Who was Leon­hard Euler?

23. September 2011
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Leonhard Euler (1707 - 1783), pastel painting by E. Handmann, 1753.

Leon­hard Euler (1707 — 1783), pas­tel paint­ing by E. Hand­mann, 1753.

Leon­hard Euler was one of the great­est math­e­mati­cians of all times. He devel­oped the basics of the mod­ern the­ory of num­bers and alge­bra, the topol­ogy, the prob­a­bil­ity cal­cu­lus and com­bi­na­torics, the inte­gral cal­cu­lus, the the­ory of the dif­f­en­ren­tial equa­tion and the dif­fer­en­tial geom­e­try, the vari­a­tional cal­cu­lus and he dis­cov­ered the coher­ence between trigono­met­ri­cal func­tions and expo­nen­tial func­tions. Leonard Euler devel­oped the hydro­dy­nam­ics and flu­idic, he made the bases for the the­ory of the gyro­scope. He was a bril­liant nat­ural sci­en­tist, an excel­lent teacher and mentor.

1851 the formerly two-storey residential house of Leonard Euler was upgraded. 1858 - 1917 girl secondary school 1918 - 1958 secondary modern school, today secondary modern school with specialized teaching curriculum in history, languages and literature. St. Petersburg, Lieutenant-Schmidt-Shore 15.

1851 the for­merly two-​​storey res­i­den­tial house of Leonard Euler was upgraded. 1858 — 1917 girl sec­ondary school 1918 — 1958 sec­ondary mod­ern school, today sec­ondary mod­ern school with spe­cial­ized teach­ing cur­ricu­lum in his­tory, lan­guages and lit­er­a­ture. St. Peters­burg, Lieutenant-​​Schmidt-​​Shore 15.

It was on April 24th, 1727 when on invi­ta­tion of the Russ­ian cza­rina Katha­rina I the 19-​​year-​​old mas­ter Leon­hard Euler left his home town of Basel and set off for a bril­lant sci­en­tific career at the Acad­emy of the Sci­ences of Rus­sia. The broth­ers Bernoulli (Niko­lai and Daniel), Chris­t­ian Gold­bach and other excel­lent Euro­pean sci­en­tist already worked there.
Peter I engaged the philoso­pher and math­e­mati­cian Chris­t­ian Wolf from Mar­burg shortly before his death to unite the best heads of Europe under the seal of Acad­emy of the Sci­ences of St. Petersburg.

In May 1771 an enor­mous blaze raged through St. Peters­burg. Hun­dreds of build­ings burned down, among oth­ers the house of the grad­u­ate Leonard Euler. But the basler crafts­man Peter Grimm suc­ceeded in sav­ing the 64-​​year-​​old blind math­e­mati­cian from death by burn­ing. Thanks to his coura­geous inter­ven­tion almost all man­u­scripts of the great­est math­e­mati­cian of all times remained for the pos­ter­ity. Among oth­ers also the work “About con­tin­ued frac­tions” (1737) and “About the vibra­tions of a string” (1748). In these papers Euler for­mu­lated the­ses whose solu­tion would keep math­e­mat­ics busy for 200 years to come. Eulers work made it pos­si­ble 250 years later to air one of the most fun­da­men­tal secrets of nature – the free vibra­tions of the universe.

Euler exam­ined already free vibra­tions of an elas­tic thread with no mass occu­pied with pearls. In con­nec­tion with this task d’Alembert devel­oped his inte­gra­tion method for a sys­tem of lin­ear dif­fer­en­tial equa­tions. Start­ing out from there Daniel Bernoulli put for­ward his famous the­o­rem that the solu­tion for the prob­lem of a free vibrat­ing string can be por­trayed as a trigono­met­ri­cal series, which lead to a dis­cus­sion between Euler, d’Alembert and D. Bernoulli which spread over few decades. Later on Lan­garne pointed out more cor­rectly, how one can come to a solu­tion of the prob­lem of swing­ing string of beads to the solu­tion of the prob­lem of the vibrat­ing of a homo­ge­neous string by breach­ing the limit. Only in 1822 J. B. Fourier in solved this for­mu­la­tion com­pletely for the first time.

Mean­while, nearly insur­moutable prob­lems still arose with pearls of var­i­ous mass and irreg­u­lar dis­tri­b­u­tion. This task leads to func­tions with gaps. Accord­ing to a let­ter of Charles Her­mite of May 20th, 1893, which called to “reject the lam­en­ta­ble plague of the func­tions with­out deriva­tions in fright and fear “, T. Stielt­jes exam­ined func­tions with dis­con­ti­nu­ities and found an inte­gra­tion method of such func­tions, which led to con­tin­ued fractions.

The grave of Leonhard Euler in St. Petersburg.

The grave of Leon­hard Euler in St. Petersburg.

Euler already rec­og­nized that com­pli­cated vibrat­ing sys­tems can con­tain also such solu­tions (inte­grals) which aren’t dif­fer­en­ti­at­ing every­where and left to the math­e­mat­i­cally tal­ented pos­ter­ity an ana­lytic mon­ster – the so called non-​​analytic func­tions (this term was cho­sen by him­self). Non-​​analytic func­tions have ensured a lot of work up until the 20th cen­tury, even after the iden­tity cri­sis of the math­e­mat­ics seemed to be already overcome.

The cri­sis started, as E. du Bois Rey­mond 1875 reported for the first time about a steady, but not sub­tly dif­fer­en­tiable func­tion designed by Weier­strass, and lasted approx­i­mately till 1925. Their dom­i­nant play­ers were Can­tor, Peano, Lebesgue and Haus­dorff. As result a new branch of the math­e­mat­ics was given a birth – the frac­tal geometry.

Frac­tal comes from the Latin frac­tus and means as much as “in pieces bro­ken” and “irreg­u­lar”. So frac­tals are really incom­plete, spite­ful math­e­mat­i­cal objects. The math­e­mat­ics of the 19th cen­tury took these objects for excep­tions and there­fore looked at reg­u­lar, steady and smooth struc­tures or tried to put down frac­tal phe­nom­e­nons to such structures.

This plaque was installed 1957 in honour of the 250th birthday of Leonard Euler at his former residential house at the shore of the Neva.

This plaque was installed 1957 in hon­our of the 250th birth­day of Leonard Euler at his for­mer res­i­den­tial house at the shore of the Neva.

The the­ory of the frac­tal quan­ti­ties made it pos­si­ble to exam­ine strictly “not ana­lytic” creased, gran­u­lous or incom­plete forms qual­i­ta­tively. Soon it turnes out that frac­tal struc­tures aren’t that rare at all. In nature one dis­cov­ered more frac­tal objects than sus­pected till now. More, it seemed so as if sud­denly the uni­verse was frac­tal by nature.

Espe­cially the works of Man­del­brot placed the geom­e­try finally in a posi­tion capa­ble of describ­ing cor­rectly frac­tal math­e­mat­i­cal objects: incom­plete crys­tal lat­tices, the Brown’s move­ment of the gas mol­e­cules, sin­u­ous poly­mer giant mol­e­cules, irreg­u­lar star clus­ters, Cir­rus clouds, the sat­urn rings, the dis­tri­b­u­tion of the lunar craters, tur­bu­lences within liq­uids, bizarre shore­lines, wind­ing river courses, folded moun­tain ranges, branched forms of growth of most dif­fer­ent plant sorts, areas of islands and seas, rock for­ma­tions, geo­log­i­cal depo­si­tions, the spa­tial dis­tri­b­u­tion of raw mate­r­ial occur­rences and so on and so on.

The Leningrad math­e­mati­cians F. R. Gant­macher and M. G. Krein looked 1950 at the deflec­tion line of a vibrat­ing string with pearls as par­ti­tioned line. Just this attempt made it pos­si­ble for them to view the prob­lem in a frac­tal way with­out being con­scious of it (Mandelbrot’s clas­sic “Le Objets Frac­tals” appeared 1975, his first works from 50’s fell into the lin­guis­tics school). Only the frac­tal view put them to the posi­tion to com­pletely solve (also for the most gen­eral case) the 200 years old Euler’s prob­lem of the vibrat­ing string of beads for pearls of var­i­ous masses and irreg­u­lar dis­tri­b­u­tion. In their work “Oscillation-​​Matrixes, Oscillation-​​Cores and Small Vibra­tions of Mechan­i­cal Sys­tems” they proved, that all free vibra­tions form a finite string of beads or string a finite or infi­nite Stieltjes-​​continued frac­tion. The masses of the pearls and the sep­a­ra­tions between them are iden­ti­cal with the part denom­i­na­tors of the con­tin­ued fraction.

1955 V. P. Ter­s­kich gen­er­al­ized the (as regards con­tent frac­tal) con­tin­ued frac­tion method on vibra­tions of com­pli­cated branched chain sys­tems. The clas­sic work of T. N. Thiele, A. A. Markov, A. J. Chintchin, O. Per­ron, J. A. Mur­phy, M. R. O’Donohoe, A. N. Chovan­sky, H. S. Wall, D. I. Bod­nar, C. I. Kucmin­skaja, V. J. Skorobogat’ko and oth­ers helped to get the def­i­nite break­through for the con­tin­ued frac­tion method and made the devel­op­ment of effi­cient algo­rithms pos­si­ble for the addi­tion and mul­ti­pli­ca­tion of con­tin­ued frac­tions up till 1981.

Math­e­mat­i­cal mod­els of vibrat­ing frac­tal chain sys­tems are used with great suc­cess in the most dif­fer­ent sci­en­tific fields today. Their pop­u­lar­ity reached a high­light already in the six­ties in the elec­trotech­ni­cal engi­neer­ing. The fast devel­op­ment of the com­puter branch dur­ing the last decades have to be put down to the effi­ciency of such mod­els in the solid state physics. One dis­cov­ered vibrat­ing frac­tal chain sys­tems also in neural net­works, our genome and eco-​​systems.

The com­plete uni­verse is a vibrat­ing frac­tal chain sys­tem, what can be com­pared math­e­mat­i­cally with the free vibra­tions of a Euler’s string of beads of gigan­tic pro­por­tion. The nat­ural oscil­la­tions of mat­ter influ­ence not only the tem­po­ral course of all phys­i­cal, chem­i­cal, bio­log­i­cal and social processes, but it is also a global mor­pho­genet­i­cal fac­tor and cause for a global selec­tion process. Their fre­quency spec­trum is log­a­rith­mic fractal.

Leonard Euler left about 900 sci­en­tific work, among others:

  • Mechan­ica (1736)
  • Über Ket­ten­brüche (1737)
  • Ten­ta­men novae musi­cae (1739)
  • The­o­rie der Plan­eten­be­we­gung (1744)
  • Neue Grund­sätze der Artillerie (1745)
  • Nova theo­ria lucis et col­o­rum (1746)
  • Über die Schwingun­gen einer Saite (1748)
  • Intro­duc­tio in analysin infin­i­to­rum (1748)
  • The­o­rie des Schiff­baues (1749)
  • Insti­tu­tiones cal­culi dif­fer­en­tialis (1755)
  • Insti­tu­tiones cal­culi inte­gralis (1770)
  • Voll­ständige Anleitung zur Alge­bra (1770)
  • Let­tres · une princesse d’Allemagne sur quelques sujets de Phsique et Philoso­phie (1772)
  • Diop­trica (1771)

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