More than a Manual: Early-Modern Mathematical Instrument Books

In Elizabethan London, one of the more surprising things a wealthy owner of a beautifully illustrated folio volume could do was to take a sharp knife and cut it to pieces. John Blagrave’s 1585 Mathematical Jewel, in fact, demands nothing less.[1] This work, which introduced an elaborate instrument of Blagrave’s design for performing astronomical calculations, included woodcuts that were specifically provided in order to be cut out and used as surrogates for the brass original:

get very fine pastboord made of purpose, and then spred your paste very fine thereon, & quickly laying on this picture & clappe it streight into a presse before it bee thorowe wette with the paste (fol. ¶6v)

In at least one surviving copy this has been done in a rather idiosyncratic manner: the plates have been cut out, pasted onto board and then pasted back into the inside of the book’s front cover (Figure 1).

Figure 1. Copy of John Blagrave’s The Mathematical Jewel (London, 1585), with the plates removed, carefully cut out, pasted onto board and mounted inside the book’s front cover. Cambridge University Library LE.28.5.

We have long known that early-modern readers were active in their approach to texts, having clear goals (social, political, practical) in mind.[2] But this takes things one step further. Owners of Blagrave’s books were certainly “studied for action,” but did they read at all, or did they rather use the amended text as an astronomical instrument in its own right?

The answer is, of course, both—and this is also true of the many other instrument books being published in increasing numbers through the sixteenth and seventeenth centuries. This adds a layer of complication to our understanding of early-modern instrument books, which repudiate bookish learning in favour of what Jim Bennett calls “operative knowledge,” while at the same time containing all that was needed to conduct a practical course of instrumental learning on paper.[3] The technical tool of scale drawing was not just useful in making machines from drawings; it was also a means of facilitating paperwork. Here is Edward Worsop, writing in 1582:

Every Figurere in this treatise is drawen according to some Scale, therfore the having of scales and compasses, and applying them to those Figureres, will make the demonstrations, and proofes herin very easie to the readers thereof, though they understand little or nothing in Geometrie[4]

The genre of the instrument book—sometimes called “Usus et fabrica” after a common title—accompanied the new mathematical instruments that were developed and sold from the late fifteenth century on. My research focuses on the instrument trade and related publishing endeavours in England in the sixteenth and seventeenth centuries. Instrument books appeared in the vernacular from the 1550s, when instruments were first sold in any significant number in London. By the 1580s, an important and vibrant trade had been established and instruments were having a marked effect in a number of practical disciplines, primarily navigation.[5]

In a sense, instrument books are quite straightforward: they advise the user or owner of a given instrument on its use. Then, as now, this is an essential complement of any reasonably complex object. Yet the fact that instrument books could themselves be mutilated in order to provide the means for practice shows that more is going on here than in an instruction-booklet for, say, a new washing machine. Closely linked to the paper-and-pasteboard instruments contained within books, many also explained how to make the instruments they describe from scratch, either in wood or metal. Again Blagrave is particularly forthcoming in this respect. His Chapter 2 is headed, “What materiall this Iewel were best to be made of,” the answers being any one of a tin alloy, brasse, silver, or even horn (for the complex “rete,” that is, the fretted part that rotates over the base).

But this leads us to a fundamental point about instrument books: the instructions for making instruments in metal cannot be taken at face value. Preparing, cutting and engraving on metal—not to mention the precision required for engraving scales—were craft secrets and required extensive equipment not detailed in instrument books. So while Blagrave explains in outline how to make his instrument, his readers are left with only a paper instrument, and no real means to make one in metal. Hence in Gabriel Harvey’s own copy he has annotated the title-page with the following note:

The Instrument itself, made & sold by M. Kynvin, of London, neere Powles. A fine workman, & mie kinde friend[6]

Understanding this state of affairs involves unpicking the complex social hierarchies of Elizabethan England, and the ways in which these interacted with the new mathematical disciplines that were becoming fashionable, especially in the capital. Blagrave himself was a landowner in the Reading area and enjoyed the patronage of Sir Francis Knolles. Indeed the Mathematical Jewel itself was a bid for patronage, or at least preferment, from William Cecil (Lord Burghley), one of the most important members of the Elizabethan Court and well known as an enthusiastic supporter of practical mathematics.

For most owners of his book, meanwhile, paper was the real material of the “Mathematical Jewel”: the book contained a built-in instrument, on which the vast proportion of its potential uses could be carried out—these included calculations for the positions of planets and stars, the place of the Sun in the ecliptic, telling the time, and so on. Should a metal instrument be sought, the reader would have to go to the shop of James Kynvyn an artisan known to have made the instrument in brass. Blagrave was able to advance practical mathematics by issuing his instrument in paper with his book, to advance his own cause by dedicating the work to Lord Burghley, and to advance the Kynvyn’s career by sending them custom.

Though particularly transparent in Blagrave’s Mathematical Jewel, these social and practical functions are also present in other instrument books. Some texts, like Leonard Digges’ A Geometrical Practise Named Pantometria (1571), included paper instruments that could be cut out and used. Others were explicit in their promotion of artisans. And many (if not all) were bids for either patronage or employment of one kind or another.

Of course, not all new inventions were publicised: some teachers preferred to keep their technical knowledge private in order to retain authority in the master-pupil relationship, and to control the way in which instruments and practice related to theoretical knowledge.[7] This resulted in a major controversy in the early 1630s, when an opportunistic mathematics teacher called Richard Delamain published an account of some instruments he claimed to have invented. A rebuttal was quickly issued on behalf of the mathematician and clergyman William Oughtred, asserting Oughtred’s priority and launching an attack on Delamain’s learning and pedagogy.[8] Instruments, claimed Oughtred, should be taught after a grounding in geometry and arithmetic; Delamain responded that practice was way of acquiring theoretical knowledge, and that it also sweetened the axiomatic pill. However, this dispute should also be understood as the result of a thriving mathematical culture. Demand was growing for the calculating and astronomical instruments over which Oughtred and Delamain argued; and the instrument maker caught between the two, Elias Allen, became one of the most celebrated craftsmen of his age (Figure 2).

Figure 2. Engraved portrait of the instrument maker Elias Allen (c.1588–1653) by Wenceslaus Hollar, 1666, via Wikimedia Commons.

By the time of Allen’s death in 1653, a large number of instrument books had been published, including the first set of works written by instrument makers themselves. This signals a shift in the fortunes of craftsmen, who were increasingly independent in their activities—able to earn a living without taking on other employment, to promote their own wares, to invent and improve instruments, and, by the end of the seventeenth century, to become members of the Royal Society (achieved by the printer and instrument maker Joseph Moxon in 1678).

If this survey presents a relatively neat picture of the genre of instrument book, questions persist about their purpose and importance. As Adam Mosley has pointed out, it remains unclear exactly what role these texts played in the early-modern classroom, if indeed that was the intended site of their consumption.[9] One way to approach this issue is to focus on the fact that instrument books could, in large part, serve the functions of instruments themselves. This insight is based on a number of related observations about early-modern mathematical practice. For one thing, “practice” did not necessarily entail going out on board ship, or into the field, or even making observations at all. Writers on the construction of sundials, for example, boasted that it was possible to work through a range of increasingly complex constructions without leaving the confines of one’s study. Indeed with the practice of “reflective” dialling this was taken to an extreme, as sundials could be constructed that relied on reflected light being brought into the study itself. And because “dialling” was central to the art of cosmography, it was possible to comprehend the entire universe on paper, using only a pair of compasses and a ruler. Another important point is that working on and with paper was, as we have seen, a key part of learning about instruments. Owners of instrument books could cut up their copies; they could work directly on the diagrams they contained (Figure 3); they could calculate in the margins; or they could move between the book and a sheet of paper, on which they could build even quite advanced constructions.

Figure 3. Photograph in raking light showing extensive ‘blind stylus’ work on an astronomical diagram in Edmund Gunter’s Works (1673 edition). The blind stylus was a tool of artists and mathematical practitioners to score the page without leaving an obviously visible mark. Another technique was to oil the paper, creating a wipe-clean surface, or to prick through the diagram with a pin and trace the image onto a separate sheet. Cambridge University Library CCD.13.23.

Of course, there was much that lay beyond the confines of the margins of the instrument book. But, in seeking to understand the ways in which these texts functioned in relation to instruments themselves, and to the development of the mathematical arts, it pays to consider instrument books as more than manuals. They could stand in for instruments, serve as miniature curricula in their own right, and also serve the social interests of their authors, engravers, booksellers, and owners.

Boris Jardine is a Leverhulme Early-Career Research Fellow at the University of Cambridge (Department of History and Philosophy of Science/Whipple Museum).

This piece was pre-circulated last spring among participants of the conference “Learning by the Book: Manuals and Handbooks in the History of Knowledge,” but its publication here as part of the corresponding series was delayed until some rights issues could be clarified.


  1. See Katie Taylor, “A ‘Practique Discipline’? Mathematical Arts in John Blagrave’s The Mathematical Jewel (1585),” Journal for the History of Astronomy 41 (2010): 329–53.  ↩
  2. Lisa Jardine and Anthony Grafton, “‘Studied for Action’: How Gabriel Harvey Read His Livy,” Past & Present 129 (1990): 30–78.  ↩
  3. Jim Bennett, “Practical Geometry and Operative Knowledge,” Configurerations 6 (1998), pp. 195–222.  ↩
  4. Edward Worsop, A Discoverie of Sundrie Errours […] (London, 1582), sig. A4v.  ↩
  5. See my forthcoming article, “Instruments of Statecraft: Humphrey Cole, Elizabethan Economic Policy and the Rise of Practical Mathematics,” in Annals of Science.  ↩
  6. Copy now held at the British Library (C.60.o.7).  ↩
  7. See Stephen Johnston, “The Identity of the Mathematical Practitioner in 16th-Century England,” in Der “mathematicus”: Zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators, ed. Irmgarde Hantsche (Bochum: Brockmeyer, 1996), 93–120.  ↩
  8. Katherine Hill, “‘Juglers or Schollers?’: Negotiating the Role of a Mathematical Practitioner,” British Journal for the History of Science 31 (1998), 253–74.  ↩
  9. Adam Mosley, “Objects, Texts and Images in the History of Science,” Studies in the History & Philosophy of Science 38 (2007): 289–301, esp. 293ff.  ↩
Suggested citation: Boris Jardine, “More than a Manual: Early-Modern Mathematical Instrument Books,” History of Knowledge, October 4, 2018, https://historyofknowledge.net/2018/10/04/mathematical-instrument-books/.