The ciphers of the monks. A forgotten number-notation of the Middle Ages (Q2762635)

This book presents a poorly known numeration system which was used in the Middle Ages. The ``ciphers'', as the author calls them, are built by adding various signs corresponding to numbers (from one to nine) on one side of a stem (drawn horizontally or vertically according to variations). The main ingenious idea consists in encoding the decimals by rotating the symbols around the stem so that, for example, the new value corresponding to five will be fifty. One can further push the elementary addenda on one half of the stem and rotate around an imaginary axis perpendicular to the stem and passing through its middle and one gets in this way the hundreds and the thousands. Arbitrary large numbers cannot be coded in this way because of the small number of symmetries of the plane which leave the stem invariant. Another drawback is that calculations cannot be performed easily using this representation. NEWLINENEWLINENEWLINEThe reviewer comments that this method is very original compared to the others from ancient times. The challenge is to encode conveniently large numbers and the base of the counting system is used for this purpose. The base is in most cases ten probably due to the fact that prehistoric people did count with their hands, but the Babylonians had a sexagesimal system (whose influence is seen on our way of measuring time), the Mayas a system with base twenty. The base is then distinguished from the elementary constituents by a new symbol. Sometimes the situation is more complicated where an intermediate symbol is introduced (like a symbol for five with the Romans and the Mayas). Here the base is not encoded in a symbolic way but by geometric transformations and zero corresponds to the stem alone. NEWLINENEWLINENEWLINEThe author is concerned with the origin and the evolutions of the ciphers as well as their use in various contexts. His presentation is based on available surviving data he has collected (although he states that more objects using the ciphers may yet be discovered thus modifying the elaboration). He shows that the contribution is original because the ciphers were ignored by recent historians of mathematics or by medievalists due to its position at the boundary between these two disciplines. Moreover, it was misinterpreted already in the Renaissance by Agrippa von Nettesheim and further studies based directly or indirectly on him. This has biased their treatment and the belief was that they were of ``Chaldean'' origin for magical goals. Here the author shows convincingly that they were used for a variety of other purposes. NEWLINENEWLINENEWLINEChapter one introduces the ciphers with the main variations, explains how to read them and presents the cultural context in which they appear, mainly medieval manuscripts where they are used for foliation and pagination. NEWLINENEWLINENEWLINEChapter two, ``the English ciphers'' (with a vertical stem) concentrates on a first class. They were introduced according to Matthew Paris in the first half of the thirteenth century by John of Basingstoke after a trip to Greece where he got acquainted with them. This addresses the issue of the origins of the ciphers in Greece. A shorthand from the fourth century BC, with striking similarities although possibly used to also write words is perhaps a clue but then the history of their survival for more than 1500 years should be clarified. Some sections are devoted to analogous representations (the Scandinavian runes, ciphers in Arabic texts) but the author claims that they are not directly related; a possible Greek origin is hypothesized in some instances in the second case. NEWLINENEWLINENEWLINEChapter three focuses on the ciphers from the Cistercians (with a horizontal stem). They spread from abbeys in England to mainland Europe in a Cistercian environment. They were particularly successful in monasteries in Flanders. Several manuscripts are examined. NEWLINENEWLINENEWLINEChapter four presents a unique astrolabe featuring the ciphers from the fourteenth century in Picardy which became available only recently after a selling in an auction. This object aroused the interest of the author in the ciphers from which this book evolved. The point now is to inquire how could the ciphers be fitted into a nonreligious environment. A possible help might be found in an engraved dedication (1522) from Berselius to Amerotius, two priests and humanists. The latter was a teacher of Greek and interested in scientific instruments. The intention of Berselius can only be conjectured. Such objects were not rarities in this area (Flanders) and gifts were also a habit. On the other hand, taking account of the culture of making gifts at the time (autographs, dedication) one might infer that Berselius appreciated not only the friendship but also the object, and then ask further why. The fact that Amerotius did not mention the cipher in a book on Greek numerals that he published is inconclusive: the ciphers could have been called ``Greek'' at the time without people knowing why. The provenance and the destination of the astrolabe remains unclear. NEWLINENEWLINENEWLINEChapter five is devoted to the French vertical ciphers which had however analogies with the Cistercian appendages. Several sources are indicated. One finds also information on the use for the volume of wine-barrels as well as letters in texts. NEWLINENEWLINENEWLINEChapter six is on the ways the ciphers survived after the Middle Ages. They were first mentioned in printed form by Agrippa von Nettesheim and then subsequent publications are reviewed. Let us mention that Cardano was aware of them and even proposed a hybrid system (symbolic-geometric) to be able to count larger numbers. The ciphers were also used more recently by the Freemasons and the Nazis perhaps because they found a resonance in their lore and folklore (analogies with mason's marks on stones described in an appendix? runes). NEWLINENEWLINENEWLINEThe last chapter reviews flaws, misinterpretations and omissions in the modern scholarly treatment of the ciphers. The author also attributes contributions. Many appendices are interesting (various types of numerical notations and associated historical considerations, information of a scientific and technical nature still in a historical context and reference to a beautiful statue which belonged to Berselius). At the end one finds references for the manuscripts, the instruments cited, an impressive bibliography and a comparatively small index. NEWLINENEWLINENEWLINEThe book is pleasantly written and accessible to a broad audience which is the intention of the author. We noticed two typographical mistakes: ``he scale'' should be written ``the scale'' (p. 135) and ``Agincourt'' should be written ``Azincourt'' (p. 140). NEWLINENEWLINENEWLINELet us point out that three paths dealing with the origins of the ciphers apparently unrelated can be distinguised in this development. The runes, Ottoman military ciphers from pages 74, 75 which are described as ``strikingly similar'' and the Uighur symbols from the footnote of page 274 also very analogous. Explaining some ancient runic inscriptions using the old Turkic alphabet has been proposed elsewhere. It may be worthwhile to try to get other insights by comparing the numeration systems of these old representations (runic and Turkic). This could be difficult: ``-- as far as I know -- there is no evidence that their use (the runes) was extended to numbers'' and a short search of the reviewer on this topic did not bring results. The Turkic version of the numeration seems also difficult to track (Could the Ottoman ciphers be a later inheritance of such a hypothetic numeration system?). One should not forget the Greek artifact mentioned earlier in this perspective especially because the appendages seem to have already been complex enough to allow an even older past history leading to them. NEWLINENEWLINENEWLINEIn the conclusion the author states that this book is ``an account of a very personal journey of an antiquarian through some obscure episodes in Europe's intellectual history''. This endeavour brought scattered data and the difficulty is to connect the available evidence and deduce causalities and evolutions. This is made hard by the paucity of the surviving objects. One feels a paradox in the variability in the classes of ciphers despite or because the fact that few knew them. And moreover they spread over a vast area. NEWLINENEWLINENEWLINESo the journey is very open and it was even more pleasant in view of the variety of topics presented unravelling perspectives left aside by classical history.