5.3 Measurement of the circle and early calculus concepts

Archimedes revolutionized mathematics with his groundbreaking work on circles and early calculus concepts. He approximated pi using polygons and proved the area formula for circles, laying the foundation for future advancements in geometry and measurement.

His method of exhaustion and integration-like techniques paved the way for calculus. Archimedes' axiom and heuristic approaches in "The Method" showcased his innovative thinking, influencing mathematical development for centuries to come.

Measurement of the Circle

Pi (π) and Circumference

• Pi (π) represents the ratio of a circle's circumference to its diameter
• Archimedes approximated π between 3 10/71 and 3 1/7 using inscribed and circumscribed polygons
• Circumference of a circle calculated using the formula $C = 2πr$ or $C = πd$
• Chinese mathematician Zu Chongzhi refined π to 3.1415926 < π < 3.1415927 in the 5th century CE
• Modern computers have calculated π to trillions of decimal places

Area and Polygonal Approximations

• Area of a circle determined by the formula $A = πr^2$
• Archimedes proved this formula using the method of exhaustion
• Inscribed polygons fit inside the circle, while circumscribed polygons enclose it
• Increasing the number of sides in these polygons improves the approximation of the circle's area
• Method of exhaustion involves finding upper and lower bounds that converge to the true value

Applications and Historical Significance

• Accurate π calculations crucial for architecture (dome construction)
• Improved circular measurements advanced astronomy and navigation
• Egyptian Rhind Papyrus (c. 1650 BCE) used 3.16 as an approximation for π
• Babylonians used 3 1/8 as an approximation for π in the Old Babylonian period
• Chinese text "Zhoubi Suanjing" (c. 1000 BCE) used 3 as an approximation for π

Early Calculus Concepts

Limits and Infinitesimals

• Limits involve finding values that a function approaches as the input nears a particular value
• Concept of limits crucial for defining continuity and derivatives
• Infinitesimals represent infinitely small quantities used in early calculus development
• Archimedes implicitly used limit concepts in his method of exhaustion
• Eudoxus of Cnidus developed the method of exhaustion, laying groundwork for limit theory

Integration and the Method of Indivisibles

• Integration calculates areas under curves and volumes of solids
• Archimedes used integration-like techniques to find areas and volumes of various shapes
• Method of indivisibles, developed by Bonaventura Cavalieri, viewed areas as composed of infinitely thin lines
• Cavalieri's principle states that volumes of two solids are equal if their corresponding cross-sections have equal areas
• John Wallis refined the method of indivisibles, contributing to the development of integral calculus

Archimedes' Contributions and Axiom

• Archimedes' axiom states that for any two line segments, a multiple of the smaller can exceed the larger
• This axiom forms the basis for the concept of real numbers and continuity
• Archimedes used the method of exhaustion to calculate areas and volumes of various shapes (parabolic segments)
• His work "On the Sphere and Cylinder" applied these methods to determine surface areas and volumes of spheres
• Archimedes' "The Method" revealed his heuristic approach using mechanical analogies to discover mathematical truths