EJMT Abstract

The purpose of this paper is to introduce a geometric method using a straightedge and compass for representing the exponent x of an equation, equivalently expressed as x = ln(a)/ln(b), in the form of a continued fraction, thereby enabling its computation. Although analogous to the Euclidean algorithm, this method operates on exponents, with division carried out geometrically rather than symbolically. The exponent of the equation is determined by locating the two perpendiculars between which the magnitude b lies. In the geometric construction, perpendiculars are drawn on the hypotenuse AC and base BC of a right‑angled triangle ABC with the right angle at B, where AB = 1 and cos(C) = a with a < 1. Since the exponent, being transcendental, is not an integer, the process must be repeated for the remainder. The reciprocal of the remainder, treated geometrically, again produces a new remainder, thus continuing the geometric process. This method opens the door to using geometry as a computational tool, rather than restricting it to its traditional illustrative or grammatical role.