Ancient Greek mathematicians laid the groundwork for calculus with methods like exhaustion and tangent studies. These approaches approximated curved areas and explored lines touching curves, setting the stage for integral and differential calculus.

In the 17th century, mathematicians built on these foundations. Cavalieri's principle, Kepler's wine barrel problem, and Fermat's method of adequality pushed mathematical thinking closer to the formal development of calculus.

Ancient Greek Methods

Method of Exhaustion and Quadrature

• Method of exhaustion approximated curved areas using inscribed and circumscribed polygons
• Increased number of sides in polygons led to more accurate approximations
• Eudoxus of Cnidus developed this method in the 4th century BCE
• Archimedes refined and applied the method to calculate areas and volumes
• Quadrature involved finding a square with an area equal to a given curved figure
• Greeks used this method to solve problems like squaring the circle
• Hippocrates of Chios successfully squared lunes (crescent-shaped figures)
• Method of exhaustion laid groundwork for integral calculus

Tangent Problem

• Tangent problem focused on finding lines touching curves at a single point
• Ancient Greeks studied tangents to circles and conic sections
• Euclid's Elements included propositions on tangents to circles
• Apollonius of Perga extended tangent studies to conic sections
• Archimedes developed method of tangents for spiral curves
• Tangent problem became crucial in development of differential calculus
• Ancient Greek approaches to tangents influenced later mathematicians (Fermat, Descartes)

17th Century Developments

Cavalieri's Principle and Kepler's Wine Barrel Problem

• Cavalieri's principle stated that volumes of two solids are equal if their corresponding cross-sections have equal areas
• Bonaventura Cavalieri introduced this principle in 1635
• Principle provided a method for comparing volumes of complex shapes
• Kepler's wine barrel problem involved finding optimal dimensions for wine barrels
• Johannes Kepler tackled this problem in 1615
• Problem required calculating volumes of solids of revolution
• Kepler's work influenced development of infinitesimal methods

Torricelli's Infinite Geometric Series and Barrow's Differential Triangle

• Evangelista Torricelli discovered an infinite geometric series with a finite sum
• Series represented the volume of Gabriel's Horn (trumpet-shaped solid with finite volume but infinite surface area)
• Torricelli's work contributed to understanding of infinite processes in mathematics
• Isaac Barrow introduced the differential triangle concept
• Differential triangle related infinitesimal changes in x and y coordinates
• Barrow's work provided geometric foundation for differentiation
• Concept of differential triangle influenced Newton's development of calculus

Fermat's Contributions

Fermat's Method of Adequality

• Pierre de Fermat developed method of adequality in the 1630s
• Adequality used to find maxima, minima, and tangents to curves
• Method involved comparing two nearby points on a curve
• Fermat introduced concept of "adequal" (approximately equal) for infinitesimal differences
• Process included setting up an equation, introducing a small increment, and canceling common terms
• Method of adequality considered precursor to differential calculus
• Fermat applied this method to various problems (finding tangents to parabolas, ellipses)
• Adequality influenced later mathematicians (Newton, Leibniz) in development of calculus