Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Leonhard Euler His Life, The Man, and His Work

Walter Gautschi [email protected]

Purdue University

April 30, 2007

Walter Gautschi Leonhard Euler St. Petersburg 1727–1741

Berlin 1741–1766

St. Petersburg 1766–1783

The Man

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Outline

Basel 1707–1727

Walter Gautschi Leonhard Euler Berlin 1741–1766

St. Petersburg 1766–1783

The Man

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Outline

Basel 1707–1727

St. Petersburg 1727–1741

Walter Gautschi Leonhard Euler St. Petersburg 1766–1783

The Man

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Outline

Basel 1707–1727

St. Petersburg 1727–1741

Berlin 1741–1766

Walter Gautschi Leonhard Euler The Man

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Outline

Basel 1707–1727

St. Petersburg 1727–1741

Berlin 1741–1766

St. Petersburg 1766–1783

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Outline

Basel 1707–1727

St. Petersburg 1727–1741

Berlin 1741–1766

St. Petersburg 1766–1783

The Man

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Basel 1707–1727 Auspicious beginnings

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Parish residence and church in Riehen

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

The old university of Basel and Johann Bernoulli

Walter Gautschi Leonhard Euler • born April 15, 1707 as the first of four children • parents: Paulus Euler (1670–1745) and Margaretha Brucker (1677–1761) • early childhood at the parish residence in Riehen; home schooling • at the age of 8, sent to the Latin gymnasium in Basel; first mathematical instruction from a private tutor • 1720: enrolled at the University of Basel (propaedeutic study); freshman mathematics course from Johann Bernoulli • 1724: masters degree; first public lecture (in Latin); advanced mathematics course from Johann Bernoulli; “privatissima”

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology

Walter Gautschi Leonhard Euler • parents: Paulus Euler (1670–1745) and Margaretha Brucker (1677–1761) • early childhood at the parish residence in Riehen; home schooling • at the age of 8, sent to the Latin gymnasium in Basel; first mathematical instruction from a private tutor • 1720: enrolled at the University of Basel (propaedeutic study); freshman mathematics course from Johann Bernoulli • 1724: masters degree; first public lecture (in Latin); advanced mathematics course from Johann Bernoulli; “privatissima”

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • born April 15, 1707 as the first of four children

Walter Gautschi Leonhard Euler • early childhood at the parish residence in Riehen; home schooling • at the age of 8, sent to the Latin gymnasium in Basel; first mathematical instruction from a private tutor • 1720: enrolled at the University of Basel (propaedeutic study); freshman mathematics course from Johann Bernoulli • 1724: masters degree; first public lecture (in Latin); advanced mathematics course from Johann Bernoulli; “privatissima”

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • born April 15, 1707 as the first of four children • parents: Paulus Euler (1670–1745) and Margaretha Brucker (1677–1761)

Walter Gautschi Leonhard Euler • at the age of 8, sent to the Latin gymnasium in Basel; first mathematical instruction from a private tutor • 1720: enrolled at the University of Basel (propaedeutic study); freshman mathematics course from Johann Bernoulli • 1724: masters degree; first public lecture (in Latin); advanced mathematics course from Johann Bernoulli; “privatissima”

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • born April 15, 1707 as the first of four children • parents: Paulus Euler (1670–1745) and Margaretha Brucker (1677–1761) • early childhood at the parish residence in Riehen; home schooling

Walter Gautschi Leonhard Euler • 1720: enrolled at the University of Basel (propaedeutic study); freshman mathematics course from Johann Bernoulli • 1724: masters degree; first public lecture (in Latin); advanced mathematics course from Johann Bernoulli; “privatissima”

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • born April 15, 1707 as the first of four children • parents: Paulus Euler (1670–1745) and Margaretha Brucker (1677–1761) • early childhood at the parish residence in Riehen; home schooling • at the age of 8, sent to the Latin gymnasium in Basel; first mathematical instruction from a private tutor

Walter Gautschi Leonhard Euler • 1724: masters degree; first public lecture (in Latin); advanced mathematics course from Johann Bernoulli; “privatissima”

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • born April 15, 1707 as the first of four children • parents: Paulus Euler (1670–1745) and Margaretha Brucker (1677–1761) • early childhood at the parish residence in Riehen; home schooling • at the age of 8, sent to the Latin gymnasium in Basel; first mathematical instruction from a private tutor • 1720: enrolled at the University of Basel (propaedeutic study); freshman mathematics course from Johann Bernoulli

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • born April 15, 1707 as the first of four children • parents: Paulus Euler (1670–1745) and Margaretha Brucker (1677–1761) • early childhood at the parish residence in Riehen; home schooling • at the age of 8, sent to the Latin gymnasium in Basel; first mathematical instruction from a private tutor • 1720: enrolled at the University of Basel (propaedeutic study); freshman mathematics course from Johann Bernoulli • 1724: masters degree; first public lecture (in Latin); advanced mathematics course from Johann Bernoulli; “privatissima”

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

From Euler’s autobiography (1767)

Walter Gautschi Leonhard Euler • 1726: two papers on isochronous resp. “reciprocal” curves in the Acta Eruditorum of Leipzig

• responded to a prize question of the Paris Academy of Sciences with a memoir on the optimal placing of masts on a ship

• 1727: applied (unsuccessfully) for the physics chair at the University of Basel with a work on the theory of sound

• left Basel (for good) in April of 1727 to assume a junior appointment at the Academy of St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Early professional activities

Walter Gautschi Leonhard Euler • responded to a prize question of the Paris Academy of Sciences with a memoir on the optimal placing of masts on a ship

• 1727: applied (unsuccessfully) for the physics chair at the University of Basel with a work on the theory of sound

• left Basel (for good) in April of 1727 to assume a junior appointment at the Academy of St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Early professional activities • 1726: two papers on isochronous resp. “reciprocal” curves in the Acta Eruditorum of Leipzig

Walter Gautschi Leonhard Euler • 1727: applied (unsuccessfully) for the physics chair at the University of Basel with a work on the theory of sound

• left Basel (for good) in April of 1727 to assume a junior appointment at the Academy of St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Early professional activities • 1726: two papers on isochronous resp. “reciprocal” curves in the Acta Eruditorum of Leipzig

• responded to a prize question of the Paris Academy of Sciences with a memoir on the optimal placing of masts on a ship

Walter Gautschi Leonhard Euler • left Basel (for good) in April of 1727 to assume a junior appointment at the Academy of St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Early professional activities • 1726: two papers on isochronous resp. “reciprocal” curves in the Acta Eruditorum of Leipzig

• responded to a prize question of the Paris Academy of Sciences with a memoir on the optimal placing of masts on a ship

• 1727: applied (unsuccessfully) for the physics chair at the University of Basel with a work on the theory of sound

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Early professional activities • 1726: two papers on isochronous resp. “reciprocal” curves in the Acta Eruditorum of Leipzig

• responded to a prize question of the Paris Academy of Sciences with a memoir on the optimal placing of masts on a ship

• 1727: applied (unsuccessfully) for the physics chair at the University of Basel with a work on the theory of sound

• left Basel (for good) in April of 1727 to assume a junior appointment at the Academy of St. Petersburg

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Physical dissertation on sound

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

St. Petersburg 1727–1741 Meteoric rise to world fame and academic advancement

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

The Academy in St. Petersburg and Peter I

Walter Gautschi Leonhard Euler • groundwork for Euler’s appointment at the Academy had been laid by Johann Bernoulli and his sons Niklaus II and Daniel I, both already active at the Academy • 1731: professor of physics; ordinary member of the Academy • 1733: succeeds Daniel Bernoulli (who returns to Basel) as professor of mathematics marriage to Katharina Gsell, which brought forth 13 children of whom only 5 reached the age of adulthood • 1735: first setback in health • 1738: second setback in health; loss of the right eye • 1741: departure from St. Petersburg following political unrest after the death (1740) of the Empress Anna Ivanovna (a niece of Peter I)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Chronology

Walter Gautschi Leonhard Euler • 1731: professor of physics; ordinary member of the Academy • 1733: succeeds Daniel Bernoulli (who returns to Basel) as professor of mathematics marriage to Katharina Gsell, which brought forth 13 children of whom only 5 reached the age of adulthood • 1735: first setback in health • 1738: second setback in health; loss of the right eye • 1741: departure from St. Petersburg following political unrest after the death (1740) of the Empress Anna Ivanovna (a niece of Peter I)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Chronology • groundwork for Euler’s appointment at the Academy had been laid by Johann Bernoulli and his sons Niklaus II and Daniel I, both already active at the Academy

Walter Gautschi Leonhard Euler • 1733: succeeds Daniel Bernoulli (who returns to Basel) as professor of mathematics marriage to Katharina Gsell, which brought forth 13 children of whom only 5 reached the age of adulthood • 1735: first setback in health • 1738: second setback in health; loss of the right eye • 1741: departure from St. Petersburg following political unrest after the death (1740) of the Empress Anna Ivanovna (a niece of Peter I)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Chronology • groundwork for Euler’s appointment at the Academy had been laid by Johann Bernoulli and his sons Niklaus II and Daniel I, both already active at the Academy • 1731: professor of physics; ordinary member of the Academy

Walter Gautschi Leonhard Euler • 1735: first setback in health • 1738: second setback in health; loss of the right eye • 1741: departure from St. Petersburg following political unrest after the death (1740) of the Empress Anna Ivanovna (a niece of Peter I)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Chronology • groundwork for Euler’s appointment at the Academy had been laid by Johann Bernoulli and his sons Niklaus II and Daniel I, both already active at the Academy • 1731: professor of physics; ordinary member of the Academy • 1733: succeeds Daniel Bernoulli (who returns to Basel) as professor of mathematics marriage to Katharina Gsell, which brought forth 13 children of whom only 5 reached the age of adulthood

Walter Gautschi Leonhard Euler • 1738: second setback in health; loss of the right eye • 1741: departure from St. Petersburg following political unrest after the death (1740) of the Empress Anna Ivanovna (a niece of Peter I)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Chronology • groundwork for Euler’s appointment at the Academy had been laid by Johann Bernoulli and his sons Niklaus II and Daniel I, both already active at the Academy • 1731: professor of physics; ordinary member of the Academy • 1733: succeeds Daniel Bernoulli (who returns to Basel) as professor of mathematics marriage to Katharina Gsell, which brought forth 13 children of whom only 5 reached the age of adulthood • 1735: first setback in health

Walter Gautschi Leonhard Euler • 1741: departure from St. Petersburg following political unrest after the death (1740) of the Empress Anna Ivanovna (a niece of Peter I)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Chronology • groundwork for Euler’s appointment at the Academy had been laid by Johann Bernoulli and his sons Niklaus II and Daniel I, both already active at the Academy • 1731: professor of physics; ordinary member of the Academy • 1733: succeeds Daniel Bernoulli (who returns to Basel) as professor of mathematics marriage to Katharina Gsell, which brought forth 13 children of whom only 5 reached the age of adulthood • 1735: first setback in health • 1738: second setback in health; loss of the right eye

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Chronology • groundwork for Euler’s appointment at the Academy had been laid by Johann Bernoulli and his sons Niklaus II and Daniel I, both already active at the Academy • 1731: professor of physics; ordinary member of the Academy • 1733: succeeds Daniel Bernoulli (who returns to Basel) as professor of mathematics marriage to Katharina Gsell, which brought forth 13 children of whom only 5 reached the age of adulthood • 1735: first setback in health • 1738: second setback in health; loss of the right eye • 1741: departure from St. Petersburg following political unrest after the death (1740) of the Empress Anna Ivanovna (a niece of Peter I)

Walter Gautschi Leonhard Euler Mechanics: Analytic theory of motion (1736) • kinematics and dynamics of a mass point • in free motion (vol. I) • in constrained motion (vol. II) Music theory: Tentamen novae theoriae musicae (1739) • nature of sound • generation and perception of sound • mathematical theory of pleasantness of musical constructs • mathematical theory of temperaments Naval science (1749, written 1740–41) • principles of hydrostatics • stability theory • naval engineering and navigation (vol. II)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works

Walter Gautschi Leonhard Euler Music theory: Tentamen novae theoriae musicae (1739) • nature of sound • generation and perception of sound • mathematical theory of pleasantness of musical constructs • mathematical theory of temperaments Naval science (1749, written 1740–41) • principles of hydrostatics • stability theory • naval engineering and navigation (vol. II)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Mechanics: Analytic theory of motion (1736) • kinematics and dynamics of a mass point • in free motion (vol. I) • in constrained motion (vol. II)

Walter Gautschi Leonhard Euler Naval science (1749, written 1740–41) • principles of hydrostatics • stability theory • naval engineering and navigation (vol. II)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Mechanics: Analytic theory of motion (1736) • kinematics and dynamics of a mass point • in free motion (vol. I) • in constrained motion (vol. II) Music theory: Tentamen novae theoriae musicae (1739) • nature of sound • generation and perception of sound • mathematical theory of pleasantness of musical constructs • mathematical theory of temperaments

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Mechanics: Analytic theory of motion (1736) • kinematics and dynamics of a mass point • in free motion (vol. I) • in constrained motion (vol. II) Music theory: Tentamen novae theoriae musicae (1739) • nature of sound • generation and perception of sound • mathematical theory of pleasantness of musical constructs • mathematical theory of temperaments Naval science (1749, written 1740–41) • principles of hydrostatics • stability theory • naval engineering and navigation (vol. II)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Mechanics I,II

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Tentamen

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Naval science I,II

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Basel 1761

Walter Gautschi Leonhard Euler Selectio 1 The Basel problem (1740) 1 1 1 π2 1 + + + + ··· = 22 32 42 6 zeta function 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 4s Euler determines ζ(4), ζ(6), ζ(8), . . . , ζ(12) Later in 1750, he was able to prove 22n−1 ζ(2n) = |B |π2n (2n)! 2n

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Walter Gautschi Leonhard Euler 1 1 1 π2 1 + + + + ··· = 22 32 42 6 zeta function 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 4s Euler determines ζ(4), ζ(6), ζ(8), . . . , ζ(12) Later in 1750, he was able to prove 22n−1 ζ(2n) = |B |π2n (2n)! 2n

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 1 The Basel problem (1740)

Walter Gautschi Leonhard Euler π2 6 zeta function 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 4s Euler determines ζ(4), ζ(6), ζ(8), . . . , ζ(12) Later in 1750, he was able to prove 22n−1 ζ(2n) = |B |π2n (2n)! 2n

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 1 The Basel problem (1740) 1 1 1 1 + + + + ··· = 22 32 42

Walter Gautschi Leonhard Euler zeta function 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 4s Euler determines ζ(4), ζ(6), ζ(8), . . . , ζ(12) Later in 1750, he was able to prove 22n−1 ζ(2n) = |B |π2n (2n)! 2n

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 1 The Basel problem (1740) 1 1 1 π2 1 + + + + ··· = 22 32 42 6

Walter Gautschi Leonhard Euler Euler determines ζ(4), ζ(6), ζ(8), . . . , ζ(12) Later in 1750, he was able to prove 22n−1 ζ(2n) = |B |π2n (2n)! 2n

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 1 The Basel problem (1740) 1 1 1 π2 1 + + + + ··· = 22 32 42 6 zeta function 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 4s

Walter Gautschi Leonhard Euler Later in 1750, he was able to prove 22n−1 ζ(2n) = |B |π2n (2n)! 2n

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 1 The Basel problem (1740) 1 1 1 π2 1 + + + + ··· = 22 32 42 6 zeta function 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 4s Euler determines ζ(4), ζ(6), ζ(8), . . . , ζ(12)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 1 The Basel problem (1740) 1 1 1 π2 1 + + + + ··· = 22 32 42 6 zeta function 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 4s Euler determines ζ(4), ζ(6), ζ(8), . . . , ζ(12) Later in 1750, he was able to prove 22n−1 ζ(2n) = |B |π2n (2n)! 2n

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Euler ca. 1737

Walter Gautschi Leonhard Euler P = {2, 3, 5, 7, 11, 13, 17,...} product formula (1737) Y 1 = ζ(s) 1 − 1/ps p∈P Euler’s derivation from ζ(s) “peal away” all terms divisible by 2 2s − 1 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 5s 7s from this do the same with the prime 3, then with 5, etc.   Y ps − 1 ζ(s) = 1  ps   p∈P

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 2 Prime numbers and the zeta function

Walter Gautschi Leonhard Euler product formula (1737) Y 1 = ζ(s) 1 − 1/ps p∈P Euler’s derivation from ζ(s) “peal away” all terms divisible by 2 2s − 1 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 5s 7s from this do the same with the prime 3, then with 5, etc.   Y ps − 1 ζ(s) = 1  ps   p∈P

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 2 Prime numbers and the zeta function P = {2, 3, 5, 7, 11, 13, 17,...}

Walter Gautschi Leonhard Euler Euler’s derivation from ζ(s) “peal away” all terms divisible by 2 2s − 1 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 5s 7s from this do the same with the prime 3, then with 5, etc.   Y ps − 1 ζ(s) = 1  ps   p∈P

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 2 Prime numbers and the zeta function P = {2, 3, 5, 7, 11, 13, 17,...} product formula (1737) Y 1 = ζ(s) 1 − 1/ps p∈P

Walter Gautschi Leonhard Euler from this do the same with the prime 3, then with 5, etc.   Y ps − 1 ζ(s) = 1  ps   p∈P

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 2 Prime numbers and the zeta function P = {2, 3, 5, 7, 11, 13, 17,...} product formula (1737) Y 1 = ζ(s) 1 − 1/ps p∈P Euler’s derivation from ζ(s) “peal away” all terms divisible by 2 2s − 1 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 5s 7s

Walter Gautschi Leonhard Euler   Y ps − 1 ζ(s) = 1  ps   p∈P

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 2 Prime numbers and the zeta function P = {2, 3, 5, 7, 11, 13, 17,...} product formula (1737) Y 1 = ζ(s) 1 − 1/ps p∈P Euler’s derivation from ζ(s) “peal away” all terms divisible by 2 2s − 1 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 5s 7s from this do the same with the prime 3, then with 5, etc.

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 2 Prime numbers and the zeta function P = {2, 3, 5, 7, 11, 13, 17,...} product formula (1737) Y 1 = ζ(s) 1 − 1/ps p∈P Euler’s derivation from ζ(s) “peal away” all terms divisible by 2 2s − 1 1 1 1 ζ(s) = 1 + + + + ··· 2s 3s 5s 7s from this do the same with the prime 3, then with 5, etc.   Y ps − 1 ζ(s) = 1  ps   p∈P

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Walter Gautschi Leonhard Euler Problem: interpolate the factorials 1! = 1, 2! = 1 · 2, 3! = 1 · 2 · 3, 4! = 1 · 2 · 3 · 4,.... at noninteger values. Euler writes

1 · 2n 21−n · 3n 31−n · 4n 41−n · 5n n! = · · · ··· n + 1 n + 2 n + 3 n + 4

1 1 √ and comes up with 2 ! = 2 π. Aha! There must be calculus involved. Ingeniously, he derives Z ∞ x! = e−t tx dt = Γ(x + 1). 0 There holds Γ(x + 1) = xΓ(x), Γ(1) = 1, so that Γ(n + 1) = n!.

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 3 The gamma function (1738)

Walter Gautschi Leonhard Euler Euler writes

1 · 2n 21−n · 3n 31−n · 4n 41−n · 5n n! = · · · ··· n + 1 n + 2 n + 3 n + 4

1 1 √ and comes up with 2 ! = 2 π. Aha! There must be calculus involved. Ingeniously, he derives Z ∞ x! = e−t tx dt = Γ(x + 1). 0 There holds Γ(x + 1) = xΓ(x), Γ(1) = 1, so that Γ(n + 1) = n!.

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 3 The gamma function (1738) Problem: interpolate the factorials 1! = 1, 2! = 1 · 2, 3! = 1 · 2 · 3, 4! = 1 · 2 · 3 · 4,.... at noninteger values.

Walter Gautschi Leonhard Euler 1 1 √ and comes up with 2 ! = 2 π. Aha! There must be calculus involved. Ingeniously, he derives Z ∞ x! = e−t tx dt = Γ(x + 1). 0 There holds Γ(x + 1) = xΓ(x), Γ(1) = 1, so that Γ(n + 1) = n!.

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 3 The gamma function (1738) Problem: interpolate the factorials 1! = 1, 2! = 1 · 2, 3! = 1 · 2 · 3, 4! = 1 · 2 · 3 · 4,.... at noninteger values. Euler writes

1 · 2n 21−n · 3n 31−n · 4n 41−n · 5n n! = · · · ··· n + 1 n + 2 n + 3 n + 4

Walter Gautschi Leonhard Euler Aha! There must be calculus involved. Ingeniously, he derives Z ∞ x! = e−t tx dt = Γ(x + 1). 0 There holds Γ(x + 1) = xΓ(x), Γ(1) = 1, so that Γ(n + 1) = n!.

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 3 The gamma function (1738) Problem: interpolate the factorials 1! = 1, 2! = 1 · 2, 3! = 1 · 2 · 3, 4! = 1 · 2 · 3 · 4,.... at noninteger values. Euler writes

1 · 2n 21−n · 3n 31−n · 4n 41−n · 5n n! = · · · ··· n + 1 n + 2 n + 3 n + 4

1 1 √ and comes up with 2 ! = 2 π.

Walter Gautschi Leonhard Euler Ingeniously, he derives Z ∞ x! = e−t tx dt = Γ(x + 1). 0 There holds Γ(x + 1) = xΓ(x), Γ(1) = 1, so that Γ(n + 1) = n!.

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 3 The gamma function (1738) Problem: interpolate the factorials 1! = 1, 2! = 1 · 2, 3! = 1 · 2 · 3, 4! = 1 · 2 · 3 · 4,.... at noninteger values. Euler writes

1 · 2n 21−n · 3n 31−n · 4n 41−n · 5n n! = · · · ··· n + 1 n + 2 n + 3 n + 4

1 1 √ and comes up with 2 ! = 2 π. Aha! There must be calculus involved.

Walter Gautschi Leonhard Euler There holds Γ(x + 1) = xΓ(x), Γ(1) = 1, so that Γ(n + 1) = n!.

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 3 The gamma function (1738) Problem: interpolate the factorials 1! = 1, 2! = 1 · 2, 3! = 1 · 2 · 3, 4! = 1 · 2 · 3 · 4,.... at noninteger values. Euler writes

1 · 2n 21−n · 3n 31−n · 4n 41−n · 5n n! = · · · ··· n + 1 n + 2 n + 3 n + 4

1 1 √ and comes up with 2 ! = 2 π. Aha! There must be calculus involved. Ingeniously, he derives Z ∞ x! = e−t tx dt = Γ(x + 1). 0

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 3 The gamma function (1738) Problem: interpolate the factorials 1! = 1, 2! = 1 · 2, 3! = 1 · 2 · 3, 4! = 1 · 2 · 3 · 4,.... at noninteger values. Euler writes

1 · 2n 21−n · 3n 31−n · 4n 41−n · 5n n! = · · · ··· n + 1 n + 2 n + 3 n + 4

1 1 √ and comes up with 2 ! = 2 π. Aha! There must be calculus involved. Ingeniously, he derives Z ∞ x! = e−t tx dt = Γ(x + 1). 0 There holds Γ(x + 1) = xΓ(x), Γ(1) = 1, so that Γ(n + 1) = n!.

Walter Gautschi Leonhard Euler Auguste de Morgan (1806–1871): “[π] comes on many occasions through the window and through the door, sometimes even down the chimney.”

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Auguste de Morgan (1806–1871): “[π] comes on many occasions through the window and through the door, sometimes even down the chimney.”

Walter Gautschi Leonhard Euler Three most important constants in mathematics: π, e, γ The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Not even known whether γ is rational or not. If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973). Euler computes γ to 15 decimals. Mascheroni (in 1790) computes it to 32 decimals. Today, it is known to (only) 108 million decimals (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740)

Walter Gautschi Leonhard Euler The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Not even known whether γ is rational or not. If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973). Euler computes γ to 15 decimals. Mascheroni (in 1790) computes it to 32 decimals. Today, it is known to (only) 108 million decimals (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740) Three most important constants in mathematics: π, e, γ

Walter Gautschi Leonhard Euler Not even known whether γ is rational or not. If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973). Euler computes γ to 15 decimals. Mascheroni (in 1790) computes it to 32 decimals. Today, it is known to (only) 108 million decimals (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740) Three most important constants in mathematics: π, e, γ The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Walter Gautschi Leonhard Euler If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973). Euler computes γ to 15 decimals. Mascheroni (in 1790) computes it to 32 decimals. Today, it is known to (only) 108 million decimals (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740) Three most important constants in mathematics: π, e, γ The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Not even known whether γ is rational or not.

Walter Gautschi Leonhard Euler Euler computes γ to 15 decimals. Mascheroni (in 1790) computes it to 32 decimals. Today, it is known to (only) 108 million decimals (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740) Three most important constants in mathematics: π, e, γ The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Not even known whether γ is rational or not. If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973).

Walter Gautschi Leonhard Euler Mascheroni (in 1790) computes it to 32 decimals. Today, it is known to (only) 108 million decimals (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740) Three most important constants in mathematics: π, e, γ The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Not even known whether γ is rational or not. If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973). Euler computes γ to 15 decimals.

Walter Gautschi Leonhard Euler Today, it is known to (only) 108 million decimals (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740) Three most important constants in mathematics: π, e, γ The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Not even known whether γ is rational or not. If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973). Euler computes γ to 15 decimals. Mascheroni (in 1790) computes it to 32 decimals.

Walter Gautschi Leonhard Euler (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740) Three most important constants in mathematics: π, e, γ The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Not even known whether γ is rational or not. If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973). Euler computes γ to 15 decimals. Mascheroni (in 1790) computes it to 32 decimals. Today, it is known to (only) 108 million decimals

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 4 Euler’s constant (1740) Three most important constants in mathematics: π, e, γ The last (and most mysterious) one is Euler’s constant

 1 1 1  γ = lim 1 + + ··· + − ln n = 0.57721 ... n→∞ 2 3 n

Not even known whether γ is rational or not. If it were, say γ = p/q, then q would have to be larger than 1010,000 (R. Brent, 1973). Euler computes γ to 15 decimals. Mascheroni (in 1790) computes it to 32 decimals. Today, it is known to (only) 108 million decimals (compared to 2 × 1011 decimals of π and 50.1 billion for e)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

From Euler’s autobiography (1767)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Berlin 1741–1766 The emergence of epochal treatises

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

The Berlin Academy and Frederick II

Walter Gautschi Leonhard Euler • 1746: Berlin Academy opens its doors, with Maupertuis its president and Euler the director of the Mathematics Class; elected foreign member of the Royal Society of London • 1750: Euler’s widowed mother comes to Berlin, where she lives in Euler’s country estate together with her daughter and Euler’s children • 1752: Maupertuis returns to Paris in disgrace; Euler takes over the Academy as de facto, if not de jure, president • 1755: elected foreign member of the Paris Academy • 1763– : Euler’s relationship with Frederick II sours • 1766: Euler returns to St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology

Walter Gautschi Leonhard Euler • 1750: Euler’s widowed mother comes to Berlin, where she lives in Euler’s country estate together with her daughter and Euler’s children • 1752: Maupertuis returns to Paris in disgrace; Euler takes over the Academy as de facto, if not de jure, president • 1755: elected foreign member of the Paris Academy • 1763– : Euler’s relationship with Frederick II sours • 1766: Euler returns to St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • 1746: Berlin Academy opens its doors, with Maupertuis its president and Euler the director of the Mathematics Class; elected foreign member of the Royal Society of London

Walter Gautschi Leonhard Euler • 1752: Maupertuis returns to Paris in disgrace; Euler takes over the Academy as de facto, if not de jure, president • 1755: elected foreign member of the Paris Academy • 1763– : Euler’s relationship with Frederick II sours • 1766: Euler returns to St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • 1746: Berlin Academy opens its doors, with Maupertuis its president and Euler the director of the Mathematics Class; elected foreign member of the Royal Society of London • 1750: Euler’s widowed mother comes to Berlin, where she lives in Euler’s country estate together with her daughter and Euler’s children

Walter Gautschi Leonhard Euler • 1755: elected foreign member of the Paris Academy • 1763– : Euler’s relationship with Frederick II sours • 1766: Euler returns to St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • 1746: Berlin Academy opens its doors, with Maupertuis its president and Euler the director of the Mathematics Class; elected foreign member of the Royal Society of London • 1750: Euler’s widowed mother comes to Berlin, where she lives in Euler’s country estate together with her daughter and Euler’s children • 1752: Maupertuis returns to Paris in disgrace; Euler takes over the Academy as de facto, if not de jure, president

Walter Gautschi Leonhard Euler • 1763– : Euler’s relationship with Frederick II sours • 1766: Euler returns to St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • 1746: Berlin Academy opens its doors, with Maupertuis its president and Euler the director of the Mathematics Class; elected foreign member of the Royal Society of London • 1750: Euler’s widowed mother comes to Berlin, where she lives in Euler’s country estate together with her daughter and Euler’s children • 1752: Maupertuis returns to Paris in disgrace; Euler takes over the Academy as de facto, if not de jure, president • 1755: elected foreign member of the Paris Academy

Walter Gautschi Leonhard Euler • 1766: Euler returns to St. Petersburg

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • 1746: Berlin Academy opens its doors, with Maupertuis its president and Euler the director of the Mathematics Class; elected foreign member of the Royal Society of London • 1750: Euler’s widowed mother comes to Berlin, where she lives in Euler’s country estate together with her daughter and Euler’s children • 1752: Maupertuis returns to Paris in disgrace; Euler takes over the Academy as de facto, if not de jure, president • 1755: elected foreign member of the Paris Academy • 1763– : Euler’s relationship with Frederick II sours

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology • 1746: Berlin Academy opens its doors, with Maupertuis its president and Euler the director of the Mathematics Class; elected foreign member of the Royal Society of London • 1750: Euler’s widowed mother comes to Berlin, where she lives in Euler’s country estate together with her daughter and Euler’s children • 1752: Maupertuis returns to Paris in disgrace; Euler takes over the Academy as de facto, if not de jure, president • 1755: elected foreign member of the Paris Academy • 1763– : Euler’s relationship with Frederick II sours • 1766: Euler returns to St. Petersburg

Walter Gautschi Leonhard Euler Calculus of variations (1744) • a very general new kind of optimization problem that grew out of the brachystochrone problem • Euler (differential) equations • numerous examples from mathematics and the natural sciences Cometary and planetary trajectories (1744) Optics (1746) • theory of light and colors Artillery (1745) • vastly expanded and annotated German translation of Robins’s New principles of gunnery (1742)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works

Walter Gautschi Leonhard Euler Cometary and planetary trajectories (1744) Optics (1746) • theory of light and colors Artillery (1745) • vastly expanded and annotated German translation of Robins’s New principles of gunnery (1742)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Calculus of variations (1744) • a very general new kind of optimization problem that grew out of the brachystochrone problem • Euler (differential) equations • numerous examples from mathematics and the natural sciences

Walter Gautschi Leonhard Euler Optics (1746) • theory of light and colors Artillery (1745) • vastly expanded and annotated German translation of Robins’s New principles of gunnery (1742)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Calculus of variations (1744) • a very general new kind of optimization problem that grew out of the brachystochrone problem • Euler (differential) equations • numerous examples from mathematics and the natural sciences Cometary and planetary trajectories (1744)

Walter Gautschi Leonhard Euler Artillery (1745) • vastly expanded and annotated German translation of Robins’s New principles of gunnery (1742)

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Calculus of variations (1744) • a very general new kind of optimization problem that grew out of the brachystochrone problem • Euler (differential) equations • numerous examples from mathematics and the natural sciences Cometary and planetary trajectories (1744) Optics (1746) • theory of light and colors

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Calculus of variations (1744) • a very general new kind of optimization problem that grew out of the brachystochrone problem • Euler (differential) equations • numerous examples from mathematics and the natural sciences Cometary and planetary trajectories (1744) Optics (1746) • theory of light and colors Artillery (1745) • vastly expanded and annotated German translation of Robins’s New principles of gunnery (1742)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Calculus of variations

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Euler 1753

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Artillery

Walter Gautschi Leonhard Euler Differential calculus (1755) • calculus of differences and differentials (part I) • power series and summation formulae (part II) Integral calculus (1763, 1773) • integration of elementary functions (vol. I) • ordinary and partial differential equations (vols. II, III) Theoria motus corporum (1765) • “second mechanics”; mechanics of rigid bodies Dioptrics (1769–1771) • chromatic and spherical aberration in optical instruments Letters to a German princess (written 1760–1762) • Euler’s philosophical views on science, religion, and ethics

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Introduction to the analysis of the infinite (1748) • elementary functions • infinite series and products; continued fractions • analytic geometry: algebraic curves and surfaces

Walter Gautschi Leonhard Euler Integral calculus (1763, 1773) • integration of elementary functions (vol. I) • ordinary and partial differential equations (vols. II, III) Theoria motus corporum (1765) • “second mechanics”; mechanics of rigid bodies Dioptrics (1769–1771) • chromatic and spherical aberration in optical instruments Letters to a German princess (written 1760–1762) • Euler’s philosophical views on science, religion, and ethics

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Introduction to the analysis of the infinite (1748) • elementary functions • infinite series and products; continued fractions • analytic geometry: algebraic curves and surfaces Differential calculus (1755) • calculus of differences and differentials (part I) • power series and summation formulae (part II)

Walter Gautschi Leonhard Euler Theoria motus corporum (1765) • “second mechanics”; mechanics of rigid bodies Dioptrics (1769–1771) • chromatic and spherical aberration in optical instruments Letters to a German princess (written 1760–1762) • Euler’s philosophical views on science, religion, and ethics

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Introduction to the analysis of the infinite (1748) • elementary functions • infinite series and products; continued fractions • analytic geometry: algebraic curves and surfaces Differential calculus (1755) • calculus of differences and differentials (part I) • power series and summation formulae (part II) Integral calculus (1763, 1773) • integration of elementary functions (vol. I) • ordinary and partial differential equations (vols. II, III)

Walter Gautschi Leonhard Euler Dioptrics (1769–1771) • chromatic and spherical aberration in optical instruments Letters to a German princess (written 1760–1762) • Euler’s philosophical views on science, religion, and ethics

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Introduction to the analysis of the infinite (1748) • elementary functions • infinite series and products; continued fractions • analytic geometry: algebraic curves and surfaces Differential calculus (1755) • calculus of differences and differentials (part I) • power series and summation formulae (part II) Integral calculus (1763, 1773) • integration of elementary functions (vol. I) • ordinary and partial differential equations (vols. II, III) Theoria motus corporum (1765) • “second mechanics”; mechanics of rigid bodies

Walter Gautschi Leonhard Euler Letters to a German princess (written 1760–1762) • Euler’s philosophical views on science, religion, and ethics

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Introduction to the analysis of the infinite (1748) • elementary functions • infinite series and products; continued fractions • analytic geometry: algebraic curves and surfaces Differential calculus (1755) • calculus of differences and differentials (part I) • power series and summation formulae (part II) Integral calculus (1763, 1773) • integration of elementary functions (vol. I) • ordinary and partial differential equations (vols. II, III) Theoria motus corporum (1765) • “second mechanics”; mechanics of rigid bodies Dioptrics (1769–1771) • chromatic and spherical aberration in optical instruments

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Introduction to the analysis of the infinite (1748) • elementary functions • infinite series and products; continued fractions • analytic geometry: algebraic curves and surfaces Differential calculus (1755) • calculus of differences and differentials (part I) • power series and summation formulae (part II) Integral calculus (1763, 1773) • integration of elementary functions (vol. I) • ordinary and partial differential equations (vols. II, III) Theoria motus corporum (1765) • “second mechanics”; mechanics of rigid bodies Dioptrics (1769–1771) • chromatic and spherical aberration in optical instruments Letters to a German princess (written 1760–1762) • Euler’s philosophical views on science, religion, and ethics

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Infinitesimal analysis I,II

Differential and integral calculus

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Second mechanics and Optics

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Letters

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Walter Gautschi Leonhard Euler Selectio 5 The K¨onigsberg bridge problem (1741) connected graph • path • circuit • Eulerian path or circuit • degree of vertex Theorem (Euler) Let n be the number of vertices of odd degree. (a) If n = 0, the graph has at least one Eulerian circuit; (b) if n = 2, it has at least one Eulerian path, but no circuit; (c) if n > 2, it has neither. (n = 1 is impossible.)

K¨onigsberg bridge graph: n = 4

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Walter Gautschi Leonhard Euler connected graph • path • circuit • Eulerian path or circuit • degree of vertex Theorem (Euler) Let n be the number of vertices of odd degree. (a) If n = 0, the graph has at least one Eulerian circuit; (b) if n = 2, it has at least one Eulerian path, but no circuit; (c) if n > 2, it has neither. (n = 1 is impossible.)

K¨onigsberg bridge graph: n = 4

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 5 The K¨onigsberg bridge problem (1741)

Walter Gautschi Leonhard Euler Theorem (Euler) Let n be the number of vertices of odd degree. (a) If n = 0, the graph has at least one Eulerian circuit; (b) if n = 2, it has at least one Eulerian path, but no circuit; (c) if n > 2, it has neither. (n = 1 is impossible.)

K¨onigsberg bridge graph: n = 4

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 5 The K¨onigsberg bridge problem (1741) connected graph • path • circuit • Eulerian path or circuit • degree of vertex

Walter Gautschi Leonhard Euler (b) if n = 2, it has at least one Eulerian path, but no circuit; (c) if n > 2, it has neither. (n = 1 is impossible.)

K¨onigsberg bridge graph: n = 4

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 5 The K¨onigsberg bridge problem (1741) connected graph • path • circuit • Eulerian path or circuit • degree of vertex Theorem (Euler) Let n be the number of vertices of odd degree. (a) If n = 0, the graph has at least one Eulerian circuit;

Walter Gautschi Leonhard Euler (c) if n > 2, it has neither. (n = 1 is impossible.)

K¨onigsberg bridge graph: n = 4

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 5 The K¨onigsberg bridge problem (1741) connected graph • path • circuit • Eulerian path or circuit • degree of vertex Theorem (Euler) Let n be the number of vertices of odd degree. (a) If n = 0, the graph has at least one Eulerian circuit; (b) if n = 2, it has at least one Eulerian path, but no circuit;

Walter Gautschi Leonhard Euler (n = 1 is impossible.)

K¨onigsberg bridge graph: n = 4

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 5 The K¨onigsberg bridge problem (1741) connected graph • path • circuit • Eulerian path or circuit • degree of vertex Theorem (Euler) Let n be the number of vertices of odd degree. (a) If n = 0, the graph has at least one Eulerian circuit; (b) if n = 2, it has at least one Eulerian path, but no circuit; (c) if n > 2, it has neither.

Walter Gautschi Leonhard Euler K¨onigsberg bridge graph: n = 4

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 5 The K¨onigsberg bridge problem (1741) connected graph • path • circuit • Eulerian path or circuit • degree of vertex Theorem (Euler) Let n be the number of vertices of odd degree. (a) If n = 0, the graph has at least one Eulerian circuit; (b) if n = 2, it has at least one Eulerian path, but no circuit; (c) if n > 2, it has neither. (n = 1 is impossible.)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 5 The K¨onigsberg bridge problem (1741) connected graph • path • circuit • Eulerian path or circuit • degree of vertex Theorem (Euler) Let n be the number of vertices of odd degree. (a) If n = 0, the graph has at least one Eulerian circuit; (b) if n = 2, it has at least one Eulerian path, but no circuit; (c) if n > 2, it has neither. (n = 1 is impossible.)

K¨onigsberg bridge graph: n = 4

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 6 Euler’s buckling formula (1744)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 6 Euler’s buckling formula (1744)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 7 Euler flow (1757)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 7 Euler flow (1757)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Octahedron

V = 6 E = 12 F = 8

Walter Gautschi Leonhard Euler In a three-dimensional convex polyhedron let

V = number of vertices E = number of edges F = number of faces

Theorem (Euler)

V − E + F = 2

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 8 Euler’s polyhedral formula (1753)

Walter Gautschi Leonhard Euler Theorem (Euler)

V − E + F = 2

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 8 Euler’s polyhedral formula (1753) In a three-dimensional convex polyhedron let

V = number of vertices E = number of edges F = number of faces

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 8 Euler’s polyhedral formula (1753) In a three-dimensional convex polyhedron let

V = number of vertices E = number of edges F = number of faces

Theorem (Euler)

V − E + F = 2

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Walter Gautschi Leonhard Euler the story begins 1734 with a letter of Euler to Daniel Bernoulli, in which he wants to interpolate the logarithm from the data x 1 10 100 1000 ··· log x 0 1 2 3 ···

Euler takes S(x) = Newton’s interpolation series, but notices that, when x = 9, one gets S(9) 6= log 9, even though the series converges fast what, then, is S(x), if not the logarithm log x? Euler’s answer (1753): a detailed study of the interpolation series S(x; a) for the logarithm to base a today we know that S(x; a) is a q-analogue of the logarithm where q = 1/a

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 9 Euler and q-theory

Walter Gautschi Leonhard Euler Euler takes S(x) = Newton’s interpolation series, but notices that, when x = 9, one gets S(9) 6= log 9, even though the series converges fast what, then, is S(x), if not the logarithm log x? Euler’s answer (1753): a detailed study of the interpolation series S(x; a) for the logarithm to base a today we know that S(x; a) is a q-analogue of the logarithm where q = 1/a

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 9 Euler and q-theory the story begins 1734 with a letter of Euler to Daniel Bernoulli, in which he wants to interpolate the logarithm from the data x 1 10 100 1000 ··· log x 0 1 2 3 ···

Walter Gautschi Leonhard Euler what, then, is S(x), if not the logarithm log x? Euler’s answer (1753): a detailed study of the interpolation series S(x; a) for the logarithm to base a today we know that S(x; a) is a q-analogue of the logarithm where q = 1/a

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 9 Euler and q-theory the story begins 1734 with a letter of Euler to Daniel Bernoulli, in which he wants to interpolate the logarithm from the data x 1 10 100 1000 ··· log x 0 1 2 3 ···

Euler takes S(x) = Newton’s interpolation series, but notices that, when x = 9, one gets S(9) 6= log 9, even though the series converges fast

Walter Gautschi Leonhard Euler Euler’s answer (1753): a detailed study of the interpolation series S(x; a) for the logarithm to base a today we know that S(x; a) is a q-analogue of the logarithm where q = 1/a

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 9 Euler and q-theory the story begins 1734 with a letter of Euler to Daniel Bernoulli, in which he wants to interpolate the logarithm from the data x 1 10 100 1000 ··· log x 0 1 2 3 ···

Euler takes S(x) = Newton’s interpolation series, but notices that, when x = 9, one gets S(9) 6= log 9, even though the series converges fast what, then, is S(x), if not the logarithm log x?

Walter Gautschi Leonhard Euler today we know that S(x; a) is a q-analogue of the logarithm where q = 1/a

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 9 Euler and q-theory the story begins 1734 with a letter of Euler to Daniel Bernoulli, in which he wants to interpolate the logarithm from the data x 1 10 100 1000 ··· log x 0 1 2 3 ···

Euler takes S(x) = Newton’s interpolation series, but notices that, when x = 9, one gets S(9) 6= log 9, even though the series converges fast what, then, is S(x), if not the logarithm log x? Euler’s answer (1753): a detailed study of the interpolation series S(x; a) for the logarithm to base a

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selectio 9 Euler and q-theory the story begins 1734 with a letter of Euler to Daniel Bernoulli, in which he wants to interpolate the logarithm from the data x 1 10 100 1000 ··· log x 0 1 2 3 ···

Euler takes S(x) = Newton’s interpolation series, but notices that, when x = 9, one gets S(9) 6= log 9, even though the series converges fast what, then, is S(x), if not the logarithm log x? Euler’s answer (1753): a detailed study of the interpolation series S(x; a) for the logarithm to base a today we know that S(x; a) is a q-analogue of the logarithm where q = 1/a

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

St. Petersburg 1766–1783 The glorious final stretch

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

The Euler house and Catherine II

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Euler 1778

Walter Gautschi Leonhard Euler • 1771: Euler loses his (good) left eye following a cataract operation and becomes virtually blind; Euler’s wooden house burns down during the great St. Petersburg fire

• 1773: Euler’s wife Katharina dies

• 1776: Euler remarries

• 1783: On September 18, Euler dies of a stroke

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology

Walter Gautschi Leonhard Euler • 1773: Euler’s wife Katharina dies

• 1776: Euler remarries

• 1783: On September 18, Euler dies of a stroke

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology

• 1771: Euler loses his (good) left eye following a cataract operation and becomes virtually blind; Euler’s wooden house burns down during the great St. Petersburg fire

Walter Gautschi Leonhard Euler • 1776: Euler remarries

• 1783: On September 18, Euler dies of a stroke

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology

• 1771: Euler loses his (good) left eye following a cataract operation and becomes virtually blind; Euler’s wooden house burns down during the great St. Petersburg fire

• 1773: Euler’s wife Katharina dies

Walter Gautschi Leonhard Euler • 1783: On September 18, Euler dies of a stroke

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology

• 1771: Euler loses his (good) left eye following a cataract operation and becomes virtually blind; Euler’s wooden house burns down during the great St. Petersburg fire

• 1773: Euler’s wife Katharina dies

• 1776: Euler remarries

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Chronology

• 1771: Euler loses his (good) left eye following a cataract operation and becomes virtually blind; Euler’s wooden house burns down during the great St. Petersburg fire

• 1773: Euler’s wife Katharina dies

• 1776: Euler remarries

• 1783: On September 18, Euler dies of a stroke

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Algebra and Second lunar theory

Second theory of ships

Walter Gautschi Leonhard Euler Algebra (1770) • a work written for the absolute beginner; a prime example of Euler’s extraordinary didactic skill; it becomes a ”bestseller”, translated into all major languages Second lunar theory (1772) • a monumental work explaining the many irregularities of the moon’s orbit • Euler’s struggle with ”solving” the three-body problem (sun–earth–moon) Second theory of ships (1773) • construction and maneuvering of ships • written for people (e.g., sailors) with no, or little, mathematical knowledge

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works

Walter Gautschi Leonhard Euler Second lunar theory (1772) • a monumental work explaining the many irregularities of the moon’s orbit • Euler’s struggle with ”solving” the three-body problem (sun–earth–moon) Second theory of ships (1773) • construction and maneuvering of ships • written for people (e.g., sailors) with no, or little, mathematical knowledge

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Algebra (1770) • a work written for the absolute beginner; a prime example of Euler’s extraordinary didactic skill; it becomes a ”bestseller”, translated into all major languages

Walter Gautschi Leonhard Euler Second theory of ships (1773) • construction and maneuvering of ships • written for people (e.g., sailors) with no, or little, mathematical knowledge

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Algebra (1770) • a work written for the absolute beginner; a prime example of Euler’s extraordinary didactic skill; it becomes a ”bestseller”, translated into all major languages Second lunar theory (1772) • a monumental work explaining the many irregularities of the moon’s orbit • Euler’s struggle with ”solving” the three-body problem (sun–earth–moon)

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Major works Algebra (1770) • a work written for the absolute beginner; a prime example of Euler’s extraordinary didactic skill; it becomes a ”bestseller”, translated into all major languages Second lunar theory (1772) • a monumental work explaining the many irregularities of the moon’s orbit • Euler’s struggle with ”solving” the three-body problem (sun–earth–moon) Second theory of ships (1773) • construction and maneuvering of ships • written for people (e.g., sailors) with no, or little, mathematical knowledge

Walter Gautschi Leonhard Euler Selectio 10 Euler’s disk

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 10 Euler’s disk

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Selecta Euleriana

Selectio 10 Euler’s disk

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selection 11 Gear transmission; Euler’s tooth profile

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Selection 11 Gear transmission; Euler’s tooth profile

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

The Man

Walter Gautschi Leonhard Euler Intellect • phenomenal memory, erudite • unusual power of mental calculation • ability to concentrate on mental work under adversity Craftsmanship • superb expositor • his goal: ultimate clarity and simplicity • yet fearless and agressive in his quest for discovery

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Personality • modest, inconspicuous, uncomplicated, yet cheerful and sociable • “honesty, uncompromising rectitude—the acknowledged national virtues of Swiss people—he possessed to a superior degree” (Fuchs) • free of priority concerns • generous in acknowledging and furthering other people’s work

Walter Gautschi Leonhard Euler Craftsmanship • superb expositor • his goal: ultimate clarity and simplicity • yet fearless and agressive in his quest for discovery

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Personality • modest, inconspicuous, uncomplicated, yet cheerful and sociable • “honesty, uncompromising rectitude—the acknowledged national virtues of Swiss people—he possessed to a superior degree” (Fuchs) • free of priority concerns • generous in acknowledging and furthering other people’s work Intellect • phenomenal memory, erudite • unusual power of mental calculation • ability to concentrate on mental work under adversity

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Personality • modest, inconspicuous, uncomplicated, yet cheerful and sociable • “honesty, uncompromising rectitude—the acknowledged national virtues of Swiss people—he possessed to a superior degree” (Fuchs) • free of priority concerns • generous in acknowledging and furthering other people’s work Intellect • phenomenal memory, erudite • unusual power of mental calculation • ability to concentrate on mental work under adversity Craftsmanship • superb expositor • his goal: ultimate clarity and simplicity • yet fearless and agressive in his quest for discovery

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man

Euler memorial plaque in Riehen

Walter Gautschi Leonhard Euler LEONHARD EULER 1707–1783 mathematician, physicist, engineer, astronomer and philosopher, spent his youth in Riehen. He was a great scholar and a kind man.

Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Epilogue

Walter Gautschi Leonhard Euler Outline Basel 1707–1727 St. Petersburg 1727–1741 Berlin 1741–1766 St. Petersburg 1766–1783 The Man Epilogue

LEONHARD EULER 1707–1783 mathematician, physicist, engineer, astronomer and philosopher, spent his youth in Riehen. He was a great scholar and a kind man.

Walter Gautschi Leonhard Euler