Question: What is psychotherapy?

Psychotherapy

Self-help groups (or “meeting groups”)


Question: What is psychotherapy?

Psychotherapy

Notes


Question: What is psychotherapy?

Psychotherapy

Recommended literature


Question: What is psychotherapy?

Psychotherapy

//Encyclopedia of Psychotherapy/ed.


Question: What is psychotherapy?

Psychotherapy

B.


Question: What is psychotherapy?

Psychotherapy

D. Karvasarsky.


Question: What is psychotherapy?

Psychotherapy

— SPb., 1999.


Question: What is psychotherapy?

Psychotherapy

//Handbook on psychotherapy.


Question: What is psychotherapy?

Psychotherapy

— 1985.


Question: What is psychotherapy?

Psychotherapy

Aleksandrov A. A. Integrative psychotherapy.


Question: What is psychotherapy?

Psychotherapy

— SPb: Peter, 2009.


Question: What is psychotherapy?

Psychotherapy

— 352 p.: Il.


Question: What is psychotherapy?

Psychotherapy

— ISBN 978-5-388-00270-9.


Question: What is psychotherapy?

Psychotherapy

Aleksandrov A. A.: training manual.


Question: What is psychotherapy?

Psychotherapy

— SPb: Peter, 2004.


Question: What is psychotherapy?

Psychotherapy

Outpatient and hospital psychotherapy and medical psychology.


Question: What is psychotherapy?

Psychotherapy

Checkmate.


Question: What is psychotherapy?

Psychotherapy

1—8 All-Russian Public Professional Medical Psychotherapeutic Conferences/edited by A. AND. Appenyansky, Yu.


Question: What is psychotherapy?

Psychotherapy

P. Boyko V. N. Krasnova V. AND. Kurpatova, Y.


Question: What is psychotherapy?

Psychotherapy

S. Shevchenko.


Question: What is psychotherapy?

Psychotherapy

— (1st — 2003; 2nd — 2004; 3rd — 2005; 4th — 2006; 5th — 2007; 6th — 2008; 7th — 2009; 8th − 2010).


Question: What is psychotherapy?

Psychotherapy

See also psychomed.nm.ru.


Question: What is psychotherapy?

Psychotherapy

Appenyansky A. AND. Difficult questions of a psychotherapist//Outpatient and hospital psychotherapy and medical psychology.


Question: What is psychotherapy?

Psychotherapy

Checkmate.


Question: What is psychotherapy?

Psychotherapy

1st All-Russian.


Question: What is psychotherapy?

Psychotherapy

ls.


Question: What is psychotherapy?

Psychotherapy

professional honey.


Question: What is psychotherapy?

Psychotherapy

psychoter.


Question: What is psychotherapy?

Psychotherapy

conf.


Question: What is psychotherapy?

Psychotherapy

— MOSCOW, 2003.


Question: What is psychotherapy?

Psychotherapy

— P. 149—185.


Question: What is psychotherapy?

Psychotherapy

Burlachuk L. F., Grabskaya I. A., Kocharyan A. S. — K.: Nika-Center; M.: Aletheia, 1999.


Question: What is psychotherapy?

Psychotherapy

— 320 p. — ISBN 966-521-029-7 (Nika-Center), ISBN 5-89321-035-2 (Aletheia).


Question: What is psychotherapy?

Psychotherapy

Burlachuk L. F., Kocharyan A. S., Zhidko M. E.


Question: What is psychotherapy?

Psychotherapy

: textbook for universities.


Question: What is psychotherapy?

Psychotherapy

Zachepitsky R. A., Karvasarsky B.


Question: What is psychotherapy?

Psychotherapy

D. Questions of the ratio of conscious and unconscious forms of mental activity in the light of the experience of pathogenetic psychotherapy of neurosis//Unconscious.


Question: What is psychotherapy?

Psychotherapy

— Tbilisi, 1978.


Question: What is psychotherapy?

Psychotherapy

Makarov V. IN.


Question: What is psychotherapy?

Psychotherapy

— M.: Academic project; Yekaterinburg: Business Book, 1999.


Question: What is psychotherapy?

Psychotherapy

— 416 p.


Question: What is psychotherapy?

Psychotherapy

It's him.


Question: What is psychotherapy?

Psychotherapy

Selected lectures on psychotherapy.


Question: What is psychotherapy?

Psychotherapy

— 2nd ed.


Question: What is psychotherapy?

Psychotherapy

Rev.


Question: What is psychotherapy?

Psychotherapy

and add.


Question: What is psychotherapy?

Psychotherapy

— M.: Academic project; Yekaterinburg: Business Book, 2000.


Question: What is psychotherapy?

Psychotherapy

— 432 p. — ISBN 5-8291-0060-6 (Academic Project), ISBN 5-88687-078-4 (Business Book).


Question: What is psychotherapy?

Psychotherapy

: textbook/comp.


Question: What is psychotherapy?

Psychotherapy

L. M. Krol, Ye.


Question: What is psychotherapy?

Psychotherapy

A. Purtova.


Question: What is psychotherapy?

Psychotherapy

— M.: “Klass”, 2001.


Question: What is psychotherapy?

Psychotherapy

— 480 p. — (Library of Psychology and Psychotherapy. Vol. 90).


Question: What is psychotherapy?

Psychotherapy

— ISBN 5-86375-033-2.


Question: What is psychotherapy?

Psychotherapy

— 2nd ed.


Question: What is psychotherapy?

Psychotherapy

, Ext.


Question: What is psychotherapy?

Psychotherapy

and revised.


Question: What is psychotherapy?

Psychotherapy

— SPb., 2000.


Question: What is psychotherapy?

Psychotherapy

: textbook/ed.


Question: What is psychotherapy?

Psychotherapy

B.


Question: What is psychotherapy?

Psychotherapy

IN. Karvasarsky.


Question: What is psychotherapy?

Psychotherapy

— 2nd ed.


Question: What is psychotherapy?

Psychotherapy

Rev.


Question: What is psychotherapy?

Psychotherapy

and add.


Question: What is psychotherapy?

Psychotherapy

— SPb.: Peter, 2002.


Question: What is psychotherapy?

Psychotherapy

Psychotherapy Guide/Ed.


Question: What is psychotherapy?

Psychotherapy

IN. E.


Question: What is psychotherapy?

Psychotherapy

Rozhnova.


Question: What is psychotherapy?

Psychotherapy

— 2nd ed.


Question: What is psychotherapy?

Psychotherapy

— Tashkent: “Medicine of the UzSSR”, 1979.


Question: What is psychotherapy?

Psychotherapy

— 620 p.


Question: What is psychotherapy?

Psychotherapy

— Tashkent: “Medicine of the UzSSR”, 1985.


Question: What is psychotherapy?

Psychotherapy

— 719 p. (This edition has removed a number of articles from the previous edition.)


Question: What is psychotherapy?

Psychotherapy

Filatov A. T., Skumin V. A. Psychoprophylaxis and psychotherapy in cardiac surgery.


Question: What is psychotherapy?

Psychotherapy

— Kyiv: Health, 1985.


Question: What is psychotherapy?

Psychotherapy

— 72 p. — (Doctor of Practice).


Question: What is psychotherapy?

Psychotherapy

Chaplina G. V.//Site


Question: What is psychotherapy?

Psychotherapy

Reshetnikov M. M.//Independent Psychiatric Journal.


Question: What is psychotherapy?

Psychotherapy

— 2003.


Question: What is psychotherapy?

Psychotherapy

— No. 2.


Question: What is psychotherapy?

Psychotherapy

Zorin N. A.


Question: What is psychotherapy?

Psychotherapy

References


Question: What is psychotherapy?

Psychotherapy

Category:Methods of psychological assistance


Question: When was the Red Dwarf discovered?

Red dwarf

Red karlik — according to the Hertzsprung-Russell diagram, is a small and relatively cold main-sequence star with a spectral class M or late K.


Question: When was the Red Dwarf discovered?

Red dwarf

General characteristics


Question: When was the Red Dwarf discovered?

Red dwarf

Red dwarfs are quite different from other stars.


Question: When was the Red Dwarf discovered?

Red dwarf

, followed by brown dwarfs).


Question: When was the Red Dwarf discovered?

Red dwarf

The temperature of the photosphere of a red dwarf can reach 3500 K, which is higher than the temperature of the spiral of an incandescent lamp, so, contrary to its name, red dwarfs, similar to lamps, do not emit light red, but rather buffy-yellowish.


Question: When was the Red Dwarf discovered?

Red dwarf

This type of star emits very little light, sometimes 10,000 times smaller than the Sun.


Question: When was the Red Dwarf discovered?

Red dwarf

Due to the low rate of combustion of hydrogen, red dwarfs have a very long life expectancy - from tens of billions to tens of trillions of years (a red dwarf with a mass of 0.1 times the mass of burn 10 trillion years) [2].


Question: When was the Red Dwarf discovered?

Red dwarf

In red dwarfs, thermonuclear reactions involving helium are impossible, so they cannot turn into red giants.


Question: When was the Red Dwarf discovered?

Red dwarf

Over time, they gradually shrink and heat up until they use up their entire supply of hydrogen fuel, and gradually turn into blue dwarfs, and then into white dwarfs with a helium core.


Question: When was the Red Dwarf discovered?

Red dwarf

But not enough time has passed since the Big Bang for the red dwarfs to get off the main sequence.


Question: When was the Red Dwarf discovered?

Red dwarf

The fact that red dwarfs remain on the main sequence while other stars leave it makes it possible to determine the age of star clusters by finding the mass at which stars are forced to leave the main sequence.


Question: When was the Red Dwarf discovered?

Red dwarf

In addition, the fact that no red dwarf outside the main sequence has been found at the moment indicates that the universe is of finite age.


Question: When was the Red Dwarf discovered?

Red dwarf

Red dwarfs in the universe


Question: When was the Red Dwarf discovered?

Red dwarf

Red dwarfs are the most common star-type objects in the universe.


Question: When was the Red Dwarf discovered?

Red dwarf

Proxima Centauri, the closest star to the Sun, is a red dwarf (spectral class M5.5Ve; magnitude 11.0m), as are twenty of the next thirty closest stars.


Question: When was the Red Dwarf discovered?

Red dwarf

However, due to their low brightness, they are poorly studied.


Question: When was the Red Dwarf discovered?

Red dwarf

The problem of primary red dwarfs


Question: When was the Red Dwarf discovered?

Red dwarf

One of the mysteries of astronomy is that there are too few red dwarfs that do not contain metals at all.


Question: When was the Red Dwarf discovered?

Red dwarf

According to the Big Bang model, the first generation of stars was supposed to contain only hydrogen and helium (and very little lithium).


Question: When was the Red Dwarf discovered?

Red dwarf

If there were red dwarfs among these stars, they should be observed today, which is not happening.


Question: When was the Red Dwarf discovered?

Red dwarf

The common explanation is that low-mass stars cannot form without heavy elements.


Question: When was the Red Dwarf discovered?

Red dwarf

Since light stars undergo thermonuclear reactions involving hydrogen in the presence of metals, the early protostar with a low mass, devoid of metals, is unable to “ignite” and is forced to remain a cloud of gas until it gets more matter.


Question: When was the Red Dwarf discovered?

Red dwarf

All this supports the theory that the first stars were very massive and soon died, releasing a large amount of metals necessary for the formation of light stars.


Question: When was the Red Dwarf discovered?

Red dwarf

Life on planets with red dwarfs


Question: When was the Red Dwarf discovered?

Red dwarf

Thermonuclear reactions of red dwarfs are “economical”: nucleosynthesis in the depths of these stars is slow.


Question: When was the Red Dwarf discovered?

Red dwarf

This is due to the strong dependence of the rate of thermonuclear reactions (about the fourth power) on temperature, which is low in low-mass stars.


Question: When was the Red Dwarf discovered?

Red dwarf

Therefore, the life cycle of red dwarfs is hundreds of times longer than that of stars like the Sun.


Question: When was the Red Dwarf discovered?

Red dwarf

If a planet near a red dwarf has the simplest life, it is incomparably more likely to develop into something interesting than short-lived stars like the Sun.


Question: When was the Red Dwarf discovered?

Red dwarf

This is due to the fact that it takes billions of years to develop a highly organized life.


Question: When was the Red Dwarf discovered?

Red dwarf

Exoplanets


Question: When was the Red Dwarf discovered?

Red dwarf

In 2005, exoplanets orbiting red dwarfs were discovered.


Question: When was the Red Dwarf discovered?

Red dwarf

In terms of size, one of them is comparable to Neptune (about 17 Earth masses).


Question: When was the Red Dwarf discovered?

Red dwarf

This planet orbits only 6 million kilometers from the star, and therefore should have a surface temperature of about 150° C, despite the star's low luminosity.


Question: When was the Red Dwarf discovered?

Red dwarf

In 2006, a terrestrial planet was discovered.


Question: When was the Red Dwarf discovered?

Red dwarf

It orbits a red dwarf at a distance of 390 million kilometers and its surface temperature is -220° C.


Question: When was the Red Dwarf discovered?

Red dwarf

In 2007, planets were discovered in the habitable zone of the red dwarf Gliese 581, and in 2010, a planet was discovered in the habitable zone near Gliese 876.


Question: When was the Red Dwarf discovered?

Red dwarf

In 2014, the Earth-sized planet Kepler-186f was discovered in the habitable zone [5].


Question: When was the Red Dwarf discovered?

Red dwarf

On February 22, 2017, it was announced that seven Earth-type planets had been discovered near the red dwarf TRAPPIST-1.


Question: When was the Red Dwarf discovered?

Red dwarf

Three of them are in the habitable zone [6].


Question: When was the Red Dwarf discovered?

Red dwarf

Problems related to the climate of the planets


Question: When was the Red Dwarf discovered?

Red dwarf

Since red dwarfs are rather dim, an effective Earth orbit should be close to a star.


Question: When was the Red Dwarf discovered?

Red dwarf

But a planet that is too close to the star becomes one side of it all the time.


Question: When was the Red Dwarf discovered?

Red dwarf

This phenomenon is called tidal capture.


Question: When was the Red Dwarf discovered?

Red dwarf

It can cause temperature differences in different hemispheres (night and day), because the daytime hemisphere is always warm (maybe very hot), and at night it can approach absolute zero.


Question: When was the Red Dwarf discovered?

Red dwarf

A dense atmosphere, however, could provide some heat transfer to the shadow hemisphere, but this, in turn, would cause strong winds.


Question: When was the Red Dwarf discovered?

Red dwarf

Red dwarfs are many times more active than the Sun (the stellar wind of such stars is not much weaker than that of the Sun).


Question: When was the Red Dwarf discovered?

Red dwarf

Very powerful outbreaks can be detrimental to possible life on the planet.


Question: When was the Red Dwarf discovered?

Red dwarf

The planet's magnetic field could partially solve this problem, becoming a barrier to radiation, but planets with tidal capture in most cases cannot have it, since the lack of rotation of the planet also means no rotation of the core.


Question: When was the Red Dwarf discovered?

Red dwarf

However, the role of the magnetosphere in protecting against cosmic radiation remained overestimated for a long time, and the protective properties of the atmosphere alone could be sufficient [7].


Question: When was the Red Dwarf discovered?

Red dwarf

Typical red dwarfs


Question: When was the Red Dwarf discovered?

Red dwarf

Proxima Centauri — (M5.5 Ve) — distance 1.31 px; luminosity — 0.000 072 solar;


Question: When was the Red Dwarf discovered?

Red dwarf

Barnard's star — (M5V) — distance 1.83 px; luminosity — 0.000 450 solar;


Question: When was the Red Dwarf discovered?

Red dwarf

Wolf 359 — (DM6e) — distance 2.34 px; luminosity — 0.000,016 solar;


Question: When was the Red Dwarf discovered?

Red dwarf

Ross 154 — (DM4e) — distance 2.93 px; luminosity — 0.000 380 solar;


Question: When was the Red Dwarf discovered?

Red dwarf

Ross 248 — (DM6e) — distance 3.16 px; luminosity — 0.000 110 solar;


Question: When was the Red Dwarf discovered?

Red dwarf

Ross 128 — (dM5) — distance 3.34 px; luminosity — 0.000,080 solar;


Question: When was the Red Dwarf discovered?

Red dwarf

Gliese 581 — (M3V) — distance 6.27 px; luminosity — 0.013 solar;


Question: When was the Red Dwarf discovered?

Red dwarf

TRAPPIST-1 — (M8V) — distance 12.10 px; luminosity — 0.000 525 solar.


Question: When was the Red Dwarf discovered?

Red dwarf

See also


Question: When was the Red Dwarf discovered?

Red dwarf

Black dwarf


Question: When was the Red Dwarf discovered?

Red dwarf

White dwarf


Question: When was the Red Dwarf discovered?

Red dwarf

Brown dwarf


Question: When was the Red Dwarf discovered?

Red dwarf

Yellow dwarf


Question: When was the Red Dwarf discovered?

Red dwarf

Orange dwarf


Question: When was the Red Dwarf discovered?

Red dwarf

Dark matter


Question: When was the Red Dwarf discovered?

Red dwarf

Subbrown dwarf


Question: When was the Red Dwarf discovered?

Red dwarf

Hertzsprung—Russell diagram


Question: When was the Red Dwarf discovered?

Red dwarf

Notes


Question: When was the Red Dwarf discovered?

Red dwarf

Literature


Question: When was the Red Dwarf discovered?

Red dwarf

Alpert, Mark (Nov. 2005).


Question: When was the Red Dwarf discovered?

Red dwarf

“Red Star Rising”.


Question: When was the Red Dwarf discovered?

Red dwarf

Scientific American, p.


Question: When was the Red Dwarf discovered?

Red dwarf


Question: When was the Red Dwarf discovered?

Red dwarf

References


Question: When was the Red Dwarf discovered?

Red dwarf

— D. Montes, UCM.


Question: When was the Red Dwarf discovered?

Red dwarf

Category:Stellar evolution


Question: When was the Red Dwarf discovered?

Red dwarf

Category:Astrophysics


Question: When was the Red Dwarf discovered?

Red dwarf

Category:Star classes


Question: When was the Red Dwarf discovered?

Red dwarf


Question: When did variation first appear?

Leonard Euler

Leonard Eiler (German: Leonhard Euler; April 15, 1707 —, St. Petersburg, Russian Empire) was Swiss, German and Russian and made a fundamental contribution to the development of these sciences (as well as physics, astronomy and a number of applied sciences) [1].


Question: When did variation first appear?

Leonard Euler

Euler is the author of more than 850 works [2] (including two dozen fundamental monographs) on mathematical analysis, differential geometry, number theory, approximate calculations, celestial mechanics, mathematical physics, optics, ballistics, shipbuilding, music theory and other fields [3] [4].


Question: When did variation first appear?

Leonard Euler

He studied medicine, chemistry, botany, aeronautics, music theory, many European and ancient languages in depth.


Question: When did variation first appear?

Leonard Euler

Academician of the St. Petersburg, Berlin, Turin, Lisbon and Basel Academies of Sciences, foreign member of the Paris Academy of Sciences [5].


Question: When did variation first appear?

Leonard Euler

He spent almost half of his life in Russia, where he made a significant contribution to the development of Russian science.


Question: When did variation first appear?

Leonard Euler

In 1726, he was invited to work in St. Petersburg, where he moved a year later.


Question: When did variation first appear?

Leonard Euler

From 1726 to 1741, as well as from 1766, he was an academician of the St. Petersburg Academy of Sciences (first an adjunct and since 1731 — a professor); in 1741-1766 he worked in Berlin (remaining at the same time honorary a member of the St. Petersburg Academy) [1].


Question: When did variation first appear?

Leonard Euler

After a year in Russia, he knew Russian well and published part of his essays (especially textbooks) in Russian [C 1].


Question: When did variation first appear?

Leonard Euler

The first Russian mathematicians (S. K. Kotelnikov) and astronomers (S. ME. Rumovsky) were students of Euler.


Question: When did variation first appear?

Leonard Euler

Biography


Question: When did variation first appear?

Leonard Euler

Switzerland (1707—1727)


Question: When did variation first appear?

Leonard Euler

Leonard Euler was born in 1707 to Basel pastor Paul Euler, a friend of the Bernoulli family, and Margarita Euler, née Brooker.


Question: When did variation first appear?

Leonard Euler

Shortly after Leonard was born, the family moved to the village of Richen (an hour's walk from Basel), where Paul Euler was appointed pastor, where the boy spent his first years of childhood.


Question: When did variation first appear?

Leonard Euler

Leonard received his primary education at home under the guidance of his father (who once studied mathematics with Jacob Bernoulli) [6].


Question: When did variation first appear?

Leonard Euler

The pastor prepared his eldest son for a spiritual career, but he also studied mathematics with him, both as entertainment and for the development of logical thinking, and Leonard early showed mathematical abilities [7].


Question: When did variation first appear?

Leonard Euler

When Leonard grew up, he was transported to his grandmother in Basel, where he attended gymnasium (while continuing to study mathematics enthusiastically).


Question: When did variation first appear?

Leonard Euler

In 1720, a capable gymnasium student was allowed to attend public lectures at the University of Basel, where he attracted the attention of Professor Johann Bernoulli (younger brother of Jacob Bernoulli).


Question: When did variation first appear?

Leonard Euler

The famous scientist gave the gifted teenager mathematical articles to study, while allowing him to come to his house on Saturdays afternoons to clarify difficult places [6].


Question: When did variation first appear?

Leonard Euler

On October 20, 1720, 13-year-old Leonard Euler became a student at the Faculty of Arts at the University of Basel.


Question: When did variation first appear?

Leonard Euler

But his love for math put Leonard on a different path.


Question: When did variation first appear?

Leonard Euler

While visiting his teacher's house, Euler met and became friends with his sons Daniil and Nikolai, who, according to family tradition, also studied mathematics in depth.


Question: When did variation first appear?

Leonard Euler

In 1723, Euler received (according to the custom at the University of Basel) his first award (primam lauream) [6].


Question: When did variation first appear?

Leonard Euler

On July 8, 1724, 17-year-old Leonard Euler gave a speech in Latin comparing the philosophical views of Descartes and Newton and was awarded a Master of Arts degree [5].


Question: When did variation first appear?

Leonard Euler

Over the next two years, young Euler wrote several scientific papers.


Question: When did variation first appear?

Leonard Euler

One of them, the “Dissertation in Physics on Sound”, was submitted to the competition to replace the unexpectedly vacant post of Professor of Physics at the University of Basel (1725).


Question: When did variation first appear?

Leonard Euler

But, despite the positive review, 19-year-old Euler was considered too young to be included in the list of candidates for the professor's department.


Question: When did variation first appear?

Leonard Euler

It should be noted that the number of scientific vacancies in Switzerland was very small [5].


Question: When did variation first appear?

Leonard Euler

Therefore, brothers Daniil and Nikolai Bernoulli left for Russia, where the Academy of Sciences was being organized; they promised to take care of a position for Euler there [8].


Question: When did variation first appear?

Leonard Euler

At the beginning of winter 1726—1727. Euler received news from St. Petersburg: on the recommendation of the Bernoulli brothers, he was invited to the position of adjunct (assistant professor) in the Department of Physiology (this department was occupied by D. Bernoulli) with an annual salary of 200 rubles (Euler's letter to the President of the Academy L. L. Blumentrost dated November 9, 1726 with gratitude for admission to the Academy).


Question: When did variation first appear?

Leonard Euler

Since Johann Bernoulli was a famous doctor, in Russia it was believed that Leonard Euler, as his best student, was also a doctor.


Question: When did variation first appear?

Leonard Euler

However, Euler postponed his departure from Basel until spring, devoting the remaining months to a serious study of medical sciences [6], the deep knowledge of which he later amazed his contemporaries.


Question: When did variation first appear?

Leonard Euler

Finally, on April 5, 1727, Euler left Switzerland forever [5], although he retained Swiss (Basel) citizenship until the end of his life [9].


Question: When did variation first appear?

Leonard Euler

Russia (1727—1741)


Question: When did variation first appear?

Leonard Euler

Peter I approved the design of the St. Petersburg Academy.


Question: When did variation first appear?

Leonard Euler

The Senate issued a decree on the creation of the Academy.


Question: When did variation first appear?

Leonard Euler

Of the 22 professors and adjuncts invited in the early years, there were 8 mathematicians who also studied mechanics, physics, astronomy, cartography, shipbuilding theory, and the service of weights and measures [5].


Question: When did variation first appear?

Leonard Euler

Euler (whose route from Basel lay through Lübeck, Revel and Kronstadt) arrived in St. Petersburg on May 24, 1727; Empress Catherine I, patron saint of the Academy, died a few days earlier, and scientists were discouraged and confused.


Question: When did variation first appear?

Leonard Euler

However, Euler was helped to settle in a new place by fellow Basel residents: academicians Daniil Bernoulli and Jacob Herman; the latter, who was a professor in the Department of Higher Mathematics, was given to a young scientist a distant relative and provided him with all kinds of protection.


Question: When did variation first appear?

Leonard Euler

Euler was made an adjunct in higher mathematics (and not physiology, as originally planned), although he conducted research in the field of hydrodynamics of biological fluids in St. Petersburg, a salary of 300 rubles per year and provided a state apartment [6].


Question: When did variation first appear?

Leonard Euler

To everyone's surprise, he began to speak Russian fluently the following year after his arrival [10].


Question: When did variation first appear?

Leonard Euler

In 1728, the publication of the first Russian scientific journal “Comments of the St. Petersburg Academy of Sciences” (in Latin) began.


Question: When did variation first appear?

Leonard Euler

The second volume already contained three articles by Euler, and in subsequent years, almost every issue of the academic yearbook included several of his new works.


Question: When did variation first appear?

Leonard Euler

In total, more than 400 articles by Euler were published in this edition [5].


Question: When did variation first appear?

Leonard Euler

In September 1730, the term of contracts concluded with academicians Y.


Question: When did variation first appear?

Leonard Euler

Herman and G. B.


Question: When did variation first appear?

Leonard Euler

Bilfinger (the latter was a professor at the Department of Experimental and Theoretical Physics).


Question: When did variation first appear?

Leonard Euler

Their departments were headed by Daniil Bernoulli and Leonard Euler, respectively; the latter received an increase in salary to 400 rubles, and on January 22, 1731 - the official position of professor.


Question: When did variation first appear?

Leonard Euler

Two years later (1733), Daniil Bernoulli returned to Switzerland, and Euler, leaving the Department of Physics, took up its department, becoming an academician and professor of higher mathematics with a salary of 600 rubles (however, Daniil Bernoulli received twice as much) [6] [5].


Question: When did variation first appear?

Leonard Euler

On December 27, 1733, 26-year-old Leonard Euler married his age Katharina (German: Katharina Gsell), daughter of academic painter Georg Gsel (St. Petersburg Swiss) [6].


Question: When did variation first appear?

Leonard Euler

The newlyweds bought a house on the Neva embankment, where they settled.


Question: When did variation first appear?

Leonard Euler

Euler's family had 13 children, but 3 sons and 2 daughters survived [11].


Question: When did variation first appear?

Leonard Euler

The young professor had a lot of work: cartography, all kinds of examinations, consultations for shipbuilders and gunners, drawing up training manuals, designing fire pumps, etc. He was even required to compile horoscopes, which Euler redirected the order with all possible tact to a full-time astronomer.


Question: When did variation first appear?

Leonard Euler

A. S. Pushkin cites a romantic story: Euler allegedly compiled a horoscope for the newborn John Antonovich (1740), but the result frightened him so much that he did not show it to anyone and only after the death of the unfortunate prince told Count K. G. Razumovsky [12].


Question: When did variation first appear?

Leonard Euler

The reliability of this historical anecdote is extremely doubtful.


Question: When did variation first appear?

Leonard Euler

During the first period of his stay in Russia, he wrote more than 90 major scientific papers.


Question: When did variation first appear?

Leonard Euler

A significant part of the academic Notes is filled with Euler's works.


Question: When did variation first appear?

Leonard Euler

He made presentations at scientific seminars, gave public lectures, and participated in the implementation of various technical orders from government departments [14].


Question: When did variation first appear?

Leonard Euler

During the 1730s, Euler led the mapping of the Russian Empire, which (after Euler's departure, in 1745) ended with the publication of an atlas of the country's territory [15].


Question: When did variation first appear?

Leonard Euler

As N. AND. Fuss, in 1735, the Academy was tasked to perform an urgent and very cumbersome mathematical calculation, with a group of academicians asking for three months, and Euler undertook to complete the work in 3 days — and coped on his own; however, the overexertion did not go unnoticed: he fell ill and lost his sight in his right eye.


Question: When did variation first appear?

Leonard Euler

However, Euler himself, in one of his letters, attributed the loss of an eye to his mapping work in the Geographic Department at the Academy [6].


Question: When did variation first appear?

Leonard Euler

The two-volume work “Mechanics, or the Science of Movement, Presented Analytically”, published in 1736, brought Euler pan-European fame.


Question: When did variation first appear?

Leonard Euler

In this monograph, Euler successfully applied the methods of mathematical analysis to the general solution of problems of movement in the void and in a resisting environment [7].


Question: When did variation first appear?

Leonard Euler

One of the most important tasks of the Academy was the training of domestic personnel, for which a university and a gymnasium were created at the Academy.


Question: When did variation first appear?

Leonard Euler

Due to the acute shortage of textbooks in Russian, the Academy asked its members to compile such manuals.


Question: When did variation first appear?

Leonard Euler

Euler compiled a very solid Manual to Arithmetic in German, which was immediately translated into Russian and served as an initial textbook for more than one year.


Question: When did variation first appear?

Leonard Euler

The first part was translated in 1740 by Vasily Adodurov, the first Russian adjunct of the Academy, a student of Euler [5].


Question: When did variation first appear?

Leonard Euler

The situation deteriorated when Empress Anna Ioannovna died in 1740, and John VI was declared emperor.


Question: When did variation first appear?

Leonard Euler

“Something dangerous was foreseen,” Euler later wrote in his autobiography. “After the death of the glorious Empress Anna, with the regency that followed... the situation began to seem uncertain.”


Question: When did variation first appear?

Leonard Euler

In fact, the St. Petersburg Academy finally fell into disrepair in Anna Leopoldovna's regency [14].


Question: When did variation first appear?

Leonard Euler

Euler began to consider the option of returning to his homeland or moving to another country.


Question: When did variation first appear?

Leonard Euler

In the end, he accepted the offer of King Frederick of Prussia, who invited him on very favorable terms to the Berlin Academy, to the post of director of its Mathematical Department.


Question: When did variation first appear?

Leonard Euler

The Academy was created on the basis of the Prussian Royal Society, founded by Leibniz, but in those years was in a depressing state.


Question: When did variation first appear?

Leonard Euler

Prussia (1741-1766)


Question: When did variation first appear?

Leonard Euler

Euler submitted his resignation to the leadership of the St. Petersburg Academy [16]:


Question: When did variation first appear?

Leonard Euler

For the sake of this, I am forced, both for poor health and other circumstances, to look for a pleasant climate and accept from His Royal Majesty of Prussia the call made to me.


Question: When did variation first appear?

Leonard Euler

For the sake of that, I ask the Imperial Academy of Sciences to graciously dismiss me and provide me with the necessary pasha for my and my family passage<!


Question: When did variation first appear?

Leonard Euler

—so in the source—>port.


Question: When did variation first appear?

Leonard Euler

On May 29, 1741, permission from the Academy was obtained [17].


Question: When did variation first appear?

Leonard Euler

Euler was “released” and approved as an honorary member of the Academy with a salary of 200 rubles.


Question: When did variation first appear?

Leonard Euler

In June 1741, 34-year-old Leonard Euler arrived in Berlin with his wife, two sons and four nephews.


Question: When did variation first appear?

Leonard Euler

He spent 25 years there and published about 260 works [18].


Question: When did variation first appear?

Leonard Euler

At first, Euler was received kindly in Berlin, even invited to court balls [17].


Question: When did variation first appear?

Leonard Euler

The Marquis Condorcet recalled that shortly after moving to Berlin, Euler was invited to the court ball.


Question: When did variation first appear?

Leonard Euler

When asked by the Queen Mother why he was so laconic, Euler replied: “I come from a country where whoever talks is hung” [19].


Question: When did variation first appear?

Leonard Euler

Euler had a lot of work to do.


Question: When did variation first appear?

Leonard Euler

In addition to mathematical research, he led the observatory [18], was engaged in many practical activities, including issuing calendars (the main source of income for the Academy [18]), minting Prussian coins, laying a new water supply system, organizing pensions and lotteries [20].


Question: When did variation first appear?

Leonard Euler

In 1742, a four-volume collection of works by Johann Bernoulli was published.


Question: When did variation first appear?

Leonard Euler

Sending it from Basel to Euler to Berlin, the old scientist wrote to his student: “I devoted myself to my childhood in higher mathematics. You, my friend, will continue to grow into adulthood.”


Question: When did variation first appear?

Leonard Euler

In the Berlin period, one after another, Euler's works were published: “Introduction to the Analysis of Infinitesimal” (1748), “Marine Science” (1749), “Theory of the Motion of the Moon” (1753), “Manual on Differential Calculus” (Latin: Institutions calculi differentialis, 1755).


Question: When did variation first appear?

Leonard Euler

Numerous articles on specific issues are published in the publications of the Berlin and St. Petersburg Academies.


Question: When did variation first appear?

Leonard Euler

In 1744, Euler discovered the calculus of variations.


Question: When did variation first appear?

Leonard Euler

His works use well-thought-out terminology and mathematical symbolism, which have largely survived to this day, and the presentation is brought to the level of practical algorithms.


Question: When did variation first appear?

Leonard Euler

All the years of his stay in Germany, Euler maintained contact with Russia.


Question: When did variation first appear?

Leonard Euler

Euler participated in the publications of the St. Petersburg Academy, purchased books and tools for it, and edited the mathematical departments of Russian magazines.


Question: When did variation first appear?

Leonard Euler

Young Russian scientists sent for internships lived in his apartment, on full board, for years.


Question: When did variation first appear?

Leonard Euler

It is known about Euler's lively correspondence with M. IN. Lomonosov, in whose work he highly appreciated the “happy combination of theory and experiment” [C 2].


Question: When did variation first appear?

Leonard Euler

In 1747, he gave a favorable review to the President of the Academy of Sciences, Count K. G. Razumovsky about Lomonosov's articles on physics and chemistry, stating [C 3]:


Question: When did variation first appear?

Leonard Euler

All these dissertations are not only good, but also very excellent, because he [Lomonosov] writes about physical and chemical matters that are very necessary, which to this day the wittiest did not know and could not interpret people that he did with such success that I am quite sure of the validity of his explanations.


Question: When did variation first appear?

Leonard Euler

In this case, g. Lomonosov must be given justice, which has an excellent talent for explaining physical and chemical phenomena.


Question: When did variation first appear?

Leonard Euler

It should be wished that other Academies would be able to produce such revelations, as G. Lomonosov said.


Question: When did variation first appear?

Leonard Euler

This high grade was not prevented even by the fact that Lomonosov did not write mathematical works and did not know higher mathematics [C 4] [C 5].


Question: When did variation first appear?

Leonard Euler

His mother notified Euler of his father's death in Switzerland (1745); she soon moved in with Euler (died in 1761).


Question: When did variation first appear?

Leonard Euler

In 1753, Euler bought an estate in Charlottenburg (a suburb of Berlin) with a garden and a plot where he accommodated his large family [20].


Question: When did variation first appear?

Leonard Euler

According to contemporaries, Euler remained a modest, cheerful, extremely sympathetic person all his life, always ready to help another.


Question: When did variation first appear?

Leonard Euler

However, relations with the king did not work out: Frederick found the new mathematician unbearably boring, completely unsecular, and treated him with disdain.


Question: When did variation first appear?

Leonard Euler

Maupertuis, President of the Berlin Academy of Sciences and friend of Euler, died in 1759.


Question: When did variation first appear?

Leonard Euler

King Frederick II offered D'Alembert the post of president of the Academy, but he refused.


Question: When did variation first appear?

Leonard Euler

Friedrich, who did not like Euler, still entrusted him with the leadership of the Academy, but without the title of president [5].


Question: When did variation first appear?

Leonard Euler

During the Seven Years' War (1756—1763), Russian artillery destroyed Euler's house; upon learning about this, Field Marshal Saltykov immediately compensated for the losses, and later Empress Elizabeth sent another 4000 rubles [16].


Question: When did variation first appear?

Leonard Euler

In 1765, the Theory of the Motion of Solids was published, and a year later, Elements of the Calculus of Variations.


Question: When did variation first appear?

Leonard Euler

This is where the name of the new section of mathematics created by Euler and Lagrange first appeared.


Question: When did variation first appear?

Leonard Euler

In 1762, Catherine II ascended the Russian throne, who pursued a policy of enlightened absolutism.


Question: When did variation first appear?

Leonard Euler

Being well aware of the importance of science both for the progress of the state and for its own prestige, she carried out a number of important, favorable for science transformations in the system of public education and culture.


Question: When did variation first appear?

Leonard Euler

The Empress offered Euler the management of the mathematical class, the title of conference secretary of the Academy and a salary of 1800 rubles per year.


Question: When did variation first appear?

Leonard Euler

“And if you don't like it,” the letter said in the letter to her representative, “he favors to inform her terms, so long as he does not hesitate to come to St. Petersburg.”


Question: When did variation first appear?

Leonard Euler

Euler responded with his terms [21]:


Question: When did variation first appear?

Leonard Euler

a salary of 3000 rubles per year and the post of Vice-President of the Academy;


Question: When did variation first appear?

Leonard Euler

an apartment free from soldier's standing [22];


Question: When did variation first appear?

Leonard Euler

paid positions for his three sons, including the post of Secretary of the Academy for the Elder.


Question: When did variation first appear?

Leonard Euler

All these conditions have been accepted.


Question: When did variation first appear?

Leonard Euler

On January 6, 1766, Catherine informed Count Vorontsov [16]:


Question: When did variation first appear?

Leonard Euler

A letter to you g. Euler gave me great pleasure, because I learn from him that he wanted to join my service again.


Question: When did variation first appear?

Leonard Euler

Of course, I find him absolutely worthy of the desired title of Vice-President of the Academy of Sciences, but for this purpose some measures should be taken before I establish this title — I say I will establish it, because until now it didn't exist.


Question: When did variation first appear?

Leonard Euler

In the current state of affairs, there is no money for a salary of 3000 rubles, but for a person with such advantages as St. Euler, I will add to my academic salary from state revenue, which together will amount to the required 3,000 rubles... I am sure that my Academy will revive from the ashes from such an important acquisition, and in advance I congratulate myself on returning a great man to Russia.


Question: When did variation first appear?

Leonard Euler

Later, Euler put forward a number of other conditions (an annual pension of 1000 rubles for his wife after his death, compensation for travel costs, a place for a medical son and a rank for Euler himself).


Question: When did variation first appear?

Leonard Euler

Catherine also satisfied these conditions of Euler, with the exception of the requirement for the rank, joking: “I would give him a rank whenever he wants... (the draft letter in French is crossed out — collegiate adviser) if I would not be afraid that this rank would equate him with the many people who are not worth g. Euler. Indeed, his fame is better than rank to give him due respect” [22].


Question: When did variation first appear?

Leonard Euler

Euler filed a petition for dismissal from service to the king, but received no response.


Question: When did variation first appear?

Leonard Euler

He filed again, but Friedrich did not even want to discuss the issue of his departure.


Question: When did variation first appear?

Leonard Euler

Euler received decisive support from the persistent petitions of the Russian mission on behalf of the Empress [23].


Question: When did variation first appear?

Leonard Euler

On May 2, 1766, Friedrich finally allowed the great scientist to leave Prussia, but could not resist the evil jokes against Euler in his correspondence (for example, on July 25, he wrote to D'Alembert: “Mr. Euler , who madly loves the Ursa Major and Ursa Minor, approached the north for greater convenience in observing them”) [6].


Question: When did variation first appear?

Leonard Euler

However, Christoph, the youngest son of Euler, who served as a lieutenant colonel of artillery (German: Oberstleutnant), the king flatly refused to release from the army [7]; later, thanks to the intercession of Catherine II, he he was still able to join his father and rose to the rank of lieutenant general in the Russian army [24].


Question: When did variation first appear?

Leonard Euler

In the summer of 1766, Euler returned to Russia — now forever.


Question: When did variation first appear?

Leonard Euler

Russia Again (1766—1783)


Question: When did variation first appear?

Leonard Euler

60-year-old Euler, his family and household members (18 people in total) arrived in the Russian capital [6].


Question: When did variation first appear?

Leonard Euler

Immediately upon arrival, he was received by the Empress.


Question: When did variation first appear?

Leonard Euler

Catherine II met him as an August person and showered him with mercies: she granted 8,000 rubles for the purchase of a house on Vasilyevsky Island and for the purchase of furnishings, provided for the first time one of its cooks and instructed to prepare ideas on the reorganization of the Academy [25].


Question: When did variation first appear?

Leonard Euler

Unfortunately, after returning to St. Petersburg, Euler developed a cataract in his left eye — he stopped seeing.


Question: When did variation first appear?

Leonard Euler

Probably for this reason, he did not receive the promised post of vice-president of the Academy (which did not prevent Euler and his descendants from participating in the management of the Academy for almost a hundred years [20]).


Question: When did variation first appear?

Leonard Euler

However, blindness did not affect the scientist's performance, he only noticed that he would now be less distracted from studying mathematics [5].


Question: When did variation first appear?

Leonard Euler

Before finding a secretary, Euler dictated his work to a tailor boy who wrote everything down in German.


Question: When did variation first appear?

Leonard Euler

The number of works he published even increased; during his second stay in Russia, Euler dictated more than 400 articles and 10 books, which is more than half of his creative heritage [20].


Question: When did variation first appear?

Leonard Euler

In 1768-1770, a two-volume classical monograph “Universal Arithmetic” was published (also published under the titles “Beginnings of Algebra” and “The Complete Course of Algebra”).


Question: When did variation first appear?

Leonard Euler

Initially, this work was published in Russian (1768—1769), and the German edition was published two years later [26].


Question: When did variation first appear?

Leonard Euler

The book was translated into many languages and reprinted about 30 times (three times in Russian).


Question: When did variation first appear?

Leonard Euler

All subsequent algebra textbooks were created under the strongest influence of Euler's book [27].


Question: When did variation first appear?

Leonard Euler

In the same years, a three-volume book Dioptrica (Latin: Dioptrica, 1769—1771) on lens systems and the fundamental Integral Calculus (Latin: Institutiones calculi integralis, 1768—1770) were published, also in 3 volumes [7].


Question: When did variation first appear?

Leonard Euler

Eulerian's Letters on Various Physical and Philosophical Matters Written to a German Princess... (1768) gained immense popularity in the XVIII century, and partly in the XIX century, which withstood more than 40 publications in 10 languages (including 4 editions in Russian).


Question: When did variation first appear?

Leonard Euler

It was a popular science encyclopedia of wide coverage, written brightly and publicly available [16].


Question: When did variation first appear?

Leonard Euler

In 1771, two major events took place in Euler's life.


Question: When did variation first appear?

Leonard Euler

In May, a big fire broke out in St. Petersburg, which destroyed hundreds of buildings, including Euler's house and almost all of Euler's property.


Question: When did variation first appear?

Leonard Euler

The scientist himself was hardly saved.


Question: When did variation first appear?

Leonard Euler

All manuscripts were protected from fire; only part of the New Theory of the Motion of the Moon burned down, but it was quickly restored with the help of Euler himself, who preserved a phenomenal memory until a very old age [28].


Question: When did variation first appear?

Leonard Euler

Euler had to temporarily move to another house.


Question: When did variation first appear?

Leonard Euler

The second event: in September of the same year, at the special invitation of the Empress, the famous German ophthalmologist Baron Wentzel arrived in St. Petersburg for the treatment of Euler.


Question: When did variation first appear?

Leonard Euler

After the examination, he agreed to undergo surgery for Euler and removed the cataract from his left eye.


Question: When did variation first appear?

Leonard Euler

Euler started to see again.


Question: When did variation first appear?

Leonard Euler

The doctor ordered to protect the eye from bright light, not to write, not to read — only to gradually get used to the new state.


Question: When did variation first appear?

Leonard Euler

However, a few days after the operation, Euler removed the bandage and soon lost his sight again.


Question: When did variation first appear?

Leonard Euler

This time it is final [28].


Question: When did variation first appear?

Leonard Euler

1772: “A new theory of the motion of the moon.”


Question: When did variation first appear?

Leonard Euler

Euler finally completed his many years of work, approximately solving the problem of three bodies.


Question: When did variation first appear?

Leonard Euler

In 1773, on the recommendation of Daniil Bernoulli, Bernoulli's student, Nikolaus Fuss, came to St. Petersburg from Basel.


Question: When did variation first appear?

Leonard Euler

This was a great success for Euler.


Question: When did variation first appear?

Leonard Euler

Fuss, a gifted mathematician, took care of Euler's mathematical works immediately after his arrival.


Question: When did variation first appear?

Leonard Euler

Fuss soon married Euler's granddaughter.


Question: When did variation first appear?

Leonard Euler

In the next ten years, until his death, Euler mainly dictated his work to him, although he sometimes used the “eyes of his eldest son” and his other students [9].


Question: When did variation first appear?

Leonard Euler

In the same year 1773, Euler's wife died, with whom he lived for almost 40 years.


Question: When did variation first appear?

Leonard Euler

The death of his wife was a painful blow to a scientist who was genuinely attached to his family.


Question: When did variation first appear?

Leonard Euler

Euler soon married Salomee-Abigail, the half-sister of his late wife [5].


Question: When did variation first appear?

Leonard Euler

In 1779, Universal Spherical Trigonometry was published, which is the first complete presentation of the entire spherical trigonometry system [27].


Question: When did variation first appear?

Leonard Euler

Euler worked actively until the last days.


Question: When did variation first appear?

Leonard Euler

In September 1783, the 76-year-old scientist began to feel headaches and weakness.


Question: When did variation first appear?

Leonard Euler

On September 7 (18), in the afternoon spent with his family, talking with Academician A. AND. With a lexel about the newly discovered planet Uranus and its orbit, he suddenly felt bad.


Question: When did variation first appear?

Leonard Euler

Euler had time to say, “I'm dying,” and he lost consciousness.


Question: When did variation first appear?

Leonard Euler

A few hours later, without regaining consciousness, he died of a cerebral hemorrhage [25].


Question: When did variation first appear?

Leonard Euler

“He stopped calculating and living,” Condorcet said at a mourning meeting of the Paris Academy of Sciences (French: Il cessa de calculer et de vivre).


Question: When did variation first appear?

Leonard Euler

He was buried at the Smolensk Lutheran Cemetery in St. Petersburg.


Question: When did variation first appear?

Leonard Euler

The inscription on the monument in German read: “Here are the remains of the famous Leonard Euler, a sage and righteous man in the world. Born in Basel on April 4, 1707, died on September 7, 1783” [29].


Question: When did variation first appear?

Leonard Euler

After Euler's death, his grave was lost and was found, in an abandoned state, only in 1830 [29].


Question: When did variation first appear?

Leonard Euler

In 1837, the Academy of Sciences replaced this tombstone with a new granite tombstone (which still exists today) with the inscription in Latin “Leonard Euler — Petersburg Academy” (Latin: Leonhardo Eulero — Academia Petropolitana) [29].


Question: When did variation first appear?

Leonard Euler

During the celebration of the 250th anniversary of Euler (1957), the ashes of the great mathematician were transferred to the Necropolis of the XVIII Century at the Lazarevskoye Cemetery of the Alexander Nevsky Lavra, where it is located near the grave of M. IN. Lomonosov Moscow State University [20].


Question: When did variation first appear?

Leonard Euler

Contribution to science


Question: When did variation first appear?

Leonard Euler

Euler left the most important works on a wide variety of branches of mathematics, mechanics, physics, astronomy and a number of applied sciences [20].


Question: When did variation first appear?

Leonard Euler

Euler's knowledge was encyclopedic; in addition to mathematics, he studied botany, medicine, chemistry, music theory, many European and ancient languages in depth.


Question: When did variation first appear?

Leonard Euler

Euler willingly participated in scientific discussions, of which the most famous were [27]:


Question: When did variation first appear?

Leonard Euler

a dispute about a string;


Question: When did variation first appear?

Leonard Euler

a dispute with D'Alembert about the properties of the complex logarithm;


Question: When did variation first appear?

Leonard Euler

An argument with John Dollond about whether it is possible to create an achromatic lens.


Question: When did variation first appear?

Leonard Euler

In all these cases, Euler's position is supported by modern science.


Question: When did variation first appear?

Leonard Euler

Math


Question: When did variation first appear?

Leonard Euler

From the point of view of mathematics, the 18th century is the age of Euler [20].


Question: When did variation first appear?

Leonard Euler

If before him, advances in mathematics were scattered and not always consistent, Euler for the first time linked analysis, algebra, geometry, trigonometry, number theory and other disciplines into a single system, while adding a lot of his own discoveries [3].


Question: When did variation first appear?

Leonard Euler

A significant part of mathematics has been taught since then “according to Euler” almost unchanged [20].


Question: When did variation first appear?

Leonard Euler

Thanks to Euler, mathematics included the general theory of series, the fundamental “Euler formula” in the theory of complex numbers, the comparison operation on an integer modulo, the complete theory of continuous fractions, analytical the foundation of mechanics, numerous methods of integrating and solving differential equations, the number e, the designation i for an imaginary unit, a number of special functions and much more [20].


Question: When did variation first appear?

Leonard Euler

In fact, it was Euler who created several new mathematical disciplines — number theory, calculus of variations, theory of complex functions, differential geometry of surfaces; he laid the foundations theories of special functions.


Question: When did variation first appear?

Leonard Euler

Other areas of his work include diophantic analysis, mathematical physics, statistics, etc. [20]


Question: When did variation first appear?

Leonard Euler

Science historian Clifford Trusdell wrote: “Euler was the first scientist in Western civilization to write about mathematics in a clear and easy to read language” [30].


Question: When did variation first appear?

Leonard Euler

Biographers note that Euler was a virtuoso algorithmist.


Question: When did variation first appear?

Leonard Euler

He always tried to bring his discoveries to the level of specific computational methods and was an unrivaled master of numerical calculations himself [25].


Question: When did variation first appear?

Leonard Euler

WELL. Condorcet said that one day, two students, performing independently complex astronomical calculations, received slightly different results in the 50th sign and turned to Euler for help.


Question: When did variation first appear?

Leonard Euler

Euler did the same calculations in his mind and gave the correct result [25].


Question: When did variation first appear?

Leonard Euler

Number theory


Question: When did variation first appear?

Leonard Euler

P. L. Chebyshev wrote: “Euler laid the foundation for all the research that make up the general theory of numbers” [31].


Question: When did variation first appear?

Leonard Euler

Most 18th-century mathematicians developed analysis, but Euler carried his passion for ancient arithmetic throughout his life.


Question: When did variation first appear?

Leonard Euler

Thanks to his works, interest in number theory was revived by the end of the century [27].


Question: When did variation first appear?

Leonard Euler

Euler continued the research of Fermat, who previously expressed (under the influence of Diophant) a number of disparate hypotheses about natural numbers.


Question: When did variation first appear?

Leonard Euler

Euler strictly proved these hypotheses, significantly generalized them and combined them into a meaningful number theory [32].


Question: When did variation first appear?

Leonard Euler

He introduced the extremely important “Euler function” into mathematics and formulated the “Euler theorem” with its help [23].


Question: When did variation first appear?

Leonard Euler

He refuted Fermat's hypothesis that all numbers of the species


Question: When did variation first appear?

Leonard Euler

F


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle F_ {n} =2^ {2^ {n}} +1}


Question: When did variation first appear?

Leonard Euler

— simple; it turned out that


Question: When did variation first appear?

Leonard Euler

F


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle F_ {5}}


Question: When did variation first appear?

Leonard Euler

is divided into 641 [3].


Question: When did variation first appear?

Leonard Euler

He proved Fermat's statement about the representation of an odd prime number as the sum of two squares [32].


Question: When did variation first appear?

Leonard Euler

He gave one of the solutions to the problem of four cubes.


Question: When did variation first appear?

Leonard Euler

Proved that the Mersenne number


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle 2^ {31} -1=2147483647}


Question: When did variation first appear?

Leonard Euler

is a prime number; for almost a hundred years (until 1867) it remained the largest known prime number [33].


Question: When did variation first appear?

Leonard Euler

Euler created the basis for the theory of comparisons and quadratic deductions, specifying the solvability criterion for the latter.


Question: When did variation first appear?

Leonard Euler

Euler introduced the concept of primitive root and hypothesized that for any prime number p there is a primitive root modulo p; he could not prove this, and later Legendre and Gauss proved the theorem.


Question: When did variation first appear?

Leonard Euler

Another hypothesis of Euler, the quadratic law of reciprocity, also proved by Gauss, was of great importance in theory [32].


Question: When did variation first appear?

Leonard Euler

Euler proved Fermat's Great Theorem for


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle n=3}


Question: When did variation first appear?

Leonard Euler

et


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle n=4}


Question: When did variation first appear?

Leonard Euler

, created a complete theory of continuous fractions, studied various classes of Diophantine equations, the theory of dividing numbers into terms [34] [3].


Question: When did variation first appear?

Leonard Euler

In the problem of the number of divisions of a natural number


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler

{\ displaystyle n}


Question: When did variation first appear?

Leonard Euler

got a formula that expresses the generating function of the number of partitions


Question: When did variation first appear?

Leonard Euler

p


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle p (n)}


Question: When did variation first appear?

Leonard Euler

through an endless piece


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

p


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

x


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

k


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

x


Question: When did variation first appear?

Leonard Euler

k


Question: When did variation first appear?

Leonard Euler

{\ displaystyle\ sum _ {n=0} ^ {\ infty} p (n) x^ {n} =\ prod _ {k=1} ^ {\ infty} {\ frac {1} {1-x^ {k}}}


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

Euler defined a zeta function, the generalization of which later received the name Riemann:


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle\ zeta (s) = {\ frac {1} {1^ {s}}} + {\ frac {1} {2^ {s}}} + {\ frac {1} {3^ {s}}} +\ ldots}


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

where


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler

{\ displaystyle\ displaystyle s}


Question: When did variation first appear?

Leonard Euler

a real number (Riemann has a complex number).


Question: When did variation first appear?

Leonard Euler

Euler deduced a decomposition for her:


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

p


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

p


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler

{\ displaystyle\ zeta (s) =\ prod _ {p} {\ frac {1} {1-p^ {-s}}}


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

where the work is taken for all prime numbers


Question: When did variation first appear?

Leonard Euler

p


Question: When did variation first appear?

Leonard Euler

{\ displaystyle\ displaystyle p}


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

Thus, he discovered that in number theory it is possible to apply methods of mathematical analysis, marking the beginning of analytical number theory [35], which is based on Euler's identity and the general method generating functions [3].


Question: When did variation first appear?

Leonard Euler

Mathematical analysis


Question: When did variation first appear?

Leonard Euler

One of Euler's main merits to science is the monograph “Introduction to the Analysis of the Infinitesimal” (1748).


Question: When did variation first appear?

Leonard Euler

In 1755, the supplemented Differential Calculus was published, and in 1768-1770, three volumes of Integral Calculus were published.


Question: When did variation first appear?

Leonard Euler

Taken together, this is a fundamental course, well illustrated with examples, with well-thought-out terminology and symbolism [7].


Question: When did variation first appear?

Leonard Euler

“It is safe to say that a good half of what is now taught in the courses of higher algebra and higher analysis is in the works of Euler” (N. N. Luzin) [23].


Question: When did variation first appear?

Leonard Euler

Euler was the first to give a systematic theory of integration and the techniques used in this process.


Question: When did variation first appear?

Leonard Euler

In particular, he is the author of the classical method of integrating rational functions by decomposing them into simple fractions and a method for solving differential equations of arbitrary order with constants coefficients [23].


Question: When did variation first appear?

Leonard Euler

He introduced double integrals for the first time [1].


Question: When did variation first appear?

Leonard Euler

Euler has always paid special attention to methods for solving differential equations, both ordinary and partial derivatives, discovering and describing important classes of integrable differential equations.


Question: When did variation first appear?

Leonard Euler

He outlined Euler's “broken method” (1768), a numerical method for solving systems of ordinary differential equations.


Question: When did variation first appear?

Leonard Euler

Along with A. K. Clairaut Euler derived the conditions for the integrability of linear differential forms from two or three variables (1739) [1].


Question: When did variation first appear?

Leonard Euler

He obtained serious results in the theory of elliptic functions, including the first addition theorems for elliptic integrals (1761) [3].


Question: When did variation first appear?

Leonard Euler

For the first time, I studied the highs and lows of functions of many variables [23].


Question: When did variation first appear?

Leonard Euler

The base of natural logarithms has been known since the days of Neper and Jacob Bernoulli, but Euler has studied this important constant so deeply that it has been named after him ever since.


Question: When did variation first appear?

Leonard Euler

Another constant he studied was the Euler-Mascheroni constant.


Question: When did variation first appear?

Leonard Euler

The modern definition of exponential, logarithmic and trigonometric functions is also his merit, as well as their symbolism and generalization for a complex case [36].


Question: When did variation first appear?

Leonard Euler

Formulas, often referred to in textbooks as the “Cauchy-Riemann conditions”, would be more correctly called the “D'Alembert-Euler conditions” [37] [3].


Question: When did variation first appear?

Leonard Euler

He shares with Lagrange the honor of the discovery of the calculus of variations by writing out the Euler-Lagrange equations for the general variational problem.


Question: When did variation first appear?

Leonard Euler

In 1744, Euler published the treatise “The Method for Finding Curved Lines...” [38], the first work on the calculus of variations [5] (among other things, it contained the first systematic presentation of the theory of elastic curves and material resistance results [1]).


Question: When did variation first appear?

Leonard Euler

Euler significantly advanced the theory of series and extended it to the complex region, while obtaining the famous Euler formula, which gives a trigonometric representation of the complex number.


Question: When did variation first appear?

Leonard Euler

The mathematical world was greatly impressed by the series first summed up by Euler, including a series of inverse squares that did not succumb to anyone before him:


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

lim


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

π


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle\ zeta (2) =\ lim _ {n\ to\ infty}\ left ({\ frac {1} {1^ {2}}} + {\ frac {1}} + {\ frac {1} {3^ {2}}} +\ cdots + {\ frac {1} {n^ {2}}}\ right) = {\ frac {\ pi ^ {2}} {6}}}


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

Using series, Euler studied transcendental functions, that is, those functions that are not expressed by an algebraic equation (for example, the integral logarithm) [7].


Question: When did variation first appear?

Leonard Euler

He discovered (1729—1730) the “Euler integrals” that now have diverse applications — special functions that have entered science as the gamma function and the Euler beta function [3].


Question: When did variation first appear?

Leonard Euler

When solving the problem of oscillations of an elastic membrane (arose in connection with determining the pitch of the timpani), Euler first introduced [39] Bessel functions for any natural index in 1764 (study by F. IN. Bessel, whose name these functions now bear, dates back to 1824) [40].


Question: When did variation first appear?

Leonard Euler

From a later point of view, Euler's actions with infinite series can not always be considered correct (the analysis was justified only half a century later), but phenomenal mathematical intuition almost always gave him the right result.


Question: When did variation first appear?

Leonard Euler

At the same time, in many important respects, his understanding was ahead of its time — for example, his generalized understanding of the sum of divergent series and operations with them served as the basis of the modern theory of these series developed in the late 19th — early 20th centuries [41].


Question: When did variation first appear?

Leonard Euler

Geometry


Question: When did variation first appear?

Leonard Euler

In elementary geometry, Euler found several facts that Euclid did not note [42]:


Question: When did variation first appear?

Leonard Euler

three triangle heights intersect at one point (orthocenter);


Question: When did variation first appear?

Leonard Euler

in the triangle, the orthocenter H, the center of the described circle U and the center of gravity S (aka centroid) lie on the same line - the “Euler straight” e (see the figure on the right).


Question: When did variation first appear?

Leonard Euler

Clarification.


Question: When did variation first appear?

Leonard Euler

On the “Euler line” also lies the center of the Euler circle (the center of the circle of nine points) (see another figure);


Question: When did variation first appear?

Leonard Euler

the bases of the three heights of an arbitrary triangle, the middle of its three sides and the middle of the three segments connecting its vertices with the orthocenter lie all on one circle (the “Euler circle”);


Question: When did variation first appear?

Leonard Euler

the number of vertices (B), faces (D) and edges (P) for any convex polyhedron is related by a simple formula:


Question: When did variation first appear?

Leonard Euler

B


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

P


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

Γ


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle\ mathrm {B} -\ mathrm {P} +\ Gamma\; =\; 2}


Question: When did variation first appear?

Leonard Euler

(in the modern interpretation, the number 2 here acts [43] as the most important topological invariant of a convex polyhedron — its Eulerian characteristic, and this Euler result obtained in 1758, marked the beginning of the accumulation of topology facts [3]).


Question: When did variation first appear?

Leonard Euler

The second volume of Introduction to Infinitesimal Analysis (1748) is the world's first textbook on analytical geometry and the basics of differential geometry.


Question: When did variation first appear?

Leonard Euler

Euler gave a classification of algebraic curves of the 3rd and 4th orders, as well as surfaces of the second order [23].


Question: When did variation first appear?

Leonard Euler

The term “affine transformations” was first introduced in this book along with the theory of such transformations.


Question: When did variation first appear?

Leonard Euler

In 1732, Euler derived a general equation of geodetic lines on the surface [27].


Question: When did variation first appear?

Leonard Euler

In 1760, fundamental “Studies on the curvature of surfaces” were published.


Question: When did variation first appear?

Leonard Euler

Euler found that at each point of the smooth surface there are two normal sections with minimum and maximum radii of curvature and that their planes are mutually perpendicular.


Question: When did variation first appear?

Leonard Euler

He derived a formula for the relationship between the curvature of the surface section and the main curvatures [27].


Question: When did variation first appear?

Leonard Euler

In 1771, Euler published his work “On Bodies whose Surface Can Be Turned to a Plane”.


Question: When did variation first appear?

Leonard Euler

In this paper, the concept of an unfolding surface is introduced, that is, a surface that can be superimposed on a plane without folds and breaks.


Question: When did variation first appear?

Leonard Euler

Euler, however, gives here a very general theory of the metric, on which the entire internal geometry of the surface depends.


Question: When did variation first appear?

Leonard Euler

Later, the study of the metric became his main tool in surface theory [44].


Question: When did variation first appear?

Leonard Euler

In connection with the tasks of cartography, Euler deeply studied conformal maps, using complex analysis tools for this for the first time [27].


Question: When did variation first appear?

Leonard Euler

Combinatorics


Question: When did variation first appear?

Leonard Euler

Euler's Magic Square


Question: When did variation first appear?

Leonard Euler

Fifth-order Greco-Latin square


Question: When did variation first appear?

Leonard Euler

Euler paid much attention to the representation of natural numbers in the form of sums of a special type and formulated a number of theorems for calculating the number of partitions [34].


Question: When did variation first appear?

Leonard Euler

When solving combinatorial problems, he deeply studied the properties of combinations and permutations, and introduced Euler's numbers [45].


Question: When did variation first appear?

Leonard Euler

Euler studied algorithms for constructing magic squares by bypassing a chess horse [46].


Question: When did variation first appear?

Leonard Euler

Two of his works (1776, 1779) laid the foundation for the general theory of Latin and Greco-Latin squares, the great practical value of which was revealed after Ronald Fisher created planning methods experiment, as well as in the theory of error correcting codes [47].


Question: When did variation first appear?

Leonard Euler

Other areas of mathematics


Question: When did variation first appear?

Leonard Euler

Euler's 1736 article “Solving the Problem Related to the Geometry of Position” [48] marked the beginning of graph theory as a mathematical discipline.


Question: When did variation first appear?

Leonard Euler

The reason for the study was the problem of seven bridges of Königsberg: is it possible to walk each bridge once and return to its original place?


Question: When did variation first appear?

Leonard Euler

Euler formalized it, reducing it to the problem of the existence in the graph (the peaks of which correspond to parts of the city separated by the channels of the Pregolya River, and the edges correspond to bridges) of a cyclic route passing along each the rib exactly once (in modern terminology, the Eulerian cycle).


Question: When did variation first appear?

Leonard Euler

Solving the last problem, Euler showed that in order to have an Euler cycle in a graph, it is necessary that each vertex has its degree (the number of edges coming out of the top) to be even (and in the problem of Königsberg bridges this is not so: degrees are 3, 3, 3 and 5) [49].


Question: When did variation first appear?

Leonard Euler

Euler made a significant contribution to the theory and methods of approximate calculations [50].


Question: When did variation first appear?

Leonard Euler

For the first time, he used analytical methods in cartography [15].


Question: When did variation first appear?

Leonard Euler

He proposed a convenient method for graphically depicting relationships and operations on sets, called the “Euler Circles” (or Euler-Venn) [51].


Question: When did variation first appear?

Leonard Euler

Mechanics and physics


Question: When did variation first appear?

Leonard Euler

Many of Euler's works are devoted to various branches of mechanics and physics.


Question: When did variation first appear?

Leonard Euler

Regarding the key role of Euler at the stage of designing mechanics in exact science, K. Trusdell wrote: “Mechanics, as it is taught to engineers and mathematicians today, is largely his creation” [52].


Question: When did variation first appear?

Leonard Euler

Theoretical mechanics


Question: When did variation first appear?

Leonard Euler

In 1736, Euler's two-volume treatise Mechanics, or the Science of Motion, in an Analytical Presentation was published [53], which marked a new stage in the development of this ancient science and devoted to the dynamics of the material point .


Question: When did variation first appear?

Leonard Euler

Unlike the founders of this section of dynamics, Galileo and Newton, who used geometric methods, 29-year-old Euler proposed a regular and uniform analytical method for solving various problems of dynamics: compilation of differential equations of motion of a material object and their subsequent integration under given initial conditions [54].


Question: When did variation first appear?

Leonard Euler

The first volume of the treatise examines the movement of a free material point, in the second volume — a non-free point, and studies motion both in emptiness and in a resisting medium.


Question: When did variation first appear?

Leonard Euler

Ballistics problems and pendulum theory are considered separately.


Question: When did variation first appear?

Leonard Euler

Here, Euler first writes down the differential equation of rectilinear motion of a point, and for the general case of its curvilinear motion introduces natural equations of motion — equations in projections on the axis accompanying trihedron.


Question: When did variation first appear?

Leonard Euler

In many specific problems, he brings the integration of equations of motion to the end; in cases of movement of a point without resistance, he systematically uses the first integral of the equations of motion — the integral energy [55].


Question: When did variation first appear?

Leonard Euler

The second volume, in connection with the problem of movement of a point along an arbitrarily curved surface, describes the differential geometry of surfaces created by Euler [56].


Question: When did variation first appear?

Leonard Euler

Euler returned to the dynamics of the material point later.


Question: When did variation first appear?

Leonard Euler

In 1746, studying the movement of a material point along a moving surface, he came (simultaneously with D. Bernoulli and P. Darcy) to the theorem on changing the moment of the amount of motion.


Question: When did variation first appear?

Leonard Euler

In 1765, Euler, using K. McLaren, the idea of decomposing velocities and forces along three fixed coordinate axes, for the first time writes down the differential equations of motion of a material point in projections on Cartesian fixed axes [54].


Question: When did variation first appear?

Leonard Euler

The last result was published by Euler in his second fundamental treatise on analytical dynamics, the book “Theory of the Motion of Solids” [57] (1765).


Question: When did variation first appear?

Leonard Euler

Its main content, however, is devoted to another branch of mechanics — the dynamics of a solid state, the founder of which Euler became.


Question: When did variation first appear?

Leonard Euler

The treatise, in particular, contains the derivation of a system of six differential equations of motion of a free solid body [55].


Question: When did variation first appear?

Leonard Euler

Of great importance for statics is the theorem set out in § 620 of the treatise on reducing a system of forces applied to a solid body to two forces.


Question: When did variation first appear?

Leonard Euler

By projecting conditions for the equality of these forces to zero on the coordinate axes, Euler for the first time obtains the equilibrium equations of a solid body under the action of an arbitrary spatial system of forces [55].


Question: When did variation first appear?

Leonard Euler

The 1765 treatise also sets out a number of fundamental results of Euler related to the kinematics of a solid state (in the 18th century, kinematics was not yet singled out as a separate section of mechanics).


Question: When did variation first appear?

Leonard Euler

Among them, we will highlight Euler's formulas for the distribution of the velocities of points of an absolutely solid body (the vector equivalent of these formulas is the Euler kinematic formula) [C 6] and Euler's kinematic equations that give expression of derivatives of Euler angles (introduced by him in 1748; in mechanics they are used to set the orientation of a solid body) through projections of angular velocity on the coordinate axis [58] [59].


Question: When did variation first appear?

Leonard Euler

In addition to this treatise, two earlier works by Euler are important for the dynamics of a solid state: “Studies on the mechanical cognition of bodies” [60] and “The rotational motion of solids around variable axis” [61], which were submitted to the Berlin Academy of Sciences in 1758, but published in its Notes later (in the same 1765 as the treatise).


Question: When did variation first appear?

Leonard Euler

In them: the theory of moments of inertia has been developed (in particular, the “Huygens-Steiner theorem” has been proved for the first time); it is established that any solid body with a fixed point has at least three axes free rotation [C 7]; dynamic Euler equations describing the dynamics of a solid body with a fixed point are obtained; an analytical solution of these equations is given in the case of equality to zero of the main moment of external forces (Euler's case) is one of three general cases of integrability in the problem of the dynamics of a heavy solid body with a fixed point [62] [63].


Question: When did variation first appear?

Leonard Euler

In the article “General formulas for the arbitrary movement of a solid body” [64] (1775) Euler formulates and proves the fundamental theorem of rotation of Euler, according to which arbitrary displacement is absolutely a solid body with a fixed point is a rotation at a certain angle around a particular axis passing through a fixed point [65].


Question: When did variation first appear?

Leonard Euler

Euler is credited with the analytical design of the principle of least action (proposed in 1744 — in a very fuzzy form — by P. L. Maupertuis), a correct understanding of the conditions for the applicability of the principle and its first proof (carried out in the same 1744 [66] for the case of one material point moving under the action of a central forces) [54].


Question: When did variation first appear?

Leonard Euler

The action here (we are talking about the so-called shortened action [67], and not the Hamilton action) in relation to the system of material points is understood as the integral


Question: When did variation first appear?

Leonard Euler

W


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

A


Question: When did variation first appear?

Leonard Euler

B


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

i


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler

m


Question: When did variation first appear?

Leonard Euler

i


Question: When did variation first appear?

Leonard Euler

v


Question: When did variation first appear?

Leonard Euler

i


Question: When did variation first appear?

Leonard Euler

d


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler

i


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

{\ displaystyle W\; =\;\ int\ limits _ {A} ^ {B}\,\ sum _ {i=1} ^ {n} m_ {_ {i}} v_ {_ {i}}\,\ mathrm {d} s_ {_ {i}}\,\,}


Question: When did variation first appear?

Leonard Euler

where


Question: When did variation first appear?

Leonard Euler

A


Question: When did variation first appear?

Leonard Euler

{\ displaystyle A}


Question: When did variation first appear?

Leonard Euler

et


Question: When did variation first appear?

Leonard Euler

B


Question: When did variation first appear?

Leonard Euler

{\ displaystyle B}


Question: When did variation first appear?

Leonard Euler

— two system configurations,


Question: When did variation first appear?

Leonard Euler

m


Question: When did variation first appear?

Leonard Euler

i


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

v


Question: When did variation first appear?

Leonard Euler

i


Question: When did variation first appear?

Leonard Euler

{\ displaystyle m_ {i},\; v_ {i}}


Question: When did variation first appear?

Leonard Euler

et


Question: When did variation first appear?

Leonard Euler

d


Question: When did variation first appear?

Leonard Euler

s


Question: When did variation first appear?

Leonard Euler

i


Question: When did variation first appear?

Leonard Euler

{\ displaystyle\ mathrm {d} s_ {i}}


Question: When did variation first appear?

Leonard Euler

— mass, algebraic velocity and trajectory arc element, respectively


Question: When did variation first appear?

Leonard Euler

i


Question: When did variation first appear?

Leonard Euler

{\ displaystyle i}


Question: When did variation first appear?

Leonard Euler

th point,


Question: When did variation first appear?

Leonard Euler

n


Question: When did variation first appear?

Leonard Euler

{\ displaystyle n}


Question: When did variation first appear?

Leonard Euler

— number of points [68].


Question: When did variation first appear?

Leonard Euler

As a result, the Maupertuis-Euler principle entered science [55], the first in a series of integral variational principles of mechanics; later this principle was generalized by J. L. Lagrange, and now it is usually interpreted [68] [69] as one of the forms (the Maupertuis-Euler form, considered along with the Lagrange form and the Jacobi form) of the Maupertuis-Lagrange principle.


Question: When did variation first appear?

Leonard Euler

Despite his decisive contribution, in the discussion that arose around the principle of least action, Euler strongly defended Maupertuis's priority and pointed to the fundamental importance of this principle in mechanics [70].


Question: When did variation first appear?

Leonard Euler

This idea attracted the attention of physicists, who in the 19th and 20th centuries found out the fundamental role of variational principles in nature and applied the variational approach in many sections of their science [71].


Question: When did variation first appear?

Leonard Euler

Machine mechanics


Question: When did variation first appear?

Leonard Euler

A number of Euler's works are devoted to issues of machine mechanics.


Question: When did variation first appear?

Leonard Euler

In his memoir “On the Most Beneficial Use of Simple and Complex Machines” (1747), Euler proposed to study machines not at rest, but in a state of motion [2].


Question: When did variation first appear?

Leonard Euler

This new, “dynamic” approach was substantiated and developed by Euler in his memoir “On Machines in General” [72] (1753); in it, for the first time in the history of science [2], he pointed out three components of the machine that in the 19th century were are defined as engine, transmission and actuator.


Question: When did variation first appear?

Leonard Euler

In his memoir “Principles of Machine Theory” [73] (1763), Euler showed that when calculating the dynamic characteristics of machines in the case of their accelerated movement, it is necessary to take into account not only the resistance forces and the inertia of the useful loads, but also the inertia of all components of the machine, and gives (in relation to hydraulic motors) an example of such a calculation [2].


Question: When did variation first appear?

Leonard Euler

Euler was also engaged in applied issues of the theory of mechanisms and machines: the theory of hydraulic machines and windmills, the study of friction of machine parts, issues of profiling gears (here he substantiated and developed the analytical theory of involute engagement).


Question: When did variation first appear?

Leonard Euler

In 1765, he laid the foundations for the theory of friction of flexible cables and received, in particular, the Euler formula for determining the tension of the cable [74], which is still used in solving a number of practical problems (for example, when calculating mechanisms with flexible links) [2].


Question: When did variation first appear?

Leonard Euler

Continuous media mechanics


Question: When did variation first appear?

Leonard Euler

The name of Euler is also associated with the consistent introduction into mechanics of the idea of a continuum, according to which the material body is represented, abstracting from its molecular or atomic structure, in the form of continuous continuous medium [75].


Question: When did variation first appear?

Leonard Euler

The continuum model was introduced by Euler [54] in his memoir Discovery of a New Principle of Mechanics [76] (reported in 1750 to the Berlin Academy of Sciences and published in its Memoirs two years later).


Question: When did variation first appear?

Leonard Euler

The author of the memoir was based on the principle of Euler's material particles, a position that is still given in many textbooks of mechanics and physics (often without mentioning the name of Euler): a continuous body with any degree of accuracy can be modeled by a system of material points, breaking it mentally into fairly small particles and interpreting each of them as a material point.


Question: When did variation first appear?

Leonard Euler

Based on this principle, it is possible to obtain certain dynamic relations for a solid body by writing down their analogues for individual material particles (according to Euler, “Taurus”) and respectfully summing (replacing at the same time, summation at all points by integration according to the volume of the area occupied by the body) [77] [55].


Question: When did variation first appear?

Leonard Euler

This approach allowed Euler to do without the use of such means of modern integral calculus (such as the Stieltjes integral) that were not yet known in the 18th century [54].


Question: When did variation first appear?

Leonard Euler

Based on this principle, Euler obtained — applying to the elementary material volume the theorem on changing the amount of motion — the first law of Euler's motion (later the second law of motion appeared Euler is the result of applying the theorem on changing the moment of the amount of motion) [52].


Question: When did variation first appear?

Leonard Euler

Euler's laws of motion were actually the basic laws of motion of continuum mechanics; to move to the currently used general equations of motion of such media, only the expression was missing surface forces through the stress tensor (this was done by O. Cauchy in the 1820s) [78].