Piano Sonata No. 17 in D Minor, “The Tempest”, Allegretto, Third Movement. Ludwig van Beethoven, 1801, played by Glenn Gould.

Carl Friedrich !

Gauss!!

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The Prince of Mathematics

Other Famous German RomanticsComposers: !• Ludwig van Beethoven !• Franz Liszt!• Johannes Brahms!• Franz Schubert!!

Johann van Goethe’s Faust!The Brothers Grimm!!Scientists: !• William Herschel!• Alexander von Humboldt!• Jean-Baptiste Lamarck!• Wilhelm Weber

(German) Romanticism

• Reaction and Response to Enlightenment!!• “Reason is the enumeration of qualities already

known; imagination is the perception of the value of those qualities, both separately and as a whole. Reason respects the differences, and imagination the similitudes of things. Reason is to imagination as the instrument to the agent, as the body to the spirit, as the shadow to the substance.” – Percy Shelley, English Romantic Poet, in A Defense of Poetry, 1821

Wanderer above the Sea of Fog (German: Der Wanderer über dem Nebelmeer), 1818, Caspar David Friedrich

Themes

• Individualism!• Imagination!• Nature!• Preference for the

values and language of the common people over the elite

Gauss: a Romantic Genius• Work in both pure mathematics and applied mathematics

– the natural world as a path to mathematical discoveries!– Rough timeline of work in Applied Mathematics: !• 1800s and 1810s: Astronomy !• 1820s: Geodesics!• 1830s: Physics!

• Rose from humble origins!• Creativity and imagination despite depression, feelings of

being misunderstood by society

Gauss: Personal Life

• Born in Brunswick, Germany in 1777 to poor, working-class parents!

• Recognized as genius early!• Studied at Gottingen University

until 1799 under the sponsorship of the Duke of Brunswick!

• Marries Johanna Ostoff in 1805; she dies after the death of their second child in 1809; Gauss remarried, but often felt misunderstood by family despite their support!

• Depression!• Dies in Gottingen in 1855 after a

long incredibly productive life The Gottingen University and its library in 1815!

Early Contributions to Pure Mathematics!

• Groundbreaking mathematical discovery while still in school: the construction of a regular 17-gon by ruler and compass - the most major advance in this field since the time of Greek mathematics!

• Publishes Disquisitiones Arithmeticae in the summer of 1801.!

• Fundamental Theorem of Algebra: a polynomial equation of degree n has n complex roots (when counted with multiplicities).

Modular Arithmetic• Before we discuss Theorema Aureum, a few words about modular

arithmetic as developed by Gauss in Disquisitiones Arithmeticae. !• In this system numbers "wrap around" upon reaching a certain value -

the modulus.

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• For example, arithmetic modulo 12 is a familiar “clock arithmetic” where there are only 12 possible hours.!

• 15 is the same as 3, 10 is the same as -2!• Instead of saying “the same”, we say

congruent and write 15 ≡ 3 mod 12!• 9 + 4 ≡ 1 mod 12!• 4 x 7 ≡ 4 mod 12!• 2 - 5 ≡ 9 mod 12

Quadratic residues

• A number p is called a quadratic residue modulo a number q if p is a perfect square modulo q.!!

• !• 0, 1, 2 and 4 are quadratic residues modulo 7!• 3, 5, and 6 are not quadratic residues modulo 7!• If q is any odd prime number then, excepting 0, exactly

half of the remainders modulo q are quadratic residues. !• What are quadratic residues modulo 5?

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x 0 1 2 3 4 5 6

x2 0 1 4 2 2 4 1

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• Theorema Aureum (The law of quadratic reciprocity). Suppose p and q are odd prime numbers. Then p is a quadratic residue modulo q if, and only if, q is a quadratic residue modulo p. There is one exception: if p and q are both congruent to 3 modulo 4 then p is a quadratic residue modulo q if, and only if, q is not a quadratic residue modulo p. !

• For example, is 7 a quadratic residue modulo 619? !• Both are prime numbers congruent to 3 modulo 4. So the behavior of 7

modulo 619 is opposite to the behavior of 619 modulo 7.!• 619=88*7+3 ≡ 3 mod 7 and 3 is not a quadratic residue modulo 7.

Therefore, 7 is a quadratic residue modulo 619! !• So is 5 a quadratic residue modulo 961748941?

Theorema Aureum

The law of quadratic reciprocity was discovered experimentally by Legendre, Lagrange, and Euler. Neither of them could prove it!!!Here is how Gauss wrote about his discovery: !!“In arithmetic the most elegant theorems frequently arise experimentally as the result of a more or less unexpected stroke of good fortune, while their proofs lie so deeply embedded in the darkness that they elude all attempts and defeat the sharpest inquiries … I discovered this theorem independently in 1795 at a time when I was totally ignorant of what had been achieved in higher arithmetic, and consequently had not the slightest aid from the literature on the subject. For a whole year this theorem tormented me and absorbed my greatest efforts until at last I obtained a proof.”!!This proof was published by Gauss in 1801 in Disquisitiones and was followed by seven others in an interval of 17 years.

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Astronomy• Director of the

Gottingen Observatory from 1807 onward!

• Determining the orbit of Ceres ! The Theoria Motus (1809)!– celestial mechanics!– landmark work on

how to estimate planetary orbitals and then refine the estimations

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Geodesie• 1818: Gauss asked to carry out geodesic survey of Hanover!• Invention of the heliotrope (reflects the sun’s rays using

mirrors and a small telescope)

!• Laying foundations of geodesy and cartography influenced

Gauss’ studies of the field of pure mathematics we now call Differential Geometry.!

• In Disquisitiones generales circa superficies curva (1828), Gauss introduced the notion of (Gaussian) curvature and proved Theorema Egregium: the curvature of the surface doesn’t change when it is bended. !

• Awareness of non-Euclidian geometry. The fifth postulate of Euclid states that there exists one, and only one, parallel line to any given line through any given point. For two thousand years mathematicians tried to prove this postulate. In other words, this property was viewed as inevitable and intrinsic to our perception of space. Gauss was aware that non-Euclidean geometry was possible but never published his ideas. This was accomplished later by Lobachevsky and Bolyai, whose work Gauss admired.

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Magnetism• 1820: basic phenomenon of electromagnetism discovered by Oersted!• 1822-1826: Ampère determines an explanation of electromagnetism in

terms of molecular currents!• 1831: the discovery of induced currents by Faraday!• Major german scientist and visionary Alexander van Humboldt

suggests a collaboration between Gauss and young physicist Wilhelm Weber which was very fruitful and lasted many years.!• They developed a system of units for electricity and magnetism and

designed magnetic observatories to measure Earth’s magnetic field.

• The invention of the electromagnetic telegraph (1833)

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“To me Gauss is like the highest peak amidst our Bavarian mountains as it appears to a spectator from the north. From the east the gradually ascending foothills culminate in the one gigantic colossus, which falls away steeply into the lowlands of a new formation, into which its spurs project for many for many miles and in which the water that streams from it begets new life.” !- Felix Klein, Development of mathematics in the Nineteenth Century

Further reading• D. Kehlmann, Measuring the World!• F. Klein, Development of Mathematics in the 19th Century!• The MacTutor History of Mathematics, http://www-

history.mcs.st-and.ac.uk/!• C.F. Gauss, Disquisitiones Arithmeticae

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