Book Reviews

RBM: A Journal of Rare Books, Manuscripts, and Cultural Heritage reviews books, reports, new periodicals, databases, websites, blogs, and other electronic resources, as well as exhibition, book, and auction catalogs pertaining directly and indirectly to the fields of rare book librarianship, manuscripts curatorship, archives management, and special collections administration. Publishers, librarians, and archivists are asked to send appropriate publications for review or notice to the Reviews Editor.

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Reading Mathematics in Early Modern Europe: Studies in the Production, Collection, and Use of Mathematical Books. Philip Beeley, Yelda Nasifoglu, and Benjamin Wardaugh, eds. New York, NY: Routledge, 2021. Hardcover, 348p. $160 (ISBN 978-0-367-60925-2). Ebook, 348p. $44.05 (ISBN 978-1-003-10255-7).

Mathematics in print was not a good business proposition, at least not for printers in London in the seventeenth century. In his chapter, “‘A Design Inchoate’: Edward Bernard’s Planned Edition of Euclid and Its Scholarly Afterlife in Late Seventeenth-Century Oxford,” Philip Beeley details the winding path of a proposed edition of Euclid’s Elements, showing just how many hands, how many years, and how much convincing it took for a project of clear academic importance to be realized in a final, printed form. Beeley, along with Benjamin Wardaugh and Yelda Nasifoglu, is one of the editors of Reading Mathematics in Early Modern Europe: Studies in the Production, Collection, and Use of Mathematical Books, an engrossing new book published last year by Routledge. The editors write, “By defining and illuminating the distinctive world of early modern mathematical reading, this volume seeks to close the gap between the history of mathematics as a history of texts and history of mathematics as part of the broader history of human culture” (i)—an ambitious academic project, and one that is realized quite successfully here. Each of the 11 chapters in the book is an essay that supports that argument from a different angle. Some focus on the history of specific mathematical concepts and on their textual transmission, and evolution, over time. Others zoom in on historic figures in the field and the textual artifacts they left behind. All of the chapters engage with the transmission of mathematics at the material level in some way and are grounded in the analysis of early modern texts, diagrams, and the material evidence of readers’ interactions with them.

Vincenzo De Risi’s chapter, “Did Euclid Prove Elements I, 1? The Early Modern Debate on Intersections and Continuity,” traces the study of the proof of Euclid’s Elements I, 1 in print, which changed over time as, among other things, early modern readers’ understanding of the mathematical concept of continuity and the role of diagrams in the study of geometry changed. Robert Goulding’s “Numbers and Paths: Henry Savile’s Manuscript Treatises on the Euclidean Theory of Proportion” considers the problem of understanding ratios and proportion in the sixteenth century through Savile’s notes in his own copy of Euclid’s Elements. Goulding also writes extensively about Savile’s letters to his contemporary, Gian Vincenzo Pinelli, preserved in the Biblioteca Ambrosiana in Milan, using these pieces of textual evidence to trace how Savile grappled with the ancient Greek understanding of ratio and was able to arrive at, and to articulate, a new understanding of the concept for early modern mathematicians. Richard Oosterhoff’s “Tutor, Antiquarian, and Almost a Practitioner: Brian Twyne’s Readings of Mathematics” provides the reader with a deep dive into the notebooks of Brian Twyne, Oxford’s first Keeper of the Archives, who was also a Fellow of Corpus Christi College, an avid historian of Oxford itself, and an amateur mathematician. Oosterhoff focuses on two subsets of those notebooks: teaching materials, which Twyne labeled “schediasticorum libri,” and which retain their original form; and manuscript materials of a wider variety (letters, reading notes, excerpts from contemporary writers) that were rebound much later (152). Through all of the book’s 11 chapters, the common thread of a close examination of a wide variety of archival sources helps to weave together a strong argument for “the history of mathematics as a history of texts,” but also as a history of all of the human activities around the creation of and engagement with those texts (i).

While each essay takes a different path to arrive at the book’s central ideas, there are many interesting points of intersection from chapter to chapter. Euclid’s Elements—its translations and mistranslations, its diagrams surviving from ancient sources and reworked by early modern ones, editions planned but never realized—is ever-present. After Euclid, Henry Savile is another figure whose historic hand helps tie several chapters together. Yelda Nasifoglu’s fascinating chapter, “Reading by Drawing: The Changing Nature of Mathematical Diagrams in Seventeenth-Century England,” brings the reader back to Henry Savile, an important sixteenth-century mathematician first introduced in this book by Robert Goulding. Nasifoglu returns to Savile, who endowed professorships at Oxford in geometry and astronomy, and his Savilian statutes that established the corresponding curricula for those subjects. Her examination of various teaching tools extant in the Bodleian Library’s Savilian Collections—manuscripts, paper instruments, large-scale drawings, posters—helps illustrate not only how mathematics was taught, but also how it was understood in seventeenth-century England. Henry Savile and the teaching of mathematics at the university level is again discussed by Mordechai Feingold in his chapter, “Reading Mathematics in the English Collegiate-Humanist Universities.” Feingold engages Robert Goulding and his chapter directly, taking issue with Goulding’s assessment of the state of mathematics at Oxford upon Henry Savile’s arrival. And Savile is again the central focus of William Poole’s “The Origin and Development of the Savilian Library,” an interesting and detailed history of the evolution of the collection first started by Henry Savile in service of the professorships in mathematics that he endowed.

The presence of early modern readers’ annotations in all manner of mathematical books is another interesting point of intersection shared by several chapters. The last three chapters of the book all engage with specific copies of works that have been annotated, in some cases heavily and by multiple hands. In Boris Jardine’s “Instrumental Reading: Towards a Typology of Use in Early Modern Practical Mathematics Texts,” Jardine focuses on the annotations in a single copy of Leonard and Thomas Digges’ Pantometria, printed in a second edition in 1591 and held by the University of Cambridge. This copy was annotated by at least six different hands, and Jardine seeks to establish a typology of use based on how these different readers interacted with the same text. Kevin Tracey also focuses on multiple hands in a single book, John Seller’s A Pocket Book, containing several choice collections: in Arithmetick, Astronomy, Geometry, Surveying, Dialling, Navigation, Astrology, Geography, Measuring, Gageing, etc. Tracey’s chapter, “‘Several Choice Collections’ in Geometry, Astronomy, and Chronology: Using and Collecting Mathematics in Early Modern England,” considers the London Science Museum’s copy of this work and makes some interesting observations about a work that was intended for a very different audience than many of the other publications discussed in this book.

Annotations also serve as important pieces of evidence in several chapters that focus on how early modern readers dealt with errors—of print and of logic. Returning to Boris Jardine’s “Instrumental Reading,” Jardine counts “corrective reading” as its own typology of use. He writes, “Corrective reading begins with the everyday practice of attending to errata, and in this case stretches to the extremes of correction (even of the errata list itself) that can only signal some intention beyond reading, either social, bibliographic, or commercial” (261). Benjamin Wardhaugh attempts to quantify just how common corrective reading was in his chapter, “‘The Admonitions of a Good-Natured Reader’: Marks of Use in Georgian Mathematical Textbooks.” Wardhaugh analyzed 366 copies of popular mathematical books from eighteenth-century Britain. And while he acknowledged the limitations of his study, specifically problems of what kinds of copies survive and why, more than 80 percent of the books he examined included marks of “straightforward correction” (232). In Renée Raphael’s “Interpreting Mathematical Error: Tycho’s Problematic Diagram and Readers’ Responses,” Raphael discusses how three different contemporaries of Tycho Brahe interpreted a very different sort of error. In a printed diagram in his 1588 work De mundi aetherei recentioribus phaenomenis liber secundus, Tycho made several significant errors in a diagram representing a comet that prompted Galileo, Johannes Kepler, and Scipio Chiaramonti (among others) to print their own responses to the work. Here, Tycho’s errors were likely not those of a printer, but of the author himself. Raphael argues that reading contemporary critical responses to Tycho’s diagram, and the text that accompanied it, allows us to reconstruct how those contemporaries read and attempted to understand Tycho’s work.

The central importance of annotations to many of Reading Mathematics in Early Modern Europe’s chapters speaks to the central importance of readers in the understanding of the history of mathematics. It also speaks to the importance of the mathematical archive writ large as a rich resource, which many of the book’s contributors make deep use of. As a reader of this text, I found the book to be quite readable, even to someone who is not a historian of math. As someone who works in special collections librarianship, I also found the discussion and use of collections like the Biblioteca Ambrosiana and the Bodleian Library to be quite compelling. There is a lot here for readers interested in early modern print and academic culture, as well as those who work in fields related to mathematics—astronomy and architecture in particular. And while the book seeks to take in the state of mathematical texts across early modern Europe, the scope is admittedly a little narrower than that, with a heavy focus on England and on Oxford in particular. That having been said, Reading Mathematics invites further study at the intersection of the histories of print and math, perhaps across a broader swath of space and time. As the book itself argues, the history of mathematics is written by readers as much as it is by those being read.—Lena Newman, Avery Architectural & Fine Arts Library, Columbia University

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