On the Definitions of the Trigonometric Functions (Classic Reprint)

A fresh look at how trigonometric functions are defined and related to space-analysis.

This edition presents Alexander Macfarlane’s exploration of a systematic notation for the partial products of vectors and how these ideas extend beyond circles to space, planes, and more general curves.

The work revisits historical definitions and builds a logical framework that connects traditional circular functions with broader analytical concepts. It argues that trigonometry is not merely an application of space-analysis but a fundamental element of it, capable of encompassing elliptic, hyperbolic, and other functions within a unified approach. The discussion includes the evolution of symbols, the treatment of area and line relationships, and the careful treatment of signs and definitions to support higher mathematics.

• Historical origins of sine, cosine, tangent, and related lines in a geometric setup
• Introduction of a consistent notation for products of two vectors and for area-based relationships
• How space-analysis generalizes trigonometry beyond the circle to ellipses and other curves
• Connections between geometric definitions and their analytical expressions

Ideal for readers who enjoy the history and foundations of trigonometry, and for those exploring advanced mathematical analysis.