No zeros among the friars counting casks

The Ciphers of the Monks? A misleading title if there ever was one. Even the subtitle, A Forgotten Number-Notation of the Middle Ages , is misleading, for the book is also about medieval sundials, astrolabes, the markings on such mundane objects as weighing scales and wine and oil casks.

Fifty-five pages are devoted to astrolabes alone, from their basic principle to their design, how to use them and the detailed description of some astrolabes in museums and private collections, accompanied by a wealth of photographs.

Astrolabes, wine casks, weighing scales, medieval monks' manuscripts - what could these have in common? The answer is the "ciphers", as David King calls them. Not ciphers in the cryptological sense, but "numerals", "digits" or "number systems". The link is, at times, tenuous - non-existent even - between, say, the Cistercian ciphers, the "merchants' marks on goods from the 13th and 14th centuries recovered from a shipwreck in Gdansk" and (next page) "Markings on wine-barrels in Schafthausen in a lithograph by Daniel Lindmayer dated 1582". Does it matter? Not one jot.

The author does not claim that the monks' original cipher and the wine-barrel markings are in any way related. He even warns us that the resemblance of the basic monks' ciphers with runes is "purely fortuitous". Healthy scepticism; the sign of a true scholar.

So why those wine-barrel markings? The preface gives it away: "As a specialist in the history of medieval Islamic science... I feel very much like a visitor to the European Middle Ages and early Renaissance." In his research, King chanced upon evidence for these forgotten European number systems of medieval times, and, naturally I should think, set about finding whatever he could about them. He even had an extraordinary piece of luck in finding, in 1999 in southern France, Roman scales marked in yet another "cipher", of which he gives complete photographs. Again, these were nothing to do with the "monks' ciphers" proper, but they are fascinating nevertheless.

But back to the monks' cipher. What is it? A system for representing numbers up to 9,999, much like ours, except that each "digit" was tagged to one extremity of a horizontal line: four positions (upper left, lower left, upper right, lower right), to each of which was assigned a multiplier (1, 10, 100, 1,000), a system that nicely bypassed the need for a "zero".

It was used for numbering the pages of manuscripts, of course, but also, astonishingly, for producing concordances. It is barely conceivable to us that a concordance, a list of every word in a text each with its place of occurrence, could be produced by hand, without a computer. But, amazingly, the Cistercian monks did produce concordances of the biblical scriptures.

Page 97 of this book shows a sample, with 55 occurrences of the word "aqua", each with what we would now call a four-digit pointer. Expressed in Roman numerals, that would be a fine mess. In the monks' system, each pointer occupied precisely the same width, and we have a neat table of rows, each with six pointers. The next page shows another sample, this time for "Deus" (God) and "Diabolus" (Devil), again with six pointers per line in six neat columns.

There is hardly a page of this book without an illustration: a reproduction of a page of a manuscript, a photograph of an astrolabe, a sundial, a statue even. There is even a short mention of Bishop John Wilkins' Essay towards a Real Character and a Philosophical Language , an early attempt at designing a universal language.

No, this book is not about some monks' number system. It is an encyclopedia of European medieval number systems, with the occasional mention of others, such as those of the Ottoman Empire; and it delves into cryptography, too, insofar as the "monks' ciphers" were also used for encryption (as was the Ottoman system). In addition, there is the full tutorial on astrolabes, their design, and how to use them.

As must be obvious by now, I was charmed by this book. Of course, I tried to fault it. Honestly, I failed. Certainly the reproduction on page 183 is smudged, too small and about useless, but contrast it with the fine photograph of a "wooden calendar from Pfroment" in Bavaria on page 323, and the photograph just below of the sundial in Gelnhausen near Frankfurt. The reproduction on page 317 of a page of a 15th-century Byzantine manuscript is quite dreadful, and so are a few others, such as the photograph on page 288. But so what? The rest of this book is so rich and informative, the photographs and illustrations so entrancing that I would call it a gem, a real marvel.

Jacques B. M. Guy is a computer scientist with a long-standing interest in decipherment. He holds a PhD in linguistics from the Australian National University.