
CHAPTER 6 Leonardo da Vinci Leonardo da Vinci was born in April 15, 1452, in Vinci, a small community near Florence, Italy. Very few people do not have heard about him. He was a genius: he worked as a scientist, architect, engineer, painter, sculptor, and inventor. He made investigations in many sciences, in which he was a pioneer. His extraordinary vision led him to create many inventions, later developed in the Twentieth Century. Among them, the autos, planes, helicopters, submarines, as well as modern machinery for times of war and peace. In the Arts, he left to the humanity the Mona Lisa and The Last Supper, among a great inventory of paintings of immeasurable value. His notebooks were handwritten backwards. Therefore, to read them, it is required a mirror. His drawings of the human body, animals, and plants, reveal his mastery and knowledge in anatomy. Among his drawings, there is one that is known the world over. He shows in the drawing one man’s figure in two positions, circumscribed by a circle, and circumscribed by a square. The purpose of the illustration was to show the typical human proportions. Leonardo’s drawing is based on the theories of Marcus Vitruvius (a Roman authority on Architecture) in relation to the proportion of the human body. According to Vitruvius, the measurements of a human body are arranged by Nature. Both a circle and a square can circumscribe them. According to Leonardo, if you set your legs so far apart as to take a fourteenth part from your height, and you open and raise your arms until you touch the line of the crown of the head with your middle fingers, the center of the circle which circumscribed the man’s figure will be the navel. The space between the legs will form an equilateral triangle. When the square circumscribes the body resulting from the fact that the span of his outstretched arms is equal to his height, its fulcrum is displaced from the navel to the "os sacrum". He demonstrates this by drawing a square around the man’s figure. The horizontal sides of the square are defined by the top of the head of the human’s figure in the drawing and his foot’s sole at the base line. The vertical lines of the square are defined by the extremes of his hand middle’s fingers, when his arms are extended in the horizontal position. The penis begins at the center of the man’s figure.
Illustration of Typical Human Proportion Drawing by Leonardo da Vinci Leonardo’s drawing is based on the theories of the Roman Architect, Marcus Vitruvius, about the application of the Golden Section, or Divine Proportion, to the proportions of the human body. Although Leonardo explained the geometrical relations of his drawing, he did not explain how he traced it. However, he left the guidelines and numerical scales used to trace the drawing. These could imply that the arrangement of the circle and the square were traced using determined numerical values. Nevertheless, as I will demonstrate in my example, that the arrangement of the circle and the square can be drawn geometrically, without the use of scales and measurements. In that case, Leonardo only needed to draw the human figure. In addition, I will show that the circle and square configuration traced by Leonardo da Vinci for his drawing, is a variation of the geometric configuration I developed for the Great Pyramid (see figure 94). In this manner, it can be better understood the characteristics he mentioned about the human’s proportions in his drawing. Figure 94, as it will be remembered, shows the basic geometric configuration traced with my geometric process to explain the Great Pyramid’s design. It consists of the configuration of a circle, a triangle and a square, arranged in a special configuration by means of a geometrical process. Remember, this configuration can be traced using just a straightedge and a drawing compass; there is no need for measurements. All parts are proportional.
Figure 94 The triangle HQT symbolizes the vertical sectional plane of the Great Pyramid, as seen through the center of its faces. The square between points I, II, III, and IV, represents the vertical projection of the base of the Great Pyramid as viewed in the vertical plane. I will show that with some additional geometric lines to this figure, it will be feasible to construct the configuration that is shown in Leonardo da Vinci’s drawing to explain the proportions of the human body. As shown in figure 95, trace a horizontal line to join the intersection points (a) and (b), between the circumference of the circle and the vertical lines that establishes the vertical sides of the square. Extend down the vertical lines of the sides of the square to intercept the horizontal line of the base line (dc), which is tangent to the circle in point K. Identify points (d) and (c) at their intersection. It will create a rectangle between the points I, II, c and d.
Figure 95 If the radius of the circle is made equal to one (1), the dimensions in the geometric configuration can be calculated. In this manner, they can be compared with those shown in Leonardo’s drawing, and examine if they are in accordance with his indications. As shown in figure 96, inscribe the circle with center O, in a square (between the point’s n, m, o, and s). Trace a line from point (n) and another from point (m) (upper corners of the square) to point K at the center of the base line. Extend the line (ab), at both ends, to intercept the lines nK and mK, at points (e) and (h) respectively. From points (e) and (h), trace the vertical lines (eu) and (hg) to intercept the base line of the square. When the circle circumscribes the human figure, line (ab) identifies the horizontal distance between the extremes of his extended and raised arm’s middle fingers, at the head level. The height of the man’ figure is represented by the length of line (bc), which is also equal to (hg), (eu) and (ad).
Figure 96 Figure 97 Enlarged section of the corner of the square
On the contrary, when the square circumscribe the human figure in the erect position, the line (eh) identifies the horizontal distance between the extremes of his hands middle’s fingers, with his arms extended in a horizontal position. As a reference, figure 98 shows the numeric relation of the measures created in the figure’s configuration. In figure 99 are shown the same measurements, but expressed in terms of the golden number. Figure 100 shows the geometric proportions, as produced in Leonardo’s drawing, when the human figure, in two different positions are superimposed over the traced drawing, in a relative scale.
Figure 98
Figure 99 As already explained, according to Leonardo’s notebooks about the human proportions, the center of the circle is located at the navel of the human figure. In the drawing, points (a) and (b) represent the horizontal distance between the extremes of his extended and raised arm’s middle fingers, to the top of his head. Since the radius of the circle is equal to one (1), its diameter is equivalent to (2). The distance between his extended arms, from point (a) to point (b), is equal to the sides of the square (I, II, III, and IV), that is, b = 2 / Öf = 1.572303.
Human figures Leonardo used in his drawing of proportions
Figure 100 The distance from the man’s navel to the top of his head is equal to 1 / f = 0.61804, or the inverse of the golden number. From the top of his head to the upper part of the circumference of the circle, the distance is equivalent to the inverse of the golden number squared, that is, 1 / f² = 0.38197. It can be observed in the figure that f + (1 / f²) = 2. This means that f (the man’s height) plus 1 / f² (space over his head) is equal to 2, or to the circle’s diameter. When the man’ figure is standing over the base line, and his arms are horizontally extended, the figure configuration changes. His arms in the horizontal position reach a greater distance; therefore his middle fingers reach and touch the vertical lines (eu) and (hg), as shown in the figure. The horizontal sides of the square are defined by the top of the head of the man’ figure, and his foot soles at the base line. The horizontal distance between his extended arms when the figure is circumscribed by the circle is equal to b = 2 / Öf. This distance changes to the value of b = (R)(f) when the figure is circumscribed by the square. Since the height of the man’s figure is also equal to f, consequently, the square is created, as explained by Leonardo in his notes. Leonardo explains that the center of the square will coincide with the genital location of the man’s figure. The following is another geometrical method, derived from the previous method, which I developed to trace the arrangement of the circle and the square, according to Leonardo’s drawing.
Figure 101
Figure 102 As shown in figure 101, trace a square (CDEF), with sides identified as (b). Establish the midpoint of its sides, that is, points E, Q’, F, and K. Join point K to point D (corner of the square). With D as center and DQ’ as its radius, trace the arc Q’F. Identify point N at the intersection of the circumference of the arc and line KD. From K, and radius KN, trace an arc to intercept the vertical sides of the square (at points L and M). Now, as shown in figure 102 (continuation of the drawing), with O as center and radius OK, trace a circle. The figure of the circle and the square just traced will be proportional and will have equal characteristics to the same figures traced by Leonardo da Vinci in his drawing. The circle’s diameter will be equal to 2, while the length of the sides of the square will be equal to b = (R)(f), This happens when the human figure changes its position from being inscribed in the circle to being inscribed in the square. In this geometric configuration, when the radius of the circle is one (1), the sides of the square are equivalent to f. For those interested in this geometric configuration, using the Pythagorean theorem, it can be established that the hypotenuse (KD)² = (b)²+ (b/2)². Solving the equation, KD = (Ö5 / 2) ( b). The radius (OK= R) of the circle is equal to KD  (b / 2). That is, R = (Ö5 / 2) (b)  (b / 2) = b (Ö5 1) / 2 , and since (Ö5 1) / 2 = 1 / f, then R = (b / f). Therefore, R (f) = b. This configuration is very interesting. I found in my studies, that it corresponds to the geometrical configuration used by Pharaoh Micerynus to build his pyramid. The drawing of the typical human proportions by Leonardo da Vinci could explain the motives of the ancient Egyptians to use the concept of the circle, the triangle and the square to build their pyramids. Although Leonardo da Vinci did not trace the figure of the triangle and the square of the base of the pyramid in his drawing, they are implicit, and are part of that special geometric configuration. I believe that if the Egyptian priests and designers knew about these geometrical concepts, and believe they were used by their Creator or God to proportion the human body and the nature around him, in the same way, they could use the configuration to obtain the perfection of their Pharaoh’s tomb, the Great Pyramid. There is an interesting fact in Leonardo’s drawing. In figure 100, where it shows the man’s figure circumscribed by the circle, it will be noticed that his genitals rest exactly over the point identified as point X in the geometrical configuration I developed for the Great Pyramid (figure 94). Besides, and as explained by Leonardo, when the figure is inscribed in the square, the center of the square also identifies the same part of the human body. As it will be remembered, in the second chapter, because of the important location, balance and security, the position of point X was selected as the ideal location to locate Khufu’s mortuary chamber. Now, symbolically, it can be added that point X represents in the geometric configuration, the place where the human existence is guarantied and secured. Most of the geometrical theories about the Great Pyramid’s design could be included in two classifications. Those theories that indicate that the geometric design has its origin in the value of Pi (p) and those that consider that its origin is based on the value of Phi (f). The results of my studies showed, clearly, that it was designed by means of a geometric configuration, based on the value of f. Leonardo’s drawing and his notes allowed me to establish its geometric relation with the configuration of the Great Pyramid. Besides, to demonstrate, as I found in my research, that the configuration that exist between the circle of the drawing and the square formed when the man is standing with his arms extended horizontally to his sides corresponds to the configuration used by pharaoh Micerynus to build his pyramid. In other words, using this configuration, the measurements of the square in relation to the circle, represents the base of his pyramid. This will be further explained in the analysis of his Pyramid in Chapter 12. Leonardo’s drawing for the proportions of the human body is very well known. Although it explains the parameters used by the Creator for the human proportions, Leonardo indicates in his notes, that to be true to nature the painter has to give different proportions to the figures, otherwise, all people in the paintings would look alike.

