Modern mathematics incorporates the insights and ideas of many brilliant mathematical scholars of different epochs. Some of these mathematicians expanded the subject by introducing radically new theories and unexplored horizons. Leonhard Euler is of one the famous and accomplished members of this elite group of mathematicians. Euler earned his reputation in history as an ingenious mathematician by exploring new dimensions in mathematics. One of his best known discoveries is his identity,
Euler was a very gifted mathematician, not only in terms of what he accomplished, but also with regard to his methods. I believe Euler was an abstract thinker, and by this I mean two things; first, that he could clearly visualize challenging mathematical concepts in his mind with few if any concrete figures or drawings to guide him, and second, that he was able to apply transcendental ideas to known mathematical subjects. His talent was seen early in his life when he was only eighteen, in his paper on the masting of ships, and more fully later in his career in 1743 when he developed the formula cited above.
On April 15, 1707 Basel, Switzerland gave birth to one of its greatest intellects, Leonhard Euler. (Bell, p. 143) Euler's father, Paul Euler, was a successful mathematician *who studied under Jakob Bernoulli (1654-1705). (Bell, p. 143) Paul Euler was Euler's first mathematics teacher, and he was also a Calvinist minister. Calvinism was an outgrowth of the Reformation initiated by Martin Luther in 1517 and became influential in Switzerland around 1546 when Protestants started ". insisting that the people - not just kings and bishops - should share in political and religious policymaking." (World Book Encyclopedia, p. 71)
Leonhard Euler attended the University of Basel, where he studied mathematics under Bernoulli's brother, Johann Bernoulli (1667-1748). At the very young age of fifteen Euler received his Bachelor's degree from that University and one year later completed his Masters degree. After earning his Masters degree, Euler came under pressure from his father to follow in his footsteps by studying theology, and entertained the idea of becoming a Calvinist minister. Although Euler ". never discarded a particle of his Calvinistic faith" he still did not wish to end his mathematical studies in favor of the study of theology. (Bell, p. 144) Nevertheless Euler did briefly halt his studies in mathematics and obeyed his father. During this period Johann Bernoulli met with Paul Euler and told him that Leonhard was a talented mathematician, not meant to be a Calvinist minister. (Bell, p. 144) Soon after this, Euler's father changed his mind and permitted his son to continue his studies in mathematics.
The world first became aware of Euler's abilities when he published a paper about the "masting of ships". Euler submitted this paper as an entry in the French Academy of Science's annual contest. In competition not only with other graduate students but with many accomplished mathematicians and scientists, he was still able to win second prize. (Muir, p. 139) Presumably this paper discussed the physics and mathematics involved in the support of the mast, "a tall vertical spar that rises from the keel of a sailing vessel to support the sail and rigging." (American Heritage Dictionary, p. 512)
I consider this paper to be an example of Euler's abstract manner of thinking because he had very limited concrete knowledge, if any, of ships while writing this paper. Due to the fact that Euler lived in Switzerland, a land-locked country, he was not given the opportunity to see ocean-going ships. (Bell, p. 144) Therefore he wrote this paper without a concrete basis, namely a ship, to illustrate his findings or to verify his results. (Muir, p. 139) Since Euler was able to write an award winning paper on the masting of ships without apparently ever seeing one, I would definitely classify him as an abstract thinker.
A few years later, in 1727, at the age of twenty, Euler was persuaded by two of his friends, Daniel Bernoulli (1700-1782) and his older brother Nicholas Bernoulli (1695- 1726), to apply for a position in the physiology department at the recently developed Imperial Russian Academy of Science in St. Petersburg. (Muir, p. 140)
At this stage of his career Euler seems to have been curiously indifferent as to his field of activity, provided only that it was something scientific. When the Bernoullis wrote of a prospective opening in the medical section of the St. Petersburg Academy, Euler flung himself into physiology at Basel and attended the lectures on medicine. (Bell, p. 144)
After quickly taking a few courses in medicine, Euler was offered a position at the Academy. (Muir, p. 140) He was pleased to be working at the Academy with his friends. Shortly after his acceptance, a vacancy in the mathematics department became available, and Euler was able to work in the field he loved again. At this time he became the Academy's chief mathematician. (Eves, p. 432) Euler stayed in St. Petersburg and worked at the Academy for fourteen years.
While working at the Academy Euler met Katharina Gsell, who came to Russia when her father, a Swiss painter, came to work for Peter the Great. (Turnbull, p. 109) In 1734 they were married, and soon started a family. They had thirteen children, but only five survived childhood. Having a big family did not interfere with Euler's work. (Bell, p. 145)
[Euler's] mind worked like lightning and was capable of intense concentration . Euler needed neither quiet nor solitude. Most of his work was done at home in the bedlam created by several small children noisily playing around his desk. Euler remained undisturbed and often rocked a baby with one hand while working out the most difficult problems with the other. He could be interrupted constantly and then easily proceed from where he had left off without losing either his train of thought or his temper. (Muir, p.141)
This description shows the magnitude of Euler's talent. His thought processes did not require long periods of intense concentration.
Since Euler had little or no problems concentrating with several interruptions, he must have been able to stop and then restart problems without losing large pieces of his progress. I believe this skill also characterizes an abstract thinker because Euler must have been able to remember the progress he made before he was interrupted without performing a time consuming review. This theory may be supported by the circumstances of Euler's life around 1766, when he became totally blind. This obstacle did not become a serious handicap for Euler or for his work. He was able to carry out long and tedious computations of fifteen digits in his head. (Turnbull, p. 111) This is a perfect example of an abstract thinker, according to my definition. At the beginning of this paper I defined an abstract thinker as a person that could visualize or in this case perform complicated mathematical concepts in their mind.
In 1741 Euler left the Russian Academy and moved to Berlin. (Turnbull, p. 110) While in Berlin, Euler published a book entitled Introductio in analysis infintiorum (Introduction to Analysis of the Infinite). This book, first published in 1784, shows some of Euler's transcendental ideas. It contains his formula, discovered in 1743, eix= cos(x)+ isin(x), which provides a link between trigonometry and exponential functions. He was "the first to link trigonometric, exponential, and logarithmic functions by his identity, making e the base for natural logarithms. This work. led to Euler's derivation of the fundamental equation ei?+1=0." (Calinger, p. 140) Using the Taylor series expansions of ex, sin(x), and cos(x), we can derive Euler's formula by substituting ix for x in ex. The computation goes as follows:
If we now make another substitution, letting x= pi, we can derive the fundamental equation:
Euler's idea of substituting the imaginary variable ix for the real variable x is an example of one of his transcendental ideas or abstract thought process. Euler does not explain why he decided to make this substitution, or why it is valid: he just simply does it.
Euler's great accomplishments do not end with his book, Introductio in analysis infintiorum. Throughout his long career, Euler wrote and published 530 books and papers. He was very productive until his death in 1783. "It is said that Euler calculated as long as he breathed. Old -past seventy -blind, and slightly deaf, he continued to produce his prodigious works." (Muir, p. 154) All of his achievements in mathematics cannot be listed and explained in one paper or even one book because Euler has contributed numerous elements to the subject of mathematics.
Euler's abstract thinking greatly influenced his early work. Without this skill or talent his career would have surely ended when he became blind, his book, Introductio in Analysis Infinitorum, would be without this transcendental ideas, including his formula, e^(ix)= cos(x)+ i sin(x), and his article on the masting of ships could have never been written without his seeing a ship. Euler's abstract reasoning enhanced his mathematical ability, and helped him achieve many of his great discovers.
Bell, E.T., Men of Mathematics, Simon and Schuster, New York, 1937.
Calinger, Ronald, Leonhard Euler: The First St. Petersburg Years (1727- 1747), Historia Mathematica 23 (1996), 121-166, Academic Press Inc., Washington D.C., 1996.
Euler, Leonhard, translated by Hewlett, John, Elements of Algebra, Springer-Verlag, New York, 1972.
Euler, Leonhard, translated by Blanton, John D., Introduction to Analysis of the Infinite, Book I and II, Springer-Verlag, New York, 1990.
Eves, Howard, An Introduction to the History of Mathematics, Saunders College Publishing, Fort Worth, 1990.
Muir, Jane, Of Men and Numbers, Dover Publications, Inc., New York.
Turnbull, H.W., The Great Mathematicians, New York.
Beiträe zu Leben und Werk, Leonhard Euler, Gedenkaband des Kantons Basel-Stadt, Birkhäuser Verlag Basel, 1983. (Not cited)
The American Heritage Dictionary, Dell Publishing, New York, 1994.
World Book Encyclopedia Vol. 3, World Book Inc., Chicago, 1994.