Euclid’s Elements provides the first known written definition of what is now called the Golden Ratio: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”
   Euclid explains a construction for sectioning a line “in extreme and mean ratio,” i.e. the Golden Ratio. Throughout the Elements, several propositions and their proofs employ the Golden Ratio. Some of these propositions show that the Golden Ratio is an irrational number.
   The name “extreme and mean ratio” was the principal term used from the 3rd century BC until about the 18th century. The modern history of the golden ratio starts with Luca Pacioli’s Divina Proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the Golden Ratio.
   The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as “about 0.6180340,” was written in 1597 by Prof. Michael Maestlin of the University of Tübingen in a letter to his former student Johannes Kepler.
   Since the twentieth century, the golden ratio has been represented by the Greek letter ϕ