The Calculus of Variants

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Oxford University PressLondon Edinbu gh Glasgow Copenhagen

New Yo k Toronto Melbourne Cape ToumB om ba y C al cu tt a M ad a s S h4 ng h4 i

Humphrey Milford Publisher the UNIVERSITY

TCALCULUS OF V RI NTS

t Essay o extual Criticism

y GREG

OXFORDT THE L RENDQN PRESS

9 7


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It is not going too far to saythat the announcementt ha t p hy si ci st s woul d h ave in future to study thetheory oftensors created a veritable panic among themwhen the verification of Einstein s predictions wasfirst announced A N. WHITEHEAD

PREFACETHE s ub je ct c on si de re d in the following p a g e s ~under the rather pretentious title of the Calculus of

Variants, has been the central problem of textualcriticism at any rate since the establishlnent of theg eneal og ica l m eth od . I am not here concerned toinquire whether that problem is completely soluble,t ho ug h I have been unable to avoid the questiona l t o g e t h e r ~ bu t only to suggest the use of morerigorous and in the e nd si mp le r m et ho ds of ap-proach. A considerablegain in ease and certainty can,I believe, be attained by a p arti al s ub st it ut io n offormal rules for the continuous application of reason;and I have been driven to seek it because in prac-tice I always myself feel considerable uncertainty asto what can and what cannot be legitimately inferredfrom a p ar ti cu la r set of variants, and observationl ea ds m e t o doubt whe th er t hi s i s a p ec ul ia r fai li ngof my own.Th e whole matter is, of course, at b ot to m o ne offormal logic, and the necessary foundations are fullyset forth by Russell and Whitehead in those sectionsof rncip Mathe J latica which d ea l w ith the an-cestral relation see Pt 11, Sect. E, 9 O 9 7 ~ inVol. i; a ls o I nt ro d. sect. VI I and Appx B in thesecond edition . No d ou bt , m os t of w hat is sig-nificant in the present essay could be e xp re ss ed i ntheir symbolism by anyone sufficiently trained toits use. This, however, I am not; nor do Iknow whether full symbolic treatment of m y a rgu -ment would result in any practical convenience.Perhaps it would no t be possible to sa y till theexperiment had b ee n t rie d. M ean wh il e, I am acutelyconscious that, compar ed with w ha t the method


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might ach ieve in abler hands the present attemptis as b rb r cel rent to the modern logic of Peanoand WittgensteinI wish at t he out se t t o make it clear that there isnothing esoteric or mysterious about my so calledCalculus: i t aims at nothing bu t defining and makingprecise for formal use the logical rules which textualcritics have always applied. It is quite incapable ofproducing any results that could not have be enattained by the traditional methods; only it aims atachieving them with less labour and greater certainty.Perhaps its chief merit if it has any at all will befound in the endeavour to give precision to termsand modes of inference which are frequently employed with quite astonishing looseness. Theworking of it out has done so much to clear my ownmind on the subject that I cannot bu t hope that itsstudy may be of some assistance to others.The Calculus was not constructed v cuo out ofmere superfluity of naughtiness bu t grew out of anat tempt to determine the relation of the manuscriptsof t he Chester Plays and the present essay beganas a secti on of an introduction to the pageant ofAntichrist in that cycle. It soon however becamedisproportionate and now appears in separateform. I hope before long to publish my editionof the play in which the method here described willfind specific application.It may be well to add that I am aware that aboutthe middle of the eighteenth century Lagrange andEuler evolved a branch of mathematics known asthe Calculus of Variations. I t does not touch theproblem discussed in the following pages.Miss St Clare Byrne has very kindly read thproofs for me.

, i

,i. .

VI

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ref ce

CONTENTSGeneral Notions: Descent and VariationRecording VariantsTypes of Variants .Principles of VariationResolution of VariantsMethod and Limitations of the Calculus

Note OnCollateral GroupsNote B On Conflat ion .Note C On Some Common ErrorsDiagrams of Typical FamiliesIndex

I141821

343

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THEC LCULUS OF V RI NTS

General Notions escent n VariationI we exclude th e possibility of memorial trans-mission, all manuscripts of a given work ar e derived(by transcription) from a single original.Th e whole collection formed by th e original

together with all i ts descendants, in th e particularrelat ion in which t he y s ta nd to one another, con-sti tutes a family,3 which, l ike other families, ha s agenealogica l t ree. Such a tree is t he s um of all th elines of descent of th e various manuscripts; a line0 es ent being a series whose consecutive terms ar e1 Memorial is a bet ter and generally rather wider term than

oral.2 In order to simplify exposition so far as possible I havedeliberately narrowed the field explicitly covered. To haveincluded memorial transmission would have necessitated some-what different and more complicated definitions. On the otherhand there is no need to exclude d ic tat ion, which is a mereincident of transcription. may affect t he character of t hevariants, not the principle of variation. Of course, print can besubstitutedfor manuscript, againwithoutalteration of the p r i n i ~ l e sinvolved, though in practice the problems that arise are generallydifferent . There seems no need to exclude from transcriptionrevis ion of the work by the author or another , but it could easilybe done by formally postulating that such a recension constituteda different work. Th e case has not been explicitly considered inwhat follows.S Th e term family is often applied in a merely extensionalsense to mean either the manuscripts of a work generally, or thoseof a particular branch. Here, however, it will always be used toinclude the genetic relation. Of course, if the inferential manu-scripts (in the sense later defined) are specified, the relation isgiven, since they are merely the formal expression of that r e l t i o n ~


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2 The Calculus Variantslinked together b y t he relation of parent an d child exemplar and t ra ns crip t). Norma ll y th e lines ofdescent ar e d iv erg en t i n a d ownward, c on ve rg en t i nan upward, direction.1In practice, however, we seldom have immediateknowledge of a whole family. What we find givenis a se t of ~ x t C . n t manuscripts two or rnore of whichma y belong to o ne l in e of descent, bu t which ar emore often no t directly linked by th e ancestral relat io n) fro m who se res em bl an ce s and differences wear e able, by a logical or quasi-logical process, to inferth e former existence of a n um be r of w ha t m ay becalled inferential manuscripts. An 1Jferenti al manzt-

s c ~ z p t is a node of th e g en ea lo gi ca l t re e, a p oi nt atw hich so me line of descent branches. O f course,th e farthest that this process of inference can takeus is b ack to th e archetype of all th e extant manuscripts. This may no t be identical with th e original

This distinguishes the genealogy of a manuscript or anyparthenogenic) family from that of a human family orany in whichsexual generation obtains). In the former the genetic relation isalways one-one or one-many, in the latter many-one or many-many.We have , however, i n t he c as e o f c on fl at io n, a p he no me no n i nmanuscript genealogy analogous to sexual generation, and givingr is e t o a man y- on e r el at io n. C on fl at io n is outside the purviewof the present essay, bu t a f w r emar ks o n t he s ub je ct will bef ou nd i n Not e 2 An inferential manuscript is t he l at es t exc lu si ve c ommo nancestor as subsequently defined) of s ome g ro up of extantman us cr ip ts . I pr ef er t he t er m i nf er e nt ia l t o t he mor e fa mil iar

, hypothetical because this latter has often a wider extension thanis he re desirable. W e are, namely, a t t im es a bl e t o co nj ec tu rethe existence of hypothetical manuscripts that are in fact internodal or ultranodal) points, intermediate between or anterior to)extant or inferential manuscripts, but which do not themselvesmark divisions in the line of descent. I doubt, however, whetherthe inference in these cases is strictly logical, or, at least, whetheri t is b as ed o n e vi de nc e of whi ch t he c al cu lu s c an t ak e a cc ou nt .Be this as it may, I have deliberately excluded such manuscripts,often including the original , from the definition of inferentialmanuscripts, relegating them, however regretfully, to the limbo ofwhat I have called the potential.

Ge1teral Notions: Descent 3postulated at th e s tar t in practice it probably seldomis : bu t no t only can th e methods here contemplatedt ak e u s no f ar th er , they cannot even t hrow l ig ht onth e question whether a ny th ing lies b ey on d. Conn ec te d w it h t hi s as ce rt ai na bl e class o f e xt an t an dinferential manuscripts, there is, of course, an indefinite number of others which probably once existedbu t whose identitycan now be bu t seldom, an d thenonly vaguely, apprehended. These ma y be called

p ~ t e n t a l m a n u s c r i j ~ s They have n o i nt eres t for ush er e b ey on d th e fact that th e discovery of a newextant manuscript will generally raise certain of themto inferential rank. We shall, therefore, define th ef mily as consisting of th e se t of all extant manuscripts together with their archetype and the otherinferential manuscripts needed to explain and expresstheir m ut ua l relation. S ho ul d i t ever be desirable to Inake more explicit th e distinction between th efamily as here defined an d th e wider conception withw hich w e s ta rt ed , th e former ma y conveniently bestyled th e logical, th e latter th e potentialf milyIn connexion with th e genealogy of manuscriptsseveral notions require definition.By ancestor of a m an us cr ip t we m ea n an y earliermanuscript in th e s al ne l in e of descent. It shouldbe observed that ancestor by itself is indefinite;w e c an no t in g en er al s pe ak of the ancestor, bu t onlyof nancestor, of a manuscript. Th e notion becomesdefinite, however, when we speak ofTh e ?0test ancestor of a m an us crip t, whic h is, ofcourse, its immediate parent.S im il ar n ot io ns a pp ly to groups of manuscripts,bu t th e i nd efin it e fo rm is so u ni mp orta nt that it isbest disregarded, an d we d efin e t he ~ o o ancestorof a group as the l at es t man uscr ipt which is anancestor of every member of th e group, that is th e

1 Their possible existence will always be i gn or ed i n f or maldiscussion.


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4 The Calculus arz antsl at es t manuscript common to the several lines ofdescent. I t should be observed that any and everygroup.of manuscripts selected from a family has ofne.cc:ssIty a common ancestor, otherwise by ou r?rlglnal postulate excluding memorial ~ r a n s m i s s i o nItS members could not all preserve th e same work.Th e most important notion of all is that of th eexclusive common ancestor of a group, that is, th elatest ancestor that is common to th e group and tono other extan t manuscript. This, i t will be observed, is not something different from, bu t a particular case of, th e common ancestor. It followsthat it does not always exist for any particular group;b ut a t th e same time the commonancestorcan alwaysbe made th e exclusive common ancestor by addingto th e group th e other manuscripts derived from it,where these ar e known.Members of one family bu t of different lines ofdescent ar e called collaterals An y group of manuscripts of a given work will therefore be of one of three

~ y p e s . l.t will b e a n , ~ ~ r a l group if th e manuscriptscomprlse.s belong to a Singfe-line of descent, thatIS, are all hnked by the ancestral relation. It will bea . f o l ' - a t e r o ; ~ group if the manuscripts all belong todIfferent hnes of descent. Lastly, it will be a mixedg oup if i t is nei ther purely ancestral nor purelycollateral. A collateral group may, of course, include or consist of inferential manuscripts, an d th eextant manuscripts of a work may form a mixedgroup. If th e collection of all extant manuscripts is

1 Th e qualification is formally necessary, since otherwise wecould not in general speak of the exclusive common ancestor ofa group of extant manuscripts alone, which is generally just whatwe want to do. At the same t ime i t is not intended to confine,and does confine, g r o ~ p to extant manuscripts. It isoften convement and qmte legItimate to speak of the exclusivec o ~ m o n a n c ~ s t o r of a group of, or including, inferential manuSCrIpts; for m such a case these are really no more thansymbols for the groups of extant manuscripts derived from them.

General Notions: Descent 5~ o n a t e r a l it . m ~ y be called a terminal group thatone COnSISting of manuscripts each of whichIS th e end of some line of descent. All termnal

~ ~ : 1 ' ~ c : i f ~ ~ ar e both extant and mutually collateral,bu t neither e xt an t n or collateral manuscripts arenecessarily terminal.Gi-yen anum b er of manuscripts of a work, whichwe wdl call A, B, C, D, . . . , the ir common ancestoran d their exclusive common ancestor may for convenience be written A ABCD ... an d xA ABCD respectively. We may also, if we so desire, use th esymbols A an d xA by themselves to mean respectIvely th e common ancestor and th e exclusive

a n ~ e s t o r of some group in question.. AgaIn: gIven xA BC, say { and also xA ABCe. xA A{3 , say Q wemay express these data in th esIngle formulaxA A(BC). Or, givenxA CD, say Y,a n ~ also xA ABCD i e. xA AB,.,), say Q fromwhIch B, and ar e independently derived, wemay wnte xA A) B) CD). On t he o th er hand ifin the latter ~ a s e , we had xA AB, say 3, we s h o ~ l d :of c O t ~ r s e ~ wnte xA AB) CD).2ThIS sImple convention of puttingxA ABC+xA BC xA A BC)enables us to express the relation of an y number of

w.ord common only serves to indicate that we arespeakmg WIth reference to a group an d not an individual whentherefore the group is. explicit it b ~ c o m e s superfluous, and isconsequently dropped m the symbohsm. In using the symbolsby themselves, however, should be remembered that they areonly strictly applicable to groups.

2 For the d e ~ n i t i o n of independent derivation see below, p. 7.It would occasIOnally be convenient to write xA AB CD) whereAB s h o ~ l d mean (AB) or (A)(B) , but i t is doubtful ~ h e t h e rthe occasIOnal convenience of an indeterminate formula wouldcompensate for the confusion its introduction might cause. Ishall throughout use roman capitals to indicate extant manuscriptsand small G r ~ e ~ letters to indicate inferential ones. In the fewcases w ~ e r e IS necessary t? distinguish b e t ~ e e ? manuscriptsand theIr readmgs, I shall mdlcate the latter by ItalIc capitals.


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6 The Calculus Variantsmanuscripts in symbolic form. Le t us suppose,t ak ing the example I shall use throughout, that awork is preserved in six extant manuscripts, namelyA, B, C, D, E, and F, and that no two of thesebelong to th e same line of descent.1 Then, if, forexample, there exist only xA EF, xA CD, an dxA CDEF, besides of necessity A ABCDEF, wecan completely define the family by the formula x)A [A][B][ CD EF ]. Here we write C x) inste ad o f C to indicate that it is only significant forth e several sub-groups, for xA has no meaning inconnexion with the sum of extant manuscripts. Th eslight formal distinction serves to indicate that weare consider ing a comprehensive relat ion: everyformula beginning with C x)A defines a completefamily, one, that is, comprising all extant and consequently also all inferential manuscripts.

We may occasionally wish to assert th e existenceof xA of some group in respect to some largergroupwhich, however, does not include all extant manuscripts, without implying anything as to those excluded. This may be done by writing, for example, CDEF)xA EF,which confines the field of the statement to th e group CDEF among ex tant manuscripts, and leaves open th e question whether A EFis also an ancestor of A or B or not.I t remains to observe that derivation is of twotypes, independent and successive. In one sense,and in connexion with particular groups, this is, ofcourse, obvious. Derivation in th e line of descent isnecessarily successive, while any number of manu-

1 To this condition of collaterality I shall return later; seep. and Note A. . For the sake of clearness, and for convenience of reference,a number of typical families of six manuscripts are exhibiteddiagrammatically on pp. 60-1, each accompanied by the formula Ithat defines it. The families represented are, of course, only aselection from those theoretically possible. For brevity I shallspeak ofthe formula as being, not merely as defining, the family.

General Notions Descent scripts are independently derived from their immediate parent. But there is a less obvious, and derivative, though for ou r purpose more important, sense, inwhich th e terms may be applied to whole families oreven to collateral especially terminal groups. An dhere it should be observed that even the independentderivation of several manuscripts from their immediate source is successive in so far as a child succeedsits parent, while without some independent derivationno collaterals could come into existence. It followsthat th e definitions will depend on degree. J,. tl.de::l

j J . ~ t derz'f q, on is found throughout any collateralgroup { r no selection from which does xA exist;that is, in th e case of our six terminal manuscripts,only in the family x)A A) B) CXO) E) F), in whichsuccession is reduced to a single generation i. e.genetic step . . $ . ~ f . & . ~ s s i v . . d e , r . ~ : ' { I , q , o , is the antith.esisof independent bu t is less easily defined. It mIghtappear sufficient to recognize as successively derivedany family in which there were never more than twomanuscripts independently derived from a commonsource. This, however, would not give a uniqueresult, such as is desirable. We can obtain this byadding the condition tha t, of each pai r of independently derived manuscripts, one at least shallbe terminal. This is satisfied only by th e family x)A A{B [C DEF ]}.l Fo r this, however, an equivalent and preferable definition is to be found in th efact that all the inferential manuscripts form anancestral group. I t is to this type, therefore, thatwe shall confine the term successive derivation. Th elooser type resulting from the definition first considered, and satisfied by x)A {AB}{C [D EF ]}and x)A [ AB CJ [D EF ] and various otherfamilies, may be described as quas'i-successve.

It is, of course, also satisfied by x)AC{[ ABC D]E}F, butthe two families are identical so long as A, B, C,... remainvariables.


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8 The Calculus VariantsLastly in those cases which, without being purelyindependent, involve the derivation of at least threemanuscripts, extant or inferential, from a commonparent, such as the families x)A AB) CD) EF) and x)A A{B [(C)(D)(EF)]}, we may recognize the derivation as quasi-independent. The importance of thenotions of independent and successive derivationlies in their relation to the corresponding forms ofvariation and divergence.1

process. of transc.ription is characterized, byvarlatlon, and It IS only In the process of transcription that variant readings arise. 2Such variation may be assumed to be universal,every transcription introducing some variants. Thisis obviously not necessarily true, but i t agrees withexperience in all but the shortest texts. Moreoveran operatIon that produces no effect may safely beignored, and, should there be such a thing as anabsolutely faithful transcript, we shall be led intono error if we treat it as identical with its exemplar,Most variants are spontaneous, that is to say thatthey are not in any way conditioned by variation in1 A few formal antitheses, out of many, may be noted. derivation is purely independent, then, in the formula definingthe family, there are the maximum number of brackets, theseof. the.same order, and.there is only a single generation;If denvatIOn IS purely succeSSIve, then there are the maximumnumber of brackets of different orders, no two pairs are of the

same order, and there are the maximum number of generationsn a ~ e l y less th3;n number of t e r ~ i n ~ l manuscripts.ThIS IS not hIstOrIcally true, but It IS a convenient andinnocent assumption. Many variants in extant manuscripts havearisen through an al terat ion being made in an ancestor afterthe original scribe had completed his work. In such a casetranscripts made before the alteration will have one readingthose p lade after i t another . But in order to render the s t t ~ment in the text rigorous we only need to postulate tha t thealteration of a manuscript is equivalent to transcription, and,therefore, that the manuscript in its original state is not identicalwith, but the parent of, the same when altered.

General Notions: Varlatlon 9the exemplar: on the other hand some are so condit ioned, since a slip in one transcription of ten leadsto emendation (correct or not) in the next . But wemar safely assume t h ~ t in ,no transcr ipt are allvarIants thus predetermined; Indeed this almost ofnecessity follows from our former a s s ~ m p t i o n Therealso follows from it, at least in suitable cases,an-.other a 1d m?re ex.treme inference, namely, that, ofthe yanants Introduced in any transcript, some willperslst. hrough subsequent transcriptions, whileothers ~ t l } undergo further variation. Since, in any

t r a n s c r l p t l o ~ ~ ~ l y a small proportion of the readingsuI dergo V a r I ~ t l o n the former part of this propositionw,lll be r e ~ d t l y allowed. The latter part is less obVI0US, but Itwdl be observed that the variations introduced in the course of any transcription themselvesform textual field, over which, if it is sufficientlyextenSive, the assumption of universal variation willo p ~ r a t i ~ e M o r e ~ v e r the principle of predetermlnatl?n.wtll make thIS field more particularly subjectto varlatl0n.We require then, the following postulates

U n l ~ e r s . a l .varz atolZ namely, that every act oftranscrlptIon Introduces some variants;That spontaneous varz at-ion is more widely effectivetha? dete:mined v a r ~ a H o n and consequently tha t thevarlants tntroduced In any transcription are never allpredetermined; , P e r s t s ~ e n c e variation and varlatt.on varlatlon,

from W h l C ~ It f ~ l l o w s that, of the, features peculiarto any manuscnpt, provided they ar e sufficientlynumerous, some are transmitted unal te red to itsdescendants while others are further modified.1I t should be observed that the term variation

1 Critics h a v ~ s?metimes tacitlyassumed the further postulateof conS anl Vartatlon, namely, that every transcription introducesa P I ? r o ~ I m a ; t e l y the same number of variations in any given text.ThIS 15 qUIte contrary to experience and leads to erroneous results(see Note C). c


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The Calculus o Varl:antsis used, strictly speaking, in two somewhat differentsenses, or at least is applied to two differentcases. There is the variation of a descendant froman ancestor, and there is the variation of two colla-terals from one another. Th e former may be calledvertcal varz aton, the latter horizontal varz atz on.Th e former is fundamental, the latter derivative;for, of course, the variation between two collateralmanuscripts is merely the effect observable i f theyare extant) of the variation of one or b ot h o f t he mfrom thei r source. Horizontal variation is thedatum, vertical the end, of textual criticism. It maybe not ed that horizontal variation always impliesvertical variation in at lea st o ne line o f d esce nt; butvertical variation only leads of necessity to horizontalvariation if it occurs within the limits of the logicalfamily. In the complete potential family all lines ofde sce nt m ay pas s t hr ou gh t he manus cr ipt i n whichvariation arose. In other words, the readings ofany c olla te ra l g ro up a re e vide nc e of the readingof the archetype only, not ofany earlier manuscript which is obvious though i t se ems t o b e s ometi mesforgotten.Just a s, in any family tree, different lines of descentare seen to be divergent in a downward direction, sothe text, in any line of descent, becomes increasinglyd iv er ge nt b oth from the original arid from tha t ofany other line of descent, nleasuring divergence bythe number of variants. This is presumably alwaystrue. We might proceed to argue that the numberof variants between a manuscript and any ancestorwas t he s um of all the variants introduced in the in-tervening transcriptions, and that the num be r o fvariants between any collaterals was the sum of thevariants introduced in the transcriptions interveningbetween them and the ir latest co mmon a nc esto r.But this would only be t ru e so long as t he var ia ntsintro du ce d wer e the mselve s d iv er ge nt. This is

General Notlons: Varlatlon no t always so. J 1 _ ~ a r ~ a t i o n in transcription m y-accidentally, and often does intentionally, restore the

~ e a d i r i g of an e a ~ l i e r ancestor. ~ l s o two indepen-dent transcriptions may alter a partiCular reading illthe same way. In either case, the second variation,instead of increasing the divergence of the texts,._reduces it. Thus by the side of the normal divergentvar;[atiolt we must recognize, in successful emenda-tion and in the chance coincidence of error, two formsof what may be called convergent variation lHorizontal variation gives riseamong collaterals togrouping that is, t o the arrangement of the manu-scripts into groups according as t he y agree or differin r espe ct to p art ic ul ar readings. By a gro uping,we u nd er st an d a l is t of all the extant manuscripts or of some selection of them) divided on thisprinciple into two or mo re g rou ps e ac h of one ornlore manuscripts. But since it is o nly by a str etchof language that a sin gle m an uscr ip t c an be calleda group, we may describe as true groupngs thoseincluding at l ea st two g ro up s ea ch of two or moremanuscripts, and we shall find in the sequel thatthese alone are significant. We shall a lso see lateron that the only groupings that can be regarded asfun ment l a re t ho se that divide the manuscriptsinto two g ro up s only. In s uc h g ro up ing s we m ayspeak of the two groups as the two sides of thegrouping, and each as the complement of the other.More generally, the complement of any group is, ofcourse, the group of all the other manuscripts inquestion.Different variants will group the manuscripts indifferent ways. Should it be found that allpo ssib le a rr an ge me nts o cc ur with muc h the samefrequency the grouping in general may be described

1 \Vith d iv er ge nc e c on si de re d, n ot i n r el at io n t o t he t ex t a s awhole, bu t as the degree of variation in particular variants, wshall be concerned later see p. 30).


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The Calculus Variantsas random and, of course, no inference as to thereIai1on 6 f the manuscripts is possible. To be of anyevidential value the groupings must be cons/anicertain arrangements occurring to the exclusion ofothers with at l ea st s uc h reg ul arit y as to suggestthat exceptions may be due to chance. When thegroupings are constant, some inferences can alwaysbe made, bu t the results will be contradictory unlessthe groupings are not only constant bu t also tt-sistent The conditions required for consistency arenot altogether easy to define formally, bu t for funda-mental groupings they appear to be satisfied if andonly if given any two constant groups, either these ortheir complements are either mutually exclusive orone wholly includes the other. Groups are, ofcourse, constant or consistent i f they o cc ur i n con-s tant or consistent groupings.) The rule comes tothis, that whi le o ne or more manuscripts may passfrom one side of a g ro up in g to the other withoutrendering it inconsistent, th ose on o pp os it e s id esmust not exchange places.The grouping of the manuscripts may be con-sidered either wit h referen ce t o a p arti cu la r v aria nt ,or generallywith reference to several, or all, variants.Th e generalized grouping is, of course, t he sum ofthe particular groupings, bu t it is of a much morecomplicated nature, since, no t only are the constituent fundamental) groups no longer confined to two, bu ttheir rel at io n is n o longer one of simple opposition.I t is clear that we s hal l r eq ui re a symbolism forvariation in someways parallel to that already adoptedto express ancestry. But the development of thisrequires to be dealt wit h i n greater detail and mus tbe postponed to later sections of this essay.1Meanwhile the parallelism just mentioned betweenvariation and descent suggests a very importantobservation. It is, namely, necessary to distinguish

1 See in particular p. 23 and further p 44.

General Notions: Variation clearly between two different meanings of the termgroup . The groups we have just been consideringare what may be called variatonal groups that ismerely groups of manuscripts having certain readingsin common. S in ce it is s uc h g ro up s that will mainlyoccupy our a tt en ti on , I s ha ll by group always meana variational group unless some other is expresslyindicated. But besides thesethere are geneticgroupsor branches of the family tree, characterized by thepossession of an exclusive common ancestor.1 Th etwo are, of course, related. Thus if the manuscriptsA, B and C constitute a genetic group, this willgive rise to v aria nt s in whi ch ABC will be opposedto DEF. On t he o ther hand, if we find t ha t t hevariants habitually divide the manuscripts into thetwo g ro up s ABC and DEF then t he se will besignificant constant groups; but, though both mayalso be genetic groups, that either ABC or DEFshould be such will suffice to account for the facts.Th e process of determining the relationship of themanuscripts consists in inferring from the variationalthe corresponding genetic groups.I t is the object of the Calculus of Variants tofacilitate this process by s ub st it ut in g, s o far a s maybe convenient, the use of symbols and formal rulesfor the continuous application of reason, thereby notonly economizing mental effort, bu t avoiding, it ishoped, certain confusions of thought which, as experi-ence shows, are liable to occur.

? At first sight it might appear sufficient to postulate t heeXIstence of BCas t he c ondi ti on of ABC orming a genet icgroup. Certainly, such a group would be, in some sense, genetic.But it would not be a complete genetic group, and it isnecessary to include completeness in the notion, since otherwiseany selection from the manuscripts would form a genetic group.The definition adopted for xA renders it unnecessary to includeinferential manuscripts in order to secure completeness.


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In these formulas it would be more usual to write s o : and, so A , but there is 110 necessity to do so.

ecordz ng Varz ants a fragmentary manuscript Z available in parts, then,where this is so, must become + Z or Inconnexion with this latter case a small formal pointdeserves mention. Should th e reading of Z differfrom that of all t he ot he r manuscripts, might,though correct, be slightly misleading to writex y y x ~ Z yixy Zand no confusion can arise if we pu t instead

yxxy ZWhenever, therefore, Z is explicitly mentioned ~ maybe substituted for an d we shall, for instance, write

xxyy ~ :xyyx EZ :xyxy F.Should occasion arise, such symbols as ~ i F ormay, of course, be used to indicate th e unspecified manuscripts of th e collections ABCDZ and

ABDEFYZ respectively.What has been said so far applies where variantsare quoted without reference to any particular text,an d must, of course, be followed, when handlingformulas for th e purposes of discussion. A few wordsmay be added on the conventions best suited torecording variants in connexion with a printed text.A common practice is always to give first the readingo f t ha t text, and to separate it from what follows bya single bracket. Thus for example, we might findxyyx] ~ yxxy EF orxyyx] A yxxy

In the former of these examples we may omit thesince the reading of t he t ex t may he assumed tobe t ha t o f all the unspecified manuscripts, and writesimply xyyx]yxxy E F, without ambiguity. On the other hand we cannot

4 ecording Variants

Given once more our six extant manuscripts A, B,C D, E and F i t will always be possible to quotetheir variant readings according to . ~ o m e suchformula as . .xyyx AB C yxxy DEF orxxYY AB : xyyx CD : xyxy EFin each case giving the words that replace oneanother in th e different groups. 1 hemost frequentformula will very likely be of the type

xyyxABCDE j txxyFwhich suggests th e need of some symbol to indicateC the rest . P u t t i n g ~ for C th e sum of th e un specifiedmanuscripts , the last-mentioned formula becomes

xyyx ~ :yxxy F.It will be best, as a rule, to reserve ~ for th e largestgroup in any formula; 2 thus we shall cont inue towrite the first pair above a s before, bu t we sha llhave, for instance,

yxxy EF an dxxyy xyyx DE : xyx F.Th e meaning of ~ is defined in relat ion to th emanuscripts that ar e generally available throughoutthe text. Should one or more of these show lacunas,then, in the passages affected, some modification ofth e symbol must be used. Fo r instance, if a stanza,say, is omitted in E and F then, in quoting variants

between the other manuscripts in these lines, we mustreplace ~ by th e qualified symbol ~ - EF more conveniently written ~ E F Similarly, should there be Where minor differences of spelling and so forthare neglected,the words (unless normalized) should be quoted in the exact formin which they occur in the first of the manuscripts cited. Occasionally, however, in discussion it is convenient todesignate some smaller group b y ~ It does not generally seemworth while to substitute in the case of two manuscripts only,even when these constitute the largest group.


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The alculus of Variantssimilarly omit th e A in th e second example sincewhere there is no specified group can h a ~ e nomeaning. And. eyen were we t.o replace byBCDEF the omission of A would st be undesirable.This applies to cases in which th e printed text isa critical. one. Where it r e p r o d ~ c e s exactly a singlemanuscript, fact of a reading appearing beforeth e bracket IS ~ q u i v a l e n t to the specification of thatmanuscript, and in that case may be used withoutambiguity. even if the letter indicating the particularmanuscript be omitted. It is, however, doubtfulwhether anything is gained by th e omission.Th e reading before the bracket C taken down)from the printed text, is called a C l e ~ I l a . Lemmasnlust agree exactly with th e text.2 Some differenceof practi;e exists as . to C taking down punctuationalong With the reading. In th e case of a criticaltext in which the punctuation is th e editor's andtherefore of no critical value, it is best neglectedaltogether. On the other hand, where the textr e p r o d u ~ e s ~ x a c t l y (or approx mately) a, particularmanuscript, It may often be Important to recorddifferences of punctuation found elsewhere. In thiscase it is sometimes held essential that , since avar iant may consist in, the absence or addition ofpunctuation, should be taken down' in all cases.This is a mistake. Punctuation need only be C takendown' when it is i tself in question. W he n t ha t isso, the variant consists in th e difference of punctua-

1 Or generally, provided the exceptions are clearly m a ~ k e dA. slight exception to this generally rigid rule is that diacritics .used In the t ~ x t n e ~ not necessarily,be retained in the lemma.For e x ~ m p l e If certam letters are printed in italic to indicate thee x p a n ~ I ~ n ~ f a ~ b r ~ v i a t i o n it may, be advisable to neglectthe d l s ~ m c t l O n In takIng d?wn . So with brackets indicatingmuttlatlon and so forth, whIch tend ' to cause,confusion in thelemma. An editor must settle these points for himself. but he llust settle them consistently and with appreciation of'what ismvolved.

ecording Variants 17tion recorded; th e preceding word is merely addedto indicate th e position; no ambiguity can arise.Some editors omit lemmas altogether, relying onthe sense to show which word or words in the textth e var iant is intended to replace. In the handsof an editor of approved competence rara avis interris the practice is unobjectionable,bu t it demandsa degree of skill and vigilance to sa y the leastuncommon, and is certainly not to be generallyrecommended. Instances could be cited in whichan apparatus critcus has been rendered largelyworthless through the lax use of this method. Inparticular th e warning is desirable that, where thevariant consists of an omission or an addition, thecontext on both sides should be given. Further , ifvariants of punctuation are taken into account at all, will here be necessary to' r egard a point as anintegral p ar t o f t he preceding word, and in all casesto quote th e one along with the other, else therewill be no logical means of recording its absence.I t should be added that no directions can be ofuniversal application. Editors will always have toadopt special conventions to meet particular needs.A matte r of some importance, that falls for discussion here, is the degree of coll tion that is, theminuteness of the variants of which notice is taken.Needless to say, this should be constant throughouts far as possible. Otherwise it is a matter ofchoice. If we confine our attention to the moreimportant variants, we can be fairly certain, providedwe are dealing with the work of a naIve scribe, thatthe readings are meant to be those of th e exemplar,and are evidence of the descent of th e manuscriptin which they occur. On the other hand, if the work

I may mention that I have attempted the method myself andabandoned it owing to the difficulty of avoiding ambiguity.2 course, some dhoc device of annotation may be adopted,and t ~ I S may be the best means of meeting thedifficulty, if it doesnot anse too often.

D


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he alculus of Variantsis a short one, we risk limiting the field too much toeliminate the operation of chance: moreover, thereare other dangers connected with conflation whichwill be touched on elsewhere. If, however , wemake our collation very detai led, we are met withdifficulties of another sort For, whereas, in majormatters, a scribe will, as a rule, follow his exemplar,in the minor points of spelling and grammatical

form he will be largely led by his own fancy. Consequently, the more minu te we make our collation,the greater t he number of non-evidential variantswe shall be recording, and the greater the risk ofchance coincidences between manuscripts. I t isobvious that transcripts of different exemplars bythe same scr ibe will show marked resemblance inthe minor readings and marked differences in themajor; whereas transcripts of the same exemplarby different scribes may agree almost throughoutin important matters and ye t will differ widely indetail. Th e degree of collation desirable must bedecided in every case in relat ion to the character ofthe work contemplated.

ypes of VariantsThe formulas expressing the variation of themanuscripts in respect to particular readings conform to a number of definite typ s There are, to

begin with, two main classes, the simple and thecomplex. In simple variants the formula definestwo alternative groups, to one or other of whichevery manuscript belongs. In complex varants thegroups are more than two in number. The simpleclass comprises two types: type I in which onegroup consists of a single manuscript, and type 2in which either gr oup consists of two or more In Note B.

ypes of ri nt 9manuscripts. Th e complex class comprises types3, 4, 5, , according as there are 3, 4, 5, g.roupsin the formula. Thus the number of types IS thesame as the nunlber of manuscripts. For example,in the case of our six terminal manuscripts we shallfind the following types:S {Type A, F c.unp e Type 2 AB EF ABC : DEF c.

rType 3 ~ : A : F ~ : D E : F A B : C D : E F c .Com- Type 4 A : E : F AB: C: D : EF c.plex lType 5 AB : C : D : E : F c.Type 6 A : B : C : D : E : FIt will be observed that several different formulas

are possible under each type except the highest :these may be called the d ~ f f e r e n t forms o f t he t ~ p eThey comprise all, the different arrangements Intowhich the manuscripts can fall, and any form may,and usually will, be exemplified in a number ofinstances.Since every manuscript contains variations fromits immediate source, any reading supported by onemanuscript alone may ~ a v e originated in that ~ a n u -script, and such a reading t h ~ r e f o r e cannot, w ~ t h o u tfurther analysis, throw any light on the relation.ofthe manuscripts of the collateral group. To establishsuch a reading as original it would have to beshown, not only that the reading correct, bu tthat it could not be due to emendation. 1 To proveeither of these is strictly impossible, and though inindividual cases the probability may be great, and

Even if this were admitted, to prove for the other manuscriptsa common and independent derivation, it ,,:ould. be necessary oassume that their common error had not ansen I n d e p ~ n d e n t l y Inthe course of various transcriptions. This assumption Ye dohabitually make, and although.it is not necessarily c o r ~ e c t In anyindividual case, without i t no mference as to the relatIon of themanuscripts would be possible.


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20 The Calculus of Variantsrepeated cases give rise to moral certainty, this ishardly a mathematical notion, an d therefore findsno p lace in th e calculus.1

t follows, the re fore , that only t ho se v ar ia nt swhich give rise to at l ea st t wo g ro up s of more than0;te ~ n u s c r i p t .each can be described as genetically)stgnijicant varzants An d o nl y t ho se which giverise to groups all of which ar e of more than on emanuscript c an b e described as completely significantBy significant groups we shall understand truegroups i e. of two or more manuscripts) arisinofront significant variants. b

t will be observed that if th e number of manuscripts a nd t yp es is n types I n an d n - I ca n neveras they stand be significant, an d that the highesttype that ca n be completely significant is n/2 if n iseven or n 1}/2, if n is odd. Also, t ha t t he numberof possible forms of type I is n an d o f t yp e n is ITh e possibilities of the i nt er med ia te t yp e s ma y beleft to mathematicians to determine.

ma y be asked why th e question of significanceha s been allowed to divide th e class of simplevariants into two types whereas th e distinction ha sn ot b een carried through th e conlplex class. Thereason is that the point ha s no t th e same importancein th e h ig h er t yp es as in th e lower. For thougha type- variant can never be significant, we shallshortly see that w he re m or e than two groups ariseth e reading of a single manuscript ma y show anaffinity that will enable th e variant to be reducedt o o ne completely significant.

Though t h ~ m a t ~ e r lies e y o n ~ my present theme 1 maypomt out a cunous dIfficulty that anses when we attempt to infermanuscript relation from supposed originality. To show that areading is original two main lines of argument are available: thatthe.reading is itselfsatisfactory, and that it explains the origin oft he erroneous alternative. But, as a rule, the easier it is toexplain how an error arose, the less val id the assumption that i tonly arose once. Thus the more likelyit is that one alternative iscorrect , the less cer ta in i t is t ha t the o ther point s to commonderivation.

21Principles of Variation

From w ha t h as b ee n said in th e previous sectionit will be clear that where three manuscripts onlyar e concerned, no merely formal process throwlight on th e relationship between them. l t h e ~ th ereadings will be all d i v ~ r g e n . t or else th e vanantswill be of type I, and SInce, In t he l at te r ~ a s e th ereading of the single divergent m a ~ u ~ c r I p talways. theoretically at least) be unorigInal, It WIllnever be possible to estab li sh a common. source foran y pair of manuscripts to t.he exclusion of . th.ethird. Given three manuscnpts therefore , I t ISimpossible e it he r t o p ro ve or disprove independent derivation. This fact , which call th e am-bigztz ty three texts we sha ll find meet us everyturn of th e discussion, an d it largely determInes th enature o f t he calculus.But though type-I variants whatever may be theirindividual interest ar e of no use for o ur p re se ntpurpose, e it he r t he absence of higher types, o r t heabsence of particular forms of the owest, ma y beof evidential value. Suppose that In t he t ex t preserved in ou r si x manuscripts, none bu t type-Ivariants occur. A t first this m ight seem to implythat all th e manuscripts were independently derivedfrom a common ancestor, but owing to a complication no t unlike that mentioned above, th e correctinference is that at least all bu t on e of the manuscripts are so d e r i ~ e d . A couple f e x a ~ p l e s willmake this clear. Suppose there ~ x l s t s xA ABa t he n t he writing of a will have Introduced certaIn

v ~ r i a n t s a nd s om e of them will according to thep o s t u l a t ~ of th e persis:ence ~ a r i a t ~ o n have survived in both derivatIves, gIVIng rise to type-2variants o f t h e form 2: : AB. Consequently th eabsence of such variants disproves th e existenceof xA A B. Suppose, however t ha t th ere exists


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22 The alculus VariantsxA BCDEF, say { then variants arising in { andsurviving in all derivatives will give rise only totype-I variants of the form : A.I Consequently theabsence of variants of higher type or of significantvariants generally does not disprove the existenceof xA of all the manuscripts except one.It follows from the postulate of universal variationthat in any collateral group all forms of type-Ivariants must occur. Consequently the absence ofone or more forms of this type is inconsistent withthe assumption that our six extant manuscripts forma collateral group. This might seem to offer a meansof eliminating ancestral elements from any collectionand thus replacing our assumption by a logicaldemonstration. Unfortunately, a consideration ofwhat has been said above shows t ha t t he converseis not necessarily true, that the presence of all formsof type-I variants does not prove that the collectionis collateral throughout. Thus the elimination ofdescendants cannot be effected by the sole use of thecalculus, and the further consideration of the questionmust necessarily be postponed.2Type-2 variants are a very different story. If wehave a variant AB: CD, then one or other readingmust differ from th at of the archetype, and one orother group must be genetic: there can be no question of al l four manuscr ip ts being independent lyder ived. Dif ferent forms of type-2 variants willdivide up our collection in different ways, and thesedivisions will correspond to the ramifications of thefamily tree. It is clear that some symbolisQ l canreadily be devised that will represent the completegrouping afforded by the variants, and that the

1 In this instance i t should be observed that,. while the appearance of A proves that A is not identical with { unless weknow that our extant manuscripts are in fact col la terals as weassume them to be , we cannot tell whether is an ancestoror a collateral of cf. p. SI .2 See Note A.

rinciples Variation 23correspondence between this and the ancestralsymbolism already adopted will give the basis ofsuch inference as can be drawn from varia tional inthe direction of genetic grouping. Provided theyare numerous enough, type-2 variants afford us allthe evidence of which we can in the calculus makeuse. Not only so, but , although individual variantsof higher type often look very tempting as a basisof inference, it is exceedingly difficult to devise anyformal method of dea ling with a number of them;the relationships are too complex to be readilyamenable to rule.When, however, we pass to the consideration ofthese variants of higher type a very important factemerges. For, if we bear in mind that an actualvar iant, however complex, can only ari se throughvariation in individual acts of transcription, and thatat any point in the text a single act of transcriptioncan only give rise to a single varia tion, it will beapparent that it is only such varia tion as we see intype 2 that is fundamentally significant.It will be convenient to consider this matter morefully in connexion with the several varieties of variation, namely independent and successive, and for thispurpose to deve lop somewhat our symbolism ofvariation. Thus, if for instance, we constantly findthe grouping DEF and also the grouping EF,it is proposed to express th is double fact by thecompounded formula D EF . Similarly, if wefind only the constant groupings ~ :AB and 2:: EFin variants of type 2 that is, if such groups as CDand DEF are absent , we shall wri te 2:: C D EF .Thus, just as in the case of genetic groups we put

xA DEF + xA E F xA D EF),so now in the case of variational groups we put

~ : D E F : : E F ~ : D E F .Th e parallelism of the formulas reflects the relation


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4 The Calculus o Variantsbetween the two kin ds of grouping: the formeralways implies the latter, though the lat te r does no tnecessarily imply the former. I t should also benoted, as part of the parallelism, that since geneticgroups mark s in gl e s te ps i n d eriv at io n, o nl y s im pl evariants can be compounded.Turning now to the varieties of variation, weobserve that in any par ti cul ar p assag e a singlevariation must, of course, arise at o ne p arti cu la rp oin t in the family tree. Further when, in thecourse of the various transcriptions that generatethe family, a p ar ticular passage is subjected torepeated variation at different points, these pointsmay li e i n the same or in different lines of descent.Thus a complex variant always implies that variationhas t aken place at more th an one point,l and itsnature will be determined by the rel at io n i n whichthose points s tand to one another.Variation being primarily vertical, successive andindependent variation are notions that apply in thefirst instance to the relation between manuscriptsand their sources. Thus in the simplest cas,es,successive variation is seen where variation occursin successive acts of transcription, thC\t is, w here amanuscript v ar ies a r ead in g which a ros e throughvariation in its parent; and independent variation iss ee n w he re v ar ia ti on o ccu rs i nd ep en de nt ly in t woacts of t ra ns crip ti on p erfo rm ed o n o ne e xe mp la r,that is, where two manuscripts both depart from thereading oftheir p aren t. M ore g e ne ra ll y, we s h al l saythat successive variation is multiple variation occurring

Historical ly, mu lt ip le vari atio n d oes n ot always l ead t ocomplex variants. only two ma nus cr ipt s a re ext ant , t he reobviously can be no complex variants, yet both may have alteredt he r ea di ng of t he sour ce . Simila rly, i n a single l ine of d e s c ~ n tany number of var ia ti ons may occ ur , a nd yet only t he t e ~ m m lreading survive. But since the calculus can only ta ke c o g l z a n ~ eof extant readings, the converse of the statement 10 t he t ext ISfor our purposes, also true, where enough manuscripts survive.

Princzples o Variation i n a s in gl e l in e of descent, while independentv ri tz onis that occurring in different lines. Of course, wewish to speak of the variation between extant manuscripts, and there will be no objection to applyingthe terms as defined, in accordance with the mannerin which the va rian ts arose, s o lo ng as we are nott empted to confuse the derivation of the readingswith the derivation of the manuscripts. Whether, and i f s o h ow far, i t is po ss ib le , fro m thenatu re o f the variants themselves, to ascertainwhether they arose successively or independently, isa p ro bl em that will engage our a tt en ti on whe n, inthe n ext section, we come to discuss variationaldivergence. Meanwhile, assuming the nature of thevariation to be given, le t u s c on si de r t he g en es is ofcomplex variants.Taking first the case where variation is independent, le t us suppose that there exist xA AB, say axA CD say 1 and xA EF say E and that Cl Y and a re i nd ep en de nt ly d eriv ed fro m the archetype;assumptions expressed by the formula

x A AB) CD) EF).Th e only constant groupings in type 2 that can occurin such a family are AB, CD and EF i.e. : CD) EF) when compounded), due to variation in a 1 and respectively, and it is m erel ythrough the chance concurrence of two of thesevariations that the complex grouping AB : CD : EFcan arise. I t is evident, therefore, that the correctway to regard this type-3 variant is as the productof two variants of type 2 t ho ug h wit ho ut k no wi ng

we are given three readings A :B :C a nd a re to ld t ha t Band vary successively from A what is meant, of course, is thatC is derived from B and B from A B ut i f the statement ismade) in resp ect to a pa rt ic ul ar variant, a bo ut t he manuscriptsA : B : C t he n what is m ea nt is st ill t he sa me, namely, t ha t t hereading of C is derived from t ha t o f B , a nd t he re adi ng o f B fromt ha t of A not t ha t C is derive d from E, or B from A


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The alculus Varz antswhich reading is original, there is no t el li ng in whi chtwo inferential manuscripts variation occurred.On th e o th er h an d, t o take a case in which variation is successive, consider th e family

x)A A{B [C DEF)J).H er e t he forms of type-2 variants ar e different fromwhat we found before, the compounded formula being

: C [D EF)]. Th e variant : E F will result fromvariation in xA E F, sa y E, an d : AB fro m v aria t io n i n xA CDEF, sa y Y and the complex groupingAB : CD : E F can only ar is e through chance concurrence of variation in Yand E in succession.Now consider th e family x)A [A] [BJ[ CD) EF)J,i n whi ch we have only xA CD, say Y xA EF, sa y E,an d xA C DE F, sa y J Th e constant groupings willbe : CD) EF), as in ou r first example. B ut h er e

th e complex variant AB : CD : E F ma y be du e toth e concurrence of independent variation in Y an d E,or else to th e concurrence of successive variation ina nd e it he r Y or E.From this it will appear that independent an dsuccessive variation may be combined in a singlecomplex. Thus, in th e family

x)A [A] [BC][D EF)],we might mee t with t he groupi ng A : BC : D : E F,due to independent variation inxA BC andxA DE F,an d var iat ion in xA EF successive to that inxA DEF. Bu t it will be noticed that the samegrouping would ar ise from var iati on in A, xA BC,an d D, when would be independent throughout.Thus while it is strictly true that every variant ofcomplex type is t he p ro du ct o f two or more simplevariants, it is no t p os sib le e ven g iv en th e familyrelation) to resolve variants into their factors withoutsuch a specificknowledge of their individual characteras can o nly be o bt ai ne d f rom a study of th e actual

Principles Variatz on 7:eadings. It follows that in a na ly si s t he g re at es tImportance attaches to th e consideration of how th eorigin of a var ia nt may r ev ea l i ts el f in th e divergence of th e rea di ng s t o whi ch i t g iv es rise.Before, h o w e ~ e r p ~ s s i n g in. th e n ex t section) toth e further conSIderatIon of this problem, it will benecessary to discuss t he t re a tme nt of defective an dredundant g r o u p i n ~ since, though th ey are of no

g r e a ~ conseq :lence In the?1selves, t he former acquiresconSIderable Importance In conneXlon with th e formalaspect of resolution.Le t u s s up po se . th e scribe of some manuscript,F to have o mItt ed a p as sa ge , a nd , at a c er ta in

p O l n ~ In that passage, t he gro up AB C to have oner ea dm g a nd D E another. Then it is pretty clear~ h a t th e absence of any c orre sp on di ng rea di ng in FIS of t he n at ur e of a v aria nt , an d t hat th e formulawIll have to be D E : F, since we do no t knowwith which group th e exemplar of F agreed, nor thatF would n ot h av e varied individually ha d it repro?uced th e passage. It is, perhaps, less obvious that,If the p a s s ~ g e h as b een l os t from F through subse

q u e ~ t m ~ t t l a t l o n th e absence of any correspondingreadmg IS equally of t he n at ur e of a v aria nt . Ye tit will be observed that h ere o ur ignorance is th es ~ m e an d that th e cause of ou r i gn or an ce is in~ l t f f r n t s uch c as es we agreed to substituteIn place of a nd we s ha ll c on se qu en tl y wri te th eformula : D E. f he question will occur: What isth e importance of distinguishina between and ?N d f b F.,ow, 1 ,Instea DE , we were s im pl y t o wri te: DE , we should imply a g rou pin g ABCF : DE .B ut t he re may be a constant grouping ABCD : EF ,a nd the tw o are, ac co rd in g t o ou r definition, incon

s l ~ t n t Bu t we do not. know th at t he reading ofF s exemplar agreed WIth AB C rather than withD E, and we, t h e r e f o r ~ have no w ~ r r a n t for assumingthat th e real groupIng contradIcted th e grouping


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The alculus 0/ VariantsABCD : EF (with which ABC : DEF is perfectlyconsistent). Thus the importance of the distinctionbetween and is that without it we risk theinvention of anomalous groupings. At the sametime, so long as we retain the variants in whichi t occurs are all in effect of complex type, and of avar iety in which no resolut ion is possible. It istherefore highly desirable to restore wherever, itis legitimate to do so. This is always the case whenthe resultant variant is of type that is when (apartfrom the defection) the only divergent reading is thatof a single manuscript. E is a defect ive grouping which conceals one of the following, viz.

: E : F, EF, E.Of these alternatives we may always safely choosethe last, since from it no inference can be drawn,land consequently in place of E we are alwaY atl iberty to write E. On the other hand, whenever (apar t from the defection) the grouping issignificant, that is whenever there are at least tworeadings each supportedby at least two manuscripts,to substitute for the qualified symbol. must alwaysalter the meaning in an effective manner. Whethert here is any objection to this will depend on theresultant grouping. Given a constant and con-

I The objection to E : F is that it assumes a specificvariation of F at this point. There is no great harm in this;but , s ince every transcript is more of ten correct than not , theassumption is contrary to probability. The bearing of this willappear when we consider the evidential value of individual manuscripts. : EF, a type-2 variant, of course involves significantgrouping.

Of course, if we knew that the reading preserved by E wasoriginal, i t would be by no means indifferent whether we wrote: E or : E, since in thatcase : E would imply the existenceof xA ABCDF which might be quite untrue so far as the inclusion of F is concerned. Thus when we come to constructa critical text we must be careful no t to reject the reading o f Ein the variant : E on the ground tha t in the variant E itcannot be original.

rinczples 0/ Variation 9sistent grouping ABC : DEF, it would be illegitimate to substitute an unqualified the formulaCEF, since it would indicate the inconsistentgrouping ABD: CEF; ,but, the creat ion of suchgroupings being the only danger of substitution,there would be no objection, in the same circumstances, to writing an unqualified in the formula: DEF. And i t will be observed that we canalways (on sufficient data) get rid of the qualificationby writing the formula the other way round. Thus,in the case supposed above, : CE F is equivalentto : AB, and this may safely be written : AB,since this is consistent with ABC : DEF. All which,of course, comes to the same as saying that wherea manuscript is defective we are at liberty to assumethat its readings, if preserved, would conform to theconstant groupings found elsewhere.1

Redundant grouping can be dealtwith much morebriefly. We previously agreed that. in portions ofthe text for which an additional manuscript Z becameavailable, we would substitute for : the qualifiedsymbol In regard to this only two observation sappear necessary. So far as the relationship of theoriginal manuscripts is concerned we can simplyleave Z out of consideration, just as we are forced toleave lost manuscripts throughout. As regards therelationship of Z to the originalmanuscripts, evidencecan, of course, only be obtained from those portionsof the text where Z is present. In these, therefore,simply becomes a new subject to precisely theI Cases may arise in which, whatever reading we assume forthe defective manuscript, the grouping will remain anomalous.But in these the anomaly is independent of the defection, andwe must merely be careful no t to assume a reading tha t willincrease it. This does not mean that Z can throw no light on the relationship of the original manuscripts. Its position with respect tothem may be such tha t its presence will resolve the ambiguity ofthe three (original) texts.


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3 The Calculus Variantssame rules as the old, and such a further qualifica-tion as may be .t reated in exactly the same wayas

Resolution VariantsWe have seen that variants of complex types ariseas the product of those of simple types, into which,therefore, they may, on sufficient evidence, beresolved. But in the examples we took to illustratethis fact we assumed that the relation of the manu-scripts was known, and then considered howcomplexvariants might arise. In practice, of course, weapproach the problem from the other end, and seekto infer the relation of the manuscripts from theform of the variants. If, therefore, we are to resolvevariants of complex types, it must be on the basis of

the intrinsic nature of the readings, and not on thatof any assumption regarding the derivation of themanuscripts. The question consequently arises,whether in fact complex var iants do exhibi t pecu-liarities of such a kind as to enable us to form anyopinion as to the ir mode of origin. I do not thinkit will be questioned tha t to some ex tent this is so,but the discussion will necessitate judgements re-specting the character of individual variants of a kindwe have not hitherto contemplated. So far we have.merely been concerned with variation, in whichreadings are treated as being simply either the sameor different. It is now necessary to introduce thenotion of dz vergence which takes account not only ofthe fact, but also of the degree, of difference. tthe same time, it is important to observe that thejudgements in question not involve the correctnessor originality of the various readings, but only theirrelatiwe resemblance.We have previously met with the idea of diver-gence of lines of descent and the progressive

Resolution Variants 31divergence of the text in different lines of descent.These may for convenience be called derivationaland textual divergence respectively. Textual diver-gence we saw to consist in the generally increasingnumber of variants. It finds expression in the com-pounded variational formulas, and since, as we alsosaw, these take account of simple variants only, it isclear that textual divergence would be unaffected bythe total absence of complex variants. The notionwith which weare here concerned, namely variationaldivergence which will.always be meant when the term,divergence is used without qualification, appliesonly to complex variants, and to these only indi-vidually. It does not lead direcdy or formally totextual divergence, since this is the generalizednotion of simple var iat ion (as shown in the com-pounded formulas), but one might regard variationaldivergence as the natural compounding of simplevariants in the manuscripts themselves. The onlyreason for introducing the notion specifically is inorder to allow of an analysis that will make possiblethe resolution of complex variants.In the case of variation we regarded it as in-different whether A was said to vary from B or Bfrom andwe wrote (generally) {or the largestgroup, without suggestion as to o r i ~ i n a l i ~ y . So, invariational divergence, when a particular reading issupported by more manuscripts than any of its rivals(when, that is, it appears in collation as the readingof we are at ,liberty to regard it as basis, andthe others as divergent from it. But it is, ,ofcourse,evident tha t, behind this var iat ional divergence,there lies what may be called genetic diflwgence that

In discussion, just as it is sometimes convenient to use fora smaller group, i t is permissible to speak of the divergence ofa larger group. It is of course, the order not the size of thegroups that is important, and it is often convenit nt to useindependently of size to indicate the point of view.


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32 The Calculus of Var1:antsis, the divergence of readings, not merely in theorder of their likeness, bu t in the sense in whichthey have actually arisen. We shall not be explicitlyconcerned with this notion, but it is sometimeshelpful to bear the distinction in mind.

In what cases do complex variants show suchpeculiarities as to enable us to form some opinion asto how they arose? or rather: In what cases do thereadings possess a clear order of similarity? I t is,properly speaking, the second of these questions thatwe must ask, since to say that one reading arosefrom another would involve a judgement of originality. And we must observe that it is not mereviolence of variation that is in question. If onereading makes an army march four parasangs,another five, and another six, there is nothing inthis to establish any order of similarity; the variations may have occurred in any manner. I thinkthat it will be found that in all instances in which acomplete order emerges the readings themselves (asdistinct from the variant) are in a manner complex.They comprise, namely, two or more part s orelements, and i t is these that are separately variedin the different manuscripts, rather than the readingas a whole. This will become clearer in the sequel.The investigation of the types of divergence is soclosely bound up with the possibilities of resolutionthat the two must be considered together. Infollows, will, I think, help to make the a rgumen tclec lr if I take concrete examples in illustration, butfor safety s sake I will add, in parentheses after eachexample, a purely conventional symbolic formula,which shall express the same variational relation,but which, being otherwise meaningless, will run norisk of introducing irrelevant ideas.We may begin pur discussion of divergence andthe possibilities of resolution by observing thatcer ta in complex var iants are only apparent being

Resolution of Variants 33due to mere chance convenience in the manner ofrecord. For instance, i t may be desirable to givea collation in the formthou ar t AB : thou . s CD : he is EFxy yz AB : xy . . zy CD :yx zy EF .

But here it is evident that the readings could berecorded with equal correctness in the formthou : he EF and is : ar t ABxy yx EF and zy yz AB),

and in this case, therefore , the resolution of theoriginal type-3 formula AB : CD : EF as: EF . :.AB),where each factor is of type 2 is obvious. It doesnot follow that this resolution preserves the fullsignificance of the variants, but i t preserves all thatthe calculus can deal with, and a study of the symbolicformulas suggests that any further inference wemight be inclined to draw from the particular example might well prove incorre.ct. It is only afterapparent cases of this nature have been eliminatedthat the real problem of resolution presents itself.Before passing on we may observe that this example also illustrates another point. If we startwith the reading thou . . . art then t/lou is mustbe a variant. of this, and he . a further variantof thou . s and the divergence w i l ~ be successive.If on th : other hand we start with the readingthou . S then both thou ar t and he . smust be direct variants of this and neither of theother, and the divergence .will be independent.Consequently, i f the grouping had been, for instance,

thou . . . is thou . . . ar t E hI . . s F1 For example, we might be tempted to say that because thereadinKof CD is erroneous it cannot be original. But this is notso:. it may very l i k ~ l y p r s ~ v an error of the archetype forwhIch the more plausIble readmgs of AB and EF are alternativeemendations.


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4 The Calculus Variantsthe 9ivergence would have been classed as inde-pendent ; but i f i t h ad been for instance either

thou . ar t : thou . is E : he isF orhe . is : thou . s E : thou art Fthe divergence would have been classed as suc-cessive. Thus the distinction between independentand successive divergence is seen to depend uponthe point of view and remains merely formal untilthe introduction of the notion of originality givesone point of view preference over another. Whenoeit also follows that if resolution can be effected inone case i t can also be effected in the other.Although of course the majority of complexvariants are not apparent in the sense of the examplewe have been discussing and are incapable of beingquoted except in their own proper types a considera-tion of that example may suggest a method ofdealing with them. Take for instance the readings

composed AB : composing CD : reposlng EFxyyx AB : xyxy CD yxxy EF .

These can be quoted in no other form ye t the caseis essentially similar to t he former. But though wecannot here immediately resolve the formulaAB : CD : Fas : EF . : AB

we can by first considering only those readingsthat agree in the ending and differ in the beginningof the word write the first factor as : EFand then by considering only those that agree inthe beginning and differ in the ending write thesecond factor as : AB. And thi s method isgenerally applicable even where there is no possibilityof splitting up the readings into component elements.Thus if we take the readingssay AB : tell CD : sng EF xxAB : xy CD : xz EF

then by first neglecting AB we can write : EF

Resolution Variants 5and next by neglecting EF we can write : ABjust as before. There fore a complex var iant suchas AB: CD : EF can always be represented as

: EP . : AB or as the product of somesimilar pair of factors and i t is upon the handling ofthese factors that possibility of resolution depends.This use of the qualified symbols ~ A B ~ E F c isof course slightly different from that with which weare already familiar. Here in writing ~ A B we takethe group AB as hypothetically absent in order forthe moment to disregard it The conditions underwhich ~ A B ~ E F and so forth can be replaced by theunqualified symbol are generally the same as beforebut we shall find as we proceed that the fact thatthe manuscripts indicated are not really bu t onlyhypothetically absent allows us to take certainliberties which would not be generally permissible.Here then we have what may be called theprinciple resolut t on. It consists simply of takingthe differences that go to build up the complexvariants one at a time. The possibility or legitimacyof so doing follows of course from the fact that as wesaw before all textual divergence is generated stepby step through variation in individual manuscripts.But since we are now tracing the process backwardscertain ambiguities arise which were not previouslyapparent ambiguities that find formal expression inthe presence of the qualified symbols. Whether theresolutions are effective or merely formal dependsupon the possibility of removing these qualifications.There is one class of cases in which this is alwayspossible. From what has been said above i t is c lea rthat such a variant as : E : F becomes whenfactorized : E . : F . But in our previousdiscussion of defective groupswe saw that in variantsof type I the qualification of the symbol may alwaysbe omitted. It follows that : E : F may alwaysbe written : E . : F . Thus variants in which


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6 The Calculus of Variantsthere is only one t rue group can always be resolvedat sight without further inquiry.l In other casesthe qualification will no t automatically disappearand i t is only where through the operation of specialconditions to be considered later the qualificationcan be removed from all fac to rs that an effectiveresolution can be obtained.I t will have already become apparent that weshall have to consider three main types of divergence namely independent successive and indeterminate. But since the distinction between the firsttwo of these depends merely upon the point of viewa more fundamenta l difference is that of indeter-minate divergenceon the one hand in ~ h i c h no orderof s imilarity can be established and determnatedivergence, in which the order is complete on theother. Independent and successive divergence arethen seen as varieties of the latter. Of these wewill now take examples and consider what can beach ieved in the way of resolution. We shall ofcourse have to t ake completely significant complexvariants and these for our collection of six manuscripts only will be of the form AB : CD : EF. Inthis I sha ll use the symbol to- indicate the pointof view from which I wish to regard it writing

~ : CD : EF or ~ : AB : EF or : AB : CD atwill.The formal nature of the distinction betweensuccessive and independent divergence makes itconvenient to take a si ngl e exanl ple to illustrateboth determinate varieties. Le t this be affordedby th e readings

B o you I t ll xyyxCD o you I say xyxyEF I say to you yxxy).

Here the order is completely determinate the first But see p. 40 note 2.

Resolution of Variants 7and last readings resemble one another less thaneither does the second that is the second is inter mediate between the extremes of the other two.Let us now look at this from the point of view ofCD writing the formula AB : EF. If thereading of thi s group is original those of AB andEF must be independent ly derived from it ; thereading of l\ cannot possibly have arisen throughtha t o f F nor th e reading o f Fthrough that o fAB. Thus the instance is one of independent diver-gence And since the two variations arose sepa-rately each might have occurred without the otherand it is therefore evident that the complex variant~ : AB : EF is merely the product of the two simplevariants ~ : AB and EF into which it mayconsequently be resolved. A formal proof of thiswhich makes no assumption as to originality is asfollows. ~ : AB : EF when factorized becomes: AB . : EF . But in the first factor EFis not really absent and s ince i ts reading is closerto that of CD than to that of AB it .can neverhave formed a group with AB in opposition to theother manuscripts and we may therefore safelyreplace 2 :EF by 2 : The same argument appliesmutatz s mutandis, to the second factor likewise inwhich may equally be replaced b y ~ It followsthat where divergence is independent resolution isalways legitimate and looked at from such a pointof view as t o make the divergence independent thevariant AB : CD : EF in the particular exampleselected becomes : A . : E F .Le t us now look at-the same readings from thepoint of view of AB writing this a s ~ If thisgroup preserves the original reading it is c lear thatthat of CD mus t have a ris en from it and that ofEF by fur ther variation from that of CD. Thusthe i ns tance is one of successive variatz on. Here itis evident that the complex variant ~ : CD : E F is


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38 The Calculus Variantsthe product of the simple variant : CDEF anda further variation of EF from CD. But : CDEFis of course equivalent to : AB and since varia-t ion in EF might have occurred without the previousvariation in CDEF the variant CD : EF is equiva-lent to : EF . The formal proof would be thus:

: CD : E F when factorized becomes: CD . : EF .In the second factor since the readings of ABand CD are closer than those of AB and EFthe lat te r pair cannot have formed a group apartfrom CD and w may therefore safely replace

b y ~ Th e opposite is the case in the firstfactor which cannot therefore be simplified sodirectly. But : CD is equivalent to : Band since the readings of EF and CD are closerthan those of EF and AB we may now replace

b y ~ Thus : CD . : EF is equiva-lent to : AB . : EF .Since looked at from the point of view of EFthe case stands on the same foo ting as it doeslooked at from that of AB the resolution is ofcourse the same. Consequently where divergenceis successive resolution is likewise always legitimateand looked at from such a point of view as to makethe divergence successive th e variant AB : CD : EFin the particular example selected becomes: AB : EF .Thus as we should have expected given certainreadings showing a completely determinate order ofdivergence the resolution of the complex variantwill be the same whether we look upon it as inde-pendent or divergent. Th e point of view will beimmaterial. In practice then we may adop t t hesimple rul e t o pu t for the intermediate readingand construct our two factors by opposing w ththe two extremes. But it must no t be supposedfrom this that when the point of view is given it

Resolution Variants 39is indifferent for the purpose of resolution whetherthe divergence is independent or successive. It isevident indeed that the reverse is the case. Let usindicate the point of view by in t he var iant : CD : EF then if the divergence is successivethe resolution will be : AB . : EF whereasif the divergence is independent the resolution willbe : CD . : E F .Next as an instance of z ndeterminate dvergencele t us t ake the readings

To you 1 te ll xyyx)CD To you I say xyyy)EF oy u I sing xyyz).Here no two are more closely similar than anyother two so that no order of divergence can beestablished. Supposing to preserve the originalreading there is nothing to show whether those ofCD and EF arose from it independently or onethrough the other and if so which through which.Hence the term indeterminate. CD : EF whenfactorized becomes ~ E F CD . : EF butsince we know nothing of the affinities of the read-ings it is impossible to get rid of the qualifiedsymbols and no effective resolution can be obtained.Where therefore divergence is indeterminate sig-nificant variants of complex type are insoluble.Cases sometimes ar ise in which though thereseems to be some order of divergence it is notcomplete and may be called sem-determinate. Asan example suppose that we take the readings

amorous AB : lovin/ CD : llt/and E F xyxx B : xyxy CD : xyyx EF .H e re the re can be no doubt t ha t t he readings ofCD and EF resemble one ano ther more closelythan e ither of them resembles tha t of AB but atthe same time it is no t possible to say that any one


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The alculus of Variantsis intermediate between the other two. If we suppose that ofAB to be original, we m ay reas onablyassume that those of CD a nd E F d id not ari se fromit independently, but one through the other, thoughthere is nothing to show which was the earlier.We m ay call this quasi success ive d ivergence. If wetry to resolve the v ar iant , we shal l find that oursecond factor CD : EF is ambiguous, representingeither : CD or : EF or, in the formal proof, weshall find that neither ~ E F CD nor : EF willsimplify). If on the other hand we suppose oneof the othe r rea dings to be original, say that of EFt he n t ho se o f AB and CD m ay have a risen from itindependently, or that of AB may have arisenthrough t ha t of CD. We m ay call this quasz inde-pendent d ivergence but it is not distinguishable, sofar as resolution is concerned, from t he indeterminate type. Our results maybe summarized as follows. Vlhenever the proposed factors of a complex varia nt a reof type the resolution is always legitimate. 2 When,however, one or both of the proposed factors areof type 2 no resolution is legitimate if the divergence is indeterminate or only semi-deterlninate),but i f a complete ord er of similarity between thereadings can be established, that is, if the divergenceis determine d a s e ithe r successive or independent,

1 In someparticularcases resolution would appear to be possible.Thus when divergence is quasi-successive it would seem legitimatet o r esol ve suc h a var ia nt as : DE : F as : DEF . : F), a ndsuc h a var ia nt a s : D : EF as DEF : D). On the otherhand, when divergence is quasi-independent, it would still seemlegitimate to resolve D : EF as D) . EF), but not: DE : F as either DEF . : F) or DE . : F).But i n a ny ca se t he se par ti cula r var iet ie s, e ven i f t he or et ic al lysoluble, are hardly worth considering.

2 Where, however, the divergence is determinate these casesa re best t re at ed i n t he same way as t he re st . the divergence isindependent the result will be the same, bu t it will differ if thedivergence is successive.

Resolution of Variantsresolution is always legitimate, though it will differin the two cases. In practice, whenever divergenceis det ermi nat e, i t is con ven ient t o look at i t fromsuch a po int of view as to mak e independent andto resolve accordingly.The discussion above has proceeded on the supposition that our complex variants were of type They may, of course, be of t ha t o r any higher type.As we p ro ceed the questions of divergence andresolution naturally become more complicated, butas no fresh principles seem to be involved, thematter need not here be pursued, especially asinstances where the divergence is completelydeterminate in variants of higher type than 3 arena turally rare. Thus we see tha t not a ll varia nts of cOlnplextype are immediately soluble, but this res ults notfrom a nything in the nature of variation, but simplyfrom the fact that divergence is sometimes indeterminate, that is t o say, that it d o e ~ not always appearfrom the nature of the readings whether, assumingsome one t o be original, t he variations b y whichthe others a rose were succ ess ive or independent.One or o th er t hey must h av e been , an d as soon aswe know which, all variants can be resolved.Wherever divergence is d e t ~ r m i n a t e there is adefinite o r d ~ r a mong the readings . This may bewritten Z when the reading Y is intermediatebetween the other two, and in respect to a particularvariant the order of the nlanuscripts may be deduced

1 We saw befo re t ha t t o p ro du ce c om pl et el y d et er mi na tedivergence in a variant of type 3 it would appear that the readingmust contain two independently variable elements. It would,therefore, seem t ha t a type-4 va ria nt would n ee d th ree s uche le me nt s, a nd so on. But t he re may be c ompl ic at ions.I It is well t o i nsist upon t hi s, since i t is somet imes t empt ingto suppose that insoluble variants point to independent variation.Th is is no t so. It is only l ac k of evidence, never the nature ofderivation, that makes variants insoluble.

G


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4 The Calculus of Varz antsfrom that of their readings and written X Y Z.This order is of course the same as Z Y X. Fromthe point of view of Y the divergence is indepen-dent from those of X and Z successive. When thepoint of view is determined by the establishment oforiginality direction is introduced. Thus if thereading of X is original the ordered series X Y Zwill become the directed series X > Y > Z if tha t o fY then X < Y > Z. H ere i t must be clearly under-stood that the direction applies primari ly to thereadings and only indirectly to t he manuscripts. Ifwe wr ite X>Y>Z what we mean is that thereading of Z is derived from that prese rved in Yand the reading of Y from that preserved in X no tthat Y is itself derived from X or Z from Y. Ifthe descent of the manuscripts themselves wereintended we might wri te X y

Ord er a nd direction are notions that appl y toindividual readings. It is clear that in some sensethey must also apply to several or all the readingsof the different manuscripts and thus to the manu-scripts themselves. However the generalization ofthese notions presents very considerable difficultiesand since the knowledge that could be gainedit does not differ in nature or extent from that to bederived from a collect ion of simple variants therewould be no object in t rying to substitute it for thefar more convenient method of the compoundedvariational formula and the possibilities need no tbe further considered.At the end of this lengthy discussion it may seeml ike a bad joke i f I add that the question of resolu-tion is after all of secondary importance. Ye t thisis ; in a manner t rue. It seems probable that inmost cases the natural variants of type 2 will provesufficient to establish the manuscript relation andthat no material accession of evidence will resul tfrom the resolution of complex types. On th e other

Resolution Variants 43hand i t may well happen in the case of some textsp r e ~ e r v e in a large number of manuscripts thatVarIatIOn has advanced to such a point that practi-cally all variants a re o f complex type. And in .anycase for the complete elucidation of the problemresolut ion will a lways be necessary. There is adoubl e reason for this. In the first place we requireto make sure that the complex variants especiallyif relatively numerous do not conceal any anomalousgroupings such as would either invalidate ourinferences as to relationship or reveal the presenceof conflation as I shall explain later. And furtherresolut ion is needed to enable us to ascertaintotal number of cases in which a manuscript isnecessarily in error and so to place it among itsfellows in order of merit another question that willengage our attention before we have done with thecalculus.Method nd Limitations the Calculus

When the variants have all been recorded andwherever possible resolved intb their simple factorsthe next step is to sor t t he m into similar classeseach comprising a single form alone. If all is wellthese classes will be constant that is to say that outof all possible forms of type 2 variants l certain oneswill predominate to th e practical exclusion of others:f u r t ~ e r m o r e the predominant classes will be mutuallyconsIstent.I f all is not well that is i f the grouping is through-ou t random or if inconsistent forms are of frequentoccurrence the relationship of the manuscripts cannotbe accounted for on the hypothesis of simple tran-scription; some sort of conflation has somewhereto be assumed i e. we must suppose that at some

1 All forms of type I should occur otherwise the collection ofmanuscripts examined w not be purely collateral.


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The Calculus Variantspoint th e genetic relation has been, not one-one orone-many, bu t many-one). This is a matter thatlies beyond th e scope of th e calculus an d cannot beproperly. d i ~ c u s s e d in t he se pages: since, however,th e appltcatlon of th e calculus to th e problem raisessome rather interestingspeculations, I have venturedto touch on th e matter in a final note. 1Th e e xt en t t o which inconsistent grouping maybe e xpe ct ed t o occur when

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