With Ptolemy trigonometric tables, at least in terms of chords of circles, begin to be computed.
Heron certainly 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. Of course if A B AB A B has length 1 and A C = x AC = x A C = x where C C C divides A B AB A B in the golden ratio, then we can use simple algebra to find x x x.ġ x = x 1 − x \large\frac\normalsize 2 √ 5 + 1 = 1. Up to this time the golden ratio seems to have been considered as a geometrical property and there is no obvious sign that any attempt was made to associate a number with the ratio. The golden ratio enters into the constructions. He is the author of what has been called Book XIV of Euclid's Elements, a work which deals with inscribing regular solids in a sphere. Hypsicles, around 150 BC, wrote on regular polyhedra. Heath claims later in the same work that the construction of a pentagon using the isosceles triangle method referred to above was known to the Pythagoreans so there is a fair amount of evidence to suggest that this is where the study of the golden ratio began. This idea that Plato began the study of as a subject in itself is not in the least inconsistent with the supposition that the problem of Euclid II, 11 was solved by the Pythagoreans. Heath writes in his edition of Euclid's Elements:. Eudoxus certainly attended lectures by Plato so it is entirely reasonable that he might work on topics suggested during these lectures. Many believe that by 'section' Proclus means 'golden ratio'. multiplied the number of propositions concerning the section which had their origin in Plato, employing the method of analysis for their solution. Proclus, writing in the fifth century AD, claims:-Įudoxus.
Now some historians believe that Book II of The Elements covers material originally studied by Theodorus of Cyrene while others attribute the material to Pythagoras, or at least to the Pythagoreans. Of course nobody believes that Euclid's Elements represents original work so there is the question of who studied the golden ratio before Euclid. Similarly construct F F F and draw the pentagon A E B D F AEBDF A E B D F. Note that the line passes through C C C, the point dividing A B AB A B in the golden ratio. Then bisect the angle A D B ADB A D B with the line D E DE D E meeting the circle at E E E. Now starting with such a triangle A B D ABD A B D draw a circle through A, B A, B A, B and D D D. The triangle A B D ABD A B D has the property that its base angles are double its vertex angle. Mark D D D on the circle so that A C = C D = B D AC = CD = BD A C = C D = B D. Then draw a circle with centre A A A radius A B AB A B. This is done by taking a line A B AB A B and marking C C C on the line in the golden ratio. Here is how the golden ratio comes into the construction of a pentagon.įirst construct an isosceles triangle whose base angles are double the vertex angle.
As well as constructions to divide a line in the golden ratio, Euclid gives applications such as the construction of a regular pentagon, an icosahedron and a dodecahedron. The definition appears in Book VI but there is a construction given in Book II, Theorem 11, concerning areas which is solved by dividing a line in the golden ratio. Euclid, in The Elements, says that the line A B AB A B is divided in extreme and mean ratio by C C C if A B : A C = A C : C B AB:AC = AC:CB A B : A C = A C : C B.Īlthough Euclid does not use the term, we shall call this the golden ratio.
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