Early History of Quadratic Equations

Mathematical Discussion

Quadratic equations were solved using geometric and arithmetic methods long before symbolic algebra. The Babylonians (c. 2000 BCE) solved these types of problems with methods resembling completing the square. The Indian Sulba Sutras (c. 600 BCE) provided step-by-step geometric solutions. In Euclid’s Elements (c. 300 BCE), these problems are approached through abstract geometric constructions. Diophantus (c. 200 CE) introduced an algebraic approach, using numerical solutions and symbols for unknowns, marking a shift toward abstract algebra.

Historical Perspective

Quadratic equations emerged from practical needs such as land measurement and construction. Babylonian clay tablets recorded step-by-step concrete solutions. In the Sulba Sutras, these problems were solved with geometric methods for the purpose of constructing altars. Euclid formalized quadratic problem-solving within a deductive framework, and Diophantus’ Arithmetica represented an early step toward algebraic notation. His work influenced later developments in Islamic and European mathematics.

Peer-Reviewed Articles

• Allaire, Patricia R., and Robert E. Bradley. "Geometric Approaches to Quadratic Equations from Other Times and Places." The Mathematics Teacher, vol. 94, no. 4, 2001, pp. 308–319.
  JSTOR
• McMillan, Robert D. "Babylonian Quadratics." The Mathematics Teacher, vol. 77, no. 1, 1984, pp. 63–65.
  JSTOR
• Gandz, Solomon. "The Origin and Development of the Quadratic Equations in Babylonian, Greek, and Early Arabic Algebra." Osiris, vol. 3, 1937, pp. 405–557.
  JSTOR

Textbooks

• Katz, Victor J. A History of Mathematics: An Introduction. Pearson, 2009.
  A comprehensive textbook covering the evolution of quadratic equations in Babylonian, Greek, and Indian mathematics.
• Smith, David Eugene. History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics. Ginn and Company, 1923.
  This text discusses the historical development of algebra, including quadratic equations, from Babylonian to Greek traditions.
  Available for free on the Internet Archive.