PUL & MARBAS present: What does the square root of two have to do with medieval European page design?

March 21, 2022

On March 1, Princeton University Library and MARBAS launched an exciting new series of events exploring Princeton’s vast holdings of rare books, manuscripts, and archives. Twice monthly, Princeton experts offer a hands-on presentation on their objects of research held in Firestone Library’s Special Collections.


Will Noel, John T. Maltsberger III ’55 Associate University Librarian for Rare Books and Special Collections, inaugurated the series with an interactive presentation involving Pythagoras, facsimiles, scissors, measuring tapes, and Siri, all of which  helped attendees understand the relationship between the square root of two and medieval page design.

If you are confusedly scratching your head wondering how the two are connected, do not fret: the explanation is simpler than it may seem.

In medieval Europe, paper mills produced their sheets in eight standard sizes with snazzy names such as Imperial (48 x 68 cm) and Chancery (31.5 x 46 cm). The sizes were not exact, but diverged slightly depending on the paper mold that formed them. Paper molds also added other important features to the paper: chain lines and watermarks. These are patterns imprinted on the paper as it dried in the mold. Chain lines are vertical and widely spaced, while watermarks come in multiple designs, including lettering, images of animals and abstract symbols.

Chain lines and a watermark of a horse. Source: Kelly Tuttle from Upenn Libraries Ms. Codex 1951

DIY Instructions: Grab a few pieces of paper. You are now a fifteenth-century paper manufacturer. Let’s assume the paper in front of you was hand-produced in a mold. Keep the paper in landscape mode, and add vertical chain lines every two inches (no need for them to be perfectly straight or aligned). Your paper is of high quality and you want people to be able to recognize that it came from your manufacture, so add a watermark on the right half of the paper. You decide how big or ornate it should be. If you need inspiration, check out the Watermarks of the Middle Ages (https://www.wzma.at/).

To assemble a manuscript, the sheets of paper would be folded in half, or cut in half and then folded depending on the format and size of the envisioned end product. Sheets would be halved multiple times if the booklet was to be pocket-sized, but could simply be folded once if it was meant to sit on a shelf.

DIY instructions: your paper manufacture failed, but you are left with a large stack of paper sheets. You decide to reinvent yourself as a book producer. Take the paper and fold it in half. Make sure that the watermark is on the outer side of the folded page facing you.
In front of you now lies the basic element for producing a booklet. If you feel inspired, you can fold a few more pieces of paper and slide them into each other to create a booklet. If the paper you grabbed is letter format (11 x 8.5 inches), the size of your booklet will be eight and a half inches in height by five and a half inches in width. 

Paper sheets come in different categories and manuscripts in different formats. But all manuscript formats are constrained by the category of paper used. If the paper sheet is Imperial (48 x 68 cm), the manuscript will be proportional to the sheet’s original size. Folded once, the manuscript’s pages would measure forty-eight by thirty-four centimeters (called a folio format), while if the sheet was first cut in half and then folded, it would result in pages measuring thirty-four by twenty-four centimeters (called a quarto format). 

Why is this important? Manuscript pages could be the same size, but produced with paper sheets of different categories. For example, a Royal quarto (30 x 21 cm) is approximately the same size as a Chancery folio (31.5 x 23 cm). How, then, can we tell which paper sheet was used at the outset of the production? Chain lines and watermarks to the rescue. A Royal quarto is a Royal sheet folded twice. The chain lines, which in the sheet were vertical, are now horizontal. The watermark, which was on the right side of the sheet, now lies across the fold. A Chancery folio is a Chancery sheet folded once. Its chain lines are vertical and the watermark is found in its full splendor in the middle of the page.

DIY instructions: A friend wants to support your new endeavor and commissions you to produce a pocket-sized booklet destined to become a Book of Hours—a decorated collection of texts, prayers and psalms. (Perhaps they had seen someone with this tiny exemplar.) Take one of your folded paper sheets. Fold it again, keeping the watermark on the outside. Then fold it again to make it even more portable. Your chain lines should now be vertical; the watermark is quartered over four pages. 

You may be thinking: now I understand the process, but when I have a manuscript in front of me, how do I know how it was made?

Visualizing Format
The basic paper-folding sequence. Source: https://commons.princeton.edu/rbwg/visualizing-format/

Grab a tape measure and open the Needham Calculator. Measure the size of the pages of your manuscript in centimeters and enter it in the calculator. Raise one of the pages to determine whether the chain lines are vertical or horizontal (pro tip: if the library allows it, use the flashlight on your phone) and select the correct option. Hit calculate, and the calculator will provide you with the closest possible options for the category and format of paper used to make your manuscript.

So where does the square root of two come in? It is what all the paper categories have in common: their ratio of height to width is approximately the square root of two. Why is that? Making medieval books was about folding paper. To have consistent proportions when folded, every time it is folded, the paper has to conform to the ratio of the square root two. Any paper resulting by folding a sheet with a ratio of the square root of two will have the same ratio! It is invariant regardless of how many times paper is folded. The magic of the square root of two.

Are you interested in learning more about the world and features of manuscripts? Join us for our next event on March 31 at 6pm in Firestone Library. Eve Krakowski and Marina Rustow will present Princeton’s first Judaeo-Arabic manuscript.