300 años Euler

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Euler cumple300 años

Basado en la conferencia del prof:V. Frederick Rickey

Vernor Arguedas

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La vida de Euler

• Basel 1707-1727 20

• Petersburg I 1727-1741 14

• Berlin 1741-1766 25

• Petersburg II 1766-1783 17 ____

76

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• Euler nace el 15 de

abril de1707 enBasel-Suiza

• Hijo de un ministroluterano, Paul Euler

• Paul Euler estudiómatemáticas conJakob Bernoulli

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LEONHARD EULER 1707 - 1783 matemático, físico, ingeniero, astrónomo y

filósofo pasó los primeros años enRiehen. El fue un gran sabio y un hombre de un gran corazón.

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Paul Euler le enseñóálgebra a su hijo

usando la edición del

año 1553 de ChristoffRudolff escrita en

1525 Coss , el primerlibro alemán dedicado

exclusivamente alálgebra.

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• A los 7 Euler regresó aBasel para asistir a una

escuela en latin

• No se enseñabamatemáticas

• Su tutor fue JohannesBurckhardt, “magni

Euleri praeceptor inmathematicis”

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• Entró a la Universidad de Basel (fundadaen 1460) en 1720, edad 13.

• 19 profesores, con cerca de 100estudiantes

Johann I Bernoulli era uno de los profesores

• Johann II Bernoulli era estudiante

• Euler se hizo amigo de sus hermanosDaniel I and Nicolaus II

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I soon found the opportunity tobecome acquainted with the

famous professor JohannBernoulli [who advised me] tolook at more difficultmathematics books and work

through them with greatdiligence [and visit him everySaturday]. When he resolvedone of my objections, ten

others disappeared, which iscertainly the best method ofmaking auspicious progress inthe mathematical sciences.

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Research in Basel by Euler

• The brachistochrone problem in resistingmedia

• The masting of ships won an accessit in

the 1728 Paris Prize competition

• An essay on the physics of sound, writtenin an unsuccessful attempt to become

professor of physics

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• St. Petersburg I

• 1727 – 1741

• Nicolaus II andDaniel I Bernoulligot Euler invited

• Also there:

– Jakob Hermann

– Christian Goldbach

– F. C. Meyer

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Vue sur le quai de l'Université et l'Académie, depuis le quai opposé

Saint-Petersbourg n'est-elle pas la Venise du Nord?

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Euler about 1737, age 30

• Painting by J. Brucker

• 1737 mezzotint bySokolov

• Black below andabove right eye

• Fluid around eye isinfected

• “Eye will shrink andbecome a raisin”

• Ask yourophthalmologist

• Thanks to Florence Fasanelli

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Research in Petersburg 1727 - 41

• Published 55 papers; wrote 35 more

– Started research in number theory

• Wrote 4 books:

– Mechanica, 1736

– Rechen-Kunst, 1738

– Tentamen novae theoriae musicae, 1739

– Scientia navalis, 2 vols, 1749

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• An arithmetic book

• Assumes the studentsare highly literate

• Covers surprisinglylittle material

• E17, E35

• 1738, 1740

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June 7th 1742,Christian Goldbach

makes aconjecture:

"Es scheinet

wenigstens, daß eine jede Zahl, die größer ist als 2,

ein aggregatum

trium numerorum primorum sey."

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“Mediation on experiments made recently

on the firing of cannon.”

Euler’s first paper on cannon, E853, written 1727,

published 1862.

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A polemic against Euler by Robins, 1739

Too algebraical and uses infinitesimals

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From Teacher to Professor ?

• Robins hoped to be the first professor ofmathematics at Woolwich

• Planned a course on fortifications and

gunnery

• Politics intervened

• Mr. Derham served 1741 -1743.

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Mathematics at Woolwich, 1741

• That the second Master shall teach the Science of Arithmetic, together withthe principles of Algebra and the Elements of Geometry, under thedirection of the Chief Master.

• That the chief Master shall further instruct the hearers in Trigonometry andthe Elements of the Conick Sections.

• To which he shall add the Principles of Practical Geometry andMechanics, applied to raising and transporting great Burthens;

• With the Knowledge of Mensuration, and Levelling, and its Application tothe bringing of water and the draining of Morasses;

• And lastly, shall teach Fortification in all its parts.

But no calculus

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• 1742

• Preface – 55 pages

• Ch I: Internal ballistics

– 65 pages

• Ch2: External ballistics

– 30 pages

• Total: 150 pp.

F d i k th G t

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Euler1745

• Frederick the Greatasks about the bestbook on gunnery

• Euler magnanimouslyrecognizes Robins

• Robins sets a researchprogram for Euler

• Euler adds Annotations

2400

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• Euler returns to

gunnery in E217

• Presented 1752

• Published 1755• Translated in 1777

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Euler translated into English, 1777

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From Euler’s preface:

Some are of the opinion that fluxions areapplicable only in such subtilespeculations as can be of no practical use

. . .

But what has been just now said of artilleryis sufficient to remove this prejudice

. . .

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It may be affirmed, that things whichdepend on mathematics cannot beexplained in all their circumstances without

the help of fluxions, and even this sublimepart of mathematics has met withdifficulties which it has not fully mastered.

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Translations of Euler’s Observations upon the new principles of gunnery

translation by Hugh Brown – p. 276 and p. 303 – 28 pages

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7 Postulates for motion of a projectile in a vacuum

Postulate 2: If the Parabola, in which the Body moves,

be terminated on a horizontal Plain, then the Vertex of

the Parabola will be equally distant from its two

Extremities.

Postulate 4: If a Body be projected in different Angles,

but with the same Velocity, then its greatest horizontal

Range will be, when it is projected in an Angle of 45º

with the horizon.

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Euler’s Annotations

• Analytically derives the equations ofmotion from the fundamental principles ofmotion in a vacuum

• Confirms each of the 7 postulates for thetrajectory of a projectile in a vacuum

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Robins gives experimental evidence to confute thepostulates posed in Proposition V.

For example, according to postulate 5 in Prop V:

– A musket ball ¾ of an inch in diameter that has

an initial velocity of 1700 feet per second at anangle of 45º should have a horizontal range ofabout 17 miles according to the fifth postulate.

– Actual range: Less than half a mile.

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Date Pages

Robins 1742 150

Robins / Euler

1745 720

Euler 1777 423

Robins /

Euler

1922 409

Editions of New Principles of Gunnery

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– Differentials to fluxions

– German to English

– Leibnizian to Newtonian notation

1745 to 1777 is a triple translation:

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Mathematics at Woolwich, 1772

1. The Elements of Euclid2. Trigonometry applied to Fortification, and the

Mensuration of Superficies and Solids3. Conic Sections.

4. Mechanics applied to the raising andtransporting heavy bodies, together with theuse of the lever pulley, wheel, wedge andscrew, &c.

5. The Laws of Motion and Resistance,Projectiles, and Fluxions.

Now some calculus !

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• Question:

– What was the impact of ballistics onmathematics?

• Answer:

– Calculus was taught to engineers and

artillerists.

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Euler’s Berlin House

Behrenstraße 21/22

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Research in Berlin 1741 - 66

• 275 papers published; 105 more written

• Wrote 14 books, including

– Methodus inveniendi, 1744

– Theoria motuum planetarum et cometarum, 1744

– Rescue of divine revelation from objections offreethinkers, 1747

– Théorie général de la dioptrique, 1765

– Lettres à une Princesse d’Allemagne, 1768

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In addition, Euler supervised the

• Library

• Observatory

• Botanical garden

• Scientificpublications

• Calendars and

maps• Various financial

matters

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Euler’s Calculus Books

• 1748 Introductio in analysin infinitorum399

402

• 1755 Institutiones calculi differentialis676

• 1768 Institutiones calculi integralis462

542

508

_____

2982

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Euler creates trig functions in 1739

Solve y k4d 4 y

dx40.

Factor 1 k4 p4 0 :

1 k p 1 kp 1 k2 p2

The solution is :

yxk C

xk D E Cos

x

kF Sin

x

k

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Often I have considered the fact

that most of the difficulties whichblock the progress of studentstrying to learn analysis stem from

this: that although they understand

little of ordinary algebra, still they

attempt this more subtle art.

From the preface of the Introductio

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Chapter 1: Functions

A change of Ontology:

Study functions

not curves

VIII T i

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VIII. Trig

Functions

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He showed a new algorithm which hefound for circular quantities, for which its

introduction provided for an entirerevolution in the science of calculations,and after having found the utility in thecalculus of sine, for which he is truly theauthor . . .

Eulogy by Nicolas Fuss, 1783

Si 1

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• Sinus totus = 1

• π is “clearly” irrational

• Value ofπ

from deLagny

• Note error in 113th decimal place

• “scribam π”

• W. W. Rouse Balldiscovered (1894) the

use ofπ

in Wm

Jones1706.

• Arcs not angles

• Notation: sin. A. z

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Research in Petersburg 1766 - 83

• More than 400 papers

– And, yes, he was blind !

• 4 more books:

– Dioptrica, 1768

– Vollstandige Anleitung zur Algebra, 1770

– Theoria motuum lunaea, 1772

– Théorie Complete de la Construction et de laManoeuvre des Vaissaux, 1773

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• Euler's method,

invented in 1768 inTHIS house

• The address isNaberezhnayaLeitenanta Shmidta No.

15, St. Petersburg.

Euler is buried in the Lavra

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Euler is buried in the LavraCemetery in St. Petersburg.

"Leonhardo EuleroAcademia MetropolitanaMDCCLXXXVII."

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Read Euler,

read Euler,he is our teacher in everything.

Laplace

as quoted by Libri, 1846

T b i d th 300th i

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To be issued on the 300th anniversary

of Euler’s birth, April 15, 1707

Un comentario

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Un comentariofinal

• Para los Bernoulli, al igual que para Euler,Lagrange y Laplace, la matemática era ante todo un instrumento para la

comprensión del cosmos. Esto se pone de manifiesto en las grandes obras de la época: las dos grandes obras

matemáticas en que culminó el siglo XVII fueron el Horologium oscillatorium de Huygens y los Principia de Newton.

Las dos grandes obras culminantes del siglo siguiente

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g g gfueron la Méchanique analytique de Lagrange y la Mechanique celeste de Laplace.

Esta identificación virtual de la matemática y la mecánica no fue esencialmente rota por los notables descubrimientos de Euler y otros en la teoría de los números, ni por Lagrange

en el álgebra.

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Decía Struik

• La vida de este matemático del siglo XVIIIestuvo dedicada casi exclusivamente altrabajo en los diferentes campos de las

matemáticas, tanto puras como aplicadas.• Aún cuando perdió un ojo en 1735 y el

otro ojo en 1776, nada pudo interrumpir su

enorme productividad.

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• Euler ya ciego, ayudado por su memoriafenomenal continuó dictando susdescubrimientos. Durante su vida su

producción alcanzó 560 libros y artículos.Dejó al morir muchos manuscritos, quefueron publicados por la Academia de

Ciencias de San Petersburgo

300 años Euler

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