Lives of Archimedes and Carl Friedrich Gauss, Term Paper

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lives of Archimedes and Carl Friedrich Gauss, two of the greatest mathematicians of all time, through a point by point comparison of their childhood and education, mathematical contributions and the influence their work has on the science of mathematics.

Childhood and Education

Archimedes (287 BC to 212 BC) lived most of his life in Syracuse, Greece. This son of an astronomer and mathematician was born into a distinguished family and was able to comfortably devote his life to mathematical research.

Carl Friedrich Gauss (1777-1855) was born into a humble German family. His early mathematical promise marked him as a prodigy and eventually earned him admission to university.

Major Mathematical Ideas

The mathematical work of Archimedes centered on the theoretical, particularly geometry. His greatest mathematical contribution involved measuring areas and segments of plane and conic sections.

Gauss's work centered on number theory. Unlike Archimedes, Gauss also used ventured into applied mathematics like astronomy and geodetic research.

Influence on Mathematics

Archimedes's mathematical treatises the work of Arabic mathematicians in the eighth and ninth centuries. Centuries later, translations of his writings contributed significantly to the work of physicists such as Johannes Kepler and Galileo, as well as mathematicians like Rene Descartes and Pierre de Fermat.

Gauss's interest in gravitation eventually formed the basis of the modern theory of potential. Like Archimedes, he made significant contributions to physics, including Weber's theory on the conservation of energy. His work in electromagnetism led to the development of the telegraph and still influences the development of modern telecommunications.

Conclusion

The work of Archimedes and Gauss revolutionized mathematics and continues to make significant contributions to modern mathematical theory and its applications.

Archimedes and Carl Friedrich Gauss are two of the greatest mathematicians who ever lived. Though their lives were separated by 2,000 years, their writings and discoveries have made definitive contributions to the science of mathematics.

This paper looks at the lives of these two brilliant mathematicians by comparing their childhood and education, their mathematical contributions and the enduring influence their work continues to have on mathematics today.

Childhood and Education

Archimedes, the most famous mathematician of antiquity, was born c. 287 BC in Syracuse, the principal city-state in Sicily. He father, the astronomer Phidias, was a good friend and adviser of King Hieron II of Syracuse. The young Archimedes studied in Alexandria under Euclid. He eventually returned to Syracuse and pursued his own theoretical mathematical research (Boyer 120-121).

Far more details survive about the life of Archimedes than about any other ancient scientist, but scholars disagree on which details are fact and which are anecdotal. The most famous Archimedes story centers on how he determined the proportion of gold and silver in a crown made for Hieron through measuring water displacement. Since he supposedly made the discovery while in the bathtub, the excited Archimedes ran naked through the streets of Syracuse shouting "Eureka!" (Muir 20).

Almost as famous are the stories about his death. Though the catapults and cranes he designed delayed the fall of Syracuse, the city was eventually captured by the Roman general Marcus Claudius Marcellus in the autumn of 212 or spring of 211 BC. According to legend, Archimedes was at work on mathematical diagrams when Roman soldiers stormed his chambers. The mathematician exclaimed "Don't disturb my diagrams!" And was stabbed by enraged enemy soldiers (Riley 44).

Carl Friedrich Gauss was born over 2,000 years after Archimedes, on April 30, 1777. Gauss was the only son of poor parents in New Brunswick, now part of Germany. However, he showed advanced mathematical abilities at an early age. At three, the young Gauss corrected mistakes in his father's summation figures. At seven, Gauss astounded his elementary school teachers by summing the integers from 1 to 100 instantly. The young prodigy realized that the sum was 50 pairs of numbers, with each pair summing to 101. This was the same technique employed more than 2,000 years ago by Pythagoras (Muir 157-159).

Unlike Archimedes, who was the son of an astronomer and mathematician, Gauss did not receive any encouragement from his father. In fact, the elder Gauss, a laborer and gardener, tried to push his son into the weaving trade. However, the Duke of Brunswick, became Gauss's benefactor. Gauss eventually went t study at Gottingen University, then later returned to Brunswick for his degree. A stipend from the Duke allowed Gauss to devote himself to his doctoral research at the University of Helmstedt (Muir 159).

Gauss's personal life was filled with tragedy.

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His first wife died in childbirth and eventually remarried. He had a total of four children and supported his family as director of the Gottingen Observatory. In the seclusion of the observatory, Gauss continued his mathematical studies and wrote many of his research (Bell 244-245).

Unlike Archimedes, Gauss lived long enough to be honored for his work. Because of his work, the city of Gottingen granted him honorary citizenship. Many academies and learned societies elected Gauss as a fellow based on his many writings in mathematics, physics, geodesy and astronomy. He also received several invitations to become a professor with many prestigious universities. However, Gauss decided to remain in Gottingen until his death on February 23, 1855 ("Gauss").

Major Ideas

Unlike Gauss, Archimedes was hardly interested in the mechanical applications of his work. Instead, he concentrated on theory, particularly in the fields of geometry, in statics and hydrostatics and in numerical methods (Riley 44).

Archimedes's greatest contribution to geometry was related to volume. He proved that the volume of a sphere inscribed in a cylinder is only two-thirds the volume of a cylinder. This discovery solved many geometric measuring problems of his day. In addition, Archimedes also developed the formulas for computing the areas of ellipses and parabola segments. By developing the formula for computing the area of a circle, Archimedes also developed a method for calculating pi. (Bell 30).

Archimedes also developed a method for finding square roots centuries before Hindu mathematicians invented periodic continued fractions. Because Greek and Roman numerical systems proved inadequate in calculating large numbers, Archimedes also developed his own numerical notation system. His mechanical principles involving levers also allowed him to calculate the areas and centers of several irregularly shaped flat polygons (Bell 31).

If Archimedes is the Father of Mathematics, Gauss is the science's modern day prince. The former's work was pioneering in the sense that he had to develop new mathematical concepts. In Archimedes's time, the concept of pi touched on the idea of infinitesimal analysis. This was during a time when the notation of numbers itself was inadequate. Archimedes had to devise his own system of notation to express numbers to infinity.

Gauss, on the other hand, built on the work of mathematicians before him. However, his contributions to number theory are pioneering in their own way. At age 24, he published the Disquisitiones Arithmeticae, his formulations of new concepts and methods in numbers theory. This book remains one of the most important treatises in the history of mathematics and formed the foundation of the modern arithmetical theory of algebraic numbers (Boyer 500-503).

Gauss, unlike Archimedes, also used his mathematical gifts for the more practical mathematical fields, such as astronomy and geodetic research. He developed a technique for calculating orbital components, a technique that allowed astronomers to locate asteroids in the sky.

At around 1820, he turned his attention to geodesy -- the mathematical determination of the shape and size of the Earth's surface. He invented the heliotrope, a measuring instrument using sunlight for increased accuracy, to help surveyors arrive at more exact measurements (Muir 175).

He also worked in the field of probability, denoting probability in the form of a bell-shaped curve. This curve is now known as the Gaussian error curve or, more commonly, the bell curve. This graphical indication of variations is now a basic method of shoring statistical distributions (Muir 173).

In 1830, Gauss began to dabble in mathematical investigations of physical problems, such as the conditions under which fluid remains at rest. His inquiry into capillary action led Gauss to devise new mathematical formulations involving interactions among the fluid particles and the force of gravity, work that had significant implications for the development of the law of conservation of energy. Gauss also worked closely with the physicist Wilhelm Weber studying terrestrial magnetism. Unfortunately for the field of mathematics, Gauss's interest in the science began to wane when he was in his seventies. He stopped teaching and instead took up new pursuits, such as languages. At the time of his death, he spoke eight different languages fluently (Muir 181).

Influence on Mathematics

Because his work was so far ahead of its time, the influence of Archimedes's work in antiquity was very small. Several centuries passed until Arabic mathematicians rediscovered his work in the late seventh or eighth centuries. The great mathematical works of Arab mathematicians during the early medieval period were built on.....

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