Leonardo
Squared the Circle! -- Da Vinci’s Secret Solution in the Vitruvian Man Decoded
By
Tom Pastorello
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.
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From Your 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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