page_title: Apollonius of Perga
section_title: Geometrical algebra
hierarchical_section_title: Apollonius of Perga / Ideas attributed to Apollonius by other writers / Methods of Apollonius / Geometrical algebra
caption_attribution_description: Русский: Пифагоровы штаны
context_page_description: Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.
  Apollonius worked on many other topics, including astronomy. Most of the work has not survived except in fragmentary references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance.
context_section_description: Heath goes on to use the term geometrical algebra for the methods of the entire golden age. The term is “not inappropriately” called that, he says. Today the term has been resurrected for use in other senses (see under geometric algebra). Heath was using it as it had been defined by Henry Burchard Fine in 1890 or before. Fine applies it to La Géométrie of René Descartes, the first full-blown work of analytic geometry. Establishing as a precondition that “two algebras are formally identical whose fundamental operations are formally the same,” Fine says that Descartes’ work “is not ... mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; ....”
  For example, in Apollonius a line segment AB (the line between Point A and Point B) is also the numerical length of the segment. It can have any length. AB therefore becomes the same as an algebraic variable, such as x (the unknown), to which any value might be assigned; e.g., x=3.
  Variables are defined in Apollonius by such word statements as “let AB be the distance from any point on the section to the diameter,” a practice that continues in algebra today. Every student of basic algebra must learn to convert “word problems” to algebraic variables and equations, to which the rules of algebra apply in solving for x. Apollonius had no such rules. His solutions are geometric.
  Relationships not readily amenable to pictorial solutions were beyond his grasp; however, his repertory of pictorial solutions came from a pool of complex geometric solutions generally not known (or required) today. One well-known exception is the indispensable Pythagorean Theorem, even now represented by a right triangle with squares on its sides illustrating an expression such as a² + b² = c². The Greek geometers called those terms “the square on AB,” etc. Similarly, the area of a rectangle formed by AB and CD was "the rectangle on AB and CD."
  These concepts gave the Greek geometers algebraic access to linear functions and quadratic functions, which latter the conic sections are. They contain powers of 1 or 2 respectively. Apollonius had not much use for cubes (featured in solid geometry), even though a cone is a solid. His interest was in conic sections, which are plane figures. Powers of 4 and up were beyond visualization, requiring a degree of abstraction not available in geometry, but ready at hand in algebra.
caption: Visual form of the Pythagorean theorem as the ancient Greeks saw it. The blue square is the sum of the other two squares.
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