Leonard Euler

AMathematician

For allSeasons

By A Chaparral High School Teacher

Leonhard Euler

lived from 1707 to 1783

Euler made large bounds in modern analytic geometry and trigonometry. He made decisive and formative contributions to geometry, calculus and number theory.

Euler was born in Basel, Switzerland on

April 15, 1707

(to hear his name correctly pronounced click here )

What do we know about Switzerland?

•Full country name: Swiss Confederation•Area: 41,295 sq km (16,105 sq mi)•Population: 7.3 million•Capital city: Bern (pop 130,000)•People: 74% German, 20% French, 4% Italian & 1% Romansch•Language: German, French, Italian & Romansch•Religion: 49% Roman Catholic & 48% Protestant

Since the dominant language is German, can you guess how his name is pronounced?

You heard right, “Oiler”!

• His father wanted his son to follow him into the church and sent him to the University of Basel to prepare for the ministry.

• He entered the University in 1720, at the age of 14, first to obtain a general education before going on to more advanced studies.

• Johann Bernoulli soon discovered Euler's great potential for mathematics in private tuition that Euler himself engineered.

Euler:

. Euler's own account given in his unpublished autobiographical writings, is as follows:-

... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli… True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand ...

•In 1723 (at the age of 16) he completed his Master’s degree. •He finished his studies within the next few years.

•In 1727 he began teaching at the St. Petersburg Academy of Sciences.

•He became professor of physics in 1730

•He became senior chair of mathematics in 1733

In 1738 he lost the sight in his right eye.

Euler’s comment upon this occurance is stated to be:

"Now I will have less distraction."

• Euler was director of mathematics and physics in the Berlin Academy from 1741-1766 (when he lost the sight of his other eye).

• He published over 800 different books and papers on pure and applied mathematics, physics and astronomy.

NOW…………..

Let’s look at some of Euler’s accomplishments!

There is a town in Russia whose name today is Kaliningrad.

In the 1700s, it was in Prussia and its name was Königsberg.

Here's a true story.

Once upon a time (OK, in the 18th century), in a place far away (well, Prussia), there once was a town called Königsberg. The townspeople enjoyed walking through their town and passing over the seven bridges spanning the river which passed through town. Over the years, a controversy arose and eventually word of it reached Euler…

The problem is stated as follows: In the town of Königsberg in Prussia there is an island called Kneiphhof, with the two branches of the river Pregel flowing around it. There are 7 bridges crossing the two branches. [as shown below] The question is whether a person can plan a walk in such a way that he will cross each of these bridges once but not more than once.

So, what did he find out?

• It was impossible to do that!• Euler went on to formulate a general

theory which solved this particular problem and created a new branch of mathematics called graph theory.

• To find out more try:paths

•Euler contributed to every mathematical field that existed at the time. He standardized modern mathematics notation

when he used symbols such as f(x), e, i and sin(x). He was the first person to represent trigonometric values as ratios and prove that e is an irrational number.

•His invention of the calculus of variations led to the general method to solve maximum and minimum value problems.

•He was also one of the first people to recognize that infinite series had to be convergent to be used safely.

•In physics, he developed the general equations for hydrodynamics and for motion.

Possibly his most impressive work was his approximation of the three-body problem of the sun, earth and moon, which he solved while completely blind and performing all the computations in his head.

WOW!

. Euler was able to relate the number of faces (F), vertices (V) and edges (E) of a polyhedron by the following equation,

F + V = E + 2

from which one can derive that there are only five regular polyhdera.

In fact, Germany created a stamp that commemorates this formula discovery:

Euler also discovered the formula

•eix = cos x + i sin x

Which happens to be commemorated on a Swiss stamp:

Euler’s importance to society is reflected in his appearance not just on stamps but on currency as well!

So, as you can see Euler’s contributions to mathematics are broad and varied. His impact can still be felt in algebra, geometry, calculus and other branches of mathematics such as topology and graph theory.

Indeed, it is difficult to find a mathematician, modern or historical, who was as prolific as Leonhard Euler,

but I challenge you to try!

Project Assessment: PowerPoint Presentation

Layout (15 points possible)

      Is your presentation easy to view and attractive? (6 points)

      Does your presentation avoid clutter yet utilize the space on the slide? (3 points)

      Are slide transitions used appropriately? (3 points)

      Are graphics, animations, sounds, and other multimedia used for a purpose? (3 points)

 

Organization (15 points possible)

      Do titles and other important elements stand out on the slide? (3 points)

      Do slides progress in a logical order? (3 points)

      Do elements of slides appear in the proper order? (3 points)

      Are bullets or other organizing aids used appropriately? (3 points)

      Are there appropriate title, outline, and conclusion slides? (3 points)

Content (15 points possible)

      Does your presentation effectively summarize important information? (3 points)

      Are words spelled correctly and is grammar used properly? (3 points)

      Does your show contain at least 10 content slides? (3 points)

      Does your presentation contain at least 5 multimedia images, sounds, animations, videos, etc.? (3 points)

      Is author and source information included? (3 points)

 

Classroom Presentation (5 points possible)

      Is the pace of your presentation a comfortable one for your audience? (3 points)

      Do the presenters speak clearly, confidently, and with appropriate eye contact? (2 points)

 

TOTAL (50 points possible)