AMA1D01C – Europe since Renaissance

Dr Joseph Lee, Dr Louis Leung

Hong Kong Polytechnic University

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance 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 St Andrews. http://www-history.mcs.st-and.ac.uk/

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

I Simmons, G. Differential Equations with Historical Notes. McGraw-Hill, 1991.

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

I Renaissance - a period of transition (1350-1550) when Europe transformed from a feudal and ecclesiastical (dominated by the 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 a wave of Greek scholars to Italy

I Economic factors: success of the Italian merchant republics

I After Renaissance came the Age of Enlightenment and the Scientific Revolution, where human reason was emphasized over the authority of the Church

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

Bertrand Russell on early Renaissance: “Many of them [Italians of the Renaissance period] had still the reverence for authority that medieval philosophers had had, but they substiuted the authority of the ancients for that of the Church. This was, of course, a step towards emancipation, since the ancients disagreed with each other, and individual judgement was required to decide which of them to follow. But very few Italians of the fifteenth century would have dared to hold an opinion for which no authority could be found either in antiquity or in the teaching of the Church.” (B. Russell, History of Western Philosophy)

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

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

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), G¨odel (1906-1978)

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

I Major advances in geometry, analysis, algebra

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

I Bertrand Russell and Kurt G¨odel

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

I The Julian calendar, introduced under Julius Caesar in 46 BC, 1 has the average year at 365 4 days I The mean tropical year, however, is approximately 365.2422 days. I Drifting seasons (relative to the calendar) means drifting Easter (first Sunday after the first full moon after the vernal equinox) I In 1580 Pope Gregory XIII called for calendar reform I 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 years I Example: 2000 was a leap year but 1900, 1800 and 1700 were not 365·400+97 I Average length of year 400 = 365.2425

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

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

I If adopted, the average length of a year is 365·4000+969 4000 = 365.24225 I The proposal is not yet officially adopted. (Plus it’s not something we have to worry about for another 2000 years.)

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

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

Ren´eDescartes (1596-1650)

I Applied methods of algebra to geometry

I Cartesian coordinates

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

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

Ren´eDescartes

I “While we thus reject all of which we can entertain the smallest doubt, and even imagine that it is false, we easily indeed 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 that we are not while we doubt of the truth of these things; for there is a repugnance in conceiving that what thinks does not exist at the very time when it thinks. Accordingly, the knowledge, I think, therefore I am, is the first and most certain that occurs to one who philosophizes orderly.” http: //www.gutenberg.org/cache/epub/4391/pg4391.txt

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance 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) and geometry

I Most famous for Fermat’s Last Theorem

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance Fermat’s Last Theorem

I Fermat is most well known for his “Last Theorem” n n n I Fermat’s Last Theorem: x + y = z has no integer solutions 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 margin is too small to contain.”

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance Fermat’s Last Theorem

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

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

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

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

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance Fermat’s Last Theorem

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

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance 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 his father, who was a tax collector

I Pascal’s Wager (from Pens´ees,“Thoughts”): It is the rational thing to wager for (i.e., to bet on) the existence of God.

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

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

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

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

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance 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, with the sun at one focus.

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

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

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

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

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

Isaac Newton

I Studied at Cambridge

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

I Difficult to assess his influence on contemporaries because of his hesitation to publish

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

Isaac Newton

I Explained Kepler’s laws using gravitation following an inverse-square law 1 I I.e., the force of gravitation is proportional to r 2 , where r is distance

I The inverse square law is what we would expect if the force of gravity is evenly distributed on the surface of an imaginary sphere

I He called quantities changing with time fluents and their rates of change fluxions, denoted byx ˙ , a notation which is still used today, especially in physics

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

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

I Born in Leipzig, Germany

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

I Invented calculus the same time as Newton R I Introduced the notations d and which we are familiar with today

I Also introduced the terms calculus differntialis and calculus integralis π 1 1 1 1 I The formula 4 = 1 − 3 + 5 − 7 + 9 ... is called the Leibniz formula (one of the many things named after him) but it is believed that James Gregory discovered it earlier

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

Jacob Bernoulli (1654-1705)

I Solved the Brachistochrone and the Tautochrone problems

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

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

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

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

I Posed by Johann Bernoulli

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

I Five mathematicians (including Johann himself) provided solutions

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

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

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

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

I l’Hˆopital’ssolutions was not published until 1988

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

Johann Bernoulli (1667-1748) ix I Discovered e = cos x + i sin x

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

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

Daniel Bernoulli (1700-1782)

I Johann’s son

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

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

Leonhard Euler (1701-1783)

I Son of a pastor in Switzerland and studied theology at the University 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 the mathematical 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

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

Leonhard Euler (1701-1783)

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

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

Leonhard Euler (1701-1783) 1 1 1 π2 I Correctly determined that 1 + 22 + 32 + 42 + ... = 6 I Developed a theory of differential equations dy x 2 2 I Example: from dx = − y we get x + y = C

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

Contributions to number theory:

I Introduced the phi function ϕ(n), which is the number of positive integers smaller than n and relatively prime with n

I Application: Modern-day encryption

I If n is a prime number, for example, then ϕ(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 common factor with 20. The latter set has 8 elements.

I Therefore ϕ(20) = 8.

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

More Phi-function examples:

I ϕ(13) = 12

I ϕ(30) = 8

I ϕ(40) = 16

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

Contributions to number theory:

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

I Proved (1760): If gcd(a,n)=1, then n divides the difference aϕ(n) − 1.

I Proved (1770): It is impossible to have non-trivial integer solutions to x3 + y 3 = z3. 25 I Showed (1732): 2 + 1 is not prime. (Fermat showed that 22n + 1, for n = 1, 2, 3, 4, were all primes.

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

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

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

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

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

I Found a change of variable formula for double integration

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

Leonhard Euler (1701-1783)

I K¨oningsberg (now Kaliningrad) Bridges Problem: Can a person take a walk so that each bridge is crossed exactly once?

I Euler proved it was not possible.

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance K¨oningsberg

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

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance K¨oningsberg

Figure: K¨oningsberg Briges Problem. Schematic.

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

Leonhard Euler (1701-1783)

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

I If an island has an odd number of bridges connecting to it, n+1 the region must appear in the sequence 2 times 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 required path does not exist.

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance Non-commutativity

Figure: If an island has an odd number of bridges connecting to it, the n+1 region must appear in the sequence 2 times

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

Leonhard Euler (1701-1783)

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 no attention to volumes or distances

I For example, a coffee mug and a donut are considered equivalent

I https://www.youtube.com/watch?time_continue=45&v= 9NlqYr6-TpA

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance 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 study at the University of G¨ottingen

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

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

I His doctoral dissertation, at the University of Helmst¨adt,gave the first substantial proof of the Fundamental Theorem of Algebra

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

Carl Friedrich Gauss (1777-1855)

I Fundamental Theorem of Algebra: Every polynomial with complex coefficients (which may or may not be real) has at least one complex root (which may or may not be real).

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

I Duke Ferdinand died in the 1806 when he led Prussian troops to fight Napoleon, Gauss’ friends helped him secure the position of director of the observatory at G¨ottingen.He held this poistion until his death

I Said to be the last “complete mathematician”, as after the mid-19th century different fields of mathematics became so specialized that only mathematicians within one field would have complete understanding of the field

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

Carl Friedrich Gauss (1777-1855)

I Proved (1796): The regular 17-gon is constructible by compass 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 calculated Ceres’ orbit, and at the end of the year the asteroid appeared where Gauss predicted

I Later calculated the orbits of many other asteroids

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

Carl Friedrich Gauss (1777-1855)

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”): The Gaussian curvature is preserved by local isometries.

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

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

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

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

Carl Friedrich Gauss (1777-1855)

I Introduced the notation for congruence: a ≡ b mod c (c divides a − b)

I Method of least squares

I Prime distribution: ∞ Z n dx X k!n n n 2n π(n) ≈ ≈ ≈ + + +... ln x (ln n)k+1 ln n (ln n)2 (ln n)3 2 k=0

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

Pierre Simon de Laplace (1749-1827)

I Wrote M´ecaniqueC´eleste (“Celestial Mechanics”)

I Also Th´eorie Analytique des Probabilit´es (“Analytic Theory of Probability”)

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

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

Joseph Louis Lagrange (1736-1813)

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

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

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

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

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

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

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

I An equation can be solved by radicals if any root can be expressed in terms of the coefficients using +, −, ×, ÷ and roots

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

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

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance Non-commutativity

Figure: AB 6= BA

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance 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 purely romantic affair to a staged political assassination

I Major contributions were in the field of algebra.

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

Bernhard Riemann (1826-1866)

I Got his doctoral degree at G¨ottingenin 1851

I Died at the age of 40

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

Bernhard Riemann (1826-1866)

I Introduced what we today call “Riemannian geometry”, spaces which may be curved but is not embedded in another space (imagine a 2-dimensional sphere living by itself, without the 3-dimensional space it sits in)

I Lay the foundation for, for example, general relativity

Dr Joseph Lee, Dr Louis Leung AMA1D01C – Europe since Renaissance 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 twin primes.

I Proposed by French mathematician Alphonse de Polignac in 1849.

I Zhang Yitang (2013): There exists an integer n smaller than 70, 000, 000 such that there are infinitely many pairs of primes p, q, where q = p + n.

I 70, 000, 000 is much bigger than 2, but the distance is nothing when compared to the distance between infinity and any real number.

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