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ARCHIMEDES
APOLLONIU
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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 $*
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– 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.
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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.
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:. 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.
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•
OnFloatingBodies
– (he la#s ofhydrostatics"3ncludingArchimedes2Principle9 Anyob<ect" #hollyor partially
immersed in a
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• 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
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– (he ratio of thecircumference of any circle toits diameter is less than : but
greater than : .
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• 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.
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•
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
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• 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
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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 %
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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
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• sed the double/ cone" andsho#ed that the conics could bedescribed by intersections #ith
more arbitrary planes.
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C$nics
• !eveloped methods verysimilar to those of analyticgeometry" using for a?es a
diameter and a tangent.
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C$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
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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
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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
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•
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.
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4athematical
4ethods in*ellenistic
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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.
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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.
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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.
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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..
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*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$G of a circle 'steps- and ,NG%
of a
circle 'halfsteps-.
&abylonians rst introduced coordinates into the s)y
is
also )no#n as the Ecliptic ystem.
(he coordinate along the ecliptic is called thelongitude.
(he coordinate perpendicular is called the latitude
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(he basic element in*ipparchus2 trigonometry #as thechord subtending a given arc 'orcentral angle- in a circle of ?edradius.
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*ipparchus used the same measure for the
radius of the
circle. circumference = 2πR, π the se?agisimal
appro?imation 3;8,30
*e calculated the radius 0 as 60 x 360/ 2π = 6,0,0/6;! = "!,8 = 3#38$
3n a circle of radius" the measure of an
angle 'length cut oO on the circumference divided by theradius- equals its
radian measure.
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(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'()
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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.
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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.
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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.
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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.
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he Iinal Chapters of 8ree) 4athematics
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)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.
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3#e !ast $* t#e 4ree,"at#e"atians ere9
• icomachus of 8erasa• !iophantus of
Ale?andria• Pappus• *ypatia
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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
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PEN3A4ONALNUM)ERS
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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"$7...etc-
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:. Har"$nic r$$rti$n 7Qgreatest term is to the smallest
as the diOerence bet#een thegreat and mean terms is thediOerence bet#een the mean andthe smallest termsQ 'for e?ample9:"G"; are harmonic proportionsince ;9: H ';G-9 'G:-
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!iophantus #rote arithmetica#hich is divided into ,: boo)s. *e
introduced symbolic abbreviationsand dealt #ith po#ers higher than:" a brea) from the traditional
8ree) usage.
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He a!s$ 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
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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.
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Hyatia':55G,5-#as the ,st
#omen
mathematician of#hom any detailsare )no# and herviolent death
mar)ed theeOective end ofthe 8ree)mathematiciantradition in
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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
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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
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