SYNOPSIS

The drafting of the two diagrams which Vitruvius describes, the square and circle encompassing the Human body requires knowledge of geometry.

To draw the iconic sketch Leonardo Da Vinci, not only required knowledge of geometry, but that of physiology as well.

There is a simple explanation of how they were conceived and drawn some 1500 years apart.

ORIGINAL TEXT; VITRUVIUS, BOOK 3, CHAPTER 1

TITLE; ON SYMMETRY: IN TEMPLES AND IN THE HUMAN BODY

1) The design of a temple depends on symmetry, the principles of which must be most carefully observed by the architect. They are due to proportion, in Greek άναλογία. Proportion is a correspondence among the measures of an entire work, and of the whole to a certain part as standard. From this result the principles of symmetry. Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of those of a well shaped man.

2) For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is the tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils is one third of it; the nose from the underside of the nostrils to a line between the eyebrows is the same; from there the lowest roots of the hair is also a third, comprising the forehead. The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth. The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown.

3) Similarly, in the members of a temple there ought to be the greatest harmony in the symmetrical relations of the different parts to the general magnitude of the whole. Then again, in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centred at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described there-from. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.

4) Therefore, since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule, that in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme, Hence, while transmitting to us the proper arrangements for buildings of all kinds, they were particularly careful to do so in the case of temples of the gods, buildings in which merits and faults usually last forever.

ORIGINAL TEXT; LEONARDO DA VINCI DIAGRAM

TITLE; VITRUVIAN MAN, THE NOTEBOOKS 1

*Vitruvius the architect, says in his work on architecture that the measurements of the human body are distributed by Nature as follows that it is 4 fingers make a palm, and 4 palms make 1 foot; 6 palms make a cubit; 4 cubits make a man’s height. And 4 cubits make one pace and 24 palms make a man; and these measures he used in his buildings. If you open your legs so much as to decrease your height 1/14 and spread and raise your arms till your middle fingers touch the level of the top of your head you must know that the centre of the outspread limbs will be in the navel and the space between the legs will be an equilateral triangle. The length of a man’s outspread arms is equal to his height.*

From the roots of the hair to the bottom of the chin is the tenth of a man’s height; from the bottom of the chin to the top of his head is one eighth of his height; from the top of the breast to the top of the head will be one sixth of a man. From the top of the breast to the roots of the hair will be the seventh part of the whole man. From the nipples to the top of the head will be the fourth part of a man. The greatest width of the shoulders contains in itself the fourth part of a man. The elbow to the tip of the hand will be the fifth part of a man; and from the elbow to the angle of the armpit will be the eighth part of the man. The whole hand will be the tenth part of the man; the beginning of the genitals marks the middle of the man. The foot is the seventh part of the man. From the sole of the foot to below the knee will be the fourth part of the man. From below the knee to the beginning of the genitals will be the fourth part of the man. The distance from the bottom of the chin to the nose and from the roots of the hair to the eyebrows is, in each case the same, and like the ear, a third of the face.

COMMENT

RmVt2D01

It will be very apparent from a reading of the above two texts that although they deal with the same subject they vary considerably in content and measurements given. Leonardo quotes many more anatomical measurements than Vitruvius and varies the proportional length, no doubt from his knowledge gained by the anatomical studies he had carried out.

Thus the first diagram indicates two human forms as described by Vitruvius encapsulated in a square, RmVt2D01A, and a circle, RmVt2D01B, as he describes, with appended measurements.

It should be noted that I have chosen to use a figure of 1800mm height comprising therefore 4 cubits of 450mm and 6 ‘feet’ of 300mm.

VITRUVIUS; HUMAN FORM

RmVt2D02&D03

The body as described by Vitruvius is encompassed in two geometrical forms, the square and circle as illustrated. The square is easily drawn and the pivot point of the arms at the shoulder located accordingly.

The circle can be found by pivoting the legs and arms at their natural joints within the body and hence it will be a known dimension centred on

the navel (12), “The Centre of The Body” as described by Vitruvius.

Within the square the body naturally has the overall height and outstretched arms as the same dimension. The arms naturally align across the top of the breast (point 4), which is also the natural pivot point of the socket for the shoulder joint.

The legs however pivoting at the hips produce the shortening of the overall height as shown by the diagrams, from the closed to open position. Thus a new pivot point (A) is established above the navel.

But using the navel as the centre point, the positions of the soles of the feet when they rotate are automatically located at any given angle.

The legs actually have a natural maximum spread of 60^{0} (see Leonardo) and thus form an equilateral triangle as diagram RmVt2D02 illustrates.

However for the arms to accord with the statement of Vitruvius they must be below the horizontal by 22 1/20 or 1/16th of a circle as Diagram RmVt2D01B illustrated.

By combining both diagrams the overall geometry of the human body as described by Vitruvius is apparent and the central pivot points for the arms and legs can be established. The basic geometry is indicated on Diagram RmVt2D03.

From this combined diagram it is immediately apparent that the arms and legs have pivot points which are very much part of the “system”, in that the arms pivot from Point 5, The Mid-Breast, and the legs from a point which is one third of a cubit above the navel (12) denoted point A.

The legs rise by only 90mm or 1/20th of the height above the sole of the foot position.

The full circles of the arms meet precisely at the upper part of the leg circle and form a “vesica piscis” at the overall head line precisely 900mm apart.

The legs from the navel form an equilateral triangle (60^{0}) and point to the arm/breast width of 450mm, that is one cubit, a quarter of the natural height.

VITRUVIUS, HIS BODY FORM

Thus the human form as described by Vitruvius is an idealised concept arrived at by Greek Sculptors or originally by Egyptian artists who had a complete set of proportions to be used when drawing the human form on temple walls.

The Greek sculptors’ idealism found in the “Kouros”, a sculpture of a young man used as a funerary statue is probably the simplest example of these proportions.

In the *Rowland/Howe text, Commentary, Book 3, page 188* there is an explanation of the possible source for the text by Vitruvius. It in all probability arose from the work of *Polycleitus of Argos*, and his “Kanon”, which was not only the title of a statue but also a treatise written in the third quarter of the 5th century BCE. This “Kanon” demonstrated the application of ‘symmetria’, or a theoretical ideal system of proportion to be used when sculpting the human figure.

Thus some 400 years later Vitruvius read and copied the data into his own text.

RmVt2D04

RmVt2D05

LEONARDO DA VINCI; HUMAN FORM – Diagrams RmVt2D04 and RmVt2D05

The text accompanying the iconic diagram by Leonardo da Vinci accepts the same basic assumption as Vitruvius, that the length of a man’s outstretched arms to the finger-tips is equal to his height when standing feet together.

But it is only when, with the legs spread and the height reduced, by according to Leonardo 1/14th part, that the navel is the centre of the outspread limbs.

Leonardo clearly states, *”the beginnings of the genitals marks the middle of man”.*

He also clearly states, *“the foot is the seventh part of a man”.*

Thus we can analyse the geometrical construct of the “Vitruvian Man” diagram and establish how it aligns to the original measures quoted by Vitruvius.

VITRUVIAN MAN; LEONARDO DA VINCI

Fortunately the diagram has the square and circle clearly drawn and hence the geometry of the whole format can easily be explored.

The first angle noted is that of the outstretched arms to a line across the top of the head, part of the circle diagram. This angle is 25 degrees and was probably used because of its close relation to the division of a circle it so gives.

Thus it is probably arrived at as follows; 360 degrees divided by 25 degrees provides for 14.4 parts i.e. illustrating the 1/14th part, the reduced height, and then 360 divided by 14 of course gives 25.7 degrees. (But the 1/14th part is in fact derived from a different set of figures as is explained later.)

The first geometric form is therefore the triangle centred upon the navel to the outstretched arms as Diagram RmVt2D04A illustrates, which is repeated when the arms are raised, but this time the top of the head forms the centre of a lozenge shape so formed.

From the circle point where the finger tips touch also the square, it is then possible to use the same 25^{0} angle downwards towards the centre line between the feet, thus forming a perfect diamond shape as Diagram RmVt2D04B illustrates.

When using the navel position and the same angle 25^{0}, applied to the horizontal arms, it denotes a point perhaps slightly above the middle finger of the left hand, but the fingers of the hands are not in fact horizontal or therefore level. Thus the line is actually across the top of the shoulders to the thumb level of the hands, but by turning the hands through 90^{0}, that is, to their natural position, there would be a completely level alignment.

This is no doubt a presentational diagram, drawn such that the parts of the arm can be seen and thus the divisions of the body properly shown.

The next coincident points are at the feet on the circle where the angular line at 250 from the navel meets a 60^{0}line formed by an equilateral triangle from the genitals, the middle of man. This point is also equidistant between the centre line and square along the 1/14th height line.

But just how did Leonardo da Vinci assess the 1/14th height alteration in the first place. This is no more than a simple geometrical or trignometrical fact, that of the 60^{0} equilateral triangle set in a square.

The square is 2 cubits by 2 cubits from the base line of the feet and thus commences at the mid point of the body of man. This is also the point from where the equilateral triangle is drawn, but of course being 300 either side of the centre line it cuts the sides of the square above the base line. The residual distance is thus 1/14th part of the height of man and it is from this geometry that Leonardo da Vinci assessed the height change caused by the opening of the legs, not from any physiological measurement.

DISCUSSION

It is therefore reasonable to opine that Leonardo da Vinci assessed the geometry of the human form and used that geometry to draw the square and circle inter-linking as has been shown.

But, he has also chosen to draw the human figure in a slightly stylised manner, that is, he chose to turn the hands to face the viewer thus ensuring the wrist and fingers were visible, and he has turned the left leg through 90^{0} to ensure the length of the foot was shown as well as the knee joint.

This adjustment of the actual diagram which would have had the hands horizontal and the legs set facing front has lead to a slight error creeping into the presentation of the diagram.

Therefore we must be cautious in the points we choose to use and assess the geometrical form of the whole rather than try to make each point on the iconic diagram fit a perfect idealism.

THE ICONIC DIAGRAM

The actual diagram by Leonardo da Vinci has the same basic geometry just discussed drawn there-on, and adjacent to it the whole geometric assessment in its simplest form is shown.

This forma indicates the lines which Leonardo would have used to draw the final sketch.

As indicated the 1/14th part, the shortening of the stature is purely geometric and not physiological.

The centre of man is closer to the genitals than the navel, but as we are dealing with an “ideal” perhaps it is inappropriate to over dissect the diagram.

The words of Leonardo Da Vinci may suffice; “Proportion is found not only in numbers and measurements, but also in sounds, landscapes, times and places, and in any form of power at all”.

(Manuscript K 49r, held at the Institut de France, Paris.)

CONCLUSION

Vitruvius copied the text of Polycleitus of Argos, which fortunately had survived some 400 years. But it is evident he did not draw the figures or he would have recognised the problem with the circle.

Leonardo took the basic text and improved it by his knowledge of physiology and resolved the problem of the square and circle by resorting to a geometrical solution, 60^{0} and 1/14th.

Leonardo first drew a proper geometrical diagram, but was then required to produce a slightly amended diagram such that the arms and legs could be properly displayed. But, in so doing he slightly compromised the exactness of the geometry within his final diagram, as has been indicated.

Regardless of the foregoing it is a masterpiece and the most iconic sketch of Man.