5.3 Measurement of the circle and early calculus concepts
3 min read•august 9, 2024
revolutionized mathematics with his groundbreaking work on circles and early calculus concepts. He approximated using polygons and proved the formula for circles, laying the foundation for future advancements in geometry and measurement.
His and integration-like techniques paved the way for calculus. Archimedes' axiom and heuristic approaches in "" 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 to its diameter
Archimedes approximated π between 3 10/71 and 3 1/7 using inscribed and
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=πr2
Archimedes proved this formula using the method of exhaustion
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
(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 "" (c. 1000 BCE) used 3 as an approximation for π
Early Calculus Concepts
Limits and Infinitesimals
involve finding values that a function approaches as the input nears a particular value
Concept of limits crucial for defining and derivatives
represent infinitely small quantities used in early calculus development
Archimedes implicitly used limit concepts in his method of exhaustion
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
, developed by Bonaventura Cavalieri, viewed areas as composed of infinitely thin lines
states that volumes of two solids are equal if their corresponding cross-sections have equal areas
refined the method of indivisibles, contributing to the development of
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 "" 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
Key Terms to Review (22)
Archimedes: Archimedes was an ancient Greek mathematician, physicist, engineer, and inventor who made significant contributions to mathematics, particularly in the fields of geometry and calculus. His work laid the groundwork for the understanding of the measurement of circles and the early concepts of calculus, influencing later mathematicians and scientists.
Area: Area is the measure of the amount of space within a two-dimensional shape or figure, typically expressed in square units. Understanding area is essential for calculating quantities in various contexts, such as determining the size of land or the surface of a circle, which is particularly relevant to early mathematical concepts and calculus. The calculation of area laid foundational principles that would later evolve into more complex mathematical ideas.
Cavalieri's Principle: Cavalieri's Principle states that if two regions in a plane or space have the same height and every cross-section at a given height has the same area, then the volumes of these regions are equal. This principle connects to various mathematical concepts, allowing for the comparison of different shapes based on their cross-sectional areas, which is a foundational idea in early calculus and geometry.
Circumference: Circumference is the distance around the edge of a circle, representing a crucial measurement in understanding circular shapes. It is commonly calculated using the formula $$C = 2\pi r$$, where $$C$$ is the circumference and $$r$$ is the radius. This concept is foundational for more advanced topics, including the early calculus concepts that involve limits and the calculation of areas and volumes.
Circumscribed polygons: Circumscribed polygons are polygons that are drawn around a circle, such that all the vertices of the polygon touch the circumference of the circle. This relationship between the polygon and the circle is significant in understanding geometric properties and measurements, including calculating areas and perimeters, which are fundamental concepts when studying the circle and the early ideas leading to calculus.
Continuity: Continuity refers to the property of a function that, intuitively, means the function does not have any sudden jumps or breaks. This concept is essential for understanding limits, derivatives, and integrals in calculus, as it ensures that a function behaves predictably in a neighborhood around any given point. In mathematical analysis, continuity becomes a fundamental requirement for many theorems and principles, linking it closely to the foundational aspects of early calculus and the later rigorization efforts in analysis and set theory.
Derivative: A derivative represents the rate at which a function is changing at any given point, providing a way to understand how small changes in input affect changes in output. This concept is foundational in calculus, connecting deeply to the measurement of motion and change, particularly in relation to curves, such as those seen in the measurement of circles. The derivative not only allows for the determination of slopes of tangent lines but also forms the basis for understanding acceleration, optimization problems, and more advanced mathematical concepts.
Egyptian Rhind Papyrus: The Egyptian Rhind Papyrus is an ancient mathematical text from around 1650 BCE that provides a glimpse into the mathematical knowledge and practices of ancient Egypt. It contains a variety of mathematical problems, including those related to the measurement of circles, which directly contributes to understanding early calculus concepts through the approximation of pi.
Eudoxus of Cnidus: Eudoxus of Cnidus was an ancient Greek mathematician and astronomer, known for his contributions to geometry and the method of exhaustion, which laid groundwork for calculus concepts. His work focused on the measurement of geometric figures, particularly circles, and he is recognized for developing a rigorous approach to calculating areas and volumes, influencing later mathematicians and early calculus ideas.
Euler's number: Euler's number, denoted as 'e', is an important mathematical constant approximately equal to 2.71828. It is the base of natural logarithms and arises in various contexts, particularly in calculus and the study of exponential growth and decay. Its unique properties make it essential in connecting the measurement of circles, growth rates, and the foundational concepts of early calculus.
Golden ratio: The golden ratio, often denoted by the Greek letter phi (φ), is a special number approximately equal to 1.6180339887. It is defined mathematically as the ratio between two quantities where the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. This fascinating ratio appears in various aspects of nature, art, and mathematics, connecting geometric measurements and aesthetic principles.
Infinitesimals: Infinitesimals are quantities that are so small that they approach zero, but never actually reach it. This concept is crucial in early calculus as it allows for the understanding of limits and the calculation of derivatives and integrals. Infinitesimals provide a way to rigorously describe changes in values that are exceedingly small, which is fundamental to measuring curves and areas, such as those involving circles.
Inscribed Polygons: Inscribed polygons are geometric figures where all vertices lie on the circumference of a circle. This concept is essential in understanding the relationship between polygons and circles, particularly in the measurement of circular areas and the development of early calculus concepts, as these polygons can approximate the area of the circle as more sides are added.
Integral calculus: Integral calculus is a branch of mathematics focused on the concept of integration, which involves finding the area under curves and determining accumulated quantities. It plays a crucial role in connecting various mathematical concepts, such as limits, derivatives, and functions, while providing tools for solving real-world problems related to motion, area, and volume.
John Wallis: John Wallis was a prominent 17th-century English mathematician known for his contributions to the early development of calculus and his work on the measurement of curves, particularly circles. His innovative ideas laid the groundwork for integral calculus and advanced the understanding of infinite series, making significant connections between geometry and algebra.
Limits: Limits refer to the fundamental concept in calculus that describes the behavior of a function as its input approaches a particular value. This idea is crucial for understanding continuity, derivatives, and integrals, as it helps mathematicians define and analyze how functions behave at specific points or over intervals. The concept of limits was pivotal in the development of early calculus, laying the groundwork for rigorous mathematical analysis in later theories.
Method of exhaustion: The method of exhaustion is a mathematical technique used to find the area of shapes by approximating them with a series of inscribed or circumscribed figures and taking limits. This approach allows for precise calculations of areas and volumes, laying the groundwork for later developments in calculus. It essentially breaks down complex shapes into simpler parts, helping mathematicians to derive their properties more systematically.
Method of indivisibles: The method of indivisibles is a mathematical technique used to calculate areas and volumes by considering them as an infinite number of infinitesimally small parts or 'indivisibles'. This method was a precursor to integral calculus and played a vital role in the measurement of curves, particularly in determining the area of circles and other shapes. By breaking down shapes into these tiny components, early mathematicians could approximate values and understand properties that were previously difficult to quantify.
On the sphere and cylinder: On the sphere and cylinder refers to the geometric study of shapes and their properties, focusing on how circles relate to spheres and cylinders. This concept is crucial in understanding the measurement of circular objects, which leads to early calculus concepts that deal with area, volume, and surface calculations of these three-dimensional shapes.
Pi: Pi is a mathematical constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. This irrational number plays a fundamental role in geometry and trigonometry, especially in calculations involving circles and circular shapes. It is also essential in various mathematical concepts, linking measurement, calculus, and the contributions of ancient mathematicians.
The method: The method refers to systematic approaches used in mathematics to solve problems, derive formulas, and understand geometric figures, particularly in relation to the measurement of circles and the development of early calculus concepts. This includes techniques such as the method of exhaustion, which approximates areas and volumes, and early calculus approaches that rely on limits and infinitesimals. Understanding this term is crucial for grasping how mathematicians historically tackled complex problems and laid the groundwork for modern mathematics.
Zhoubi suanjing: The Zhoubi Suanjing, also known as the 'The Nine Chapters on the Mathematical Art', is an ancient Chinese mathematical text that is considered one of the earliest works on mathematics in China, dating back to around the 1st century BCE. This text includes methodologies for measurement and geometry, particularly focusing on calculating the area and circumference of circles, reflecting early concepts that relate closely to calculus.