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Euler's Mathematical Contributions

The Work of One of the Greatest Minds in Mathematical History

Jun 1, 2008

Leonard Euler was surely the most prolific and important mathematician of the eighteenth century, and perhaps of all time, with accomplishments far too numerous to count.

In his 76 years on Earth during the midst of the 18th century, Leonard Euler worked in almost every imaginable area of mathematics (including some which had not yet become popularized), effectively pushing the world forward into the next stage of numerical thought. The constant "e" is named for Euler.

This work continued, unabated, throughout Euler's life, despite several major relocations and the onset of near-complete blindness. Euler was one of the most prolific mathematicians of all time and also left his mark on such diverse sciences as physiology and physics.

The following are examples of Euler's achievements in calculus, function notation, number theory, topology and topics outside of mathematics.

Calculus

Building upon the work of Newton and Leibnitz decades earlier (the "co-creators" of the calculus), some of Euler's most famous work was contained in his 1755 Institutiones Calculi Differentialis in which he further expanded upon the function notation which he himself had introduced in 1748.

Euler's vast and agile mind was able to comprehend some of the more complicated aspects of the new theories of analysis, enabling him to solve some of the great calculus problems of the day, as well as to simplify things in such a way that they could be later solved far more easily by students of mathematics. He helped immeasurably to install the mathematical formalism that is still in use today.

One of the great formulas to come out of Euler's calculus has become known as Euler's Identity: eip + 1 = 0, which is considered to be one of the most beautifully elegant equations ever derived from the fundamental constants.

Euler's Constant

Euler's number, "e", is one of the most important (if not the most important) fundamental constants in mathematics. It forms the base of the "natural" logarithm. It is an irrational number (meaning that it has no exact value, much like pi and can continue to be written forever), and while no one is exactly sure why, it is incredibly useful in describing a great many real-world phenomena, such as the continuous growth of compound interest and other exponential functions.

Number Theory

Euler's phenomenal mind took him on several tangents through the world of number theory (a mathematical subject which deals with the relationship of numbers with each other, and in grouping and categorizing them).

Even as a blind old man, Euler discovered many new and interesting ways to deal with infinite series of numbers, along with advancements in prime number theory and proofs regarding the divergence of the reciprocal of prime numbers. This was in a time far before modern day computers, calculators and even primitive adding machines, another wonderful testament to the true abilities of his mind.

Topology

Before he was thirty, Euler was able to solve a famous problem called "The Bridges of Konigsberg." The problem revolved around a river which flowed through Konigsberg, Prussia, forming two large islands, connected to the mainland and each other by a total of seven bridges. Euler was able to find proof that there existed no possible route which traversed all seven bridges only once.

While the subject had not yet come into his own, this was truly one of the first great problems in the area of "topology," which deals with the quantification and classification of objects in a geometrical sense. Euler's development of the idea of an "Eulerian Circuit" (a path which would allow each point to be crossed only once, nonexistent in the problem above) can still be used today in such modern mathematical forms as in knot theory.

Outside of Math

Euler's achievements extended far beyond mathematics, although this is where most of his work took him. He also was able to undertake problems which led to advancements in practical engineering, e.g. the Euler-Bernoulli Beam Equation and astronomy, using his mathematical skills to calculate the motion of heavenly bodies according to Newton's theories to a great accuracy.

Even theology was not out of Euler's reach, perhaps best (and most humorously) exemplified by his famous statement in a debate with a prominent atheist: "Sir, (a + bn)/z = x, therefore God exists - reply!"

Leonard Euler's impact on the thinking world of the eighteenth century extends far beyond the limited examples found here, of course. Thousands of pages of his work have survived the past three centuries, a great deal of which are filled with new, original, and altogether brilliant ideas. A search throughout history finds one very hard-pressed to find a more revolutionary figure in the history of mathematics than Euler.