics of the text. And in the oldest manuscripts, wefind sets of such arbitrary symbols used as indexes to indicate where marginal glossesand insertions are to go. The notation was thus on hand for the present use of variablesin logical schemata; unfortunately, it was not appreciated earlier.The text of the key, above the figure, written with compendia, is: artia artiakis,perittaperittakis,artiaperittakis.ib, c, e. Meno 82B. A rather crude simplification of the 'final figure' design of thestandard scholia. It is interesting primarily because of its bearing on the adventures ofcod. Vienna F. Its awkward marginal location in F suggests that it is a later addition(from our photographs, hands or inks are impossible to determine here). The identityof the F design and Florence Laur. i suggests that the figure originated with i, then wascopied at some later date by someone comparing i with F. On the other hand, it maywell be that the version in u is the oldest; correction ofi from u is a definite possibility.This confirms the reciprocal comparison and correction of F and i: it is helpful inworking out the exact and complex detail of the sub-family of manuscripts derivingfrom Paris B plus other components. Schneider had already called attention to theagreement of u and F in the Republic;Dodds has studied the relations, for the text of theGorgias, n detail for corrections from F via a second hand in Paris B; the present figuresuggests that u or i may prove a source of later corrections in F.Id. Timaeus 53C. The unexpected use of six elementary triangles, joined katadiametron o form the isosceles triangular faces of the elements, led to the presentclarifying sketches, which also appear (in a later hand, presumably) in Paris A. Theconstruction is discussed, and the most plausible explanation put forward to date isgiven in F. M. Cornford, Plato's Cosmology.if. Same. A routine illustration of the combination of four isosceles triangular facesto form the tetrahedral 'molecule' of fire. Its occurrence along with the much morepuzzling plane synthesis of the equilateral triangle suggests that scholars had lost theirfamiliarity with solid geometry at this point in the Platonic tradition.Ig. Theaetetus147. A diagram to illustrate the achievement of Theaetetus andSocrates in their work with 'roots' and 'surds' (cf. Euclid xi, 9; my discussion, PMI,p. 38, fig. I7, and Taliaferro in New Scholasticism,I957, 261). Clearly, one of thesefigures is a squarenumber, with integral roots as its sides, while the other is a productofunequal actors that has an irrational square root. The numerals along the sides aretherefore probably meant to be nu (50) for the square and long side of the rectangle,and iota (I o) for the shorter side of the latter. (Iota, perhaps with a light top strokerepresenting a prime mark, to indicate a numeral, rather than gamma,for two reasons.First, gammawould involve a mixing of uncial and minuscule letters, which is not foundin other mathematical figures (see PMI). An exception is SN 22 above, the 'cipher'from cod. Ambros. r.) Second, with iota the rectangle is drawn approximately to scale,whereas a gammawould make it represent a rectangle with sides in ratio of 3:50 by afigure with sides whose measured lengths are about I:5.hy these are the numbers chosen is a question. Probably the reference to 'squareson diagonals of the pempad' in the 'Nuptial Number' (Rep. 546A) inspired the 50 x 50square. Perhaps Theodorus's separate proofs for the roots 'from 3 to 17' suggestedthree as the short side of the rectangle.

Ih. Protagoras330Aff. This scholion, from one of Bessarion's manuscripts, is particu-


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PLATO SCHOLIA 9larly interesting because it singles out one of the logically most peculiar arguments inthe dialogue. When Socrates and Protagoras agree (I) that 'justice itself is just, ifanything is', (2) that this also holds for piety, (3) that justice is pious, then (4) thatpiety is just, the stage is set for a refutation of Protagorasand for a standing logicalpuzzle for Western scholarship. Bessarion (the scholion seems to be his) seems tosuperimpose the three steps of argument with arcs and lines indicating inclusion ridentical xtension.

The text is: dikaiosyne diahosiotes hosiosThe problem raised by Socrates is: whether piety and justice are distinct parts ofvirtue, or rather identical. The former is Protagoras'sopinion.(I) 'justiceis surelyjust'-agreed (top line in figure)(2) 'and piety is pious'-agreed (bottom line)(3) 'justice,furthermore, s pious'--eventually admitted (left-rightdiagonal)(4) 'and piety is just'-eventually admitted (right-leftdiagonal)(5) 'so these arenotdistinctpartsof virtue'-(arcs at sidesof figure)Protagoras evades this by a series of digressiveresponses; but when discussionisresumed,he choosescourages against 'the rest' as a distinctivepart of 'virtue'.The self-predicabilityof the forms has been discussed at great length by recentscholars:Vlastos, Sellars,and Wedberg, for example. It is therefore nterestingto findBessarionfastening on this 'logically odd' inference, and trying to elucidate it withsimple geometry.Ii. Phaedrus45C. Extension of class-inclusionarcsand arbelusdesignto schematizePlato's argumentfor the immortality of the soul.The text runs:

psyche athanaton aphthartonaeikinetonautokinetonIn some versions, the asymmetricaltransitiveinclusion relation is to be read fromright to left, in othersfrom left to right. (This again representsa failureto standardizegeometrical conventions, because the use of terms from natural language makes itpossible to interpret without such standardization.) For the argument itself, seeHackforth'scomments in his translationof the Phaedrus.2 a, b. Timaeus35. Complex use of arbelus design to indicate ratios of the scaleused in the construction of the world-soul. (This use of the arbelus in ratio theoryoccurs in the older scholia in a Gorgias cholion, transposed [in a later hand] to theTimaeusn ParisA.) Interestingas showing the independence of the Tiibingen scholiafrom those of A, B, W, F, and Y, and the copying of Tiibingen (for these seem to be inthe original hand) by cod. Mon. gr. 237-The largerversionis evidently based on the TimaeusLocris since, (i) it extends thetotal compass of the scale from I to 36, instead of I to 32; and (2) it introducesthe leimma s the TimaeusLocrisdoes). An exact and detailed account of the 'scales'ofthe Timaeusand TimaeusLocris s given in A. E. Taylor's A Commentaryn Plato'sTimaeus.The numbers are:

32 24 18 i6 12 9 8

Ic 6 LOxcrtoC~v I~--


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1o ROBERT S. BRUMBAUGH2c. Astronomy/astrology.nd papersof ParisD. One of a pairof fixedwheels,usedto compute he date of Easter. One of the fewcasesof interactionbetweenourPlatomanuscriptsnd symbolismromanother radition. (Forthe text of ParisD,see Klibansky ndLabowsky, latoLatinusII.)3a. Republic11A. The 'divided ine'redesigneds a tree.This is a redesignwhichquite evidently ancies tselfan improvementn Plato'soriginal igure(noscholiast anmisstheconstructionirectionsor'a line divided ..into fourparts' n the Republic).n one respect, t is a new use of the treedesign, nwhichprogressive ranches unfrombeing t the top to purebecomingt the bottom.(Sofar,standarduses have beeneither or successiveenerationsrgenus-species-variety)subdivisions.)The presenttree looks enough like some of RaymondLull's trees(FrancesYates,'TheArtof RamonLull',thisJournal,XVII, I954,pls. 15, 16, 17a;'RamonLullandJohn ScotusErigena',hisJournal,XXIII, I960,pl. 4), to suggestadirect nfluence.3b. Same. 04A. 'Wheel' cholion.centre:psyche

border:aphe, kouon,psis, smosis,eusisThis, too, is a new design. The centralsoul, surrounded y its senses,suggestssomethingikethe traditional herubwith head andwings.3c, e, d. Laws7ooB,864B,888E. A sampleof the new scholiafrom CardinalBessarion'smanuscript.CardinalBessarion'sibrary,nowin the BibliotecaMarciana n Venice, ncludedanumberof Platomanuscripts.Theseare cod.Ven. Marc.gr.590 (Schanz'sM; a copyof ViennaY); I84 (Bekker's i, copiedby Rhosus orBessarion);86 (corrected ndannotatedby Bessarion);187 (copiedby Bessarion);188 (alsocopiedby him); and189 (Bekker's igma). Some notionof the renewedattention o thegeometric choliacanbe had froma comparativeist of both newand old scholiaof this type in thesemanuscripts. Numbersor the scholia eterare from our list in the previousarticle,PartII.)ViennaY: 9, 9a, 85, 64, 65-Venice590: 5, 6, 7, 9, 21, 30, 85.VeniceI86: 2, 3, 7, 8, 9, I0, 25, 30, 35, 85; plus6 newSophistigures.Venice187:48, 49a.Venice 184:25, 26, 27, 28, 29, 30.VeniceI88: 57, 57a,57b,58, 60o,61, 62, 63; plus7 new Laws igures.The most significantact for our presentpurposes,however, s the new scholia.In additionto the Protagorasigure llustratedabove, and a Rep.511 'divided ine'construction,hereare thirteenothersall of whichseemthe workof Bessarion.Thesmallchartabovesuggestshatmygeneral haracterizationf Bessarionsappreciativeof the 'geometricalogic'tradition an be madesharper.Althoughhistorically e heldto compatibilitybetweenPlato and Aristotle,he was not willing to superimpose'syllogism' iagrams nhisPlatonicexts. His owndesigns rethechiand tree. (Noticethatwe do not findschol.vet. 6 or 21 repeatedn Ven. I86.) Hissupplementso thetradition are of two types: in the Laws, where there are multiple and complex distinc-tions set out from time to time as 'topical outlines' he sets up tree and division schematain the margin. In two other cases, where the text is peculiarly puzzling, either forlogical or mathematical reasons (the Protagoraspassage, cited above, and the Republic'divided line') he tries to clarify them by geometrical schemata. In the Sophist,wherethe six initial inadequate definitions are set up in the text, he puts all six, in tree form,in his margins. From the structure of the Sophistone might well guess that he did thisboth because the text clearly described such classification figures and because he was


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PLATO SCHOLIA I1puzzled by their dialectical function. (In fact, this has not yet been satisfactorilyexplained.) 53f, g. Astrological/astronomical symbolism. Drawings, from my notes, of twosequences of digits that form part of the astrological material on the end papers of cod.Ambros D 56 sup. (The last two symbols in the first sequence are not given here; thesecond sequence is complete.) This scholion needs further study by someone expert inastrology; it clearly represents a second case (see 2c, p. 9) of an intersection of Platonicsymbolism with that of the astronomical-astrological tradition. The originatorseems to be using archaic numeral designs as a kind of cipher for his key series. (Forexample, the design of five in the first sequence comes later than its stabilization inprinted books; the seven next to it is in the style of the thirteenth century; the secondcharacter of the second sequence resembles a very early Arabic numeral four; whilethe sixth character of the first sequence is an alternative our of standard thirteenth-century design.) I have not identified the first, third, or sixth symbols in the secondsequence.

5 Mr. Frederick Oscanyan has suggested,in an unpublished paper, that the sixdivisions are sarcastically applying epithetsto individual Sophists whom Plato's readerscould recognize. In that case, the genus-genealogy resemblance of tree designs be-comes highly relevant background. Since

we do not usually, today, insult someone bycommenting on his own inferioroccupationalstatus nor that of his paternal ancestors(though 'son of a sea-cook' remains oneinstance of this type), we have missed thehumour that Plato intended to build into hislogic lesson.

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Symbolism in the Plato Scholia III Robert S. Brumbaugh