Expression denoting a harmonic proportional ratio thought to have originated in the circle of Pythagoras (6th century BC). Expressed as a geometric ratio, the golden section holds that a:b = b:a+b, or the shorter is to the longer as the longer is to the sum of the shorter and longer. Among early mathematicians the division of a line or a rectangle according to this system was known as ‘extreme’ and ‘mean’ ratio. With the invention of algebra it became possible to express the ratio as φ (Phi, the initial letter of Phidias’s name, a term that gained wide acceptance only at the beginning of the 20th century), where φ = 1+square root of 5, divided by 2 (1+√[5]/2). The numerical value of φ is 1.61803 in a positive solution, and its negative reciprocal is 0.61803.