In Celebration of Euler's 300th Birthday

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In Celebration of Euler's 300th Birthday

• V. Frederick Rickey
• West Point

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Eulers Life

• Basel 1707-1727 20
• Petersburg I 1727-1741 14
• Berlin 1741-1766 25
• Petersburg II 1766-1783 17
• ____
• 76

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• Euler born 15 April 1707 in Basel
• Son of a Lutheran minister, Paul Euler
• Paul Euler studied mathematics with Jakob
  Bernoulli

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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.
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• Paul Euler taught algebra to his son using
  Michael Stifels 1553 edition of Christoff
  Rudolffs 1525 Coss, the first German book
  entirely devoted to algebra.

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• At age 7 Euler returned to Basel to attend a
  Latin school
• No mathematics taught
• Tutored by Johannes Burckhardt, magni Euleri
  praeceptor in mathematicis

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• 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

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• 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.

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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

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• St. Petersburg I
• 1727 1741
• Nicolaus II and Daniel I Bernoulli got 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 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

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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 students are highly literate
• Covers surprisingly little material
• E17, E35
• 1738, 1740

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• 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."  

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Mediation on experiments made recently on the
firing of cannon. Eulers first paper on
cannon, E853, written 1727, published 1862.
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A polemic against Euler by Robins, 1739Too
algebraical and uses infinitesimals
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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.

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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

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• 1742
• Preface
• 55 pages
• Ch I Internal ballistics
• 65 pages
• Ch2 External ballistics
• 30 pages
• Total 150 pp.

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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
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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 Eulers 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
• . . .

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• 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.

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Translations of Eulers 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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Eulers 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

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Prop VI

• 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.

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Editions of New Principles of Gunnery
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1745 to 1777 is a triple translation

• Differentials to fluxions
• German to English
• Leibnizian to Newtonian notation

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Mathematics at Woolwich, 1772

• The Elements of Euclid
• Trigonometry applied to Fortification, and the
  Mensuration of Superficies and Solids
• Conic Sections.
• Mechanics applied to the raising and transporting
  heavy bodies, together with the use of the lever
  pulley, wheel, wedge and screw, c.
• The Laws of Motion and Resistance, Projectiles,
  and Fluxions.
• Now some calculus!

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• Question
• What was the impact of ballistics on mathematics?
• Answer
• Calculus was taught to engineers and
  artillerists.

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Eulers Berlin HouseBehrenstraße 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 of
  freethinkers, 1747
• Théorie général de la dioptrique, 1765
• Lettres à une Princesse dAllemagne, 1768

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In addition, Euler supervised the

• Library
• Observatory
• Botanical garden
• Scientific publications
• Calendars and maps
• Various financial matters

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Eulers Calculus Books

• 1748 Introductio in analysin infinitorum
• 399
• 402
• 1755 Institutiones calculi differentialis
• 676
• 1768 Institutiones calculi integralis
• 462
• 542
• 508
• _____
• 2982

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Euler creates trig functions in 1739
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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
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Chapter 1 Functions

• A change of Ontology
• Study functions
• not curves

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VIII. Trig Functions
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• 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

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• Sinus totus 1
• p is clearly irrational
• Value of p from de Lagny
• Note error in 113th decimal place
• scribam p
• W. W. Rouse Ball discovered (1894) the use of p
  in Wm Jones 1706.
• 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 la
  Manoeuvre des Vaissaux, 1773

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• Euler's method, invented in 1768 in THIS house
• The address is Naberezhnaya Leitenanta Shmidta
  No. 15, St. Petersburg.

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Euler is buried in the Lavra Cemetery in St.
Petersburg.
"Leonhardo EuleroAcademia MetropolitanaMDCCLXXXV
II."
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Read Euler, read Euler, he is our teacher in
everything.

• Laplace
• as quoted by Libri, 1846

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To be issued on the 300th anniversary of Eulers
birth, April 15, 1707