History of Calculus CURVES AREAS VOLUMES

METHOD OF EXHAUSTION Discovered by Antiphon Inscribe a shape with multiple polygons whose area converge to the area of the shape

Eudoxus Eudoxus rigorously developed Antiphon’s Method of Exhaustion  For the calculation of the volume of the pyramid and cone. Archimedes notes that Eudoxus was the first to prove that the cone and the pyramid are one-third respectively of the cylinder and prism with the same base and height.

Archimedes Archimedes used the method of exhaustion to compute the area inside a circle

Other results obtained by Archimedes using the method of exhaustion Archimedes cont... Other results obtained by Archimedes using the method of exhaustion The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height; The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes; The volume of a sphere is 4 times that of a cone having a base and height of the same radius; The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter; The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length; Use of the method of exhaustion also led to the successful evaluation of a geometric series (for the first time). Estimations of Pi were being worked out in ancient Greece

Liu Hui (3rd Century AD) Liu provided a detailed step-by-step description of an iterative algorithm to calculate pi to any required accuracy based on bisecting polygons; he calculated pi to between 3.141024 and 3.142708 with a 96-gon; he suggested that 3.14 was a good enough approximation, and expressed pi as 157/50; he admitted that this number was a bit small. Later he invented an ingenious quick method to improve on it, and obtained π ≈ 3.1416 with only a 96-gon, with an accuracy comparable to that from a 1536-gon. Look at MS Word notes for quick method.

Gregoire de Saint-Vincent Method of Exhaustion was expanded by Gregoire de Saint-Vincent when he discovered that the area under a rectangular hyperbola (i.e. a curve given by xy = k) is the same over [a,b] as over [c,d] when a/b = c/d. Although a circle-squarer he is known for the numerous theorems which he discovered in his search for the impossible; Jean-Étienne Montucla ingeniously remarks that "no one ever squared the circle with so much ability or (except for his principal object) with so much success."