How did analytic geometry connect algebra with geometry?

Analytic geometry connected algebra and geometry by establishing a systematic method to study geometric properties through algebraic operations on symbols defined within a coordinate system. Developed independently by Pierre de Fermat and René Descartes in the 17th century, this approach allowed mathematicians to represent geometric curves as algebraic equations.123

The connection functions through the following key principles:

• Coordinate Systems: By using the Cartesian coordinate system, points in a plane are represented by pairs of real numbers $(x,y)$. This maps geometric space directly onto numerical values.4
• Algebraic Representation: Geometric objects, such as lines and circles, are characterized by polynomial equations (e.g., $x^2+y^2=1$ for a circle). This allows properties like tangents and enclosed areas to be derived using algebraic manipulation rather than purely geometric construction.23
• Symbolic Flexibility: Descartes introduced a unit length as a reference for geometric constructions, enabling algebraic operations—such as finding square roots—to be performed on line segments.2

This synthesis transformed geometry from a field of isolated, difficult problems into one where a vast class of curves could be analyzed routinely using the power of algebra.3