Medieval European interest in mathematics was driven by
concerns quite different from those of modern mathematicians. One driving
element was the belief that mathematics provided the key to understanding the
created order of nature, frequently justified by Plato's Timaeus and the
biblical passage (in the Book of Wisdom) that God had ordered all things in
measure, and number, and weight.
Boethius provided a place for mathematics in the curriculum
in the 6th century when he coined the term quadrivium to describe the study of
arithmetic, geometry, astronomy, and music. He wrote De institutione
arithmetica, a free translation from the Greek of Nicomachus's Introduction to
Arithmetic; De institutione musica, also derived from Greek sources; and a
series of excerpts from Euclid's Elements. His works were theoretical, rather
than practical, and were the basis of mathematical study until the recovery of
Greek and Arabic mathematical works.
In the 12th century, European scholars traveled to Spain and
Sicily seeking scientific Arabic texts, including al-Khwārizmī's The
Compendious Book on Calculation by Completion and Balancing, translated into
Latin by Robert of Chester, and the complete text of Euclid's Elements,
translated in various versions by Adelard of Bath, Herman of Carinthia, and
Gerard of Cremona.
See also: Latin translations of the 12th century
These new sources sparked a renewal of mathematics.
Fibonacci, writing in the Liber Abaci, in 1202 and updated in 1254, produced
the first significant mathematics in Europe since the time of Eratosthenes, a
gap of more than a thousand years. The work introduced Hindu-Arabic numerals to
Europe, and discussed many other mathematical problems.
The 14th century saw the development of new mathematical
concepts to investigate a wide range of problems. One important contribution
was development of mathematics of local motion.
Thomas Bradwardine proposed that speed (V) increases in
arithmetic proportion as the ratio of force (F) to resistance (R) increases in
geometric proportion. Bradwardine expressed this by a series of specific
examples, but although the logarithm had not yet been conceived, we can express
his conclusion anachronistically by writing: V = log (F/R). Bradwardine's
analysis is an example of transferring a mathematical technique used by
al-Kindi and Arnald of Villanova to quantify the nature of compound medicines
to a different physical problem.
One of the 14th-century Oxford Calculators, William
Heytesbury, lacking differential calculus and the concept of limits, proposed
to measure instantaneous speed "by the path that would be described by [a
body] if... it were moved uniformly at the same degree of speed with which it
is moved in that given instant".
Heytesbury and others mathematically determined the distance
covered by a body undergoing uniformly accelerated motion (today solved by
integration), stating that "a moving body uniformly acquiring or losing
that increment [of speed] will traverse in some given time a [distance]
completely equal to that which it would traverse if it were moving continuously
through the same time with the mean degree [of speed]".
Nicole Oresme at the University of Paris and the Italian
Giovanni di Casali independently provided graphical demonstrations of this
relationship, asserting that the area under the line depicting the constant
acceleration, represented the total distance traveled.In a later mathematical
commentary on Euclid's Elements, Oresme made a more detailed general analysis
in which he demonstrated that a body will acquire in each successive increment
of time an increment of any quality that increases as the odd numbers. Since
Euclid had demonstrated the sum of the odd numbers are the square numbers, the
total quality acquired by the body increases as the square of the time
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