The Illustrated Method of Archimedes: Utilizing the Law of the Lever to Calculate Areas, Volumes, and Centers of Gravity

In 1906 Johan Ludwig Heiberg (1854-1928), a Danish philologist and historian of science, discovered a previously unknown text of Archimedes (287-212 B.C.). It was a letter addressed to Eratosthenes (285-194 B.C.), the famous Greek scholar and head librarian of the Great Library of Alexandria. In it, Archimedes presented a heuristic method for calculating areas, volumes and centers of gravity of geometric figures utilizing the law of the lever. This book presents the essence of Archimedes’s method, concentrating on the physical aspects of his calculations. Figures illustrate all levers in equilibrium, and the postulates he utilized are emphasized. The mathematics is kept to the minimum necessary for the proofs. The definition of the center of gravity of rigid bodies is presented, together with its experimental and theoretical determinations. The law of the lever is discussed in detail. The main results obtained by Archimedes concerning the circle and sphere are also discussed. The book describes the lemmas utilized by Archimedes. The main portion of the book sets out the physical demonstrations of theorems I (area of a parabolic segment), II (volume of a sphere) and V (center of gravity of a segment of a paraboloid of revolution). The importance of these three theorems is discussed. There is a bibliography at the end of this book.