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#### The Calculus of Calculus: W. W. Greg and the Mathematics of Everyman Editions by JOSEPH A. DANE and ROSEMARY A. ROBERTS

There are four recorded sixteenth-century copies of Everyman, each from a different edition: two by John Skot (1528, STC 10606) (1535, STC 10606.5), and two by Richard Pynson (1515, STC 10604) (1526, STC 10604.5). The two Skot copies are complete. The first Pynson copy is a four-leaf fragment (including the colophon); the second Pynson copy lacks signature A (12 leaves). In 1910, in the final volume of his critical editions of these fragments, W. W. Greg considered what these surviving copies indicate about the popularity of this text; how many editions might have been produced in the early sixteenth century? Since no two of the four surviving copies are from the same edition, what can we say about the total number of editions that were produced? Greg's note is as follows:

It is obvious that, if no more than 4 editions are printed, it is very unlikely that, of 4 surviving copies, each should belong to a different edition (in point of fact the chance is only 3/32 or about 1 in 11), and that as the number of editions printed increases so does the probability of such an occurrence. There must therefore be a point (a particular number of editions) at which the chance approximates most nearly to 1/2. That number is 10, for which the actual chance is 1/2 + 1/250. Ten, therefore, is the smallest number of editions which make the actually occurring arrangement as likely as not to occur.[1]

Greg attributes the solution to this problem and the mathematics to J. E. Littlewood, professor of mathematics at Trinity College, Cambridge. But Greg does not make explicit the nature of the precise problem he presented to Littlewood, nor what assumptions he included with that problem.

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The purpose of the present paper is to reconstruct those assumptions, and to deal more generally with the implications of the use of probability in such bibliographical problems. We will deal with the question in two parts, based on the two sets of assumptions that Greg and his mathematician seem to have used. In part 1, we will first reconstruct the mathematical model that was used by Greg and Littlewood. The assumptions specific to the various calculations are as follows, and Greg and Littlewood must at some point have discussed these explicitly:

1. The total number of books is “large” (say, “over 100”; Greg's calculations are not applicable if the hypothetical editions consist of only one or two members);

2. All editions have approximately the same number of books.

We will then relax these restrictions in order to see to what extent what might be called the “Printers of the Mind” situation (under which all printers print the same number of book-copies in each edition) operates in the real world of printing (where printers produce editions of different sizes).

In part 2, we will deal explicitly with a second set of assumptions—assumptions that are necessary to construct the mathematical model. These are the following:

1. All book-copies under consideration have an equal chance of surviving;

2. Such book-copies survive independently of each other;

3. Each book-copy either survives or it does not survive.

Such fundamental assumptions were perhaps never articulated by Greg and Littlewood in their discussions, and Greg's note contains not even an allusion to them. As a mathematician, Littlewood might have found such assumptions so basic as hardly to deserve mention; but for bibliography, they are of course extremely problematic.

Our expectations were mathematical and bibliographical. Mathematically, we expected that the solution to the problem would depend on fairly restrictive assumptions of regularity: uniform edition size, and minimum edition size. These expectations were not entirely accurate, and there are interesting implications here easily brought out through the use of a modern calculator that might well have escaped Greg and Littlewood. Bibliographically, we expected to find that material and quantitative evidence such as that at issue here can only support pre- existent assumptions—that any mathematical model would do no more than reflect the set of initial bibliographical assumptions brought to it. Again, the mathematics suggests that this expectation also needs modification.

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Part 1: The Greg-Littlewood Solution

According to Greg, the likelihood that any four surviving book copies will belong to four separate editions is 3/32. If we take extreme cases (for example, edition sizes of one), we see that this figure does not apply to all cases: if each print-run is one, then the chances that four extant copies belong to four separate editions is 100% regardless of the number of editions. These figures, therefore, seem to be based on at least one unstated assumption: (1) that the total number of book-copies is sufficiently “large.” A second assumption is also necessary here—(2) that each edition is the same size. This is the restriction that we will relax later in the discussion below.[2]

Let there be n editions with k1, k2, k3,..., kn the number of book-copies in each edition. We assume uniform edition-size, that is: k1 = k2 = k3 =... =kn. We will call this edition-size k. The probability that four surviving book-copies will come from different editions is:

In this formula, is the number of ways to choose the 4 editions from n editions; is the number of ways to choose 1 of the k book-copies from each of the four chosen editions; and is the number of ways to choose 4 book-copies from the nk available. Now:

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As long as nk, the total number of book-copies, is “large” (about 100) this is approximately:

 (n-1)(n-2)(n-3) n3

For values of n, the number of editions, ranging from 4 to 10, we get the following values for the probability that 4 copies belong to different editions. This is clearly the same series and the same solution presented to Greg by Littlewood:

 No. of editions: 4 5 6 7 8 9 10 Probability: 3/32 24/125 60/216 120/343 210/512 336/729 504/1000

As the number of editions increases beyond 10, the probability increases and can be made arbitrarily close to one. Greg chose to focus on the number of editions needed to make the probability exceed 1/2—we call this the “magic number.” Clearly, however, it is possible for the four surviving book-copies to come from different editions with any number of editions (as long as there are four or more) and Greg's choice of 1/2 is somewhat arbitrary.

Since the figures in the above series are exactly Greg's figures (504/1000 = 1/2 + 1/250), it appears that Greg and Littlewood simply assumed for these calculations that all editions were the same size. Based on generally accepted edition sizes (in the early twentieth century, these might have been estimated at between 500 and 1500 for pamphlets such as Everyman), it is interesting to see what the definition of “large” must be in order for Greg's math to work. Contrary to our expectations, the formula is not particularly sensitive to variation in edition size, and works quite well for any real world estimates.

Assuming edition sizes of 1000, the specific probabilities of four surviving copies belonging to four separate editions are 3.005/32 (if there are only four editions) and 5.043/10 (if there are ten editions); the number of editions required to produce a greater than 50% chance that four copies will belong to four separate editions is ten. Greg's figures are nearly the same. Assuming an edition size of 500 yields the following corresponding probabilities: 3.009/32 and 5.046/10. There is no significant change. If we assume 100 book-copies per edition (perhaps the lowest figure anyone might accept), the corresponding fractions are 3.045/32 and 5.070/10. Again, there is little change. At 10 book-copies per edition, there is again, much to our surprise, little change in these fractions: 3.501/32 and 5.355/10. To attain more than a 50% chance that the four copies will belong to four separate editions still requires ten editions.

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Under the assumption of editions of uniform size, then, the calculations of Greg and Littlewood seem to hold for any range of edition-size that bibliographers would accept. The probability is not particularly sensitive to edition size—a result we found somewhat surprising.

Now let us relax the operative assumption here of uniform edition size (that is, that k1 = k2 = k3 =... = kn); we doubt Greg and Littlewood went this far. The mathematics is slightly more complex, but the formulae are simple variants of the formulae above.

Again let there be n editions with k1,... kn the number of book-copies in each edition. It is convenient to call the average edition size k. Then the total number of book-copies is nkİ. Suppose that the four surviving books come from editions 1, 2, 3, and 4. The probability of this is:

Clearly this depends on k1, k2, k3, and k4—the number of book-copies in editions 1, 2, 3, and 4. If we calculate the probability of the four surviving books coming from editions 1, 2, 3, and 5, this will depend on the number of book-copies in editions 1, 2, 3, and 5, and will be a different value if k4 and k5 are different. To find the probability that the four surviving books come from different editions (the problem as stated by Greg), we must add up the probabilities of the books coming from every possible subset of four of the n editions. And to calculate this, we need to know how many book-copies there are in each edition. There is no getting around this.[3]

The formula above for editions 1, 2, 3, and 4 can be written:

If the total number of book-copies nkİ is “large,” (that is to say if 1- 1/nkİ, 1-2/nk, etc. are each nearly equal to 1), then the formula is approximately:

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The value of this probability clearly depends on how close the edition sizes are to the average size although as above, it is not particularly sensitive to the average size of a print run. If all edition sizes are approximately equal, then the probability that the four surviving books come from different editions will be close to the probabilities given by Greg and calculated above.

By assuming here that edition sizes are approximately the same, the formula is the same as the formula we used above, since in this case, every edition size will be close to the average and each of the factors (k1/k̄),(k2/k̄),(k3/k̄),(k4/k̄) will be approximately 1. The question remains: how close do the number of book-copies per edition have to be to the average?

To answer this requires some calculations that are very easy with a modern calculator, but would have been tedious and time-consuming for anyone in 1910. Perhaps Littlewood did not mention this particular aspect of the problem to Greg.

Let us consider an average edition size of 1000, and editions of 500, 750, 1250, 1500. The probability that four book-copies would come from different editions if only four editions were involved is, instead of 3/32 (Greg's figure), 2.113/32. For, say, eight editions, two of each size, we get 182/512, instead of Greg's 210/512. With twelve editions, three of each size, the probability would be 907/1728 instead of 990/1728 (the probability one would get using Greg's restricted formula). What we call the “magic number”—the number where the probability exceeds 1/2—increases to about 12 with print runs of these sizes.

It is interesting, of course, that if we assume a low average print run (say, 100) and use specific print runs in the same ratio as above (50, 75, 125, and 150), these figures do not change significantly. Again, the formula is not the least sensitive to the size of the average print-run. However, it is extremely sensitive to deviations from average: if, for example, we substitute the figures 90, 95, 105, 110 for those above, we get results much closer to those in Greg's calculations: 3.007/32; the implied “magic number” for sets of editions that differ from this average in these ratios would still be 10.

To take a more extreme example, but one that can be understood without resorting to the calculator, consider editions of only two sizes, 1000 and 1.[4] Assume further that for every edition of size 1000 there is an edition of size 1. In this case, the small editions will be nearly irrele-

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vant. The chances of four survivors being from different editions will be about 3/32 for eight rather than for four editions (4 of size 1000 and 4 irrelevant editions of size 1). The “magic number” will, instead of 10, be 20.

These calculations can be continued ad infinitum or ad nauseam. The cases above assume a regular distribution of edition size from average, but there is of course no need to do that. And we close with an even more extreme version of the last example. Let us suppose, for example, that there are an unknown number of editions—that one of them again has 1000 book-copies, and that the others are tentative runs with 1 copy each. You will need no calculator to see that the chances of four copies coming from four different editions if there are only a few of these trial editions is nearly 0. The calculator, for what it is worth, tells us that we would need 1594 such editions to reach the “magic number.” If we assume that these small editions contain, say, 10 book-copies, the magic number is 165.

In conclusion, Greg's calculations do not require print-runs that bibliographers would consider unusual; the notion of “large” required mathematically does not seem bibliographically problematic. But they do require an assumption of uniformity—that print-runs are close to average. And it is difficult to see how a bibliographical argument could be constructed to provide solid support for such an assumption. What is also interesting (and rather unnerving to one of us, who began this study believing that Greg had overestimated the number of early Everyman editions) is that relaxing some of Greg's unstated restrictions produces in many cases larger estimates of numbers of editions—precisely the point Greg seemed to be trying to make! Does this mean that STC—by any standards a monumental achievement of enumerative bibliography—grossly misrepresents the total number of editions that were actually produced before 1640? and are we to imagine dozens of lost editions of Everyman?

Part 2: General Assumptions

More than twenty years ago, Robert Potter argued convincingly that the popularity of Everyman is to some extent illusory—a function of its later reception in the last two centuries, and not a fact of its early reception.[5] Obviously, Greg's estimate of ten contemporary editions of the play disputes such an argument, as indeed does our own skeptical chal-

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lenge to Greg's mathematics: when we relax Greg's overly restrictive assumptions of uniformity, we seem to end up with an even more popular and widely disseminated Everyman. How can this be?

Greg, interested in discovering evidence of a popular Everyman, might have welcomed this result, as we initially did not. But this result forces us to look at a second set of assumptions—assumptions so basic to the problem that Greg never mentions them:

1. All book-copies under consideration have an equal chance of surviving;

2. Book-copies survive independently of each other;

3. Each book-copy either survives or it does not.

Any bibliographer who looks at these assumptions knows that they simply do not apply to the actual history of books and the survival of book-copies. Although bibliographers occasionally speak of the “accidental” survival of book-copies, there is no such thing as “accidental” in the mathematical sense, that is to say, “random.” In fact, a staple of bibliographical study and the study of provenance is the articulation of specific reasons for survival. Let us take each of these assumptions in turn in their more general form.

1. Book-copies have an equal chance of surviving. They do not. Some early books are printed on vellum and are thus more likely to survive as, say, binding material in other books than those printed on paper (early Donatus grammars are the most obvious example); some are large and luxurious (the Gutenberg Bible has an extremely high survival rate). Some are beautifully illustrated; some are by Shakespeare. There is no end to such particulars, and let us consider only some factors that might affect the survival of copies of Everyman. Everyman is formally a play, and early printed copies of English plays acquired value in a way that other early English books did not: they were formal predecessors to Shakespeare.[6] The history of the reception of Everyman shows this clearly, with the introduction of the play text into the dramatic canon in the eighteenth century and the surfacing of other copies in the nineteenth.[7] There are other reasons: it was printed in black-letter by Pyn-

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son, and such books acquired value as they were incorporated into the antiquarian book trade. Copies were obviously thrown away as valueless in the sixteenth century; but they were less likely to be thrown away in the nineteenth or twentieth centuries; all four copies now are contained in major repositories, and the names that come up in the provenance histories are those of the great English collectors.[8] The survival of any particular book-copy, thus, is dependent on specific external factors, and the disappearance (or surfacing) of book-copies does not follow a random pattern.

Fragments of sixteenth-century English printing, incorporated as binding material in other books, can still be found on the shelves of antiquarian book dealers. Those we have run across have been “Statutes” and we did not record them. And perhaps some unknown sixteenth-century statute has escaped bibliographical history through our negligence. But are we to check the shelves of these same dealers more diligently, imagining that if one sixteenth- century printed text can be found, then any other sixteenth-century printed text might be found as well? Perhaps the pastedowns and flyleaves of that next book will have speakers' names printed in the margins! An undiscovered Hickescorner? A fifth copy of Everyman? comparable to the Bodleian Pynson fragment? At this point in our musings, the real world intrudes. We know that in any real world bookshop, we will have no such luck, and the bookshops containing Everyman fragments are at best “Bookshops of the Mind.” The dealer who ignored the sixteenth-century statute will not ignore an Everyman fragment or an unrecorded speech from King Lear. Books and fragments of books simply do not enter and leave the book trade in random ways.

2. Book-copies survive independently of each other. They do not. If a printer did not sell all his stock, the remaining books would have been destroyed as a unit, not independently. In addition, collectors do not buy and preserve early books at random. Some, like Richard Heber and Henry Clay Folger, deliberately sought out multiple copies of the same edition. Others, like Henry E. Huntington, systematically sold off such duplicates.[9] The survival of any one book-copy is thus potentially de-

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pendent on the survival of other book-copies involved in the problem, and forms of such dependence operate at all stages in the histories of these copies.

3. Each book-copy either survives or it does not. That is to say, when flipping a coin, the coin comes up heads, or it comes up tails; there are no “leaners.” This at first glance seems obvious, but in early printing, it is not at all obvious. Let us consider the copies here. The two Skot copies are complete, and bear all the marks of editions in any bibliographical sense. The second Pynson copy (STC 10604.5), while incomplete, is missing only sig. A, and most catalogues even give its collation as [A6]. But the status of the first Pynson copy (STC 10604) is not at all so clear: it is a binding fragment, consisting of a single leaf. There are no marks of use in this fragment, that is to say, no sign that it was ever read as a book.[10]

In bibliographical language, this fragment seems to be “printer's waste” (waste produced at the print shop from materials never incorporated into books), not “binder's waste” (waste from books that have been used and abandoned).[11] The difference in this case is crucial: “binder's waste” is from books that were distributed and read in the real world —it is thus evidence of both book-copies and the edition that produced those book-copies. But “printer's waste” only implies what might be called a “printer's project”; it implies nothing about whether that project was ever realized: we in fact do not know whether such a thing as this “first Pynson edition” of Everyman ever existed.

Furthermore, catalogues dealing with such fragments do not record them in ways consistent with the ways in which they record other books. For incunables, fragments of many early gramamr books are assigned numbers in the Gesamtkatalog der Wiegendrucke and in union catalogues such as the British Library Incunable Short-Title Catalogue on the basis of location of particular fragments: the roughly 400 catalogue

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entries for the Donatus Ars minor in GW and in ISTC refer to 400 sets of fragments, each in a specific location. The number of editions these might represent is in GW taken up in the introductory note only, where the numbered entries are organized according to presumed editions.[12] Such fragments need to be distinguished from fragments that show evidence of having been incorporated within specific book-copies (for example, grammatical fragments with some marginalia, or owner's marks or signs of use). We are not aware, however, that attempts to sort these out as editions have specifically used such evidence; unmarked fragments seem to have the same bibliographical status as marked ones.

The Pynson fragment provides that mathematically malevolent example of a book that we cannot say exists or does not exist. We do not know for certain whether there is such an edition, even though we have an apparent fragment of a book-copy here. And thus, we cannot say whether we have a survival or not. And we cannot thus assume that bibliographical catalogues themselves reflect in any systematic way the actual way that book-copies survive. Catalogue decisions reflect the exigencies of cataloguing, not the reality of book-copies circulating in the real world. We are forced to deal with the fragment as a unique individual, not a representative one.[13]

Conclusion

In 1910, Everyman was still in the process of being canonized, a process Potter argues was abetted by the establishment in the twentieth century of the genre, the “Morality Play,” that Everyman seemed to exemplify. Greg spent considerable effort in editing Everyman and must have felt that what he had in the surviving copies was evidence of an important text, and at least a moderately popular one (surely printed more than four times); that is, he must have assumed that popularity or even historical import was a function of “number of editions,” particularly the number of “ordinary” editions. As a mathematician, Littlewood could do little more than give back to Greg a version of the information that Greg provided him: here is an important, moderately popular English text, and for such texts, it would not seem unreasonable

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to assume multiple editions of uniform size. The moderately popular text Greg defined for Littlewood then received its mathematical underpinnings.

Yet other assumptions were also plausible: Everyman might not be a popular text at all, as evidenced by the absence of any contemporary allusion to it; and the four surviving copies might be anomalous, more an index of Shakespeare's popularity than the popularity of the text itself (they survived as “Shakespeareana”). What appear to be published and disseminated editions might actually be trial editions, aborted editions, and proof-sheets. Perhaps the pattern of survival reflects only the peculiar tastes of early collectors. Such considerations might have led Greg to various conclusions, but they undermine any form of mathematical modelling. In order for Greg to claim mathematical support for any conclusion (and thus to enhance his status as a rigorous bibliographical “calculator”), such inconvenient particularities would have to be ignored.

[1]

W. W. Greg, “Everyman,” from the Fragments of the two Editions by Pynson preserved in the Bodleian Library and the British Museum together with Critical Apparatus, Materialien zur Kunde des alteren englischen Dramas, Bd. 28 (Louvain, 1910), p. 35, n. 2.

[2]

Note that the mathematics here requires an oversimplified version of the relation of print-runs of specific editions to members or products of those print-runs. For purposes of discussion, we are using only two terms here: edition (an hypothesized print-run) and book-copies (particular members of these print-runs). For easily accessible discussion of the mathematics employed here, see, e.g., Carol Ash, The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (And Everyone Else!) (New York, 1993), esp. chap. 1: “Basic Probability”; Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, 2nd ed. (Providence, 1997).

[3]

The general formula could be written as follows:

 Σ all subsets (i1, i2, i3, i4) of the editions 1, 2, ..., n 24⋅ki1⋅ki2⋅ki3⋅ki4 nk̄(nk̄-1)(nk̄-2)(nk̄- 3)

[4]

See note 2 above and discussion below; the simplified definition of “edition” used for purposes of calculation here does not distinguish editions from impressions and issues or even from trial sheets; thus, edition sizes of 1 cannot be discounted as purely hypothetical.

[5]

Robert Potter, The English Morality Play: Origins, History and Influence of a Dramatic Tradition (London, 1975); see esp. pp. 222- 245.

[6]

The more complete of the two Pynson copies seems to have been in Garrick's collection, although Greg himself wavered on this point, and in 1910 denied it; Greg “Everyman”, p. 35. The four copies were known by the early nineteenth century and are mentioned in J. Payne Collier, The History of English Dramatic Poetry to the Time of Shakespeare and Annals of the Stage, 3 vols. (London, 1831), 2:310-312.

[7]

For the history of provenance, see Greg, “Everyman”, pp. 34-35. The first edition of Everyman after its eighteenth-century rediscovery is by Thomas Hawkins, The Origin of the English Drama, Illustrated in its various Species, viz. Mystery, morality, tragedy, and comedy, 3 vols. (Oxford, 1773) (vol. 1); the prefatory essay for Everyman is taken directly from the second edition of Thomas Percy's Reliques. The increasing import of the play to English histories of drama is sketched in part through various editions of Dodsley's multi- volume anthology Old Plays. Everyman is still not included in the third edition of 1825 (with additions by Collier) but first added, from Hawkins, to the fourth edition of 1874, edited by Hazlitt. It is not mentioned in the influential discussion of moralities by Thomas Warton, History of English Poetry, 3 vols. (London, 1778-81), e.g., 2:360-366.

[8]

Copy A (now at the Huntington Library)—Dibdin, Spencer, Heber, Britwell; copy B (British Library)—Huth; copy C (Bodleian Library)—Douce; copy D (British Library) (Garrick? William Herbert). Greg, “Everyman”, 34-35.

[9]

Sales of duplicates in book history are notorious, and objections were made (e.g., in the sale of German incunables) to such practices as early as the mid- nineteenth century. See Bettina Wagner, “The Bodleian Incunables from Bavarian Monasteries,” Bodleian Library Record, 15 (1995): 90-107; P. R. Harris, A History of the British Museum Library, 1753-1973 (London, 1998) 42, 70-71. For sales of Huntington duplicates, see Anderson Galleries sales of March 1916-June 1925.

[10]

We thank Alexandra Gillespie of Corpus Christi, Oxford, for her notes on this fragment.

[11]

The first use of this distinction and its importance in determining the early history of books is by Henry Bradshaw, “List of the Founts of Type and Woodcut Devices Used by Printers in Holland in the Fifteenth Century” (= Memorandum 3) June 1871; in Henry Bradshaw, Collected Papers (Cambridge, 1889), pp. 262-263. See further, Paul Needham, “Fragments in Books: Dutch Prototypography in the Van Ess Library,” in “So Precious a Foundation”: The Library of Leander van Ess at the Burke Library of Union Theological Seminary in the City of New York, ed. Milton McC. Gatch (New York, 1996), 93-110; E. Ph. Goldschmidt, Gothic and Renaissance Bookbindings: Exemplified and illustrated from the author's collection, 2 vols. (London, 1928), esp. 1:119-121, on binder's waste, printer's waste, and “bookseller's waste.”

[12]

Each set of fragments in turn poses its own unique set of problems; see for example, Joseph A. Dane, “Note on the Huntington Library and Pierpont Morgan Library Fragments of the pseudo-Donatus, Ars minor (Rudimenta grammatices) (GW 8995, GW 8996),” Papers of the Bibliographical Society of America 94 (2000): 275-282.

[13]

We would need to consider many aspects of its individuality: the fragment includes a colophon, which might balance its lack of marginalia. Since colophons are generally assumed to have been printed late in a print-run, we might want to assume that there is a good likelihood that this is part of a complete edition, not a mere trial sheet. But as far as we know, there is no way to quantify these speculations.