Mideval Europe
12 
century
The Europeans learned Arabic in the 12 
century. All
mathematics and astronomy was written in Arabic.
By the end of the 12 century the best
mathematics was done in Christian Italy.
During this century there was a spate of translations of
Arabic works to Latin. Later
Example. Elements in Arabic 
Latin in 1142 by
Adelard of Bath (ca. 1075-1160). He also translated Al-Khwarizmi's
astronomical tables (Arabic Latin) in 1126 and in 1155 translated
Ptolemy's Almgest (Greek Latin) (The world background at this time was the crusades.)
Genealogy of our digits. Following Karl Menninger, Zahlwort und Ziffer (Göttingen: Vanderheock & Ruprecht, 1957-1958, 2 vols.), Vol. II, p. 233.
Gherard of Cremona
Born: 1114 in Cremona, Italy
Died: 1187 in Toledo, Spain
Gherard's name is sometimes written as Gerard. He went to
Toledo, Spain
to learn Arabic so he could read Ptolemy's Almagest since
no Latin translations existed at that time. He remained there for the rest of his life.
Gherard made translations of Ptolemy (1175) and of
Euclid from Arabic. Some of these translations from Arabic became more popular than the (often
earlier) translations from Greek.
In making translations of other Arabic work
he translated the Arabic word for sine into the Latin sinus, from where our
sine function comes.
He also translated .
Adelard of Bath
Born: 1075 in Bath, England
Died: 1160
aka Robert of Chester
During this period (12 century) the Hindu
numerals became known to Latin readers by Adelard of Bath.
Adelard studied and taught in France and travelled in Italy, Syria and Palestine before returning
to Bath.
He became a teacher of the future King Henry II.
Adelard made a Latin translation of Euclid's Elements from
Arabic sources which was
for centuries the chief geometry textbook in the West.
He also translated al'Khwarizmi's
tables, wrote on the abacus and on the astrolabe.
His Quaestiones naturales consists of 76
scientific discussions based on Arabic science.
AbacistsvsAlgorists
Adaption of the new system was very slow.
Leonardo Pisano Fibonacci
Born: 1170 in (probably) Pisa (now in Italy)
Died: 1250 in (possibly) Pisa (now in Italy)
Fibonacci or Leonard of Pisa,
played an important role
in reviving ancient mathematics
and made significant contributions of his own.
Leonardo Pisano is better known by
his nickname Fibonacci. He
played an important role in reviving ancient mathematics and made
significant contributions of his own.
Fibonacci was born in Italy but was educated in North Africa
where his father held a diplomatic post. He travelled widely
with his father, recognising and the enormous advantages of
the mathematical systems used in these countries.
Fibonacci Liber abaci (Book of the Abacus), published in
1202 after his return to Italy, is
based on bits of arithmetic and algebra that Fibonacci had accumulated during his travels. Liber abaci introduced the Hindu-Arabic place-valued
decimal system and the use of Arabic numerals into Europe.
Liber abaci did not appear in print until the 19
century.
A problem in Liber abaci led to the introduction of the
Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered
today. The Fibonacci Quarterly is a modern journal devoted to
studying mathematics related to this sequence.
Fibonacci's other books of major importance are
Practica geometriae in
1220 containing a large collection of geometry and
trigonometry. Also in Liber quadratorum in 1225 he
approximates a root of a cubic obtaining an answer which in
decimal notation is correct to 9 places.
Features of Liber abaci:
``How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on.''
The sequence is given by
which obeys the recursion relation
Some of Fibonacci's results:
Theorem. (i) Every two successive terms are relatively prime.
(ii) 
.
Proof. (i) 
. If 
and 
,
then 
.#.
(ii) From we have
So, if 
exists and equals r, it follows that
ß
This is a section connection.
To show that 
converges, define 
. Clearly,
Then
, 
. If x>0 |f'|<1
Since 
, this establishes convergence.
Alternatively:
.
This is a decreasing function.
So 
. Now give a lower bound. etc,etc,etc.
Other properties:
So to get
This formula can also be used to prove that 
exists.
Also
The pascal triangle connection.
Beginning with each one (1) going down the left diagonal, sum up the diagonal entries where the diagonal slope is 1/3 (i.e. 3 cells right, 1 cell up, ...). This scheme generates the Fibnonacci sequence.
The modern, general form: Given a,b,c, and d. Let
There are many results about such sequences, some similar to those already shown.
A cubic equation. Fibonacci showed that the solution to
the cubic equation
can have no solution of the form 
, where a and b are
rational. He gives an approximation 1; 22, 7, 42, 33, 4, 40 - best to that
time, and for another 300 years. Note the use of sexigesimal numbers.
He also wrote Liber quadratorum, a brilliant work on intermediate analysis. Consider the Diophantine-like problem problem posed by Master John of Palermo. Added or subtracted from the square, the result will be the square of
a rational.
The solution of this problem appears as Proposition 17 of the 24 propositions in the work. Fibonacci's resolution is remarkably sophisticated. First, he defines the notion of congruous numbers:
numbers of the form ab(a+b)(a-b) if a+b is even or 4ab(a+b)(a-b) if a+b is odd. Such numbers he shows must be divisible by 24. Moreover, the system 
and 
has integers solutions only if
m is congruous. Next, he shows that 5 is not congruous, but 
is congruous. From this he is able to find a rational solution. Answer: 
.
Liber quadratorum makes frequent use of the identities
which had also appeared in Diophantus.
Liber abaci summary:
More Examples.
(A) If you give me a coin, we have the same.
(B) If I give you a coin, you have ten times what I have.
is the total amount.
Solution. Add x+1 to both sides of the first equation to get
So
Mideval Universities
Jordanus Nemorarius
(fl. 1220)
,
Jordanus, younger than Fibonacci, was the
founder of the mideval school of mechanics. Almost nothing about him is known, though it is believe he taught in Paris about 1220. He wrote on geometry, arithmetic and mechanics.
He solved a problem that eluded Archimedes, namely
the problem of the inclined plane,
He wrote the book Arithmetica. In it he demonstrates a mastery of the theory of proporition as well as facility with quadratic terms. Some examples of his results:
Theorem. A multiple of a perfect number is abundant. A divisor of a perfect number is deficient.
Note the rhetorical base significant for the use of letters instead of numerals for numbers.
Theorem. If a given number is divided in two parts whose difference is given, each of the parts is determined.
Theorem. If a given number is divided into two parts whose product is determined each of the parts is determined.
Theorem. If a given number is divided into however many parts, whose continued proportions are given, then each of the parts is determined.
Nicole Oresme
Born: 1323 in Allemagne, France
Died: 11 July 1382 in Lisieux, France
After studying theology in Paris, Oresme became bursar in the University of
Paris, then canon and later dean of Rouen. In 1370 he was appointed
chaplain to King Charles V and advised him on financial matters.
Oresme invented coordinate geometry before Descartes whereby he establihed the
logical equivalence between tabulated values and their graphs. He
proposed the use of a graph for plotting a variable magnitude whose
value depends on another. It is possible that Descartes was influenced
by Oresme's work since it was reprinted several times over 100 years
after its first publication.
Oresme also worked on infinite series.
Another work by Oresme contains the first use of a fractional exponent,
although, of course, not in modern notation. Oresme also opposed the
theory of a stationary Earth as proposed by Aristotle and taught motion
of the Earth, 200 years before Copernicus.
Latitude of forms. About 1361 he conceived of the idea to
visualize or picture the way things vary (function representation at an
early stage). Everything measurable, Oresme wrote, is imaginable in the manner of continuous quantity;
hence he drew a velocity-time graph for a body moving with uniform acceleration.
In this connection he used the terms latitude and longitude as we use abscissa and ordinate.
His graphical representation is akin to our analytic geometry.
His use of coordinates was not new however. (Apollonius)
He main interest was in quadratures, and therefore he missed noticing functions and functional ideas, per se.
The graphical representation of function, known as the latitude of forms was a popular topic from the time of Oresme to Galileo. His Tractatus de figuratione potentiarum et mensurarum
was printed four times between 1482 and 1515.
Oresme even suggested a three dimensional version of his latitude of forms.
Oresme generalized Bradwardinés rule of proportion to include
fractional powers giving the equivalents of our laws of exponents
suggested the use of irrational powers 
but failed due to
terminology and notation and 
Merton school - Oxford, Mean Speed Rule, 13 century - distance
travelled by an object in uniform acceleration.
He goes further. Consider this:
halves
Area 1st half : Area 2nd half = 1 : 3 (Mean Speed Rule)
thirds:
fourths:
etc. In as much as the sums of the odd integers are 
, the total
distance covered varies as the square of time
Galileo: law of motion.
Infinite series In the fourteenth century mathematicians had imagination and precision of thought, but lacked algebraic and geometric facility. Hence, they could not extend the classics. However, they ventured into new areas, among them were infinite series. Oresme did this without the horror infiniti of the Greeks. Among the many series summed was:
He produced a beautiful geometric proof.
This appearance of computing infinite sums may be illusory if taken from Oresme's viewpoint. Recall that at the time, philosophy was still very much Aristotelian, and Aristotle's conception of infinity was essentially temporal. An infinite process is one which does not end. But could this be applied to permanent objects? There were tensions.
Aristotle views that continuous objects were infinitely divisible.
But when one divides a length into halves, then one of the halves into halves, one of those quarters into halves and so on indefinitely. Are those pieces really there? Aristotle would say they are only potentially there.
Mathematicians such as William of Ockham and Gregory of Rimini maintained that they were indeed there -- but there was no last member.
This is the usual problem in dealing with the infinite.
An interpretation. What Oresme may have been doing is experimenting with moving these parts around. He conceived of two squares, each a foot in length. He divided the second square into proportional parts by means of vertical slices. These parts were then moved one on top of the other, creating a vertical tower. The total area was two square feet. Oresme's main interest, however, was in showing how a finite object could be infinite in this respect.
Robert Suiseth, (or Swineshead) (fl. ca. 1350), an English logician, better known as Calculator solved the following infinite series problem :
If throughtout the first half of a given time interval a variation continues at a certain intensity, throughout the next quarter of the interval at double this intensity, throughout the following eighth at triple the intensity and so ad infinitum; then the average intensity for the whole inteval will be the intensity of the variation during the second subinterval.
This is equivalent to saying the the sum of the series
Calculator gave a long and tedious proof, not knowing the graphical representation.
Alternate proof: (modern)
Oresme also summed
He proved the harmonic series diverges by grouping
Decline of mideval learning