The Secret Teaching of All Ages
by Manly P. Hall
Leonardo Squared the Circle! -- Da Vinci’s Secret Solution in the Vitruvian Man Decoded
Leonardo uses the Roman architect Vitruvius’ archetypal proportions of the human body and the Golden Ratio to “solve” his day’s great squaring of the circle problem – to greater than 99.8% accuracy. For all practical purposes, Leonardo’s solution, encoded in his Vitruvian Man, is definitive. His solution is also ingenious, previously undiscovered and elegant in its simplicity. (A geometric solution of 100% accuracy is not possible because the underlying mathematics involve Pi, the infinitely repeating decimal places of which, p = 3.14159265..., defy the drawing of perfectly measured geometric shapes.)
Vitruvius' De Architectura, is a source of inspiration for Leonardo's drawing in his sketchbook, the famous Vitruvian Man. This passage is also providing the Key of the composition of the ancient architecture, firmly revived in the Renaissance. The emphasize is on rationalisation of geometry, by means of small whole numbers to build the composition - in this manner Vitruvius delineates the proportion of a male human body that Leonardo so faithfully reproduced. Rationalisation as the basic rule of composition and the simplicity of an ingenious design (proportioned by means of small whole numbers) are often ignored.
Both Leonardo and Vitruvius make no mention of the Phallos as the center of the square, but it plays an important role. The side of the square is 4 cubits or 24 palms as can also be measured on the ruler beneath the square, and deduced from the Vitruvius' text. The diagonal of such a square comes close to 34 palms (33.94... exactly). If we rotate the square around Phallos for 45 degrees we get the upper point of the circle, and from this we can calculate the diameter of the circle as 29 palms (the difference between the square and the circle is 5 palms).