MATH 4400, History of Mathematics The mathematization of motion Professor: Peter Gibson October 22, 2019
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MATH 4400, History of MathematicsThe mathematization of motion
Professor: Peter Gibson
October 22, 2019
The early Renaissance
The Late Middle Ages (approximately 1300-1500) and the earlyRenaissance (the early 16th century) did not produce much newmathematics—in contrast to the arts, which flourished in the earlyRenaissance.
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Historical context
Europe took a long time to rebound from the devastation of famine andpestilence of the 14th century. Having lost a quarter or more of itspopulation during the initial outbreak of the Black Death in mid century,there was a continual population decline until roughly the middle of the15th century.
There was major political and social upheaval during this time. Inparticular, the 100 years war between France and Germany lasted from1337-1453(!).
The year 1453 also marked the Ottoman conquest of Constantinople,bringing to an end the long Byzantine (or Eastern Roman) domination ofthat city. The Ottoman Empire would endure until the First World War.
P. Gibson Math 4400 22.10.2019 3 / 32
Historical context
Europe took a long time to rebound from the devastation of famine andpestilence of the 14th century. Having lost a quarter or more of itspopulation during the initial outbreak of the Black Death in mid century,there was a continual population decline until roughly the middle of the15th century.
There was major political and social upheaval during this time. Inparticular, the 100 years war between France and Germany lasted from1337-1453(!).
The year 1453 also marked the Ottoman conquest of Constantinople,bringing to an end the long Byzantine (or Eastern Roman) domination ofthat city. The Ottoman Empire would endure until the First World War.
P. Gibson Math 4400 22.10.2019 3 / 32
Historical context
Europe took a long time to rebound from the devastation of famine andpestilence of the 14th century. Having lost a quarter or more of itspopulation during the initial outbreak of the Black Death in mid century,there was a continual population decline until roughly the middle of the15th century.
There was major political and social upheaval during this time. Inparticular, the 100 years war between France and Germany lasted from1337-1453(!).
The year 1453 also marked the Ottoman conquest of Constantinople,bringing to an end the long Byzantine (or Eastern Roman) domination ofthat city. The Ottoman Empire would endure until the First World War.
P. Gibson Math 4400 22.10.2019 3 / 32
The late 15th and early 16th centuries were a period of religious upheavalin Europe. In particular the Protestant Reformation impacted each ofEngland, France and Germany.
There was also a Counter Reformation, notably led by the Jesuits, whoestablished a system of highly-organized colleges across the globe.
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Ignatius Loyola (1491-1556)
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Ignatius Loyola (1491-1556)
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A contemporary Jesuit college, still cultivating the mathematical sciences.
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Mathematics of the early Renaissance
There were some genuinely new mathematical methods developed at thistime concerning the solution of polynomial equations.
Scipio del Ferro (died 1526) of the University of Bologna developed amethod for reducing cubic equations to basic types and solving themexplicitly.
His methods were rediscovered by Tartaglia (c.1535), and later publishedby Girolamo Cardano in 1545 in Ars Magna.
The method is known as Cardano’s, despite the fact that he was not thediscoverer.
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Mathematics of the early Renaissance
There were some genuinely new mathematical methods developed at thistime concerning the solution of polynomial equations.
Scipio del Ferro (died 1526) of the University of Bologna developed amethod for reducing cubic equations to basic types and solving themexplicitly.
His methods were rediscovered by Tartaglia (c.1535), and later publishedby Girolamo Cardano in 1545 in Ars Magna.
The method is known as Cardano’s, despite the fact that he was not thediscoverer.
P. Gibson Math 4400 22.10.2019 8 / 32
Mathematics of the early Renaissance
There were some genuinely new mathematical methods developed at thistime concerning the solution of polynomial equations.
Scipio del Ferro (died 1526) of the University of Bologna developed amethod for reducing cubic equations to basic types and solving themexplicitly.
His methods were rediscovered by Tartaglia (c.1535), and later publishedby Girolamo Cardano in 1545 in Ars Magna.
The method is known as Cardano’s, despite the fact that he was not thediscoverer.
P. Gibson Math 4400 22.10.2019 8 / 32
The general cubic equation
ax3 + bx2 + cx + d = 0
simplifies, upon division by a and substituting x = t − b3a , to
t3 + pt + q = 0
where
p =3ac − b2
3a2and q =
2b3 − 9abc + 27a2d
27a3
P. Gibson Math 4400 22.10.2019 9 / 32
The general cubic equation
ax3 + bx2 + cx + d = 0
simplifies, upon division by a and substituting x = t − b3a , to
t3 + pt + q = 0
where
p =3ac − b2
3a2and q =
2b3 − 9abc + 27a2d
27a3
P. Gibson Math 4400 22.10.2019 9 / 32
Adriaan van Roomen (1593)
Problem: Find all the zeros of the polynomial
p(x) = x45 − 45x43 + 945x41 − 12300x39 + 111150x37
−740259x35 + 3764565x33 − 14945040x31 + 46955700x29
−117679100x27 + 236030652x25 − 378658800x23 + 483841800x21
−488494125x19 + 384942375x17 − 232676280x15 + 105306075x13
−34512075x11 + 7811375x9 − 1138500x7 + 95634x5 − 3795x3 + 45x − c
Van Roomen’s problem was solved in part by a French graduate of a Jesuitcollege, Francois Viete (1540-1603).
P. Gibson Math 4400 22.10.2019 10 / 32
Adriaan van Roomen (1593)
Problem: Find all the zeros of the polynomial
p(x) = x45 − 45x43 + 945x41 − 12300x39 + 111150x37
−740259x35 + 3764565x33 − 14945040x31 + 46955700x29
−117679100x27 + 236030652x25 − 378658800x23 + 483841800x21
−488494125x19 + 384942375x17 − 232676280x15 + 105306075x13
−34512075x11 + 7811375x9 − 1138500x7 + 95634x5 − 3795x3 + 45x − c
Van Roomen’s problem was solved in part by a French graduate of a Jesuitcollege, Francois Viete (1540-1603).
P. Gibson Math 4400 22.10.2019 10 / 32
Adriaan van Roomen (1593)
Problem: Find all the zeros of the polynomial
p(x) = x45 − 45x43 + 945x41 − 12300x39 + 111150x37
−740259x35 + 3764565x33 − 14945040x31 + 46955700x29
−117679100x27 + 236030652x25 − 378658800x23 + 483841800x21
−488494125x19 + 384942375x17 − 232676280x15 + 105306075x13
−34512075x11 + 7811375x9 − 1138500x7 + 95634x5 − 3795x3 + 45x − c
Van Roomen’s problem was solved in part by a French graduate of a Jesuitcollege, Francois Viete (1540-1603).
P. Gibson Math 4400 22.10.2019 10 / 32
The 16th century
The early 16th century brought great change to the world. For the firsttime since the Ice Age, the Americas became reconnected to Afro-Eurasia.This had devastating consequences for the pre-existing cultures of theAmericas, bringing to an end the great Aztec and Incan empires. Diseasessuch as smallpox devastated indigenous North and South Americans.
New foods and products flowed from the Americas to the rest of theworld. Precious metals were extracted for world trade. New colonies wereestablished.
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The 16th century
The early 16th century brought great change to the world. For the firsttime since the Ice Age, the Americas became reconnected to Afro-Eurasia.This had devastating consequences for the pre-existing cultures of theAmericas, bringing to an end the great Aztec and Incan empires. Diseasessuch as smallpox devastated indigenous North and South Americans.
New foods and products flowed from the Americas to the rest of theworld. Precious metals were extracted for world trade. New colonies wereestablished.
P. Gibson Math 4400 22.10.2019 11 / 32
The world of knowledge also changed fundamentally. The end of the 16thcentury saw major breakthroughs in the applications of mathematics to anunderstanding of the physical world and of the heavens.
Galileo, Kepler and Descartes all left a lasting mark, both for theirparticular discoveries and—perhaps even more importantly—for their newmethods.
P. Gibson Math 4400 22.10.2019 12 / 32
The world of knowledge also changed fundamentally. The end of the 16thcentury saw major breakthroughs in the applications of mathematics to anunderstanding of the physical world and of the heavens.
Galileo, Kepler and Descartes all left a lasting mark, both for theirparticular discoveries and—perhaps even more importantly—for their newmethods.
P. Gibson Math 4400 22.10.2019 12 / 32
About Kepler
Kepler lived from 1571-1630 in the Holy Roman Empire(present day Germany, Czech Republic and Austria)
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A scene from Ulm...
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Among his works:
Mysterium cosmographicum
Dioptrice
Harmonice Mundi
Tabulae Rudolphinae
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Among his works:
Mysterium cosmographicum
Dioptrice
Harmonice Mundi
Tabulae Rudolphinae
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An important picture from Kepler’s first great work:
A beautiful, mathematical theory of the planets that turned out to be false.
P. Gibson Math 4400 22.10.2019 16 / 32
An important picture from Kepler’s first great work:
A beautiful, mathematical theory of the planets that turned out to be false.
P. Gibson Math 4400 22.10.2019 16 / 32
An important picture from Kepler’s first great work:
A beautiful, mathematical theory of the planets
that turned out to be false.
P. Gibson Math 4400 22.10.2019 16 / 32
An important picture from Kepler’s first great work:
A beautiful, mathematical theory of the planets that turned out to be false.
P. Gibson Math 4400 22.10.2019 16 / 32
From Harmonice mundi:
P. Gibson Math 4400 22.10.2019 17 / 32
From Harmonice mundi:
P. Gibson Math 4400 22.10.2019 17 / 32
Kepler’s work reflects the mathematics of antiquity (Pythagorean harmony,Platonic solids).
But he was also a bridge to modernity, in particular with respect to precisemeasurement of the stars in order to test his theories.
Kepler’s work on polytopes endures. His work on ellipses was a crucial step(as will be seen in later lectures).
P. Gibson Math 4400 22.10.2019 18 / 32
Kepler’s work reflects the mathematics of antiquity (Pythagorean harmony,Platonic solids).
But he was also a bridge to modernity, in particular with respect to precisemeasurement of the stars in order to test his theories.
Kepler’s work on polytopes endures. His work on ellipses was a crucial step(as will be seen in later lectures).
P. Gibson Math 4400 22.10.2019 18 / 32
Kepler’s work reflects the mathematics of antiquity (Pythagorean harmony,Platonic solids).
But he was also a bridge to modernity, in particular with respect to precisemeasurement of the stars in order to test his theories.
Kepler’s work on polytopes endures. His work on ellipses was a crucial step(as will be seen in later lectures).
P. Gibson Math 4400 22.10.2019 18 / 32
Galileo
Galileo lived from 1564 to 1642 in present day Italy
His work marks the transition toward modern science
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Galileo
Galileo lived from 1564 to 1642 in present day Italy
His work marks the transition toward modern science
P. Gibson Math 4400 22.10.2019 19 / 32
Galileo’s works include:
Siderius Nuncius (1610)
Discorso del flusso e reflusso del mare (1616)
Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze(1638)
We shall look closer at (an English translation of) the latter work.
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Galileo’s works include:
Siderius Nuncius (1610)
Discorso del flusso e reflusso del mare (1616)
Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze(1638)
We shall look closer at (an English translation of) the latter work.
P. Gibson Math 4400 22.10.2019 20 / 32
Galileo studied statics
II6 THE TWONEW SCIENCESOF GALII.VOhasherebeensaidthe weightof the solidBE)itselfhasbeenleftoutof consideration,or rather,the prismhasbeenassumedto bedevoidofweight. But if theweightofthe prismis tobetakenaccountofin conjuncCtionwiththeweightE,wemustadd
to theweightEonehalf that of theprismBD: so thatif, forexample,thelatter weighstwopounds and theweightE is tenpounds we musttreat theweightEas if it wereeleven
D pounds.SL_P.Why not
twelve?SALv.Theweight
E,my dearSimp-licio,hangingattheextremeendC acCtsupontheleverBCwithits_allmo-mentoftenpounds:so alsowould the
Fig.17 solid BD if sus-pendedat the samepoint exertitsfilllmomentof twopounds;but, as youknow,this solidis uniformlydistributedthrough-SecondDay consistsin a failureto seethat, in sucha beam,there mustbe equilibriumbetweenthe forcesof tensionand compressionoveranycross-section.The correctpoint of viewseemsfirstto have beenfoundby E. Mariotte in I68Oand by A. Parent in I7I3. Fortunately thiserror does not vitiate the conclusionsof the subsequentpropositionswhich deal only with proportlons--not actual strength--of beams.FollowingK. Pearson(Todhunter'sHistoryof Elasticity)onemightsaythat Galileo'smistakelay in supposingthefibresofthe strainedbeamtobe inextensible.Or, confessingthe anachronism,onemightsay that theerrorconsistedin taking the lowestfibreof thebeamas the neutralaxis.
[Tra_;.]
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He studied motion, based on physical experiment
THIRD DAY 173T_EoP,x_ I, PRo_osmoNI
The timeinwhichanyspaceis traversedby a bodystart-ingfromrestanduniformlyacceleratedisequaltothetimeinwhichthat samespacewouldbetraversedby thesamebodymovingat a uniformspeedwhosevalueisthemeanofthe highestspeedandthe speedjustbeforeaccelerationbegan.
Let us representby the lineABthe timeinwhichthe spaceCD is traversedby a bodywhichstartsfromrestat C andisuniformlyaccelerated;let the finaland highestvalueof thespeedgainedduringthe intervalABbe representedbythelineEB drawnat rightanglesto AB;drawthe lineAE, then alllinesdrawnfromequidistantpointsonABandparalleltoBEwillrepresenttheincreasingvaluesofthespeed, cbeginningwiththe instantA. Let thepointF G _. -bisecCtthe lineEB; drawFG parallelto BA,andGAparalleltoFB,thusformingaparallel-ogramAGFBwhichwillbeequalinareato thetriangleAEB,sincethe sideGFbisecCtsthesideAEat thepointI; forif theparallellinesinthetriangleAEBareextendedtoGI, thenthesum x,ofalltheparallelscontainedinthequadrilateral/isequalto thesumofthosecontainedinthetri-angleAEB;for thosein the triangleIEF areequal to thosecontainedin the triangleGIA,whilethoseincludedinthetrapeziumAIFBare icommon.Sinceeachandeveryinstantoftime_ 1_in the time-intervalABhas its corresponding jpointon the lineAB,fromwhichpointspar- ,_allelsdrawninandlimitedbythetriangleAEB Drepresentthe increasingvaluesof thegrowing Fig.47velocity,andsinceparallelscontainedwithinthe re.angle rep-resentthevaluesofaspeedwhichisnotincreasing,butconstant,it appears,inlikemanner,thatthemomenta[momenta]assumedbythemovingbodymayalsoberepresented,inthe caseof theacceleratedmotion,by the increasingparallelsof the triangleAEB;
In this he broke firmly with academic tradition.
P. Gibson Math 4400 22.10.2019 22 / 32
He studied motion, based on physical experiment
THIRD DAY 173T_EoP,x_ I, PRo_osmoNI
The timeinwhichanyspaceis traversedby a bodystart-ingfromrestanduniformlyacceleratedisequaltothetimeinwhichthat samespacewouldbetraversedby thesamebodymovingat a uniformspeedwhosevalueisthemeanofthe highestspeedandthe speedjustbeforeaccelerationbegan.
Let us representby the lineABthe timeinwhichthe spaceCD is traversedby a bodywhichstartsfromrestat C andisuniformlyaccelerated;let the finaland highestvalueof thespeedgainedduringthe intervalABbe representedbythelineEB drawnat rightanglesto AB;drawthe lineAE, then alllinesdrawnfromequidistantpointsonABandparalleltoBEwillrepresenttheincreasingvaluesofthespeed, cbeginningwiththe instantA. Let thepointF G _. -bisecCtthe lineEB; drawFG parallelto BA,andGAparalleltoFB,thusformingaparallel-ogramAGFBwhichwillbeequalinareato thetriangleAEB,sincethe sideGFbisecCtsthesideAEat thepointI; forif theparallellinesinthetriangleAEBareextendedtoGI, thenthesum x,ofalltheparallelscontainedinthequadrilateral/isequalto thesumofthosecontainedinthetri-angleAEB;for thosein the triangleIEF areequal to thosecontainedin the triangleGIA,whilethoseincludedinthetrapeziumAIFBare icommon.Sinceeachandeveryinstantoftime_ 1_in the time-intervalABhas its corresponding jpointon the lineAB,fromwhichpointspar- ,_allelsdrawninandlimitedbythetriangleAEB Drepresentthe increasingvaluesof thegrowing Fig.47velocity,andsinceparallelscontainedwithinthe re.angle rep-resentthevaluesofaspeedwhichisnotincreasing,butconstant,it appears,inlikemanner,thatthemomenta[momenta]assumedbythemovingbodymayalsoberepresented,inthe caseof theacceleratedmotion,by the increasingparallelsof the triangleAEB;
In this he broke firmly with academic tradition.
P. Gibson Math 4400 22.10.2019 22 / 32
Galileo made extensive use of “the language of the universe”
THIRD DAY z39foreCO£)F>CV:VB,and,accordingto the precedinglemma,CO>CV.Besidesthisit is clearthatthetimeofdescentalongDC is to the timealongDBCasDOCis to the sumofDOandCV.
SCHOLIUM
Fromthe precedingit is possibleto inferthat the path ofquickestdescent[lationemomniumvelocissimam]fromonepointto anotheris not the shortestpath,namely,a straightline,but the arc ofa circle.*In the quadrantBAEC,havingthe sideBCvertical,dividethe arcACinto anynumberof .equalparts,AD,DE,EF, FG,GC,andfromCdrawstraightlinesto the pointsA,D,E, F, G; _ Adrawalsothe straightlinesAD,DE,EF,FG,GC. Evidentlyde-scentalongthepathADCisquicker
[264] DthanalongACaloneor alongDCfromrestatD. Butabody,start-ingfrom rest at A,willtraverseDC morequicklythan the pathADC;while,if it startsfromrestatA, itwilltraversethepathDEC 1_in a shortertimethanDCalone.CHence descentalong the three Fig. lO3chords,ADEC,willtakelesstimethan alongthe twochordsADC. Similarly,followingdescentalongADE,thetimerequiredto traverseEFCislessthanthat neededforECalone.There-foredescentismorerapidalongthefourchordsADEFCthanalongthe threeADEC.Andfinallyabody,afterdescentalongADEF,willtraversethe twochords,FGC,morequicklythanFCalone.Therefore,alongthefivechords,ADEFGC,descentwillbemorerapidthanalongthefour,ADEFC.Consequently* It is wellknownthat the first correctsolutionfor the problemof
quickestdescent,under the conditionof a constantforcewasgivenbyJohn Bernoulli0667-I748). [Trans.]
But mathematics was not always sufficiently advanced.
P. Gibson Math 4400 22.10.2019 23 / 32
Galileo made extensive use of “the language of the universe”
THIRD DAY z39foreCO£)F>CV:VB,and,accordingto the precedinglemma,CO>CV.Besidesthisit is clearthatthetimeofdescentalongDC is to the timealongDBCasDOCis to the sumofDOandCV.
SCHOLIUM
Fromthe precedingit is possibleto inferthat the path ofquickestdescent[lationemomniumvelocissimam]fromonepointto anotheris not the shortestpath,namely,a straightline,but the arc ofa circle.*In the quadrantBAEC,havingthe sideBCvertical,dividethe arcACinto anynumberof .equalparts,AD,DE,EF, FG,GC,andfromCdrawstraightlinesto the pointsA,D,E, F, G; _ Adrawalsothe straightlinesAD,DE,EF,FG,GC. Evidentlyde-scentalongthepathADCisquicker
[264] DthanalongACaloneor alongDCfromrestatD. Butabody,start-ingfrom rest at A,willtraverseDC morequicklythan the pathADC;while,if it startsfromrestatA, itwilltraversethepathDEC 1_in a shortertimethanDCalone.CHence descentalong the three Fig. lO3chords,ADEC,willtakelesstimethan alongthe twochordsADC. Similarly,followingdescentalongADE,thetimerequiredto traverseEFCislessthanthat neededforECalone.There-foredescentismorerapidalongthefourchordsADEFCthanalongthe threeADEC.Andfinallyabody,afterdescentalongADEF,willtraversethe twochords,FGC,morequicklythanFCalone.Therefore,alongthefivechords,ADEFGC,descentwillbemorerapidthanalongthefour,ADEFC.Consequently* It is wellknownthat the first correctsolutionfor the problemof
quickestdescent,under the conditionof a constantforcewasgivenbyJohn Bernoulli0667-I748). [Trans.]
But mathematics was not always sufficiently advanced.
P. Gibson Math 4400 22.10.2019 23 / 32
Two New Sciences
Some basic considerations...
How does the author address the reader?
What is the essential content of the section on uniform motion?
What is natural acceleration?
How does Galileo reason about comparative magnitudes?
Is his reasoning rigorous? Are the results correct?
P. Gibson Math 4400 22.10.2019 24 / 32
Two New Sciences
Some basic considerations...
How does the author address the reader?
What is the essential content of the section on uniform motion?
What is natural acceleration?
How does Galileo reason about comparative magnitudes?
Is his reasoning rigorous? Are the results correct?
P. Gibson Math 4400 22.10.2019 24 / 32
Two New Sciences
Some basic considerations...
How does the author address the reader?
What is the essential content of the section on uniform motion?
What is natural acceleration?
How does Galileo reason about comparative magnitudes?
Is his reasoning rigorous? Are the results correct?
P. Gibson Math 4400 22.10.2019 24 / 32
Two New Sciences
Some basic considerations...
How does the author address the reader?
What is the essential content of the section on uniform motion?
What is natural acceleration?
How does Galileo reason about comparative magnitudes?
Is his reasoning rigorous? Are the results correct?
P. Gibson Math 4400 22.10.2019 24 / 32
Two New Sciences
Some basic considerations...
How does the author address the reader?
What is the essential content of the section on uniform motion?
What is natural acceleration?
How does Galileo reason about comparative magnitudes?
Is his reasoning rigorous? Are the results correct?
P. Gibson Math 4400 22.10.2019 24 / 32
Two New Sciences
Some basic considerations...
How does the author address the reader?
What is the essential content of the section on uniform motion?
What is natural acceleration?
How does Galileo reason about comparative magnitudes?
Is his reasoning rigorous? Are the results correct?
P. Gibson Math 4400 22.10.2019 24 / 32
What is the basic importance of Galileo’s work on motion?
...this and other facts, not few in number or less worth know-ing, I have succeeded in proving; and what I consider more impor-tant, there have been opened up to this vast and most excellentscience, of which my work is merely the beginning, ways and meansby which other minds more acute than mine will explore it’s remotecorners.
P. Gibson Math 4400 22.10.2019 25 / 32
What is the basic importance of Galileo’s work on motion?
...this and other facts, not few in number or less worth know-ing, I have succeeded in proving; and what I consider more impor-tant, there have been opened up to this vast and most excellentscience, of which my work is merely the beginning, ways and meansby which other minds more acute than mine will explore it’s remotecorners.
P. Gibson Math 4400 22.10.2019 25 / 32
From the modern perspective, Galileo’s results on motion, serve as thebasis for an analysis of frames of reference. (This requires someillustration...)
P. Gibson Math 4400 22.10.2019 26 / 32
Descartes
Rene Descartes lived from 1596-1650.
P. Gibson Math 4400 22.10.2019 27 / 32
Descartes
Rene Descartes lived from 1596-1650.
P. Gibson Math 4400 22.10.2019 27 / 32
A brief timeline...
1604-1612 Jesuit college of La Fleche in Anjou
1616 Law degree at University of Poitiers
1618 studies with Isaac Beekman in Holland
1619 enlists in the Bavarian army
1620-1628 travels widely in Europe
1628 moves to Holland
1649 moves to Stockholm
P. Gibson Math 4400 22.10.2019 28 / 32
A brief timeline...
1604-1612 Jesuit college of La Fleche in Anjou
1616 Law degree at University of Poitiers
1618 studies with Isaac Beekman in Holland
1619 enlists in the Bavarian army
1620-1628 travels widely in Europe
1628 moves to Holland
1649 moves to Stockholm
P. Gibson Math 4400 22.10.2019 28 / 32
A brief timeline...
1604-1612 Jesuit college of La Fleche in Anjou
1616 Law degree at University of Poitiers
1618 studies with Isaac Beekman in Holland
1619 enlists in the Bavarian army
1620-1628 travels widely in Europe
1628 moves to Holland
1649 moves to Stockholm
P. Gibson Math 4400 22.10.2019 28 / 32
A brief timeline...
1604-1612 Jesuit college of La Fleche in Anjou
1616 Law degree at University of Poitiers
1618 studies with Isaac Beekman in Holland
1619 enlists in the Bavarian army
1620-1628 travels widely in Europe
1628 moves to Holland
1649 moves to Stockholm
P. Gibson Math 4400 22.10.2019 28 / 32
A brief timeline...
1604-1612 Jesuit college of La Fleche in Anjou
1616 Law degree at University of Poitiers
1618 studies with Isaac Beekman in Holland
1619 enlists in the Bavarian army
1620-1628 travels widely in Europe
1628 moves to Holland
1649 moves to Stockholm
P. Gibson Math 4400 22.10.2019 28 / 32
A brief timeline...
1604-1612 Jesuit college of La Fleche in Anjou
1616 Law degree at University of Poitiers
1618 studies with Isaac Beekman in Holland
1619 enlists in the Bavarian army
1620-1628 travels widely in Europe
1628 moves to Holland
1649 moves to Stockholm
P. Gibson Math 4400 22.10.2019 28 / 32
A brief timeline...
1604-1612 Jesuit college of La Fleche in Anjou
1616 Law degree at University of Poitiers
1618 studies with Isaac Beekman in Holland
1619 enlists in the Bavarian army
1620-1628 travels widely in Europe
1628 moves to Holland
1649 moves to Stockholm
P. Gibson Math 4400 22.10.2019 28 / 32
A brief timeline...
1604-1612 Jesuit college of La Fleche in Anjou
1616 Law degree at University of Poitiers
1618 studies with Isaac Beekman in Holland
1619 enlists in the Bavarian army
1620-1628 travels widely in Europe
1628 moves to Holland
1649 moves to Stockholm
P. Gibson Math 4400 22.10.2019 28 / 32
Important works...
(1641) Meditationes de prima philosophia
(1637) Discours de la methode pour bien conduire sa raison etchercher la verite dans les sciences
I La DioptriqueI Les MeteoresI La Geometrie
P. Gibson Math 4400 22.10.2019 29 / 32
Important works...
(1641) Meditationes de prima philosophia
(1637) Discours de la methode pour bien conduire sa raison etchercher la verite dans les sciences
I La DioptriqueI Les MeteoresI La Geometrie
P. Gibson Math 4400 22.10.2019 29 / 32
Important works...
(1641) Meditationes de prima philosophia
(1637) Discours de la methode pour bien conduire sa raison etchercher la verite dans les sciences
I La DioptriqueI Les MeteoresI La Geometrie
P. Gibson Math 4400 22.10.2019 29 / 32
P. Gibson Math 4400 22.10.2019 30 / 32
P. Gibson Math 4400 22.10.2019 31 / 32
Descartes is often credited with the introduction of analytic geometry.Whether this is valid or not, his work represented a shift toward moremodern notation and rigorous abstraction.
P. Gibson Math 4400 22.10.2019 32 / 32
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