by Friar Thomas Bacon (David Moreno)

Orignally published in the July 1981, A.S. XVI issue of the Dragonflyre, a publication of the Barony of Vatavia.

The period from the fall of Rome to the Renaissance represents the nadir of European mathematics. It was the Arabs who preserved and expanded the mathematical legacy of the Greek civilization. And yet, when European mathematics exploded in the 17th century, the subject had undergone a vast transformation, which I will explore here.

The most striking aspect of the transformation was the change form geometric to algebraic methods of problem solving, from the use of drawn figures to the manipulation of numbers and symbols. The importance of this change lies in the type of problems that could be solved. Geometric algebra could handle only a limited set of problems, with classical geometry, that of the straight edge and compass, limited to an even smaller subset. The three classical problems of trisecting the angle, doubling the cube, and squaring the circle are outside the set for classical geometry and the last outside of geometric algebra as well.

The beginnings of modern algebra are found in the Hindu civilization of India around 600 to 700 AD. But it was the Arabs who put it on a firm foundation. Much of the credit goes to al-Khowarizmi, who lived in the first half of the ninth century. It is from his name that we get the word algorithm, a general procedure use to solve problems. It is from the title of his most important book “Al-jabr wa’l Muqabalah” that we get the word algebra. The importance of al-Khowarizmi lies not with any important original discovery, but like Euclid and his “Elements” in geometry, he gives the first systematic and comprehensive treatise of algebra, from which others could build on.

Al-Khowarizmi also transmitted the use of Hindu numbers, the basis of our own number system. The actual symbols used today are only distantly related to the original Hindu symbols, but it was the concept of positional notation that was the important step forward. The fist European to write about these numbers was Gerbert, later known as Pope Sylvester II, at the end of the tenth century. But the credit for introducing Hindu-Arabic numbers is usually given to Leonardo of Pisa, Gibonacci, with his 1202 book “Liber Abaci.” But adoption was slow and not complete until the 16th century. Florence had even passed a law in 1299 against their use.

Adoption was slow for two reasons. The first was that its chief advantage of ease of computation was negated by the widespread use of the abacus. Even today a skilled user of an abacus can beat someone with a hand calculator. The second problem was that of fractions. While decimal fractions were occasionally used, most fractional expressions were written in unit fractions or sexagesimals. A unit fraction is of the form of one over a number, with fractions then being written as a sum of unit fractions. Sexagesimals are fractions based on the number 60. Minutes and seconds are sexagesimal fractions of an hour. Sexagesimals were used primarily in astronomy. Decimal fractions did not come into their own until the beginning of the 17th century, with the decimal point first appearing in 1616, invented by John Napier.

One of the main breaks with Greek mathematics had to do with irrational numbers. These are numbers that can not be expressed as a ratio of two numbers. Pi and the square root of two are the best known irrationals. The Pythagoreans, the earliest school of mathematicians, had built a mystical religion around whole numbers, and the discovery of irrationals was devastating. As a result, the Greeks turned to geometry.

As irrationals could be approximated by rational numbers, the Medieval mathematicians had little fear of them. Two other types of numbers, however, gave them more trouble: negative and imaginary numbers. It was not until the mid-16th century, with the vague concept of sense, were mathematicians willing to find negative numbers plausible. Imaginary numbers, the square roots of negative numbers, were more troublesome, and were not readily accepted until the end of the 18th century. Renaissance mathematicians, nevertheless, had to struggle with them in their work on cubic equations.

The cubic equation, a mathematical expression involving an unknown number raised to the third power, was the object of much effort in the first half of the 16th century. During this time, Italian mathematicians would hold puzzle-solving contests for fame and money. And it was to the solvers of the cubic equation, Scipime del Ferro, Antonio Fior, Nicolo Fortano or Tartaglia, and their “Euclid” Giraolamo Cardano, that the greatest fame went to.

The use of letters to represent numbers had a long development period. Al-Khowarizmi wrote his equations in a form called rhetorical, that is, entirely in words, and while describing a general from or procedure would use specific numbers. Jordanus Nemorarius, a contemporary of Leonardo of Pisa, was the first to use letters to represent numbers, but his example was not generally followed until the 16th century. The form now used did not appear until the 17th century under Descartes and Leibniz.

Our current symbols for mathematical operations are relatively recent. For most of the Middle Ages, equations were usually written in rhetorical form, with the words plus and minus occasionally appearing as p and m with a bar over them. The current signs first appeared in a German book published in 1489. The equality sign was developed by Robert Recorde in 1557, but did not receive wide use until 80 years later. The greater than, less than, and multiplication signs all came out in 1631. The division symbol, used as such came last in 1659. The symbol itself is much older and had other meanings.

While mathematics has a dominant influence in modern life, during the Medieval period, theology was the queen of study. Mathematics was a minor subject, important only to merchants and tax collectors. This lack of importance of mathematics did not go unchallenged as both Robert Grosseteste and Roger Bacon championed its use. Bacon said:

Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world.

Such was the state of mathematics at the beginning of the 17th century. This century experienced a virtual explosion of mathematical thought by such great minds as Fermat, Descartes, Pascal, and Kepler. It produced analytical geometry, probability, and logarithms, ending with the crowning achievement of the co-invention of calculus by Isaac Newton and Gottfried Leibniz. The world hasn’t been the same since.

Bergamini, David. __Mathematics__. New York: Tine Inc., 1963.

Boyer, Carl B. __A History of Mathematics__. New York: John Wiley and Sons Inc., 1968.

Durant, Will, and Ariel Durant. __The Story of Civilization: The Age of Faith, A History of Medieval Civilization__. New York: Simon & Schuster, 1950.

Smith, David Eugene. __History of Mathematics__. New York: Dover Publications Inc., 1953.

Copyright © 1997 - present His Lordship Friar Thomas Bacon (David Moreno). All rights reserved.