Calculating in Medieval Europe

This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page.


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In the early thirteenth century, in the midst of a period of dramatic transformation, medieval Italy enjoyed significant commercial growth and economic stability. The country was awash in capital, which was used to finance the construction of breathtaking Gothic cathedrals and numerous universities. Much of that money came from taxes levied on profits from the extensive trading that occurred between countless merchants who traveled frequently to and from Mediterranean cities and ports to the west, and Asia Minor, Syria, and Baghdad to the east. Some intrepid adventurers even traveled so far as India and China, bringing back with them exotic supplies and novel ideas, beliefs, and methodologies. Exposure to people, languages and cultures in and from other parts of the world through trade and war gradually led to an intellectual transformation in Europe. In “Leonardo of Pisa and His Liber Quadratorum,” American Professor R. B. McClenon explains that the crusades “awakened the European peoples out of their lethargy of previous centuries” and “brought them face to face with the more advanced intellectual development of the East.” Marco Polo, McClenon reminds us, was only “the most famous among many who in those stirring days truly discovered new worlds.” In addition to being carriers of merchandise, traders and political representatives became “network specialists,” comprising a “textual network” of sorts, providing a continuous exchange of written texts covering commercial, political, religious and literary topics (Roselaar). Courts which had access to unique intellectual resources reaped financial rewards; wealthy courts, therefore, financed some scholars permanently, sponsoring those who travelled from court to court, contributing to the spread of new ideas. Knowledge was also circulated via the translation and transcription of books that were usually copied at monasteries (“Transition”). Among the most important ideas brought into Europe this way was the Hindu numeral system.



Euclid Portrait
Antonio Cifrondi, Euclid, 17th century, oil

Perhaps the most influential scholar of Western mathematics was the Greek mathematician Euclid, who lived from about 325 B.C. to about 265 B.C. His treatise on geometry, The Elements, written in thirteen books (chapters), contained everything known at that time about number theory. Notably, it includes an original geometric treatment of irrational numbers as well as the first known written definition of the “extreme and mean ratio” between the greater and lesser segments of a line, what is presently known as the Golden Ratio (Fitzpatrick). The significance of Euclid’s book, The Elements, was evidenced by the fact that only the Bible was printed and studied more at that time (“Euclid”). Another important contributor to progress was the Salerno physician Constantine, who, early in the eleventh century, traveled thirty-nine years throughout Africa, Asia, and India, learning the Oriental sciences, and whose manuscripts were used as textbooks for centuries (Smith and Karpinski). Hindu numeral forms appeared in Christian Europe long before manuscripts documented their arrival, for traders, travelers and ambassadors carried them from the East to various European markets. Most traders were rather good at transporting and applying new information and business methods but were less 7 concerned with documenting sources, so pinpointing an exact date of adoption of Hindu numeral forms remains a mystery (Smith and Karpinski).


For centuries, traders in the Muslim world and Europe used either finger arithmetic or a mechanical abacus to perform calculations. The earliest of these devices were simple boards dusted with sand on which numbers could be traced. The Hebrew word for dust, avaq, may be the origin of the name “abacus.” Later versions consisted of a flat board upon which were drawn or carved ruled lines; small pebbles were placed and moved around on these lines to indicate addition or subtraction. Since the Latin word for a pebble is calculus, this form of early calculation became known as “calculus” (Devlin, Finding 30). Medieval abaci had counters sliding along wires. Typically, an abacus had four wires, with beads on each wire representing units. An abacus was sufficient for conducting simple arithmetic operations, but users were at an enormous disadvantage when attempting to handle more complex computations. Since medieval merchants simply added and subtracted most of the time, they could manage using just finger calculations or Roman numerals. The fundamental disadvantage was, of course, the lack of a place-value system. (Livio 93-94).

Rechentisch (Holzschnitt vermutlich aus Straßburg) * Scan aus: Karl Menninger: ''Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl'', Dritte Auflage, Göttingen 1979, S. 153

Before man used written symbols for math, he used images to represent numerals and mathematical operations. Finger signs were particularly convenient and popular for calculating, but successful utilization required a great deal of practice to develop skill and dexterity. Nevertheless, because it was a reliable calculation method, the skill of calculating with fingers was preserved in various societies, passed from one generation to the next (“Calculating”). Finger calculations facilitated communication across language barriers for centuries. Ever desirous of educating the masses, scholars transcribed finger calculating representations in books; the earliest known transcription was written by the Venerable Bede (circa 673–735), an English Benedictine monk. In his book, De Ratione Temporum, he provides “a complete explanation of counting with fingers and rules for this method of calculating.” Drawings in his text show that a person using both hands is able to represent all the numbers from 1 to 9,999 (“Calculating”). Calculating with fingers remained popular even after the introduction of Indo-Arabic numerals because it enabled higher calculations as well as counting, addition and subtraction. Scholars considered finger counting so important that a calculation manual was regarded as incomplete if it did not contain drawings showing the various finger positions (“Calculating”).


Finger Counting
Venerable Bede, De Temporum Ratione (725), Paris BN, MS Lat. 7418, f. 3v

This latter fact explains why Leonardo Pisano’s first book, Liber Abaci, includes instruction on “how the numbers must be held in the hands” (Devlin, Finding 89). At the end of the first chapter he includes a sophisticated system of finger counting as an aid for calculating in the “new” place-value system. The ancient method of finger counting was not deposed by the abacus, nor was it truly pushed into the background in Central Europe until the triumph of written calculation with Indo-Arabic numerals well into the thirteenth century (“Calculating”).

The problem with the abacus or finger calculations is that these methods require considerable practice to achieve accuracy and speed. Additionally, there is no way to correct errors or check for accuracy because “neither method leaves a record of the calculation.” When trading, one needs to be able to inspect records and audit transactions regularly, explains Keith Devlin, the mathematical historian who authored Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World (31). A better method was sorely needed. In Baghdad, among the Arab scholars who studied and translated Greek and Hindu mathematical texts was a distinguished mathematician called Abu Abdallah Muhammad ibn Musa Al-Khwˆarizmˆı (ca. 780 to ca. 850). By the middle of the twelfth century, both of Al-Khwˆarizmˆı’s books had been translated into Latin by scholars. These became essential resources for Europeans who wanted to learn the new mathematics. But Al-Khwˆarizmˆı’s approach did not allow people to correct errors because it “involved cancelation and over-writing, which made it impossible to track the course of the calculation after it was completed” (Devlin, Finding 76-80). Although Al-Khwˆarizmˆı wrote about the Hindu-Arabic numerals in the ninth century, translations of his work did not appear in Europe until the twelfth century. Yet, even then, the Hindu-Arabic numerals did not completely replace the daily and commercial use of Roman numerals until the fifteenth century. When a region adopted the Hindu-Arab numerals, they were often only used by mathematicians, surveyors and scientists; this was the case even in Arab lands (Devlin, Man 42).

Arabic Numerals
Evolution of Hindu-Arabic Numerals, Source: Wikimedia

Calculating with Hindu-Arabic numerals coexisted with the use of the abacus until the early thirteenth century. Indeed, one of the reasons people may have been reluctant to discard Roman numerals altogether is because they are “well suited for use on the abacus.” Before he was made Pope Sylvester II in the tenth century, Gerbert of Aurillac was a scholar and teacher who invented a new abacus, known as the Gerbertian abacus. The counters of his abacus were marked with Hindu-Arabic numerals (“Transition”). Some twelfth-century translations of Arab treatises on algorism – the art of calculating by means of nine figures and zero (Merriam-Webster) – present calculations conducted with Roman numerals while others use Hindu-Arabic numerals. The early thirteenth century brought the appearance of some “very influential treatises on algorism” which “show a greater acquaintance with the new number system than the translations from the 12th century” (“Transition”). Yet, number forms varied widely from region to region, and nowhere in Europe were Hindu-Arabic numerals used exclusively. Not long after Al-Khwˆarizmˆı, the great Egyptian mathematician Abu Kamil (c. 850-c. 930), author of The Book of Algebra, sometimes worked on problems until he found all possible solutions; for one problem, he calculated and recorded 2676 solutions! (Sesiano). This unique, meticulous approach was adopted by Leonardo Pisano.


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