My Report in History of MATHS

7/21/2019 My Report in History of MATHS

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ARCHIMEDES

APOLLONIU



ARCHIMEDES• Facts:

– Lived in Syracuse– A!ied "at#e"atics t$ractica! r$%!e"s as

e!! as "$re t#e$retica!r$%!e"s

–Died in '(' )CE at t#e#ands $* a R$"an s$!dier durin+ t#e attac, $n

Syracuse %y t#e *$rces $*



– Plutarch claimed Archimedes

requested that a cylinderenclosing a sphere be put onhis gravestone. Cicero claims

to have found that

!ied at the age of 75" #hich

#ould put his birth at about $%7&CE 'from (he &oo) of *istories by (+et+es",$th century CE-.– (he Eure)a/ story came from

the 0oman architect 1itruvius"about a century afterArchimedes2 death.



A0C*34E!E2 603(38

La $* Lever

Postulates9

,.Equal #eights at equal distances arein equilibrium" and equal #eights atunequal distances are not inequilibrium but incline to#ard the

#eight that is at the greater distance.

$. 3f" #hen #eights at certain distances

are in equilibrium" something is addedto one of the #eights" they are not inequilibrium but incline to#ard the#eight to #hich the addition #as made.



:. imilarly" if anything is ta)en a#ayfrom one of the #eights" they are notin equilibrium but incline to#ard the

#eight from #hich nothing #as ta)en.

;. 3f magnitudes at certaindistances are in equilibrium" other

magnitudes equal to them #illalso be in equilibrium at the samedistances.



•

OnFloatingBodies

– (he la#s ofhydrostatics"3ncludingArchimedes2Principle9 Anyob<ect" #hollyor partially

immersed in a



• Measurement of theCircle

– (he area of any circleis equal to that of a

right triangle in #hichone of the legs is equal

to the radius and the



– (he ratio of thecircumference of any circle toits diameter is less than : but

greater than : .



• On the Quadrature of the

Parabola

– Proved" using a double reductio

argument and e?haustion" that thearea of a parabolic segment is of thearea of the inscribed triangle.



•

On the Sphere and theCylinder

–

ho#ed that a sphere hasa volume t#o@thirds that ofa circumscribed cylinder

'i.e." of the same heightand diameter-– ho#ed that the sphere

has an area t#o@thirds that



• The Sand Reckoner

– Archimedes calculates

that the number ofgrains of sand requiredto ll the universe is %B'in modern notation-.– 4entions that his

father #as an



APDD3

• &orn about $;$ &CE" in Perga" onthe 4editerranean coast of #hat is

no# (ur)ey.• tudied and probably #or)edmuch of his life in Ale?andria.

• 4a<or contributions include9– Astronomy – the theory of deferentcircles and epicycles

– 4athematics – the most important#or) bein Conics a #or) in %



Conics– :%F Propositions in %boo)s" 7 of #hich #ehave 'G in 8ree)" : in

Arabic-.Prior to Apollonius" conic

sections #ere described interms of the intersection of acone and a plane" but9• (he plane of intersection#as al#ays perpendicular to aside" and the verte? angle ofthe cone #as either acute



• sed the double/ cone" andsho#ed that the conics could bedescribed by intersections #ith

more arbitrary planes.



C$nics

• !eveloped methods verysimilar to those of analyticgeometry" using for a?es a

diameter and a tangent.



C$nics

• )$$, I9 0elations satised by the diametersand tangents of conics" and ho#to dra# tangents to given conics.

• )$$, II9 *o# hyperbolas are related to

their asymptotes• )$$, III9 4ore tangents• )$$, I19 3ntersections of conics

(hese are considered &asic/ by Apollonius

although he does prove ne# results" especiallyin &oo) 333.• )$$,s 1 – 1II9

– discuss normal to conics and inparticular ho# they can be dra#n froma point and propositions determining



Na"es $* t#e C$nics• (he parabola #as considered to bethe locus of points such that thesquare on the ordinate #as equal toa rectangle on the abscissa andparameter '-.• (ranslation9 the square of y #as

equal to a multiple of ?" or in other#ords H " for a parabola #ith verte?at the origin.• (hese points could actually be lotted/ in a sense b usin



Ior ellipses and hyperbolas#ith one verte? at theorigin" the equations can

be #ritten as9 H ," or letting " as9 H " #ith the J/for a hyperbola and the

/ for the ellipse. o" #ehave9



•

H for the parabola"• K for the hyperbola" and• L for the ellipse.

• Ellipsis/ refers to a deciency –leaving somethingout.

• *yperbola/ refers to an e?cess –

a thro#ingbeyond.

• Parabola/ refers to placing

beside" or a comparison.



4athematical

4ethods in*ellenistic



THE HELLENISTIC PERIOD

3n this brief section" #e sho#

#hat type of problemsmathematicians solved" and thee?tent of 8ree) 4athematicsbefore the collapse of the

mathematical #orld in about GMMC.E.



EUDOXUS

*e is famous in his #or)s in ratios and

methods of e?haustion.*e is responsible in turning the astronomyinto a mathematical science.*e is the inventor of the t#osphere model/"

the modications for the various motions ofthe sun" moon and planets.



APOLLONIUS

*e proposed the follo#ing solution9 Place the center of the sun2s orbit at apoint 'called the eccenter- displayeda#ay from the earth. 3f the sun moves

uniformly around the ne# circle 'calledthe deferent circle-" an observer on earth#ill see more than a quarter of the circleagainst the spring quadrant 'the upperright- than the summer quadrant 'the

upper left-. (he distance of E! or betterthe ratio of E! to ! is )no#n as theeccentricity of the deferent.



HIPPARCHUS AND THEBEGINNING OF TRIGONOMETRY*e carried out numerous observations ofplanetary positions" introduced a coordinate systemfor the stellar sphere" the tabulation oftrigonometric ratios.Syste" C$$rdinate

(o deal #ith the positions of the stars andplanets" one needs both a unit of measure for arcs

and angles" as #ell as" a method of specifying#here a particular body is located on the celestialsphere..



*e adopted the division of the circumference of the

circle

into :;M parts 'degrees-" along #ith the se?agisimal

division of degrees into minutes and seconds.

*e also used arcs of ,N$G of a circle 'steps- and ,NG%

of a

circle 'halfsteps-.

&abylonians rst introduced coordinates into the s)y

is

also )no#n as the Ecliptic ystem.

(he coordinate along the ecliptic is called thelongitude.

(he coordinate perpendicular is called the latitude



(he basic element in*ipparchus2 trigonometry #as thechord subtending a given arc 'orcentral angle- in a circle of ?edradius.



*ipparchus used the same measure for the

radius of the

circle. circumference = 2πR, π the se?agisimal

appro?imation 3;8,30

*e calculated the radius 0 as 60 x 360/ 2π = 6,0,0/6;! = "!,8 = 3#38$

3n a circle of radius" the measure of an

angle 'length cut oO on the circumference divided by theradius- equals its

radian measure.



(o calculate the table of chords" he

began #ith a ;Mangle.

C%&'() = C%& '60) = 3#38$ = "!,8*0+ -.C1%& 4 5 7 R2 = #862$ = 8, 2

(o calculate of the other angles" he used

t#ogeometric results"

9%& '80:() = '2R)2 < 9%&2'()



PTOLEMY AND THE

ALMAGESTClaudius Ptolemy

- *e made numerous observations of theheavens from locations near Ale?andriaand #rote several important boo)s.

- 4athemati)i ynta?is '4athematicalCollection-" a #or) in ,: boo)s thatcontained a complete mathematical

description of the 8ree) model of theuniverse #ith parameters for the variousmotions of the sun" moon and planets.



Chords Tables

: (o construct the tables of chords ofall arcs from /2+ 7 80+ '-7%4 >

/2+) 1 7? R = 60

Iirst calculation established the chord of

:; namely" the

length of decagon inscribed in a circle.



ROMAN MATHEMATICSC9% admitted that 0omans #ere not interested in

4athematics.

@7%4

*e #rote the boo) n Architecture/" architectsneeded

comprehensive liberal education 'draftsmanship

to astronomy-. 8eometry oOers many aids to architecture.



Lucius Columella: 0oman 8entleman Iarmer*e #rote that one deals #ith elds needs tobe able to

#or) out areas.&asic Iormulas 'area of squares" rectangles"triangles" circles- including the use of '/3 /0) 42 for the area of triangle of side 4, 22/! =

π -& A = ' 1)1 /# 'D/2)2 for the area ofa circle segment of base b and height h.



he Iinal Chapters of 8ree) 4athematics



)y Pt$!e"y2s ti"e

• the entire eastern

4editerranean #as part of the0oman Empire• the local rulers and romans left

the local language and cultureintact and the great Ciceroadmitted that the 0omans #erenot that interest in math.



3#e !ast $* t#e 4ree,"at#e"atians ere9

• icomachus of 8erasa• !iophantus of

Ale?andria• Pappus• *ypatia



ot much is )no# about

icomachus although it isspeculated that he studied inAle?andria. *e dealt #ith

perfect primes #here hecalculated the rst four9 ;" $%"GF; and %,$%. *e" ho#everdid not present any proof.&esides perfect numbers"icomachus also #or)ed #ith



PEN3A4ONALNUM)ERS



Nic$"ac#us used a r$$rti$n in adi0erent sense *$r" Euc!id5s- F$rNic$"ac#us t#ere as 6 tyes $*r$$rti$ns:

(- arit#"etic r$$rti$n7consecutive pair of numbers diOerby the same quantity 'for e?ample,"5"F",: ... etc diOer by G-

'- +e$"etric r$$rti$n7 Qthegreatest terms is to the ne?tgreatest as that one is to the ne?tQ

'for e?ample :"F"$7...etc-



:. Har"$nic r$$rti$n 7Qgreatest term is to the smallest

as the diOerence bet#een thegreat and mean terms is thediOerence bet#een the mean andthe smallest termsQ 'for e?ample9:"G"; are harmonic proportionsince ;9: H ';G-9 'G:-



!iophantus #rote arithmetica#hich is divided into ,: boo)s. *e

introduced symbolic abbreviationsand dealt #ith po#ers higher than:" a brea) from the traditional

8ree) usage.



He a!s$ resented s$"einterestin+ r$%!e"s:

• (o divide a give number into t#o

having a given diOerence• (o divide a given number into

t#o number such that givenfractions 'not the same- of each

number #hen added togetherproduced a given number

• (o divide a given square into t#osquares



Paus is best )no#n for his Collections"

#hich are #or)s on various topicsin mathematics. (hrough one ofhis remar)s #e learned that

#omen #here involved inmathematics in Ale?andria.



Hyatia':55G,5-#as the ,st

#omen

mathematician of#hom any detailsare )no# and herviolent death

mar)ed theeOective end ofthe 8ree)mathematiciantradition in



Hyatia #as a respected andeminent teach in Ale?andria.he #as given a thorough

education in math andphilosophy by her father" (heon of Ale?andria. !espite

having in=uential friends"#hen a rival of her friendsspread rumors about her

about practicing sorcery a



Her$n #rote a detailed#or) on indirectmeasurement. Ior instance"#e have t#o points" A and &"on one inaccessible side of ariver '#here the observer ison the other side-. 6e #antto nd the distance bet#eenA and &. *eron used similar





HA88 NA9U

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My Report in History of MATHS