Paul Euler studied mathematics with Jakob Bernoulli

LEONHARD EULER 1707 - 1783 mathematician, 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.

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

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.

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

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.

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.

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 nocalculus

1742

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

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 . . .

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.

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.

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.

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 . . .

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

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.