The document discusses medieval mathematics from the 12th-14th centuries. It provides biographies of several important medieval mathematicians including Fibonacci, who introduced the Fibonacci sequence to Western Europe and studied rabbit populations. It also discusses Nicole Oresme who proved the divergence of the harmonic series and Giovanni di Casali who analyzed accelerated motion graphically. The document notes that during this time, Europeans learned mathematics from Arabic sources that had been translated to Latin.

2. Aside from 144 being
the only square
Fibonacci number. It is
also the 12th Fibonacci
number. Note that 12 is
the square root of 144. The first seven digits of the golden
ratio (1618033) concatenated is
prime!

3. ❖Boethius and his Quadrivium
❖Nicomachu’s Introduction to Arithmetic
❖Leonardo Pisano Bigollo (Fibonacci)
❖Thomas Bradwardine
❖Nicole Oresme
❖Giovanni di Casali

4. Medieval Mathematics
much mathematics and astronomy available in
the 12th century was written in Arabic, the Europeans
learned Arabic. By the endofthe12th century the best
mathematics was done in Christian Italy. During this
century there was a spate of translations of Arabic
works to Latin. Later there were other translations.
Arabic → Spanish Arabic → Hebrew (→ Latin) Greek →
Latin.
Europe had fallen into the Dark Ages, in which
science, mathematics and almost all intellectual
endeavour stagnated. Scholastic scholars only valued
studies in the humanities, such as philosophy and
literature, and spent much of their energies quarrelling
over subtle subjects in metaphysics and theology, such
as "How many angels can stand on the point of a
needle?"

5. ❖Boethius and his Quadrivium
Boethius
was one of the most influential early medieval
philosophers. His most famous work, The Consolation
of Philosophy, was most widely translated and reproduced
secular work from the 8th century until the end of the Middle
Ages
Quadrivium
(plural: quadrivia) is the four
subjects, or arts (namely arithmetic,
geometry, music and astronomy),
taught after teaching the trivium. The
word is Latin, meaning four ways,
and its use for the four subjects has
been attributed to Boethius or
Cassiodorus in the 6th century.

6. Nicomachus of Gerasa
was an important ancient mathematician best known for his works
Introduction to Arithmetic and Manual of Harmonics in Greek. He was born in Gerasa,
in the Roman province of Syria, and was strongly influenced by Aristotle. He was a
Neopythagorean, who wrote about the mystical properties of numbers.
❖Nicomachu’s Introduction to Arithmetic

7. ❖Leonardo Pisano Bigollo (Fibonacci)
Leonardo Pisano
is better known by his nickname Fibonacci.
Fibonacci was an Italian mathematician from the
Republic of Pisa, considered to be "the most talented
Western mathematician of the Middle Ages". The
name he is commonly called, Fibonacci, was made up
in 1838 by the Franco-Italian historian Guillaume Libri
and is short for filius Bonacci .

8. Fibonacci sequence is a series of numbers in which the next number is calculated
by adding the previous two numbers. It goes 0,1,1,2,3,5,8,13,21,34,55 and so on. Though
the sequence had been described in Indian Mathematics long ago, it was Leonardo
Fibonacci who introduced the sequence to Western European mathematics. The sequence
starts with F1=1 in Leonardo Liber Abaci but it can also be extended to 0 and negative
integers like F0=0, F1=1, F2=2, F3=3, F4=4, F5=5 and so on.
Example:
The common difference of
1,3,5,7,9,11,13,15 is 2. The 2 is found by
adding the two numbers before it (1+1)
The common difference of
2,5,8,11,14,17,20,23 is 3. The 3 is found by
adding the two numbers before it (1+2).

9. The Rule
The Fibonacci Sequence can be written as a "Rule" (Sequences and Series).
First, the terms are numbered from 0 onwards like this:
So term number 6 is called x6 (which equals 8).
So we can write the rule:
The Rule is xn = xn-1 + xn-2
where:
•xn is term number "n"
•xn-1 is the previous term (n-1)
•xn-2 is the term before that (n-2)

10. Makes A Spiral
When we make squares with those widths, we get a nice spiral:
Do you see how the squares fit
neatly together?
For example 5 and 8 make 13,
8 and 13 make 21, and so on.
This spiral is found in
nature!

11. ➢ FIBONNACI SEQUENCE WAS THE SOLUTION OF A RABBIT
POPULATION PUZZLE IN LIBER ABACI
In Liber Abaci, Leonardo
considers a hypothetical situation where
there is a pair of rabbits put in the field .
They mate at the end of one month and
by the end of the second month the
female produces another pair. The rabbit
never die , mate exactly after a month
and the females always produces a pair
(one male, one female). The puzzle that
the Fibonacci posed was: how many pair
will there be in one year? If one calculates
then one will find that the number of
pairs at the end of the nth month would
be Fn or the nth Fibonacci number. Thus
the number of rabbit pairs after 12
months would be F12 or 144

12. The Fibonacci numbers occur in the sums of “shallow” diagonal in Pascal's triangle
starting with 5 , every second Fibonacci number is the length of the hypotenuse of a right
triangle with integers sides. Fibonacci number are also an example of a complete sequence.
This means that every positive integers can be written as a sum of Fibonacci numbers,
where any one number is used once at most. Fibonacci sequence is used in computer
science for several purpose like the Fibonacci search technique , which is a method of
searching a sorted array with aid from the sequence.

13. Two quantities are said to be in golden ratio if (a+b)/a=a/b where a>b>0. its value is (1=root5)/2
or 1.6180339887…Golden ratio can be found in patterns in nature like the spiral arrangement of leaves
which is why it is called divine proportion. The proportion is also said to be aesthetically pleasing due to
which several artists and architects. The Fibonacci Sequence and the golden ratio are intimately
interconnected. The ratio of consecutive Fibonacci numbers converge and golden ratio and the closed
from expression for the Fibonacci sequence involves the golden ration,
Example:

14. The Actual Value
The Golden Ratio is equal to:
1.61803398874989484820... (etc.)
The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number,
and I will tell you more about it later.
Formula
We saw above that the Golden Ratio has this property:
ab = a + ba
We can split the right-hand fraction like this:
ab = aa + ba
ab is the Golden Ratio φ, aa=1 and ba=1φ, which gets us:
φ = 1 + 1φ
So the Golden Ratio can be defined in terms of itself!
Let us test it using just a few digits of accuracy:
φ =1 + 11.618
=1 + 0.61805...
=1.61805...
With more digits we would be more accurate.

15. ➢ FIBONACCI NUMBERS CAN BE FOUND IN SEVERAL BIOGICAL
SETTING
Apart from drone bees,
Fibonacci sequence can be found in
other places in nature like branching in
trees, arrangement of leaves on a
stem, the fruitlets of a pineapple, the
flowering of artichoke, an uncurling
fern and the arrangement of pine
cone. Also on many plants, the number
of petals is a Fibonacci number. Many
plants including butter cups have 5
petals; lilies and iris have 3 petals;
some delphiniums have8;corn
marigolds have 13 petals; some asters
have 2 whereas daisies can be found
with 34,55 or even 89 petals.

16. Thomas Bradwardine, (born c. 1290—died Aug. 26, 1349,
London), archbishop of Canterbury, theologian, and mathematician.
Bradwardine studied at Merton College, Oxford, and became a proctor
there. About 1335 he moved to London, and in 1337 he was made
chancellor of St. Paul’s Cathedral. He became a royal chaplain and
confessor to King Edward III. In 1349 he was made archbishop of
Canterbury but died of the plague soon afterward during the Black
Death.
Bradwardine’s most famous work in his day was a treatise on
grace and free will entitled De causa Dei (1344), in which he so stressed
the divine concurrence with all human volition that his followers
concluded from it a universal determinism. Bradwardine also wrote
works on mathematics. In the treatise De proportionibus velocitatum in
motibus (1328), he asserted that an arithmetic increase in velocity
corresponds with a geometric increase in the original ratio of force to
resistance. This mistaken view held sway in European theories of
mechanics for almost a century.
THOMAS BRADWARDINE

17. THOMAS BRADWARDINE
Bradwardine’s
most famous work in his
day was a treatise on
grace and free will entitled
De causa Dei (1344)

18. ❖Nicole Oresme
Nicole Oresme also known as Nicolas Oresme was a
significant philosopher of the later Middle Ages. Oresme was a
determined opponent of astrology, which he attacked on religious
and scientific grounds. In De proportionibus proportionum (On
Ratio of Ratios) Oresme first fixed examined raising rational
number to rational powers before extending his work to include
irrational power.
Significantly, Oresme developed the first proof of the
divergence(is an infinite series that is not convergent)of the harmonic
series (is the divergent infinite series: σ 𝑛=1
∞
𝑛 − 1 = 1 +
1
2
+
1
3
+
1
4
+ ⋯
His proof, requiring less advanced mathematics that current “standard”
tests for divergence( for example, the integral test (is method used to
test infinite series of non-negative terms for convergence), begins by
nothing that for any n that is a power of 2, there are n/2-1 terms in the
series between 1/(n/2) and 1/n.

19. ❖Giovanni di Casali
Giovanni (or Johannes) di Casali (or da Casale;
c. 1320- after 1374) was a friar in the Franciscan Order,
a natural philosopher and a theologian , author of works
on theology and science , and a papal legate.
About 1346 he wrote a treatise De velocitate
motus alterationis( on the Velocity of the Motion of the
Alteration) which was subsequently printed in Venice in
1505. In it he presented a graphical analysis of the
motion of accelerated bodies. His teaching in
mathematics physics influenced scholars at the
University of Padula and it is believed may have
ultimately influenced the similar ideas presented over two
centuries by Galileo Galilie.

20. REFERENCES
• https://Leonardo-newtonic.com/fibonnaci-facts
• Giovanni da Casale’,Enciclopedie on line, Treccani
• Maarten van der Heijden and Bert Roest,’Franaut-j’,Franciscan
AUTHORS.13th -18th Century:A Catalogue in Progress
• Marshall Claget. The Science of Mechanics in the Middle
Ages.(Madison:Univ.of Wisconsin Pr.,1959).pp 332-3,382-
391.644