## Abstract

Declercq's rejection of an otherwise well-supported model is based on demonstrably too narrow an interpretation of the use of Bede's *De Temporum Ratione* and on questionable appreciation of predictive modeling as a complementary alternative to traditional deductive methods. Additional evidence on library holdings further supports previous conclusions regarding the survival of medieval manuscripts.

The model proposed in (*1*) is intended to predict the age distribution of the manuscripts of a text in cases in which certain simple assumptions apply. Applied to data on Bede's textbook *De Temporum Ratione* (DTR) in the time range 725 to 1500 A.D., the model estimates that about 30% of the DTR manuscripts extant in 900 survive today. A nicely fitting least-squares curve that explains 98% of the variance in the number of manuscripts (*R*^{2} = 0.98) is evidence that the model and its assumptions apply rather well in this instance [see figure 1 in (*1*)]. Three other Bede texts yield similar results, reinforcing the conclusion that losses have been far from “practically immeasurable,” as Declercq claims (*2*).

Declercq argues that there were indeterminately more than 1000 DTR copies in 900 A.D., that library holdings in general were likewise indeterminately large, that the fraction of manuscripts surviving must be indeterminately small, and hence that the predictive model applied to DTR must be fundamentally flawed. These assertions hinge on what Charlemagne meant by the word “computus” when he decreed in Article 72 of his *Admonitio Generalis* of 789 that it be studied as part of the standard school curriculum and on how completely this decree was carried out.

The meaning of *computus* was considerably broader than Declercq supposes, and the word itself was by no means so closely tied to the Easter question, as Wallis makes clear: “For the most part, Bede uses *computus* only in the general mathematical sense of `calculation,' e.g. in the title of ch. 1 of [DTR], *De computo vel loquela digitorum* [“Calculating or speaking with the fingers”],... [and] on only one occasion does he use the term to refer to a work on Easter-reckoning” (*3*). It seems reasonable to believe that Charlemagne meant *computus* in Bede's sense, inasmuch as Article 72 is concerned with the education of school-age youths, and the head of his own education system, Alcuin of York, was Bede's academic grandchild. Could not teachers who perhaps learned Easter-reckoning out of DTR in the few college- or university-equivalent schools have made do with officially sanctioned excerpts and anthologies in their own classrooms, as mentioned in Charlemagne's *Capitula* of 809, which addresses the Easter question in greater detail (*4*, *5*)?

Contemporary catalogs of early medieval libraries closely associated with Charlemagne, Alcuin, Bede, and their successors suggest that considerably fewer than 1000 copies of DTR and its forerunner, *De Temporibus* (DT), were called for, now or in 900 A.D. Hundreds of surviving book lists show that most early medieval libraries contained no more than a dozen or two volumes (*6*, *7*) and seldom contained DTR and/or DT (*6*). The most extensive catalogs of the very largest libraries list no more than a few hundred volumes (Table 1). Conservatively interpreting the uncertainties in Table 1, copies of DTR/DT are relatively rare, roughly 0.6% of the six libraries' holdings (∼0.006 ± 0.005, with 95% confidence). On this basis, of the 7000 volumes surviving from Carolingian libraries, roughly 40 (41 ± 33) of these should contain DTR and/or DT, or just under half the number that actually survive. A further indication that a substantial fraction of Carolingian DTR/DT copies survive is that the number listed on each library's catalog in Table 1 is quite similar to the number of eight- and ninth-century copies that survive from each library.

Library | Catalog date | Volumes | |||
---|---|---|---|---|---|

Total | Bede | DTR/DT | Extant 8th- and 9th-century DTR | ||

Reichenau | 822 | 415 | 17 | 2 | 3 |

St. Gall | 9th century | 362 | ≥6, ≤33 | 6 | 5 |

” | 9th century | 428 | 20 | 0 | ” |

Bobbio | 10th century | 666 | 10 | ≤1 | 1 |

Lorsch | 10th century | 590 | 24-26 | 2 or 3 | 2 |

Durham | 12th century | 546 | 11 or 12 | 1 or 2 | 0 |

Corbie | 12th century | 313 | 14 | 2 | 1 |

” | ∼1200 | 342 | 19 | 4 | ” |

Declercq is mistaken in other assertions. The sigmoid shape of the four age-distribution curves is by no means a foregone conclusion. As figure 1A in (*1*) showed, as the ratio of the death to birth parameters (μ/λ) increases from 0.05 to 0.20, the shape of the age distribution changes from sigmoid, to more or less linear, to exponential. DTR was also far from outdated by the 12th century. The Roman system of finger-counting, well established in both marketplace and cloister centuries before DTR became the standard introduction to it, remained in use even after the 16th century, when excerpts became some of the earlier technical works in print (*8*).

Declercq perpetuates misconceptions introduced by Gilman and Glaze (*9*) about the model's assumptions. Some of these have been dealt with elsewhere (*10*). There is no question that the proposed model purposely neglects many conceivably important factors, that it applies no more to every manuscript tradition than the Verhulst-Pearl equation does to every biological population, or that it can be modified to accommodate additional factors. The strength of the model is that it explains so much with so little, 98% of the variance with only three fitted constants in the case of DTR. Further quantitative analyses will help determine how widely applicable predictive modeling may be as a viable and complementary alternative to traditional ways of studying learned communication in medieval Europe.