Dr Joseph Lee, Dr Louis Leung - Hong Kong Polytechnic

AMA1D01C – Europe since Renaissance

Dr Joseph Lee, Dr Louis Leung

Hong Kong Polytechnic University

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance

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References

These notes mainly follow material from the following book:

I Katz, V. A History of Mathematics: an Introduction.

Addison-Wesley, 1998.

and also use material from the following sources:

I Burton, D. The History of Mathematics: an Introduction.

McGraw-Hill, 2011.

I MacTutor History of Mathematics Archive, University of StAndrews. http://www-history.mcs.st-and.ac.uk/

I Struik, D. A Concise History of Mathematics. G. Bell andSons, 1954.

I Simmons, G. Di↵erential Equations with Historical Notes.

McGraw-Hill, 1991.


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Europe since Renaissance

I Renaissance - a period of transition (1350-1550) when Europetransformed from a feudal and ecclesiastical (dominated bythe Church) society to one which is urban and secular

I Important event: invention of printing by Johann Gutenberg(1450). Books were now cheaper.

I Also the fall of Constantinople to the Turks in 1453 brought awave of Greek scholars to Italy

I Economic factors: success of the Italian merchant republics

I After Renaissance came the Age of Enlightenment and theScientific Revolution, where human reason was emphasizedover the authority of the Church


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Europe since Renaissance

Bertrand Russell on early Renaissance:“Many of them [Italians of the Renaissance period] had still thereverence for authority that medieval philosophers had had, butthey substiuted the authority of the ancients for that of theChurch. This was, of course, a step towards emancipation, sincethe ancients disagreed with each other, and individual judgementwas required to decide which of them to follow. But very fewItalians of the fifteenth century would have dared to hold anopinion for which no authority could be found either in antiquity orin the teaching of the Church.” (B. Russell, History of Western

Philosophy)


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Major Figures

Some of the major figures (we will not have time to cover all ofthem)

I Newton (1643-1727), Leibniz (1646-1716), The Bernoullis,Euler (1707-1783), Lagrange (1736-1813), Laplace(1749-1827), Fourier (1768-1830), Gauss (1777-1855),Cauchy (1789-1857), Abel (1802-1829), Dirichlet(1805-1859), Galois (1811-1832), Weierstrass (1815-1897),Stokes (1819-1903), Riemann (1826-1866), Dedekind(1831-1916), Cantor (1845-1918), Hilbert (1862-1943), Godel(1906-1978)


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Summary

I Major advances in geometry, analysis, algebra

I Since the 19th century, attention was given to the internallogic of mathematics

I Bertrand Russell and Kurt Godel


Calendar Reform

I The Julian calendar, introduced under Julius Caesar in 46 BC,has the average year at 3651

4days

I The mean tropical year, however, is approximately 365.2422days.

I Drifting seasons (relative to the calendar) means driftingEaster (first Sunday after the first full moon after the vernalequinox)

I In 1580 Pope Gregory XIII called for calendar reformI The days between 4 Oct and 15 Oct 1582 were dropped (i.e.,

in the year 1582, 4 Oct was followed by 15 Oct)I Years which are multiples of 100 but not of 400 are no longer

leap yearsI Example: 2000 was a leap year but 1900, 1800 and 1700 were

notI Average length of year 365·400+97

400= 365.2425


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春分 (Spring Equinox)

Calendar Reform

I John Herschel, the son of William Herschel (who discoveredUranus), proposed that years which are multiples of 4000should be common (non-leap)

I If adopted, the average length of a year is365·4000+969

4000= 365.24225

I The proposal is not yet o�cially adopted. (Plus it’s notsomething we have to worry about for another 2000 years.)

I Problem: the year is a di�cult thing to define(http://adsbit.harvard.edu//full/1992JBAA..102...40M/0000042.000.html).


約翰·赫歇爾

Descartes

Rene Descartes (1596-1650)

I Applied methods of algebra to geometry

I Cartesian coordinates

I “I think, therefore I am.” (Principia philosophiae)


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笛卡兒

我思故我在

Descartes

Rene Descartes

I “While we thus reject all of which we can entertain thesmallest doubt, and even imagine that it is false, we easilyindeed suppose that there is neither God, nor sky, nor bodies,and that we ourselves even have neither hands nor feet, nor,finally, a body; but we cannot in the same way suppose thatwe are not while we doubt of the truth of these things; forthere is a repugnance in conceiving that what thinks does notexist at the very time when it thinks. Accordingly, theknowledge, I think, therefore I am, is the first and mostcertain that occurs to one who philosophizes orderly.” http:

//www.gutenberg.org/cache/epub/4391/pg4391.txt


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Fermat

Pierre de Fermat (1601-1665)

I A mathematician and a lawyer

I Treated mathematics as a hobby, but with passion

I Did not like writing down the details of his proofs

I Worked on number theory (the study of integers) andgeometry

I Most famous for Fermat’s Last Theorem


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費馬

Fermat’s Last Theorem

I Fermat is most well known for his “Last Theorem”

I Fermat’s Last Theorem: xn + yn = z

n has no integersolutions when n is an integer greater than 2

I Found by his son Samuel in Pierre’s copy of Diophantus’Arithmetica

I “I have discovered a truly remarkable proof which this marginis too small to contain.”


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I In June 1993, Andrew Wiles claimed to have a proof.

I In December, Wiles withdrew his claim because of a problemfound in his proof.

I In October 1994, Wiles gave a proof which was accepted bythe mathematical community.

I Mathematicians who made contributions along the way: GerdFaltings, Goro Shimura, Yutaka Taniyama, Andre Weil,Richard Taylor


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Figure: Fermat’s Last Theorem on a stamp. Source:http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Fermat.html


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Pascal

Blaise Pascal (1623-1662)

I Studied geometry, probability and physics

I Communicated with Fermat on the topic of probability

I Pascal’s Triangle

I Invented a mechanical calculator (a.k.a. the Pascaline) for hisfather, who was a tax collector

I Pascal’s Wager (from Pensees, “Thoughts”): It is the rationalthing to wager for (i.e., to bet on) the existence of God.

I Summary: By wagering for God, one gets infinite reward ifGod exists but at most finite loss if he doesn’t. By wageringagainst God, one gets infinite punishment if God exists and atmost a finite gain if he doesn’t.

I https://plato.stanford.edu/entries/pascal-wager/


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帕斯卡

Pascaline

Figure: The Pascaline. Source:http://www-history.mcs.st-andrews.ac.uk/Diagrams/Pascals_machine.html


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Kepler

Kepler’s Laws

I Empirical laws without a theory

I Inspired Newton’s later mathematical formulation of gravity

I 1. The planets revolve around the sun in elliptic orbits, withthe sun at one focus.

I 2. The line joining the sun and a planet sweeps out equalareas in equal intervals of time.

I 3. The square of the orbital period of a planet is proportionalto the cube of its mean distance from the sun.


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開普勒

Kepler

Figure: Kepler’s Second Law.Source:https://www-istp.gsfc.nasa.gov/stargaze/Kep3laws.htm


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Newton

Isaac Newton

I Studied at Cambridge

I Philosophiae Naturalis Principia Mathematica.(“Mathematical Principles of Natural Philosophy”)

I Di�cult to assess his influence on contemporaries because ofhis hesitation to publish


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Newton

Isaac Newton

I Explained Kepler’s laws using gravitation following aninverse-square law

I I.e., the force of gravitation is proportional to 1

r2 , where r isdistance

I The inverse square law is what we would expect if the force ofgravity is evenly distributed on the surface of an imaginarysphere

I He called quantities changing with time fluents and their ratesof change fluxions, denoted by x , a notation which is still usedtoday, especially in physics

I Also realized that finding fluents (functions) from fluxions(derivatives) is equivalent to finding areas under curves


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Leibniz

I Born in Leipzig, Germany

I “[H]is philosophy embraced history, theology, linguistics,biology, geology, mathematics, diplomacy, and the art ofinventing.” (Struik)

I Invented calculus the same time as Newton

I Introduced the notations d andR

which we are familiar withtoday

I Also introduced the terms calculus di↵erntialis and calculus

integralis

I The formula ⇡4= 1� 1

3+ 1

5� 1

7+ 1

9. . . is called the Leibniz

formula (one of the many things named after him) but it isbelieved that James Gregory discovered it earlier


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萊布尼茲

Bernoulli

Jacob Bernoulli (1654-1705)

I Solved the Brachistochrone and the Tautochrone problems

I Wrote Ars Conjectandi (“Art of Conjecturing”), a book onthe theory of probability

I Today, experiments with yes-no outcomes (e.g., flipping acoin) are called Bernoulli trials


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雅各布·伯努利

Bernoulli

Johann Bernoulli (1667-1748)

I Younger brother of Jacob

I Solved the Brachistochrone and the Tautochrone problems

I Brachistochrone:https://www.youtube.com/watch?v=Z-qaXZeJT4s

I Tautochrone:https://www.youtube.com/watch?v=Q2q7e-ReC0A

I Both problems have cycloids as their solutions

I Cycloids: http://mathworld.wolfram.com/Cycloid.html


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约翰·伯努利

Brachistochrone

I Posed by Johann Bernoulli

I Knowing what the solution was, he challenged othermathematicians to solve it

I Five mathematicians (including Johann himself) providedsolutions

I Newton, Jacob Bernoulli, Leibniz, l’Hopital, Johann Bernoulli


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Ehrenfried Walther von TschirnhausGuillaume de l'Hôpital

Brachistochrone

I Newton’s solution was published in the January 1697 issue ofThe Philosophical Transactions of the Royal Society

I In the May 1697 issue of Acta Eruditorum (“Acts ofScholars”), solutions by Leibniz, Johann B., Jacob B. and aLatin translation of Newton’s solution could be found onpages 205, 206, 211 and 223, respectively

I l’Hopital’s solutions was not published until 1988


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Bernoulli

Johann Bernoulli (1667-1748)

I Discovered eix = cos x + i sin x

I His two sons, Nicolaus, and Daniel, were also mathematicians


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Bernoulli

Daniel Bernoulli (1700-1782)

I Johann’s son

I Most famous for Bernoulli’s Equation, an equation describingthe relation between the speed and pressure of a moving fluid


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丹尼爾·伯努利

Euler

Leonhard Euler (1701-1783)

I Son of a pastor in Switzerland and studied theology at theUniversity of Basel

I Met Johann Bernoulli there

I Joined the Academy of St. Petersburg 1730

I Three years later became chief mathematician at the academy

I Moved to the Berlin Academy in 1741 and directed themathematical division there

I Returned to St Petersburg in 1766

I Left so much unpublished material that it took the St.Petersburg Academy 47 years to publish all his manuscripts


萊昂哈德·歐拉

Euler


I Introduced the modern notation sin x , cos x , f (x), and ⌃


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Euler


I Correctly determined that 1 + 1

22+ 1

32+ 1

42+ . . . = ⇡2

6

I Developed a theory of di↵erential equations

I Example: from dydx = � x

y we get x2 + y2 = C


Euler

Contributions to number theory:

I Introduced the phi function '(n), which is the number ofpositive integers smaller than n and relatively prime with n

I Application: Modern-day encryption

I If n is a prime number, for example, then '(n) = n � 1

I Example: Let n = 20. Out of the set{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20},only the numbers {1, 3, 7, 9, 11, 13, 17, 19} have no commonfactor with 20. The latter set has 8 elements.

I Therefore '(20) = 8.


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Euler

More Phi-function examples:

I '(13) = 12

I '(30) = 8

I '(40) = 16


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Euler

Contributions to number theory:

I Proved (1751): Every prime of the form 4k + 1 can be writtenuniquely as a sum of two squares

I Proved (1760): If gcd(a,n)=1, then n divides the di↵erencea'(n) � 1.

I Proved (1770): It is impossible to have non-trivial integersolutions to x

3 + y3 = z

3.

I Showed (1732): 225

+ 1 is not prime. (Fermat showed that22

n+ 1, for n = 1, 2, 3, 4, were all primes.


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Euler

Figure: A postage stamp in honour of Euler, issued by former EastGermany. Source:http://www-groups.dcs.st-and.ac.uk/history/PictDisplay/Euler.html


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Euler

I Introduced the concept of double integrals in a paper in 1769

I Used it to find volumes of solids under graphs (which aresurfaces) of functions of two variables

I Found a change of variable formula for double integration


Euler


I Koningsberg (now Kaliningrad) Bridges Problem: Can aperson take a walk so that each bridge is crossed exactly once?

I Euler proved it was not possible.


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Königsberg

Koningsberg

Figure: Koningsberg Briges Problem. Source:http://www-history.mcs.st-andrews.ac.uk/Extras/Konigsberg.html


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Königsberg

Königsberg

Koningsberg

Figure: Koningsberg Briges Problem. Schematic.


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Königsberg

Königsberg

Euler


I We need to find a sequence of 8 letters, such that B appearsnext to A 2 times, C appears next to A 2 times, and Dappears next to A, B, and C one time each

I If an island has an odd number of bridges connecting to it,the region must appear in the sequence n+1

2times

I Therefore A, B, C, D must appear in the sequence 3, 2, 2,and 2 times, respectively

I 3 + 2 + 2 + 2 = 9 > 8, contradiction. Therefore the requiredpath does not exist.


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Non-commutativity

Figure: If an island has an odd number of bridges connecting to it, theregion must appear in the sequence n+1

2times


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Euler


I Proved (1751): v � e + f = 2

I Can be used to prove there are only five Platonic solids

I Beginning of topology, the study of shapes of spaces with noattention to volumes or distances

I For example, a co↵ee mug and a donut are consideredequivalent

I https://www.youtube.com/watch?time_continue=45&v=

9NlqYr6-TpA


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Gauss

Carl Friedrich Gauss (1777-1855)

I Born into a poor family in Brunswick, Germany

I Duke Ferdinand of Brunswick’s support allowed him to studyat the University of Gottingen

I Returned to Brunswick in 1798 and earned a poor living byprivate tutoring

I Later Duke Ferdinand granted him a fixed position so hecould focus on research

I His doctoral dissertation, at the University of Helmstadt, gavethe first substantial proof of the Fundamental Theorem ofAlgebra


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高斯

Gauss


I Fundamental Theorem of Algebra: Every polynomial withcomplex coe�cients (which may or may not be real) has atleast one complex root (which may or may not be real).

I His calculation of the orbit of the asteroid Ceres led him to ano↵er from St. Petersburg, but he declined it.

I Duke Ferdinand died in the 1806 when he led Prussian troopsto fight Napoleon, Gauss’ friends helped him secure theposition of director of the observatory at Gottingen. He heldthis poistion until his death

I Said to be the last “complete mathematician”, as after themid-19th century di↵erent fields of mathematics became sospecialized that only mathematicians within one field wouldhave complete understanding of the field


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ᨕᐟจฦṼॊ

Gauss


I Proved (1796): The regular 17-gon is constructible bycompass and straightedge

I Proved (1801): A regular p-gon, where p is prime and is equal

to 22k+ 1 for some k , is constructible by compass and

straightedge.

I 1801: Based on 41 days’ worth of data, Gauss calculatedCeres’ orbit, and at the end of the year the asteroid appearedwhere Gauss predicted

I Later calculated the orbits of many other asteroids


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Gauss


I Defined the curvature of a surface (Gaussian curvature)

I Given a small fixed area, how much area does the normal(perpendicular) vector sweeps out in the sky?

I Theorema Egregium (“The Remarkable Theorem”): TheGaussian curvature is preserved by local isometries.

I In more intuitive language, just by walking around within alocal region and making measurements (i.e., without leavingthe surface of the earth or going a full circle), we mayconclude that the earth is a curved surface


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Gauss

Figure: Gauss curvature. Source:https://starchild.gsfc.nasa.gov/docs/StarChild/questions/question35.html


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Gauss


I Introduced the notation for congruence: a ⌘ b mod c (cdivides a� b)

I Method of least squares

I Prime distribution:

⇡(n) ⇡Z n

2

dx

ln x⇡

1X

k=0

k!n

(ln n)k+1⇡ n

ln n+

n

(ln n)2+

2n

(ln n)3+. . .


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Laplace

Pierre Simon de Laplace (1749-1827)

I Wrote Mecanique Celeste (“Celestial Mechanics”)

I Also Theorie Analytique des Probabilites (“Analytic Theory of

Probability”)

I Accused of changing back and forth between republicanismand royalism for personal gains


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拉普拉斯

Lagrange

Joseph Louis Lagrange (1736-1813)

I Made contributions in calculus of variations, analyticalmechanics, number theory and algebra

I When Euler left Berlin for St. Petersberg, he recommendedLagrange to Frederick the Great

I Today points where a satellite can orbit in a constantconfiguration with the Sun and the Earth is called a Lagrangepoint (there are five of them).


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拉格朗日

Lagrange

Figure: Lagrange Points. Source:https://map.gsfc.nasa.gov/mission/observatory_l2.html


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Abel

Niels Henrik Abel (1802-1829)

I Born near Stavanger, Norway

I Died of tuberculosis at age 26

I The Abel Prize, established in 2002, was named after him


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阿貝爾

Abel

I An equation can be solved by radicals if any root can beexpressed in terms of the coe�cients using +,�,⇥,÷ androots

I Proved that the general 5th-degree polynomial cannot besolved by radicals

I A group (a set in which we can do multiplication satisfyingcertain properties) where multiplication is commutative (i.e.,AB = BA for any A, B) is called Abelian


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Non-commutativity

Figure: AB 6= BA


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Galois

Evariste Galois (1811-1832)

I Born in Bourg-la-Reine, France

I Went to Ecole Normale

I Participated in the Revolution of 1830

I Died in a duel five months before his twenty-first birthday

I Many speculations on the nature of this duel: from a purelyromantic a↵air to a staged political assassination

I Major contributions were in the field of algebra.


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伽羅華

Riemann

Bernhard Riemann (1826-1866)

I Got his doctoral degree at Gottingen in 1851

I Died at the age of 40


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黎曼

Riemann

Bernhard Riemann (1826-1866)

I Introduced what we today call “Riemannian geometry”,spaces which may be curved but is not embedded in anotherspace (imagine a 2-dimensional sphere living by itself, withoutthe 3-dimensional space it sits in)

I Lay the foundation for, for example, general relativity


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Twin Prime Conjecture

Twin primes are a pair of primes p, q such that q = p + 2.Twin Prime Conjecture: There are infinitely many pairs of twinprimes.

I Proposed by French mathematician Alphonse de Polignac in1849.

I Zhang Yitang (2013): There exists an integer n smaller than70, 000, 000 such that there are infinitely many pairs of primesp, q, where q = p + n.

I 70, 000, 000 is much bigger than 2, but the distance is nothingwhen compared to the distance between infinity and any realnumber.


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波利尼亞克

張益唐

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Dr Joseph Lee, Dr Louis Leung - Hong Kong Polytechnic