Last Modified on Saturday, August 10, 2002.A Chronology of Phi
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Note: This page uses R(n) to indicate the square root of n (this notation was commonly used before the modern day notation was adopted). This page also uses pi for the irrational number 3.14159..., phi for the golden mean (which is the irrational number 1.618...), and ^ to indicate exponents. About 300BC
Euclid, in The Elements, says that the line AB is divided in "extreme and mean ratio" (phi) by C if AB:AC = AC:CB. The Elements also deals with the idea of incommensurability, although the proof that R(2) is irrational was probably added in later editions of the work.
About 150BC
Hypsicles writes on regular polyhedra, and the golden ratio enters into his constructions.
About 50
Heron begins to compute approximate ratios, and in his work he gives approximate values for the ratio of the area of the pentagon to the area of the square of one side. He also gives a method for approximating the square root of a number (this method was probably known to the Babylonians).
About 120
Ptolemy calculates the side of a regular pentagon in terms of the radius of the circumscribed circle. He also gives a value for R(3) that is correct to 6 decimal places.
About 450
Proclus uses the term "section," although some historians later dispute that his reference to "section" refers to the golden ratio.
About 825
Al-Khwarizmi gives a problem on dividing a line of length 10 into two parts and finds a quadratic equation for the length of the smaller part of the line of length 10 divided in the golden ratio. It is not known whether Al-Khwarizmi recognized the golden mean as a special number.
About 900
Abu Kamil gives equations which arise from dividing a line of length 10 in various ways. Two of these ways are related to the golden ratio but it is unclear whether Abu Kamil was aware of this.
1202
Fibonacci produces Liber Abaci, using many Arabic sources and one of them contains the problems of Abu Kamil. Fibonacci clearly indicates his awareness of the connection between Abu Kamil's two problems and the golden ratio. In Liber Abaci he gives the lengths of the segments of a line of length 10 divided in the golden ratio as R(125) -5 and 15 - R(125). He also shows that not all types of irrationals may be generated using a compass and straight edge.
1509
Pacioli writes Divina Proportione (Divine Proportion), which is his name for the golden ratio. The book contains little new on the topic, collecting results from Euclid, Fibonacci and other sources on the golden ratio. He states (without any attempt at a proof or a reference) that the golden ratio cannot be rational. Pacioli also gave a method for the approximation of square roots (using a special case of Newton's method).
1597
The first known calculation of the golden ratio as a decimal is given in a letter written by Michael Maestlin, at the University of Tübingen, to his former student Kepler. He gives "about 0.6180340" for the length of the longer segment of a line of length 1 divided in the golden ratio. The correct value is 0.61803398874989484821...
1609
Kepler knows that the ratio of adjacent terms of the Fibonacci sequence tends to the golden ratio.
1632
Albert Girard discovers (independently of Kepler) that the ratio of adjacent terms of the Fibonacci sequence tends to the golden ratio. This is not published until 1634, two years after Albert Girard's death.
1753
Simson notes that in the Fibonacci sequence the ratio between adjacent numbers approaches the golden ratio.
1835
The first known use of the term "golden section" appears in a footnote in a later edition of Die Reine Elementar-Matematik by Martin Ohm.
About 1900
Mark Barr first uses the Greek letter, phi, for the golden ratio.
Last Modified on Saturday, August 10, 2002.