medieval European mathematics

ABOUT MEDIEVAL EUROPEAN MATHEMATICS HISTORY

2. Asidefrom144being
the only square
Fibonaccinumber.Itis
alsothe12thFibonacci
number.Notethat12is
thesquarerootof144.
The first seven digits of the golden
ratio (1618033) concatenated is
prime!

3. ❖BoethiusandhisQuadrivium
❖Nicomachu’sIntroductiontoArithmetic
❖LeonardoPisanoBigollo(Fibonacci)
❖ThomasBradwardine
❖NicoleOresme
❖GiovannidiCasali

4. MedievalMathematics
muchmathematicsandastronomyavailablein
the12thcenturywaswritteninArabic,theEuropeans
learnedArabic.Bytheendofthe12thcenturythebest
mathematicswasdoneinChristianItaly.Duringthis
centurytherewasaspateoftranslationsofArabic
workstoLatin.Latertherewereothertranslations.
Arabic→SpanishArabic→Hebrew(→Latin)Greek→
Latin.
EuropehadfallenintotheDarkAges,inwhich
science,mathematicsandalmostallintellectual
endeavourstagnated.Scholasticscholarsonlyvalued
studiesinthehumanities,suchasphilosophyand
literature,andspentmuchoftheirenergiesquarrelling
oversubtlesubjectsinmetaphysicsandtheology,such
as"Howmanyangelscanstandonthepointofa
needle?"

5. ❖BoethiusandhisQuadrivium
Boethius
wasoneofthemostinfluentialearlymedieval
philosophers.Hismostfamouswork,TheConsolation
ofPhilosophy,wasmostwidelytranslatedandreproduced
secularworkfromthe8thcenturyuntiltheendoftheMiddle
Ages
Quadrivium
(plural:quadrivia)isthefour
subjects,orarts(namelyarithmetic,
geometry,musicandastronomy),
taughtafterteachingthetrivium.The
wordisLatin,meaningfourways,
anditsuseforthefoursubjectshas
beenattributedtoBoethiusor
Cassiodorusinthe6thcentury.

6. Nicomachus of Gerasa
wasanimportantancientmathematicianbestknownforhisworks
IntroductiontoArithmeticandManualofHarmonicsinGreek.HewasborninGerasa,
intheRomanprovinceofSyria,andwasstronglyinfluencedbyAristotle.Hewasa
Neopythagorean,whowroteaboutthemysticalpropertiesofnumbers.
❖Nicomachu’sIntroductiontoArithmetic

7. ❖LeonardoPisanoBigollo(Fibonacci)
LeonardoPisano
isbetterknownbyhisnicknameFibonacci.
FibonacciwasanItalianmathematicianfromthe
RepublicofPisa,consideredtobe"themosttalented
WesternmathematicianoftheMiddleAges".The
nameheiscommonlycalled,Fibonacci,wasmadeup
in1838bytheFranco-ItalianhistorianGuillaumeLibri
andisshortforfiliusBonacci.

8. Fibonaccisequenceisaseriesofnumbersinwhichthenextnumberiscalculated
byaddingtheprevioustwonumbers.Itgoes0,1,1,2,3,5,8,13,21,34,55andsoon.Though
thesequencehadbeendescribedinIndianMathematicslongago,itwasLeonardo
FibonacciwhointroducedthesequencetoWesternEuropeanmathematics.Thesequence
startswithF1=1inLeonardoLiberAbacibutitcanalsobeextendedto0andnegative
integerslikeF0=0,F1=1,F2=2,F3=3,F4=4,F5=5andsoon.
Example:
Thecommondifferenceof
1,3,5,7,9,11,13,15is2.The2isfoundby
addingthetwonumbersbeforeit(1+1)
Thecommondifferenceof
2,5,8,11,14,17,20,23is3.The3isfoundby
addingthetwonumbersbeforeit(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 x
6(which equals 8).
So we can write the rule:
The Rule is x
n= x
n-1+ x
n-2
where:
•x
nis term number "n"
•x
n-1is the previous term (n-1)
•x
n-2is 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
InLiberAbaci,Leonardo
considersahypotheticalsituationwhere
thereisapairofrabbitsputinthefield.
Theymateattheendofonemonthand
bytheendofthesecondmonththe
femaleproducesanotherpair.Therabbit
neverdie,mateexactlyafteramonth
andthefemalesalwaysproducesapair
(onemale,onefemale).Thepuzzlethat
theFibonacciposedwas:howmanypair
willtherebeinoneyear?Ifonecalculates
thenonewillfindthatthenumberof
pairsattheendofthenthmonthwould
beFnorthenthFibonaccinumber.Thus
thenumberofrabbitpairsafter12
monthswouldbeF12or144

12. TheFibonaccinumbersoccurinthesumsof“shallow”diagonalinPascal'striangle
startingwith5,everysecondFibonaccinumberisthelengthofthehypotenuseofaright
trianglewithintegerssides.Fibonaccinumberarealsoanexampleofacompletesequence.
ThismeansthateverypositiveintegerscanbewrittenasasumofFibonaccinumbers,
whereanyonenumberisusedonceatmost.Fibonaccisequenceisusedincomputer
scienceforseveralpurposeliketheFibonaccisearchtechnique,whichisamethodof
searchingasortedarraywithaidfromthesequence.

13. Twoquantitiesaresaidtobeingoldenratioif(a+b)/a=a/bwherea>b>0.itsvalueis(1=root5)/2
or1.6180339887…Goldenratiocanbefoundinpatternsinnaturelikethespiralarrangementofleaves
whichiswhyitiscalleddivineproportion.Theproportionisalsosaidtobeaestheticallypleasingdueto
whichseveralartistsandarchitects.TheFibonacciSequenceandthegoldenratioareintimately
interconnected.TheratioofconsecutiveFibonaccinumbersconvergeandgoldenratioandtheclosed
fromexpressionfortheFibonaccisequenceinvolvesthegoldenration,
Example:

14. The Actual Value
The Golden Ratio is equalto:
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
abis 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
Apartfromdronebees,
Fibonaccisequencecanbefoundin
otherplacesinnaturelikebranchingin
trees,arrangementofleavesona
stem,thefruitletsofapineapple,the
floweringofartichoke,anuncurling
fernandthearrangementofpine
cone.Alsoonmanyplants,thenumber
ofpetalsisaFibonaccinumber.Many
plantsincludingbuttercupshave5
petals;liliesandirishave3petals;
some delphiniumshave8;corn
marigoldshave13petals;someasters
have2whereasdaisiescanbefound
with34,55oreven89petals.

16. ThomasBradwardine,(bornc.1290—diedAug.26,1349,
London),archbishopofCanterbury,theologian,andmathematician.
BradwardinestudiedatMertonCollege,Oxford,andbecameaproctor
there.About1335hemovedtoLondon,andin1337hewasmade
chancellorofSt.Paul’sCathedral.Hebecamearoyalchaplainand
confessortoKingEdwardIII.In1349hewasmadearchbishopof
CanterburybutdiedoftheplaguesoonafterwardduringtheBlack
Death.
Bradwardine’smostfamousworkinhisdaywasatreatiseon
graceandfreewillentitledDecausaDei(1344),inwhichhesostressed
thedivineconcurrencewithallhumanvolitionthathisfollowers
concludedfromitauniversaldeterminism.Bradwardinealsowrote
worksonmathematics.InthetreatiseDeproportionibusvelocitatumin
motibus(1328),heassertedthatanarithmeticincreaseinvelocity
correspondswithageometricincreaseintheoriginalratioofforceto
resistance.ThismistakenviewheldswayinEuropeantheoriesof
mechanicsforalmostacentury.
THOMAS BRADWARDINE

17. THOMAS BRADWARDINE
Bradwardine’s
mostfamousworkinhis
daywasatreatiseon
graceandfreewillentitled
DecausaDei(1344)

18. ❖NicoleOresme
NicoleOresmealsoknownasNicolasOresmewasa
significantphilosopherofthelaterMiddleAges.Oresmewasa
determinedopponentofastrology,whichheattackedonreligious
andscientificgrounds.InDeproportionibusproportionum(On
RatioofRatios)Oresmefirstfixedexaminedraisingrational
numbertorationalpowersbeforeextendinghisworktoinclude
irrationalpower.
Significantly,Oresmedevelopedthefirstproofofthe
divergence(isaninfiniteseriesthatisnotconvergent)oftheharmonic
series(isthedivergentinfiniteseries:σ
??????=1
∞
??????−1=1+
1
2
+
1
3
+
1
4
+⋯
Hisproof,requiringlessadvancedmathematicsthatcurrent“standard”
testsfordivergence(forexample,theintegraltest(ismethodusedto
testinfiniteseriesofnon-negativetermsforconvergence),beginsby
nothingthatforanynthatisapowerof2,therearen/2-1termsinthe
seriesbetween1/(n/2)and1/n.

19. ❖GiovannidiCasali
Giovanni(orJohannes)diCasali(ordaCasale;
c.1320-after1374)wasafriarintheFranciscanOrder,
anaturalphilosopherandatheologian,authorofworks
ontheologyandscience,andapapallegate.
About1346hewroteatreatiseDevelocitate
motusalterationis(ontheVelocityoftheMotionofthe
Alteration)whichwassubsequentlyprintedinVenicein
1505.Inithepresentedagraphicalanalysisofthe
motionofacceleratedbodies.Histeachingin
mathematicsphysicsinfluencedscholarsatthe
UniversityofPadulaanditisbelievedmayhave
ultimatelyinfluencedthesimilarideaspresentedovertwo
centuriesbyGalileoGalilie.

20. REFERENCES
•https://Leonardo-newtonic.com/fibonnaci-facts
•GiovannidaCasale’,Enciclopedieonline,Treccani
•MaartenvanderHeijdenandBertRoest,’Franaut-j’,Franciscan
AUTHORS.13
th
-18
th
Century:ACatalogueinProgress
•MarshallClaget.TheScienceofMechanicsintheMiddle
Ages.(Madison:Univ.ofWisconsinPr.,1959).pp332-3,382-
391.644