Biography

1720

14

... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. ... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand ...

1723

1723

1726

1726

1727

1727

1727

1726

1726

(

19

)

This decision ultimately benefited Euler, because it forced him to move from a small republic into a setting more adequate for his brilliant research and technological work.

5

1727

17

1727

Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometer Jakob Hermann, a relative; Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholar Christian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F Maier, working in trigonometry; and the astronomer and geographer J-N Delisle.

1727

1730

1730

1733

7

1734

13

... after 1730 he carried out state projects dealing with cartography, science education, magnetism, fire engines, machines, and ship building. ... The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics. He viewed these three fields as intimately interconnected. Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems.

(1736

37)

1735

1738

1740

... lost an eye and [ the other ] currently may be in the same danger.

1735

1753

1740

1738

1740

19

1741

25

I can do just what I wish [ in my research ] ... The king calls me his professor, and I think I am the happiest man in the world.

1744

... he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal ... At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.

380

(

)

(3

1768

72)

1759

1763

1766

1771

1771

(

59)

1766

(

1769)

1772

1776

.. the scientists assisting Euler were not mere secretaries; he discussed the general scheme of the works with them, and they developed his ideas, calculating tables, and sometimes compiled examples.

775

1772

250

1776

On 18 September 1783 Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain haemorrhage and uttered only "I am dying" before he lost consciousness. He died about eleven o'clock in the evening.

1783

50

(1765)

f ( x ) f (x) f ( x )

(1734)

e e e

(1727)

i i i

1

(1777)

(1755)

Δ y \Delta y Δ y

Δ 2 y \Delta ^{2} y Δ 2 y

1729

2 n + 1 2^{n} + 1 2 n + 1

n n n

2

n n n

1

2

4

8

16

1732

2 32 + 1 = 4294967297 2^{32} + 1 = 4294967297 2 3 2 + 1 = 4 2 9 4 9 6 7 2 9 7

641

ϕ ( n ) \phi(n) ϕ ( n )

k k k

1 ≤ k ≤ n 1 ≤ k ≤ n 1 ≤ k ≤ n

k k k

n n n

a a a

b b b

a 2 + b 2 a^{2} + b^{2} a 2 + b 2

4 n − 1 4n - 1 4 n − 1

1749

ζ ( 2 ) = ∑ ( 1 / n 2 ) \zeta(2) = \sum (1/n^{2}) ζ ( 2 ) = ∑ ( 1 / n 2 )

1735

ζ ( 2 ) = π 2 / 6 \zeta(2) = \pi^{2}/6 ζ ( 2 ) = π 2 / 6

ζ ( 4 ) = π 4 / 90 , ζ ( 6 ) = π 6 / 945 , ζ ( 8 ) = π 8 / 9450 , ζ ( 10 ) = π 10 / 93555 \zeta(4) = \pi^{4}/90, \zeta(6) = \pi^{6}/945, \zeta(8) = \pi^{8}/9450, \zeta(10) = \pi^{10}/93555 ζ ( 4 ) = π 4 / 9 0 , ζ ( 6 ) = π 6 / 9 4 5 , ζ ( 8 ) = π 8 / 9 4 5 0 , ζ ( 1 0 ) = π 1 0 / 9 3 5 5 5

ζ ( 12 ) = 691 π 12 / 638512875 \zeta(12) = 691\pi^{12} /638512875 ζ ( 1 2 ) = 6 9 1 π 1 2 / 6 3 8 5 1 2 8 7 5

1737

ζ ( s ) = ∑ ( 1 / n s ) = ∏ ( 1 − p − s ) − 1 \zeta(s) = \sum (1/n^{s}) = \prod (1 - p^{-s})^{-1} ζ ( s ) = ∑ ( 1 / n s ) = ∏ ( 1 − p − s ) − 1

n n n

1739

C C C

ζ ( 2 n ) = C π 2 n \zeta(2n) = C\pi^{2n} ζ ( 2 n ) = C π 2 n

1735

1 1 + 1 2 + 1 3 + . . . + 1 n − log ⁡ e n \large\frac{1}{1}

ormalsize + \large\frac{1}{2}

ormalsize + \large\frac{1}{3}

ormalsize + ... + \large\frac{1}{n}

ormalsize - \log_{e}n 1 1 ​ + 2 1 ​ + 3 1 ​ + . . . + n 1 ​ − lo g e ​ n

n n n

16

1744

π / 2 − x / 2 = sin ⁡ x + ( sin ⁡ 2 x ) / 2 + ( sin ⁡ 3 x ) / 3 + . . . \pi/2 - x/2 = \sin x + (\sin 2x)/2 + (\sin 3x)/3 + ... π / 2 − x / 2 = sin x + ( sin 2 x ) / 2 + ( sin 3 x ) / 3 + . . .

1755

8

1736

Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.

... will be publishing a book on fluxions. ... he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.

... I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy.

n = 3 n = 3 n = 3

a + b √ − 3 a + b√-3 a + b √ − 3

1748

e i x = cos ⁡ x + i sin ⁡ x e^{ix} = \cos x + i \sin x e i x = cos x + i sin x .

ln ⁡ ( − 1 ) = π i \ln(-1) = \pi i ln ( − 1 ) = π i

1727

1751

1777

1752

1755

(1768

70)

1729

Ⓣ ( A method for curves )

1740

... one of the most beautiful mathematical works ever written.

1736

The distinguishing feature of Euler's investigations in mechanics as compared to those of his predecessors is the systematic and successful application of analysis. Previously the methods of mechanics had been mostly synthetic and geometrical; they demanded too individual an approach to separate problems. Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way.

Outstanding in both theoretical and applied mechanics, it addresses Euler's intense occupation with the problem of ship propulsion. It applies variational principles to determine the optimal ship design and first established the principles of hydrostatics ... Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion.

1765

Ⓣ ( Theory of the motion of solid bodies )

1750

1752

However sublime are the researches on fluids which we owe to Messrs Bernoulli, Clairaut and d'Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this accord of their profound meditations with the simplicity of the principles from which I have drawn my two equations ...

... determination of the orbits of comets and planets by a few observations, methods of calculation of the parallax of the sun, the theory of refraction, consideration of the physical nature of comets, .... His most outstanding works, for which he won many prizes from the Paris Académie des Sciences, are concerned with celestial mechanics, which especially attracted scientists at that time.

1765

3000

300

Ⓣ ( A new musical theory )

1739

... part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing.

... for musicians too advanced in its mathematics and for mathematicians too musical.

1735

1745

20