Intellectual history

Genres of Math: Arithmetic, Algebra, and Algorithms in Ancient Egyptian Mathematics

By contributing author E.L. Meszaros

As non-native readers of Egyptian hieratic and hieroglyphics, our understanding of the mathematics recorded in these languages must necessarily go through a process of translation. Such translation is both necessary to allow us to study these problems, but also precarious. If done improperly, it can prevent us from true understanding. One way that we approach translating Egyptian math problems is by grouping them into genres, using categorization to aid in our translation by thinking about problems as algebraic or geometric equations, crafting them into algorithms, or piecing together word problems from their prose. If the process of understanding Egyptian math problems relies so heavily on translation, and translation in turn is influenced by categorization, then we must consider how our processes of categorization impact our understanding of ancient Egyptian math. 

The necessity of translation for the modern study of ancient mathematics has been the source of a great schism within the community. In an infamous 1975 paper, Unguru argued that one of the unintentional consequences of translation was the attribution of algebraic thinking to these ancient cultures. Mathematicians and historians tend to translate the word problems of ancient Iraq or Egypt into the abstracted symbolic statements we are familiar with today. This has helped us to better understand ancient mathematical ideas, but has also done a disservice to the math itself. The process of abstraction manipulated the geometry or arithmetic of ancient math into algebra, a way of examining mathematical problems that Unguru argued these ancient cultures never used (78).

Image of a fragment of the Moscow Papyrus showing problem 14 on how to calculate the volume of a frustum. The top portion shows the original hieratic, which has been translated below into Egyptian hieroglyphics.

However, others have pushed back against Unguru. Van der Waerden suggests that Unguru has misunderstood “algebra” by attributing such importance to the symbolic representation of data. Rather, van der Waerden emphasizes the convenience of symbols as a way of interpreting, analyzing, and comparing data, rather than the structural language of understanding data (205). Freudenthal similarly takes umbrage with Unguru’s understanding of what algebra is. “Symbols,” he writes, “…are not the objects of mathematics…but rather they are part of the language by which mathematical objects are represented” (192).

We can compare the strict translation of an Egyptian word problem to its algebraic translation by looking at problem 14 of the Moscow Papyrus.

Prose English translation:
Method of calculating a / ̄\.
If you are told a / ̄\ of 6 as height, of 4 as lower side, and of 2 as upper side.
You shall square these 4. 16 shall result.
You shall double 4. 8 shall result.
You shall square these 2. 4 shall result.
You shall add the 16 and the 8 and the 4. 28 shall result. 
You shall calculate  ̅3 of 6. 2 shall result.
You shall calculate 28 times 2. 56 shall result.
Look, belonging to it is 56. What has been found by you is correct. (Translation by Imhausen 33)

Algebraic Translation:
V = 6 (22 + (2*4) + 42)/3

Abstracted Algebraic Translation:
V = h (a2 + ab + b2)/3
h (height) = 6
a (base a) = 2
b (base b) = 4
V = volume

The algebraic translations are at once easier to take in but also visibly shorter, clearly missing information that the prose translation contains.

As an alternative to these translation techniques, Imhausen proposes the use of algorithms. Imhausen suggests that we translate Egyptian mathematical problems into a “defined sequence of steps” that contain only one individual instruction (of the type “add,” “subtract,” etc.) (149). These algorithms can still represent math problems in multiple ways. A numerical algorithm preserves the individual values used within Egyptian problems, while a symbolic form abstracts the actual numbers into placeholders (152). 

Numeric Algorithmic Translation:

  1. 42 = 16
  2. 4 x 2 = 8
  3. 22 = 4
  4. 16 + 8 + 4 = 28
  5.  ̅3 x 6 = 2
  6. 2 x 28 = 56

Here the first three numerical values are the given bases and height from the problem. The unfamiliar ” ̅3″ is the standard way of writing a fraction of 3, namely 1/3, in ancient Egyptian math.

Symbolic Algorithmic Translation:

  1. D22
  2. D2 x D3
  3. D32
  4. (1) + (2) + (3)
  5.  ̅3 x D1
  6. (5) x (4)

Drawing out the scaffolding of the problem by defining such algorithms allows scholars to easily compare math problems. The abstraction into symbols, the removal of extraneous information, and the sequential rendering allow us to more easily notice variation or similarity between problems (“Algorithmic Structure” 153). Imhausen suggests that identifying the substructure encoded beneath the language of presentation allows us to compare individual math problems not only with each other, to generate groups of mechanisms for solving and systems of similar problems, but also to look cross-culturally. Breaking down problems from Mesopotamia, China, and India may reveal similarities in their underlying algorithmic structures (158). 

The generation of algorithmic sequences from Egyptian word-based math problems does not solve all of our translation problems, however. Any act of translation, no matter how close it remains to the original language, is a choice that necessitates forgoing certain options. It also allows for the insertion of biases on the part of the translator themselves—or rather, such insertion is unavoidable.

In the example from the Moscow Papyrus, for example, the initial given values of the frustum are not specifically identified. The images from the original problem are missing, as are the verbs for the mathematical operations. Imhausen herself points out that this algorithmic form reduces some interesting features. The verb “double” in the original problem, for example, is replaced with “x 2” in the algorithmic translation (75). Making these changes requires us to confront the choice between algorithmic structure and staying true to the source material. “Fixing” these differences allows us to more easily compare math problems, but also presumes that we know what was intended.

The translation of Egyptian math problems into schematic algorithmic sequences is, therefore, not without its own set of problems. While Imhausen claims that they avoid some of the pitfalls of translation into algebraic equations that have so divided the community (158), algorithm interpretations are still likely to present the material in a way that differs from how ancient mathematicians thought about their own material. However, when applied carefully, such mapping may provide valid interpretations of these texts and a focal point for comparison.

Thinking about the genre of translation, the use of algebraic or geometric or algorithmic tools to interpret ancient math, is important for a number of reasons. We have already seen that the choice of genre impacts ease of understanding. Modern scholars used to thinking about math problems in an algebraic format will, unsurprisingly, read algebraic translations more easily. But these choices also impact what aspects of the original we preserve — algebraic translations lose information about the order of operations and remove the language used to present the problem.

However, paying attention to generic classification can also prevent us from reading ancient math problems with the “Western” lens. While algebraic interpretations are an artifact of modern scholarship, they are also an artifact of European scholarship. Too often the idea of geometry is put forward as an entirely Greek invention, while algebra is thought of as belonging to Renaissance Europe. By privileging these ways of thinking about ancient math problems we may be inherently white-washing native Egyptian thinking. Prioritizing algebraic interpretations, even if they aid in understanding, work to translate Egyptian math into the more familiar “Western” vernacular. Instead, scholars should work with the unfamiliar and think about these math problems without filtering them through these modern concepts.

Regardless of who one sides with in the debate between algebra and arithmetic, prose and algorithm, we must be cognizant of the fact that categorizing ancient Egyptian math is a conscious choice that influences how these problems are understood. Much like the act of translation itself, categorization is a process that is inherently influenced by the biases—intentional or otherwise—of the scholar. There may be nothing wrong with thinking about Moscow 14 in terms of an algebraic equation as long as we understand that this is an act of translation from the original and, therefore, reflects a reduced understanding of the native problem itself and incorporates aspects of the translator’s biases.

Which is all to say: tread carefully, because even numbers are not immune to the bias of translation.

E.L. Meszaros is a PhD student in the History of the Exact Sciences in Antiquity at Brown University. Her research focuses on the language used to talk about science, particularly as this language is transmitted between cultures and across time.

Think Piece

Mai-mai Sze and the I Ching

by contributing editor Erin McGuirl

“What is the I Ching?” was the title of Eliot Weinberger’s recent review of two new translations of the I Ching. It’s an excellent question, and in his review he expertly summarizes the history of the text, from its mysterious origins in the seventeenth century BCE through its introduction to European audiences in the eighteenth century, continuing into the height of the book’s popularity in the West in the mid-twentieth century. As he summarizes, the I Ching meant a lot of different things to a lot of different people, particularly in the West. Hegel thought it was a load of nonsense, Leibniz “enthusiastically found the universality of his binary system in the solid and broken lines,” and English sinologists like James Legge, Herbert Giles (both of whom translated the book) and Arthur Waley (who didn’t) were skeptical of its value as a philosophical text. In the 1950s and ’60s, artists, writers, and musicians, from Philip K. Dick to Bob Dylan to Merce Cunningham, found inspiration in the enigmatic poems they read in the Legge and Wilhelm translations. In the ’60s and early ’70s especially, the book appears all over the popular press. A search for “I Ching” or “Book of Changes” in a historical newspaper database turns up a trove of reviews of translations, interviews with artists and cultural figures, and editorials that mention the book in a variety of ways.

One can get lost in references to the I Ching in the popular press (truth be told, my research for this piece was so entertaining that it threatened to derail my writing completely). But I set out to find out what the I Ching meant to Mai-mai Sze. As JHIBlog readers well know, Sze is an enigmatic and fascinating woman whose dogged pursuit of knowledge across a wide range of subjects comes to life in the penciled notes she left in the vast collection that she bequeathed to the New York Society Library. One of very few traces of her scholarly life survives in William McGuire’s archive: a note at the bottom of a letter she sent him in 1979—thanking him for a copy of Iulian Shchutskii’s Researches on the I Ching—identifies her as a Bollingen author and “scholar of the I Ching” (Sze to William McGuire, 27 Sept. 1979. Box 47 Folder 9, William McGuire Papers, Library of Congress).

The index to her Tao of Painting—a translation of the Manual of the Mustard Seed Garden, or Jieziyuan Huazhuan—lists thirteen references to the I Ching, and one is an extended passage covering several pages. In this section, forming a major part of her introduction to the text, Sze references the Legge translations in her footnotes. This is a bit odd, as the Wilhelm translation was available by 1950, in the midst of her work on the project.

Wilhelm, Richard & Baynes, Cary F. (translator). The I Ching, or Book of Changes. New York: Pantheon Books, 1950. Volume 1. Copy in the New York Society Library, Sharaff/Sze Collection.
Wilhelm, Richard & Baynes, Cary F. (translator). The I Ching, or Book of Changes. New York: Pantheon Books, 1950. Volume 1. Copy in the New York Society Library, Sharaff/Sze Collection.

Although she does include an in-depth discussion of the I Ching and its relationship to Chinese painting in the introduction to the Tao of Painting (Bollingen, 1956), Sze seems to have turned to the I Ching in earnest quite late. A note on the title page of her copy of the two-volume set of the Baynes-Wilhelm I Ching directed me to her copy of the one-volume edition (which wasn’t published until 1968) for “notes + Chin. text.” Over a decade after the publication of the Tao of Painting, Sze studied the I Ching as closely as she studied other classics of Chinese philosophy (such as Laoze, the Confucian Analects, and the works of Mencius). I’m sad to report that her copy of the one volume Baynes-Wilhelm translation is now lost, leaving a gaping hole in the record of her interaction with this book. However, she did leave notes in the other translations that she owned, as well as in her copies of secondary sources on the I Ching in English. They reveal a bit about what she was up to.

Sze’s most heavily annotated copy of the I Ching is one of only a few English translations with the text printed alongside the original Chinese. (She also owned an edited and annotated beginner’s edition entirely in Chinese, but this contains none of her characteristic penciled notes.) The Text of the Yi king (and its appendixes) Chinese original with English translation by Z.D. Sung, published in 1935 in Shanghai, contains typical Sze marginalia and inserts. Some characters are circled, and in the English text below she makes notes in English for alternate translations. On the inserted sheet of paper, she’s drawn out characters in question and jotted down a Wade-Giles pronunciation guide, with some further explorations of a possible English translation below.

Annotations and inserts in Mai-mai Sze’s copy of the Z.D. Sung translation of the I Ching. New York Society Library, Sharaff/Sze Collection.
Annotations and inserts in Mai-mai Sze’s copy of the Z.D. Sung translation of the I Ching. New York Society Library, Sharaff/Sze Collection.

As David Hinton points out in the introduction to his new translation, the “texture of open possibility suffuses every dimension of the I Ching” because of the “wide-open grammar” of classical Chinese. The meanings of the characters are never precise. Verbs don’t indicate time with tense, and no punctuation was used, making it extremely difficult to extract a convincingly accurate English phrase from a cryptic string of graphs. To show how this works in practice, Hinton’s illustration looks much like one of Mai-mai’s annotations:

Hinton, David. I Ching: The Book of Change. New York: Farrar, Strauss, and Giroux, 2015, xvi.
Hinton, David. I Ching: The Book of Change. New York: Farrar, Strauss, and Giroux, 2015, xvi.

The original Chinese appears above English words whose meanings aren’t always closely related. This suggests that Sze was aware not only of the many possibilities for an English translation of a Chinese word, but also of the vast open territory to be covered by the translator in rendering the English into the Chinese. Her inclusion of phonetic transcriptions of the Chinese characters also indicates that she was clued into the importance of the sound of the original, which often rhymed.

The Sung translation of the text was described in the excellent 2002 annotated bibliography of the I Ching as a “convenient arrangement of the Legge translation,” as opposed to a totally new interpretation of the text in English. Sze referenced the Legge translation throughout the Tao of Painting, and kept up with new translations and secondary sources about the I Ching as they came out. Her collection includes copies of the Blofeld translation (Allen & Unwin, 1965), the previously mentioned two-volume Baynes-Wilhelm translation, and three studies of the book published by the Bollingen foundation by Richard Wilhelm, Hellmut Wilhelm, and the Russian scholar Iulian Shchutskii. All of them show the telltale signs of Sze’s intense engagement: TLS reviews are taped to the front covers, and the texts contain annotations in English and Chinese, with cross-references from one book to another.

Based on the surviving record, it seems clear that Sze’s most intense scholarly engagement with the text took place during the 1960s and 1970s; this period and the interaction were defined primarily by her engagement with the text in the original Chinese, as well as in English translations and studies published by the Bollingen Foundation. While the Bollingen angle is certainly worth investigating (particularly from a Jungian point of view), I’d rather close this piece by turning again to Hinton’s introduction and especially his discussion of how to read the I Ching. “As a poetic/philosophical text,” he writes, “it can be read like any other text, from beginning to end. However, even in this conventional reading, the book frustrates expectations of coherence. It is made up of fragmentary utterances, mysterious enough in and of themselves. And these fragments often feel quite disparate in nature: poetry alternates with philosophy, bare image with storytelling, social and political with private and spiritual, plainspoken and earnest with satire and humor” (xvii). As I’ve written previously on this blog, Mai-mai Sze’s interests were as wide ranging and complex as the I Ching that Hinton describes. Her library reveals her explorations of poetry and philosophy, visual art and literature, politics and social life, and spirituality, and I believe she saw all of these things at work in the I Ching.

With so many ellipses in the story of Sze’s life, it’s almost certain that there’s a more to this story than what I’ve been able to describe here. Like the I Ching, there’s always room for new interpretations when it comes to Mai-mai Sze. I hope these posts will inspire a new investigation.

Think Piece

Thinking About Knowledge in Motion and Social Engagement at HSS

by guest contributor Patrick Anthony

Amidst the great diversity of ideas and perspectives circulating at this year’s History of Science Society (HSS) meeting in San Francisco, two themes continue to resonate in my mind: knowledge in motion and social engagement. Indeed, the annual HSS meeting is itself an example of the far-ranging mobility of ideas and the impulse of many scholars to engage with diverse and interdisciplinary audiences. It is fitting then that some of the most palpable themes at HSS concerned the transmission and translation of scientific knowledge on the one hand and the social and political engagement of scholars on the other.

With sessions and round-tables like “Knowledge in Motion,” “Translation as Process,” and “Translation as an Epistemic Tool,” I was reminded of the text of James A. Secord’s plenary lecture at a Halifax conference titled “Circulating Knowledge” in 2004. Under the title “Knowledge in Transit,” Secord expressed his concern that a focus on localized sites of knowledge creation had lead to a “loss of direction,” suggesting in stead that we begin to view “knowledge as communication” and shift our gaze toward “patterns of circulation.” “It means thinking about statements as vectors with a direction and a medium,” Secord argued — and this is precisely how many of his colleagues were thinking at this year’s HSS meeting.

The “Knowledge in Motion” session featured a host of scholars who had indeed found direction. In his analysis of letters to and from the twentieth-century physicist Paul Dirac, Aaron Wright, for instance, sought to identify the “Principles of Correspondence.” Wright suggested that correspondence contains a unique “form of knowledge” akin to Michael Polanyi’s “tacit” or “personal knowledge,” one that allows for a freer and more metaphysical exchange of ideas than we find in published works. Also in this session, Noah Moxham reconstructed the eighteenth-century distribution circuits of the scientific periodical, the Philosophical Transactions; Barbara Di Gennaro traced pathways of knowledge concerning the origin of the “true” balsam plant through a network of Muslim, Jewish, and Christian thinkers that spanned the Eastern Mediterranean in the sixteenth-century; and Sophie Brockmann’s “Geography of Knowledge in Central America” followed roads, trade routes and correspondence networks to elucidate the role of “local ‘lived’ and ‘imagined’ landscapes” in the creation of geographic knowledge.

With an eye toward both the visual and the textual, the public and the private, scholars are currently not only concerned with how knowledge travels, but also with the ways in which knowledge is reconfigured while in transit. Studies of translation figure centrally here. In “Translation as Process,” Martina Schlünder examined the 1979 English translation of Ludwik Fleck’s Genesis and Development of a Scientific Fact (first published in German in 1935) to further Fleck’s own argument. Fleck argued that scientific knowledge is conceived, situated, and, as Schlünder phrased it, “trapped in” but “blind to” socially conditioned “thought styles.” Can the same be said for translations? Schlünder said yes, suggesting that the English translation of Genesis and Development is itself an example of a text being appropriated by a different thought collective.

Historian Lynn K. Nyhart was thinking along similar lines. In her exciting new project, presented at HSS under the title “Reproducing Science: William B. Carpenter and the British Reception of German Ideas on Generation, 1839-1854,” Nyhart set herself the task of explaining how scientific knowledge changed while en route from Germany to Britain in the mid-nineteenth century. Thus far, Nyhart has found that ideas once rooted in Germany were, by the time they reached British students by way of French and then English translations, heavily “mediated,” and in some respects “decisively Anglophilic.” I was struck by the affinities between Nyhart’s project and Nick Hopwood’s new book Haeckel’s Embryos: Images, Evolution, and Fraud, which explores the social life of a set of controversial drawings by Ernst Haeckel to argue that images, like texts, are not just mindlessly recycled, but creatively reproduced. My sense is that the red thread here — in studies of translation and circulation as in the work of Nyhart and Hopwood — is an enthusiasm for studying the mobility of ideas as they evolve through time and space.

If “translation” and “motion” were on the minds of many in San Francisco, “engagement” and “justice” seemed to be rival buzzwords. While one session, facilitated by Janet D. Stemwedel, exchanged ideas on “How to Engage with Government and Beyond Using the History of Science,” another round-table, chaired and organized by Joanna Radin and Myrna Perez Sheldon, posed the bold and noble question, “How Should the History of Science Engage with Political Activism and Social Justice?” Of the sessions I was able to attend, the most dynamic and compelling was a roundtable titled “Historians of Science in the Public Sphere.”

Chaired by Joshua Howe of Reed College, the panel featured a set of scholars working at the intersection of academia and social justice: Erik M. Conway, co-author of Merchants of Doubt: How a Handful of Scientists Obscured the Truth on Issues from Tobacco Smoke to Global Warming; Jane Maienschein, director of the Embryo Project at Arizona State University; Alice Dreger, whose work on intersex research and identity politics has courageously fused scholarship and activism; and Robert Proctor, who in 1999 became the first historian to testify against the tobacco industry, and continued to do so in over one hundred cases.

“What happens,” Howe began, “when historians of science use their craft to make the world a better place?” In their profound and inspiring answers (and in spite of their differences) the panel revealed many commonalities. Most panelists had received hate mail, and some, death threats; all expressed a deep resentment of relativism, and alternatively, for “ultra-postmodernism”; and each expressed a commitment to truth and justice. While Proctor and Maienschein set their social/academic aims within a broad “Enlightenment” tradition, Dreger spoke of relativism as a position held by those who have not had to consider justice—that is, as a privilege. Only briefly mentioned then, but certainly worth recalling here, was the gendered nature of hate mail. As Conway related, Naomi Oreskes, his co-author of Merchants of Doubt, had received thousands of hate e-mails after the book’s publication — a striking quantity beside his few.

The panel agreed that much of the value in the study of history lies in its potential to humble us. “Humility,” Dreger said simply and sardonically, “might be a good way to approach things.” But the problem with being a historian in the public sphere, Dreger continued, is that people want simple, black and white, stories of good and evil. Proctor concurred: “Complexity can get in the way,” he said, especially in the courtroom. But it was Proctor’s stirring meditations on humankind that seemed to dominate the tenor of the session. Beginning with the premise that “to be human is to want to make the world a better place,” Proctor’s principal message was that one ought to be “a human first and a historian second,” and that we ought to be wary of the evils that follow from the reversion of this relationship. He concluded with a bit of levity and the words of a Rabbi: “Nonsense is nonsense, but the history of nonsense is scholarship!”

Patrick Anthony is a first year graduate student working on the history of science and exploration at Vanderbilt University. At the HSS meeting in San Francisco, Patrick presented a poster titled „Views of Justice in Views of Nature: Mapping Alexander von Humboldt’s Cosmic Law.“ Aside from his work on Humboldt’s Romantic conception of justice, Patrick is also developing a project on Americans’ views of revolutions in Saint-Domingue and Latin America during the early national period.

Think Piece

The Bookends of Chronicles: Decisions about Time

by Madeline McMahon

At the very end of Jerome’s chronicle, after the narration of events has stopped, time is tallied up: “The whole list (canon) from Abraham until the time written above, 2,395 years. And from the flood until Abraham there are reckoned 942 years. And from Adam until the flood, 2,242 years. From Adam until the fourteenth year of Valens—that is, his sixth consulship and the second of Valentinian—all the years were 5, 579” (my translation, Latin in Helm, ed. Eusebius Werke 7). Jerome, having reached the end of historical time, suddenly retraced his steps backwards, from landmark figure to landmark event back to Adam.

Chronicles are difficult but rewarding sources. Environmental historians have found their laconic comments on crops and weather helpful, and historians and literary scholars have long debated the meaning of the “fiery dragons flying across the firmament” in the entry for 793 in the Anglo-Saxon Chronicle. Every entry can be unpacked for further meaning. In this post, I want to think about the ends of chronicles, especially that of the chronicle that became the beginning to many continuations, through the middle ages and into the early modern period: Jerome’s Latin Chronicle.

Jerome’s own chronicle is itself a continuation: he translated the Greek chronicle of Eusebius of Caesarea, which had ended in 325, in the reign of the first Christian emperor, Constantine—a Christ-like figure for Eusebius who ended persecution and ushered in a new era of Christian world history. (It should be noted that neither Eusebius nor Jerome used AD dating—those dates have been added for the convenience of the reader by modern editors.) Although Jerome had inserted details, especially Roman historical events, into his translation from the fall of Troy onwards, at the end of Eusebius’s chronicle he took over: “Until this point Eusebius Pamphili, companion to the martyrs, writes this history, to which we ourselves have appended these things following” (year 326). Jerome updated Eusebius’s chronicle, taking it up to the year 378, to the disastrous Battle of Adrianople and the death of the Emperor Valens.

First Council of Constantinople (Wikicommons)
First Council of Constantinople (Wikicommons)

Jerome’s vision of history thus had a very different shape from that of Eusebius, who had ended his work of providential history triumphantly. Jerome’s depressing ending was deliberate: he was writing in 381, yet he chose not to continue the chronicle up to the present day. In the preface to his work, he wrote: “Contented with this end-point, I have reserved the rest of the time of Gratian and Theodosius [the current emperors] for the pen of a broader history; not that I have been afraid to write freely and truly about living persons—for the fear of God drives out the fear of men—but because while the barbarians are still raging here in our land, all things are uncertain” (preface). Aside from the barbarians, Jerome was still unsure what to make of Theodosius. In 381, both were at Constantinople for the church council that the emperor had convened, and Jerome was watching that first Council of Constantinople closely to see where Theodosius’s ecclesiastical loyalties lay.

Jerome’s calculation of years from the beginning of time was thus deliberately outdated, in part to avoid treading upon delicate ecclesiastical and political terrain. But it also did precisely what Eusebius had avoided: it enumerated the years since the Creation. Of the major world chronicles composed between the third to tenth centuries, Eusebius’s is the only one that does not begin at the beginning (McKitterick, Perceptions of the Past, 9). Instead, his history starts with Abraham, in order not to give substance to those who, relying upon a particular interpretation of the book of Daniel, thought that the world would end after six thousand years (Grafton and Williams, Christianity and the Transformation of the Book, 153). While Jerome, too, began with Abraham and did not add the history between Adam and Abraham to the Chronicle, he nonetheless went back to the beginning at the end of his work.

Even in the first major Latin chronicle, we can see that the end of a chronicle asks the copyist or continuator to make decisions about the presentation of time and history. The end of a chronicle could be fraught with tension, as the continuator implicitly commented on the politics of the present as well as the intentions of the original author. In a ninth-century copy of Jerome’s chronicle, the copyist at the monastery of Reichenau in Germany decided to keep Jerome’s computations exactly: he did not alter or update the text to reflect the years that had passed (Merton College MS 315, 149v). Nonetheless, he added a list of rulers and their years to the back of the manuscript, perhaps to aid readers looking for specific events in the chronicle itself (149v – 155v). On the last page, he also added a calculation from another late antique chronicler, Africanus, to provide yet another estimate of the world’s age in late antiquity (156r). Yet this copyist was unusual: many others added their own times to the end of the chronicle they copied.

The end of Jerome’s Chronicle and the beginning of Prosper’s in Estienne’s edition (1518). Bayerische Staatsbibliothek.

When Henri Estienne printed the chronicle in 1512 and 1518, he titled it: The Chronicle of Eusebius bishop of Caesarea, which the presbyter Jerome translated into Latin with divine genius, and added with Roman eloquence until emperor Valens. To which Prosper and Matteo Palmieri have added much… (my translation; 1518 edition). In this edition, the end of Jerome’s version is announced after his account of the Battle of Adrianople and Valen’s death, and Prosper’s chronicle continues until the humanist Palmieri’s takes over. Estienne skipped Jerome’s calculation of years from the beginning of time to the end of his chronicle. Nonetheless, running alongside the text in Estienne’s edition are “anni mundi,” the years of the world, and these tally exactly with Jerome’s own computation, so that when Jerome’s chronicle ended in 5, 579, Prosper’s picked up in 5, 580. Continuators of chronicles changed and maintained the temporal schemas of their predecessors in myriad ways, and one of the best ways to see where chroniclers thought time began and ended is to look to the end of one chronicle and the beginning of the next.

Further reading:
Daniel Rosenberg and Anthony Grafton, Cartographies of Time: A History of the Timeline (New York: Princeton Architectural Press, 2010)
R. W. Burgess and M. Kulikowski, Mosaics of Time: The Latin Chronicle Traditions from the 1st Century BC to the 6th Century AD (Turnhout: Brepols, 2013)