Archimedes and Gauss: Two Great Mathematicians Compared

Introduction

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

This paper compares the lives of these two brilliant mathematicians by examining their childhood and education, their major 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. His 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 by 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 the enraged 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 Brunswick, now part of Germany. He showed advanced mathematical abilities at a very early age. At three, the young Gauss corrected mistakes in his father's summation figures. At seven, he 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, each pair summing to 101 — the same technique employed more than 2,000 years earlier by Pythagoras (Muir 157–159).

Unlike Archimedes, who was the son of an astronomer and mathematician, Gauss received no 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 to study at university, then later returned to Brunswick for his degree. A stipend from the Duke allowed Gauss to devote himself to his doctoral research (Muir 159).

Gauss's personal life was filled with tragedy. His first wife died in childbirth, and he eventually remarried. He had a total of four children and supported his family as director of the university observatory. In the seclusion of the observatory, Gauss continued his mathematical studies and produced many of his major works (Bell 244–245).

Unlike Archimedes, Gauss lived long enough to be widely honored for his accomplishments. Many academies and learned societies elected him as a fellow based on his prolific writings in mathematics, physics, geodesy, and astronomy. He also received several invitations to accept professorships at prestigious universities. However, Gauss chose to remain in Brunswick until his death on February 23, 1855 ("Gauss").

Major Mathematical 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, statics, hydrostatics, and numerical methods (Riley 44).

Archimedes's greatest contribution to geometry concerned volume. He proved that the volume of a sphere inscribed in a cylinder is only two-thirds the volume of the cylinder. This discovery resolved many geometric measuring problems of his day. Archimedes also developed formulas for computing the areas of ellipses and parabola segments. By developing the formula for computing the area of a circle, he simultaneously 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 for calculating large numbers, he devised 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. Archimedes's work was pioneering in the sense that he had to develop entirely new mathematical concepts. In his time, the concept of pi touched on the idea of infinitesimal analysis during an era when numerical notation itself was inadequate. Archimedes had to devise his own system of notation simply to express numbers approaching infinity.

Gauss, by contrast, built on the work of mathematicians before him — yet his contributions to number theory were pioneering in their own right. At age 24, he published the Disquisitiones Arithmeticae, presenting new concepts and methods in number 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 also applied his mathematical gifts to practical fields such as astronomy and geodetic research, areas that Archimedes largely ignored. Gauss developed a technique for calculating orbital components that allowed astronomers to locate asteroids in the sky.

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 that used reflected sunlight for increased accuracy, helping surveyors arrive at more precise measurements (Muir 175).

Gauss also worked in the field of probability, representing probability distributions 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 representation of variation is now a fundamental method for illustrating statistical distributions (Muir 173).

In 1830, Gauss began investigating physical problems mathematically, such as the conditions under which a fluid remains at rest. His inquiry into capillary action led him to devise new mathematical formulations involving interactions among fluid particles and the force of gravity — work with significant implications for the development of the law of conservation of energy. Gauss also collaborated closely with the physicist Wilhelm Weber on the study of terrestrial magnetism. Unfortunately for the field of mathematics, Gauss's interest in mathematical research began to wane in his seventies. He stopped teaching and took up new pursuits, including languages. At the time of his death, he spoke eight languages fluently (Muir 181).

Conclusion

The work of Archimedes and Gauss continues to make significant contributions to all fields of mathematics. Many mathematical disciplines would not even be possible without their work.

Both men faced significant obstacles to their mathematical research. Archimedes lived in a time of very limited mathematical knowledge. Even commonplace mathematical concepts today — the infinitesimal, pi, and infinity — were unknown during his era. The cumbersome Greek and Roman numeral systems were unwieldy for his computational needs, so he was compelled to devise his own notation.

Gauss, by contrast, was born into poverty and lacked his family's support, living in or close to poverty for much of his life.

Yet both men ultimately overcame these obstacles to produce research of lasting significance. Centuries after their deaths, the work of these two great minds continues to revolutionize the science of mathematics.

Works Cited

"Archimedes." Guide to the History of Calculus. Retrieved 30 November 2002.

Bell, E. T. Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré. New York and London: Simon and Schuster, 1965.

Boyer, Carl B. A History of Mathematics, 2nd ed. New York: John Wiley and Sons, 1991.

"Gauss." Guide to the History of Calculus. Retrieved 30 November 2002.

Muir, Jane. Of Men and Numbers: The Story of the Great Mathematicians. New York: Dover Publications, 1996.

Riley, Mark T. "Archimedes." Great Thinkers of the Western World. Ed. Ian P. McGreal. New York: HarperCollins Publishers, 1992.