7 - Geometric Calculus

Summary

Preliminary Remarks

During the decade 1660–1670, the discoveries of the previous quarter century on the mathematics of infinitesimals were systematized, unified, and extended. Those earlier discoveries included the integration of y = x^m/n, y = sin x or cos x; the connection of the area under the hyperbola with the logarithm; the reduction of the problem of finding arc length to that of quadrature; the method for finding the tangent to a curve; and the procedure for determining the maximum or minimum point on a curve. Before 1660, the interdependence between problems on construction of tangents and problems concerning areas under curves had been evident only in special cases, but during this decade, Isaac Barrow (1630–1677), James Gregory (1638–1675), and Isaac Newton (1642–1727) independently discovered the fundamental theorem of calculus. Gregory and Barrow stated this result as a theorem in geometry, whereas Newton, deeply influenced by Descartes's algebraic approach, gave it in a form recognizable even today. Later on, Newton adopted the geometric perspective of his Principia. It is interesting to see that evaluations of the trigonometric functions took a very simple form when performed geometrically. In the geometric calculus, one got direct visual contact with the elementary functions and their properties, whereas the abstract approach, while more general and widely applicable, gave less insight into its underpinnings.