In Celebration of Euler's 300th Birthday

V. Frederick Rickey

West Point

Euler’s Life

• Basel 1707-1727 20

• Petersburg I 1727-1741 14

• Berlin 1741-1766 25

• Petersburg II 1766-1783 17____

76

• Euler born 15 April 1707 in Basel

• Son of a Lutheran minister, Paul Euler

• Paul Euler studied mathematics with Jakob Bernoulli

LEONHARD EULER 1707 - 1783mathematician, physicist, engineer, astronomer, and philosopher spent his early years in Riehen.

He was a great savant and a kind-hearted man.

Paul Euler taught algebra to his son using Michael Stifel’s 1553 edition of Christoff Rudolff’s 1525 Coss, the first German book entirely devoted to algebra.

• At age 7 Euler returned to Basel to attend a Latin school

• No mathematics taught

• Tutored by Johannes Burckhardt, “magni Euleri praeceptor in mathematicis”

• Entered the University of Basel (founded 1460) in 1720, age 13.

• 19 professors, about 100 students

• Johann I Bernoulli was one professor

• Johann II Bernoulli was a fellow student

• Euler became friends with his brothers Daniel I and Nicolaus II

I soon found the opportunity to become acquainted with the famous professor Johann Bernoulli [who advised me] to look at more difficult mathematics books and work through them with great diligence [and visit him every Saturday]. When he resolved one of my objections, ten others disappeared, which is certainly the best method of making auspicious progress in the mathematical sciences.

Research in Basel by Euler

• The brachistochrone problem in resisting media

• The masting of ships won an accessit in the 1728 Paris Prize competition

• An essay on the physics of sound, written in an unsuccessful attempt to become professor of physics

• St. Petersburg I

• 1727 – 1741

• Nicolaus II and Daniel I Bernoulli got Euler invited

• Also there:– Jakob Hermann– Christian Goldbach– F. C. Meyer

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

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

Euler about 1737, age 30

• Painting by J. Brucker• 1737 mezzotint by

Sokolov• Black below and

above right eye• Fluid around eye is

infected• “Eye will shrink and

become a raisin”• Ask your

ophthalmologist• Thanks to Florence Fasanelli

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

• An arithmetic book

• Assumes the students are highly literate

• Covers surprisingly little material

• E17, E35• 1738, 1740

June 7th 1742, Christian Goldbach makes a conjecture:

"Es scheinet wenigstens, daß eine jede Zahl, die größer ist als 2,

ein aggregatum trium numerorum primorum sey."  

“Mediation on experiments made recently on the firing of cannon.”

Euler’s first paper on cannon, E853, written 1727,published 1862.

A polemic against Euler by Robins, 1739

Too algebraical and uses infinitesimals

From Teacher to Professor ?

• Robins hoped to be the first professor of mathematics at Woolwich

• Planned a course on fortifications and gunnery

• Politics intervened

• Mr. Derham served 1741 -1743.

Mathematics at Woolwich, 1741

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

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

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

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

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

But no calculus

• 1742

• Preface– 55 pages

• Ch I: Internal ballistics– 65 pages

• Ch2: External ballistics– 30 pages

• Total: 150 pp.

Euler 1745

• Frederick the Great asks about the best book on gunnery

• Euler magnanimously recognizes Robins

• Robins sets a research program for Euler

• Euler adds Annotations

2400

• Euler returns to gunnery in E217

• Presented 1752• Published 1755• Translated in 1777

Euler translated into English, 1777

From Euler’s preface:

Some are of the opinion that fluxions are applicable only in such subtile speculations as can be of no practical use . . .

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

. . .

It may be affirmed, that things which depend on mathematics cannot be explained in all their circumstances without the help of fluxions, and even this sublime part of mathematics has met with difficulties which it has not fully mastered.

Translations of Euler’s Observations upon the new principles of

gunnery

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

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.

Euler’s Annotations

• Analytically derives the equations of motion from the fundamental principles of motion in a vacuum

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

Robins gives experimental evidence to confute the postulates 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 an angle of 45º should have a horizontal range of about 17 miles according to the fifth postulate.

– Actual range: Less than half a mile.

Date Pages

Robins 1742 150

Robins / Euler

1745 720

Euler 1777 423

Robins /

Euler

1922 409

Editions of New Principles of Gunnery

– Differentials to fluxions

– German to English

– Leibnizian to Newtonian notation

1745 to 1777 is a triple translation:

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 and

transporting heavy bodies, together with the use of the lever pulley, wheel, wedge and screw, &c.

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

Now some calculus!

• Question: – What was the impact of ballistics on

mathematics?

• Answer: – Calculus was taught to engineers and

artillerists.

Euler’s Berlin House

Behrenstraße 21/22

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 of

freethinkers, 1747– Théorie général de la dioptrique, 1765– Lettres à une Princesse d’Allemagne, 1768

In addition, Euler supervised the

• Library• Observatory• Botanical garden• Scientific

publications• Calendars and

maps• Various financial

matters

Euler’s Calculus Books

• 1748 Introductio in analysin infinitorum399

402

• 1755 Institutiones calculi differentialis676

• 1768 Institutiones calculi integralis462

542

508

_____

2982

Euler creates trig functions in 1739

Solve y k4d4y

dx4 0.

Factor 1 k4p4 0 :1 k p1 kp1 k2p2The solution is :

y xk C

xk D E Cosx

k F Sinx

k

Often I have considered the fact that most of the difficulties which block the progress of students trying 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

Chapter 1: Functions

A change of Ontology:

Study functions

not curves

VIII. Trig Functions

He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author . . .

Eulogy by Nicolas Fuss, 1783

• Sinus totus = 1• π is “clearly” irrational• Value of π from de

Lagny• Note error in 113th

decimal place• “scribam π”• W. W. Rouse Ball

discovered (1894) the use of π in Wm Jones 1706.

• Arcs not angles• Notation: sin. A. z

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 la

Manoeuvre des Vaissaux, 1773

• Euler's method, invented in 1768 in THIS house

• The address is Naberezhnaya Leitenanta Shmidta No. 15, St. Petersburg.

Euler is buried in the Lavra Cemetery in St. Petersburg.

"Leonhardo EuleroAcademia MetropolitanaMDCCLXXXVII."

Read Euler, read Euler, he is our teacher in everything.

Laplace

as quoted by Libri, 1846

To be issued on the 300th anniversary of Euler’s birth, April 15, 1707