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The Vitruvian Man: Leonardo Da Vinci Squares the Circle

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.)

Klaus Schröer (Das Geheimnes der Proportionsstudie, Waxmann Publisher, Germany, 1998) is the first to recognize that Leonardo, in his Vitruvian Man, is attempting to square the circle, i.e., to demonstrate how a square and circle of equal area can be created. Schröer uses Leonardo’s drawing to mathematically extend infinite sets of squares and circles of unequal area from Leonardo’s initial one set of unequal square and circle, and to show that their area ratios converge to 1.00037. Schröer’s approach is unnecessarily elaborate, however, and violates the solution procedure restrictions under which Leonardo labored, viz., that the solution to the squaring of the circle employ only a straight edge and compass. Also, it should be demonstrated that an equal-area square be derived from a circle, not that the two shapes be developed simultaneously.

I have discovered Leonardo’s elegant solution, only two geometric figures removed from his drawing of the Vitruvian Man, employing only the hints encoded within this work. Note, in the first illustration -- a copy of the Vitruvian Man proportional to the original, that the square and circle that Leonardo embeds in the drawing are unequal in area. Leonardo’s square is drawn around the proportions of the man, which conform to Vitruvius’ treatise, and his circle (clearly larger than the square) is developed from the square by taking as its radius that length which forms a Golden Ratio to the side of the square. (The Golden Ratio is approximated by 1 to .618…, with its infinitely repeating decimal places.) The circle is centered at the man’s navel. The man’s arms that are parallel to the floor extend out to the edges of the square and are rotated up by Leonardo to the edge of the circle. (On either side of the man, the distance from the level hand’s tip to the rotated hand’s tip is equal to the man’s hip length line through the pubic bone.) A neckline, equal in length to the distance between the navel and pubic bone or the length of one hand (Vitruvius’ unit), is centered between the man’s neck base and chest. The neckline culminates in dots at both end points.

To the geometer, a dotted point invites the use of a compass to draw a circle. What would be the radius length of any such circle? By rotating the man’s hands from the square to the circle, Leonardo answers that very question: Use a straight edge to draw a radius line from one of the dots on the neckline through the center of the man’s arm to the hand tip on the circumference of the circle. From which neckline dot to what rotated hand does one draw a radius? Leonardo is trying to reconcile the opposites of square and circle. I took this as a hint from Leonardo to set a compass point on one dot of the neckline, extend the compass marker to the fingertip on the opposite side of the neckline dot and draw a circle, as shown in the second illustration. The circle’s circumference is adjacent to the square on the square’s inside right side. The astounding result is that the circle so drawn is equal in area to Leonardo’s square!

The equality of Leonardo’s square and the circle drawn according to my instructions can be proven to greater than 99% accuracy. Use any proportional copy of Leonardo’s Vitruvian Man, including a printout of the file copy of the accompanying illustration, to measure one side of Leonardo’s square and the radius of the circle drawn according to my instructions. Using the formula

    pr² = s²,
that expresses the equality of circle and square areas; substitute the length of Leonardo’s square for s, the radius length of my circle for r and 3.14 for p. No matter what specific lengths or consistent metrics you use, the equality will be accurate to a degree greater than 99%. For example, I used a printout of the first illustration (jpeg file size of 5.417" by 7.569", or 390 by 545 pixels, at 72 pixels per inch.) I found the length of Leonardo’s square to be 3 and 14/16” or 3.875”, and the length of a straight edge line from the left dot on the neckline to right hand tip (the radius for my circle) to be 2 and 3/16” or 2.1875”. With these numeric inputs, the above formula would read 3.14 x 2.1875² = 3.875², rendering the equation, 15.0156 sq. inches = 15.025 sq. inches. This equivalence of areas of circle and square is as close as 99.94%, in spite of the fact that the estimate of p is expressed to only two decimal places and the measuring instruments used are as imprecise as a common straight edge ruler and compass. You can check and verify these results by printing out, from a copy of its jpeg file, the accompanying illustration of the Vitruvian Man and, using the second illustration as your guide, draw the radius and circle, take measurements, and do the calculations described above. 

The calculations above and the second illustration show how one can start with Leonardo’s square and develop a circle of practically equivalent area. Technically, this is circling the square, not squaring the circle. For the latter, only one more geometric step is needed – one which builds on my added circle and Leonardo’s circle to create a square with the same area as his circle – using only straight edge. Again, the Vitruvian Man’s neckline is crucial to the process.

Take a straight edge and draw a line through the neckline that extends from the left-side circumference of Leonardo’s circle to the right-side circumference of my new circle. This line is one side of the square with the same area as Leonardo’s original circle! The third illustration shows how the line is drawn, and how it can be extended down into the shape of its square. Following the measuring procedures and formulae used above on the same proportional copy of the Vitruvian Man, the line measures out to 4 and 5/32” or 4.15625”. The area of its concomitant square, therefore, is 4.15625² or 17.274 sq. inches. In the proportional copy of da Vinci’s work, the distance from the man’s navel to the circumference of Leonardo’s circle is 2 and 11/32” or 2.34375”. The square of the radius length is 5.4932. The final step in the circle area calculation is the multiplication of radius-square by 3.14: 5.4932 x 3.14 = 17.25 sq. inches. The area of my drawn square (17.274 sq. inches) and Leonardo’s circle (17.25 sq. inches) are 99.86% comparable. Leonardo squared the circle! It is my recommendation and hope that my procedures and calculations be replicated with Leonardo’s actual drawing of the Vitruvian Man (or an exact facsimile) and with the most precise straight edge ruler and compass available.

The Vitruvian Man, per se, is not needed to square the circle. It merely facilitates application of the underlying more abstract use of Golden Ratios. In most general terms, Leonardo’s procedure starts with any circle and a square whose side is the line of which the circle’s radius is the longer segment of the Golden Ratio (square side equals circle radius plus .618 times circle radius). The square is centered laterally within the circle and set at its base. Golden Ratio intersections within the top half of circle and square lead to determining top-right or top-left “neckline points” – either one of which is the starting point for the squaring of the circle procedure described in this article. (For example, to find the left “neckline dot,” divide the circle’s horizontal diameter at the Golden Ratio point with a vertical line that leaves the short segment to the right.  In the top half of the circle, divide that vertical segment at its lower Golden ratio point with a horizontal line within the circle.  This horizontal line is the “neckline.”  Divide that portion of this horizontal line that runs from the vertical line to the square’s left side at the Golden Ratio point with a vertical line that leaves the short segment to the right.  The intersection of the latter two lines is the left “neckline dot.”)

In conclusion, Leonardo uses principles of Vitruvius’ human proportions and the Golden Ratio to allow the squaring of the circle by means of straight edge and compass to a highly accurate result. In light of Leonardo’s success, it is merely interesting to speculate as to why he chose to use the neckline in his Vitruvian Man as the key to the solution that has eluded scholars for centuries, and why he hid his solution in his drawing rather than announce the result directly. Some suggest that his Vitruvian Man is Jesus. I suggest that the neckline may be symbolic of John the Baptist’s beheading and that, therefore, the Vitruvian Man is an allusion to John the Baptist. To understand his need for secrecy, I suggest further that one read about Leonardo’s heretical beliefs involving the supremacy of John the Baptist over Jesus and the consequences that such a belief would have had were it known to church officials. (A good starting point for literature on this topic is The Templar Revelation by L. Picknett and C. Prince, Touchstone Books, NY, NY, 1997.) Leonardo’s secret encoding, within his Vitruvian Man, of his astounding solution to the squaring of the circle, may have merely masked the more explosive beliefs of his heresy.


From Former About.com Art History Guide: Thomas Pastorello, Ph.D., is Professor Emeritus, Research Methods and Statistics, College of Human Services and Health Professions at Syracuse University, Syracuse, New York. Images contained within this article are copyright of Tom Pastorello, and used with his kind permission.

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