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Vitruvian Man as rendered by Leonardo da Vinci is one of humankind’s most famous and influential images. Detailed specifications for the layout of Vitruvian Man’s body were given more than 2000 years ago by the Roman architect Vitruvius in his famous treatise The Ten Books on Architecture. Leonardo’s illustration and material found in the artist’s notebooks bring to light an inventive design concept not noticed before.
Pythagorean design is seen from a new and memorable perspective.
Figure 1: Leonardo da Vinci’s rendition of Vitruvian Man, man the microcosm
1. Preamble
1.1. This essay is one of a series of works being developed by the present writer. The works deal with famous ancient designs developed by the inventive use of certain mathematical features found in the five Platonic solids. Only basic mathematical skills (multiplication, division, addition and subtraction) are required to deal with the material.
An explanation of the way ancient writers are cited in this document is provided in Remarks on references and abbreviations in appendix 1.
Figure 2: The five Platonic solids: from left, the tetrahedron, cube, octahedron, dodecahedron, and icosahedron
Three well-known geometric shapes are involved in the construction of the five solids: the equilateral triangle, the square, and the pentagon.
Figure 3: Pentagram inside a pentagon
The pentagram is a Pythagorean symbol.
Geometry, especially geometry found in living nature, can produce startling mathematical formations that might inspire a person with a belief in what is, these days, controversially referred to as “intelligent design”. It is well known by people familiar with Plato’s most famous book, the Republic, that the philosopher recommended the study of geometry as a way of obtaining a “vision of the Form of the Good”, a mystical concept (Lee, pp. 273–4/S526c – 527c).
Figure 4: Nature’s pentagram in the base of a rose.
The rose is a Rosicrucian symbol.
1.2. The success of the novel The Da Vinci Code by Dan Brown has helped raise the profile of the Roman architect Vitruvius, author of The Ten Books on Architecture, one of the most influential books in history, particularly during the Italian Renaissance.
Little is known about the shadowy figure of Vitruvius, who lived in the first century BC, apart from what he tells about himself in his book. One thing he makes clear is his admiration for the Greek philosophers Plato and Pythagoras: see Morgan, page 195, Introduction to Book 7, and page 251, Introduction to Book 9.
In The Ten Books on Architecture Vitruvius provides, amongst many other things, rules and directions for the design of temples, theatres and war machines. A mysterious and much discussed mathematical formulation for a “well shaped man” is also given. For example, Vitruvius says the head, vertically, is one-eighth the measure of the whole body height: see Book 3, chapter 1. The formulation was adapted by Leonardo da Vinci (1452–1519) to create the now famous illustration of Vitruvian Man, the man in the square and the circle shown in figure 1 on page one.
As can be seen in Leonardo’s drawing, the measure of the outstretched arms is the same as the height of Vitruvian Man—hence the square. Vitruvius prescribes this setup. The “well shaped man”, he says, is six feet tall (Morgan pp. 72–4/Book 3.1.1).
1.3. Leonardo, until he was thirty, lived in and around Florence where Platonic, Neoplatonic and Pythagorean ideas flourished. The central idea of Pythagoreanism is that Number is the First Principle of the universe. “All is number,” said the Pythagoreans, who also considered, symbolically, odd numbers to be masculine and even numbers to be feminine.
Three is the first masculine number; two is the first feminine number. (The number one had a more elaborate and special definition.)
Six is the product of three times two.
Two of nature’s laws provide notable manifestations of the number six:
● Six, and only six, same size circles fit exactly around the circumference of an inner seventh same size circle.
Figure 5: Circles
● A cube has six square faces.
Figure 6: Cube
Compelling evidence that Leonardo and Vitruvius had more than a passing interest in Pythagorean and Platonic ideas is presented in this document and in forthcoming works by the present writer.
Leonardo is known to have had contact with leading mathematicians of his day, including Fra Luca Pacioli. In Milan, where the two men met, Pacioli collaborated with, lived with, and taught mathematics to Leonardo.
Figure 7: Portrait of Fra Luca Pacioli, controversially attributed
to Jacopo de’ Barbari, 1495 (Capodimonte, Naples)
Observe the dodecahedron in the lower left of the painting.
A dodecahedron has 12 faces, each face a pentagon.
1.4. Some numerical characteristics of the cube have significance in this essay. The three-dimensional solid has:
a) 6 faces
b) 12 edges
c) 24 angle locations (four on each of the six square faces)
The numbers can be related to the way time is measured. For example, a day contains 24 hours. It was said in ancient Egyptian, Greek and Roman times that the day contained 12 daytime hours and 12 night-time hours. Shadow clocks measured time in 6 hour periods: 6 hours sunrise to midday and 6 hours midday to sunset.
Figure 8: Reproduction of an ancient Egyptian shadow clock
1.5. Why the designs discussed in this work and in other works by the present writer have elements of concealment is partly explained by quoting a passage from Plutarch’s essay “The E at Delphi”, which is published in Moralia, Volume V, immediately following the essay on “Isis and Osiris”. Plutarch is noted for his biographic works and essays on philosophy and ethics. He was of Greek origin and lived from about 46 to 120 AD. He was a priest at Delphi, a place sacred to the god Apollo. He writes:
“When Nicander had expounded all this, my friend Theon, whom I presume you know, asked Ammonius if Logical Reason had any rights in free speech, after being spoken of in such a very insulting manner. And when Ammonius urged him to speak and come to her assistance, he said, “That the god [Apollo] is a most logical reasoner the great majority of his oracles show clearly; for surely it is the function of the same person both to solve and to invent ambiguities. Moreover, as Plato said, when an oracle was given that they should double the size of the altar at Delos (a task requiring the highest skill in geometry), it was not this that the god was enjoining, but he was urging the Greeks to study geometry. And so, in the same way, when the god gives out ambiguous oracles, he is promoting and organizing logical reasoning as indispensable for those who are to apprehend his meaning aright.” (Babbitt, pp. 209¾211/S386)
Underlined phrases are the present writer’s emphasis.
2. About Vitruvian Man
2.1. As stated earlier, Marcus Vitruvius Pollio was a Roman architect and engineer who lived in the first century BC. He is the author of the influential treatise The Ten Books on Architecture. It is in this book that his description of Vitruvian Man—the “well shaped man”—is given. Preceding the formulation for man is this famous passage, which has been rewritten and expressed in many ways by many people:
ON SYMMETRY: IN TEMPLES AND IN THE HUMAN BODY
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 àναλογία. Proportion is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected 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. (Morgan, p. 72/Book 3.1.1)
Vitruvius says that the “well shaped man” is six feet tall (Morgan, p. 74/V3.1.7) and that his outstretched arms are the same measure:
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. (Morgan, p. 73/Book 3.1.3).
The following passage from Vitruvius’s treatise is of particular relevance:
And further, as the foot is one sixth of a man’s height, the height of the body as expressed in number of feet being limited to six, they [the Greeks] held that this was the perfect number, and observed that the cubit consisted of six palms or of twenty-four fingers. (Morgan, p. 74/Book 3.1.7)
It has been noted that the architect’s formulation for man produces a rather curious looking individual. Leonardo made his interpretation more realistic.
2.2. Vitruvius says the proportions of the original Doric pillar (column) are based on the proportions of man, whose height is six times the length of his foot. The pillar’s height is six times its diameter at the base. Below is the relevant text.
Now these cities, after driving out the Carians and Lelegans, called that part of the world Ionia from their leader Ion, and there they set off precincts for the immortal gods and began to build fanes: first of all, a temple to Panionion [sic][1] Apollo such as they had seen in Achaea, calling it Doric because they had first seen that kind of temple built in the states of the Dorians.
Wishing to set up columns in that temple, but not having rules for their symmetry, and being in search of some way by which they could render them fit to bear a load and also of a satisfactory beauty of appearance, they measured the imprint of a man’s foot and compared this with his height. On finding that, in a man, the foot was one sixth of the height, they applied the same principle to the column, and reared the shaft, including the capital, to a height six times its thickness at its base. Thus the Doric column, as used in buildings, began to exhibit the proportions, strength, and beauty of the body of a man.
(Morgan, p. 103/V4.1.5–6)
Figure 9: Doric columns front the Shrine of Remembrance, Melbourne
2.3. Roman measures.
Characteristics of Roman measures have a bearing on this work.
The Roman foot (pes) was divided into 16 (4 x 4) digits (digiti) or 12 inches (unciae). It also contained 4 palms (singular palmus): see Zupko, p. 6. Vitruvius uses both inch and digit measures in The Ten Books on Architecture.
A Roman foot measured around 11 2/3 British imperial/US inches, about 296 mm: see Zupko, p. 6.
A Roman stade contained 625 Roman feet (625 is 5 x 5 x 5 x 5). The stade was also equal to 125 Roman paces (125 = 5 x 5 x 5): see Zupko, p. 6. A Roman pace measure was equal to five Roman feet.
A Roman mile was equal to eight stades: 8 = 2 x 2 x 2, that is, 2 cubed.
The Roman cubit was equal to 1½ Roman feet, which was equal to 24 digits. The cubit was also equal to 6 palms: see Rowland and Howe, pp. 189–192.
A square Roman cubit contained 576 square digits: 24 x 24 = 576.
2.4. Leonardo’s remarks on Roman measure are pertinent to the above-mentioned data. He writes in his notebooks:
The architect Vitruvius states in his work on architecture that the measurements of a man are arranged by Nature thus: – that is that four fingers make one palm, and four palms make one foot, six palms make one cubit, four cubits make once a man’s height, and four cubits make a pace, and twenty four palms make a man’s height, and these measurements are in his buildings.
(MacCurdy, pp. 225–6). [2]
Leonardo has noted the cubit, palm and digit measures in his drawing; they can be seen just below the feet of Vitruvian Man in figure 1.
3. Apollo’s pedestal
3.1. Apollo is mentioned numerous times in the The Ten Books on Architecture, notably in relation to the replacement of a cracked pedestal for a statue of the god. Academic H. J. Rose writes of Apollo:
Apollo, whatever his origin, is in his developed form the most characteristically Greek of all the gods. He is also, from the picturesque beauty with which Hellenic art and literature surrounded him, perhaps the best known today, and it is a commonplace with those writers, such as Swinburne, who love to contrast Hellenism with Christianity, to draw a sharp antithesis between him and Christ. ... His developed type in art is well known; his is the ideal male figure, which has reached its full growth, but still has all the suppleness and vigour of youth. He generally holds either a lyre or a bow. While all Greece worshipped him, and references to him are almost as numerous as those to Zeus himself, his most famous shrines in Greece proper were Delphoi on the mainland and the holy island of Delos; in Asia Minor he had many shrines, the best known being Klaros, Branchidai and Patara. (Rose, pp. 134–5)
Figure 10: Statue of Apollo
(courtesy Wikipedia)
3.2. The description of Apollo’s pedestal is given in the last book of the The Ten Books on Architecture, that is, in Book 10. Vitruvius writes that the pedestal is twelve feet long, eight feet wide and six feet high (Morgan, p. 289/Book 10.2.13). Note the six-foot high measure (the height of the “well shaped man”) in the pedestal’s dimensions.
Figure 11: Representation of Apollo’s pedestal
(measures in feet)
The volume is easily established as being 576 cubic feet: 12 x 8 x 6 = 576. The number 576, as already mentioned in sub section 2.3, is a square number: 24 x 24.
Especially note that the ratio 8:6 on the front face of the illustrated pedestal is the same as 4:3. Consequently, on the faces of the pedestal that have these dimensions (shown below), the diagonal must measure 10 feet because it is the hypotenuse, the “five” side, of a 3:4:5-proportion triangle. The 3:4:5 triangle has often been linked to Pythagoras and the Pythagoreans—for example, see Morgan, pp. 252–3/Introduction to Book 9.6–7.
The sum of 6 + 8 + 10 is 24.
Figure 12: Representation of the pedestal that includes
the 10-foot diagonal measure
3.3. It is possible to discern now why the pedestal was made to be twelve feet long: it ensures the volume becomes 576 (24 x 24) cubic feet. Vitruvius has clearly linked the number 24 to the god Apollo and to the character of Vitruvian Man. Consequently, Vitruvian Man and his horizontal and vertical measures can be presented in the following manner —
Figure 13: Measurements of Vitruvian Man
The area of the square can be expressed as:
● 36 square feet
● 16 square cubits
● 576 square palms
4. Leonardo da Vinci and Vitruvian Man
4.1. The subject of architecture greatly interested Leonardo and he was familiar with Vitruvius’s opus. As already stated, his drawing of Vitruvian Man is probably the most famous illustration of its kind in history. It is thought Leonardo drew “the well shaped man” around the year 1490.
A prolific writer and an active researcher, Leonardo’s studies were written down in notebooks and on individual sheets of paper. Texts were often illustrated with sketches. A considerable amount of this work survives. His notes and papers reflect an active, investigative mind; there is inventiveness, speculation and the occasional error—see the footnote in sub section 2.4, for example.
In the illustration of Vitruvian Man note that Leonardo has, on a vertical axis, marked lines at the base of the throat, the nipples, the genitals and the knees. On a horizontal axis there are lines at the wrists, elbows and tops of the arms. Vitruvius prescribes most of these locations.[3] In his notebooks, Leonardo describes a body feature not mentioned by Vitruvius in extant versions of his work. Leonardo writes (Quaderni vi 10 r):
The greatest thickness of a man from the breast to the spine goes eight times into the height and is equal to the space between the chin and the crown of the head. (MacCurdy, Volume I, page 224)
In sub section 2.4, Leonardo refers to the height of Vitruvian Man in terms of cubits (4) and palms (24). The focus here is on cubits.
The measure of the head (chin to crown) is 12 digits, which is half a cubit and one-eighth the height of the body. Consequently, Vitruvian Man would need a rectangular solid container 4 cubits x 4 cubits x ½ a cubit to enclose his body. The “container” is pictured below.
Figure 14: Vitruvian Man contained.
(measures in cubits)
The volume of the container is 8 cubic cubits, which is equal to the volume of a cube with edges measuring 2 cubits. The cube is illustrated below.
Figure 15: Cube with edges measuring two cubits
(length, breadth and height shown here)
The following observations about this cube are noteworthy:
1) Since the cube has 12 edges, the sum of the length of the edges is 24 cubits.
2) Each face of the cube has an area of 4 square cubits: 2 x 2 = 4.
The cube has 6 faces; therefore, the total area of the faces is 24 square cubits.
3) A cube has 24 angle locations, four on each of the six faces.
4) Each edge of the cube measures 2 cubits, which is equal to 48 digits. The sum of the twelve edges can be expressed as 576 digits and the number 576 is 24 squared: compare this with the volume of the pedestal of Apollo (576 cubic feet) detailed in sub section 3.2 and the area of a square cubit (576 square digits) mentioned in sub section 2.3.
5) Each face of the cube contains 2304 square digits: 48 x 48 = 2304 square digits. The sum of the six faces is 13,824 square digits. The number 13,824 is 24 cubed (24 x 24 x 24). A representation of the 48-digit cube is shown below.
Figure 16: Representation of the 48-digit cube
6) A reference to “24” can be found in the head of Vitruvian Man. The Roman architect, as noted earlier, establishes that the “well shaped man” is 24 palms tall: see sub sections 2.3, 2.4 and 3.3. He states in Book 3 that:
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 a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; (Morgan, p.72/Book 3.1.2)
Accordingly, the face—a tenth part of the whole height—must measure 2.4 palms. (So do the hands.) The head and face are illustrated below.
Figure 17: Head and face of Vitruvian Man
Furthermore, the distance from the top of the forehead to the crown, as can be seen in the illustration above, is 0.6 of a palm, which is equal to 2.4 digits.
It is not difficult to see why Leonardo formulated the “solid man” the way he did. Many more findings, some quite startling, are unveiled in forthcoming works, especially in relation to the hidden nature of Vitruvian Man.
5. Summary
5.1. Vitruvius has more to say on Apollo and cubes. The doubling of the size of a cube is raised by the architect in his Introduction to Book 9, passages 13 and 14.
13. Now let our attention be turned to the researches of Archytas of Tarentum and Eratosthenes of Cyrene. For they made many welcome discoveries for humanity by means of mathematics. Therefore, just as they were appreciated for their other inventions, in this matter they are most greatly admired for their inspirations: each of them used a different method to carry out what Apollo had ordered in an oracle at Delos, namely that his altar which had equal feet on all sides, be doubled, in order that the people of the island be freed from an ancient curse.
14. Archytas carried out the task by drawing a diagram of half cylinders, and Eratosthenes achieved the same objective by using a machine, the mesolabe. (Rowland and Howe, p. 108)
Note that it is Apollo who sets up the task. And compare the above quotation with Plutarch’s remarks in sub section 1.5.
5.2. All the facts presented in this essay are from authoritative sources. Apollo has been linked to the time number 24 through attributes of the cube and a square, the dimensions of a pedestal for the god, the characteristics of Vitruvian Man as provided by the Roman architect Vitruvius, and through the formulation of Leonardo’s solid man container.
Apollo has a family connection with the cube. This is known from a book written some 2500 years ago.
5.3. The Greek Herodotus (c. 490–420 B.C.) is the author of the first great narrative history produced in the ancient world: The Histories. Cicero called him “the Father of History”. Pythagoras is mentioned in the historian’s masterwork—for example, see Marincola, p. 245/Book 4.95.
Some remarks about Greek measure are pertinent here. A Greek stade[4] contained 600 Greek feet or 400 (20 x 20) Greek cubits or 2400 Greek palms.
The Greek foot and cubit had the same number of divisions as the Roman foot and cubit: the Greek foot contained 16 digits (daktyloi) or 4 palms (palaste). The Greek cubit contained 24 digits or 6 palms (Dilke, p. 26).
Of Apollo Herodotus writes:
The Egyptians have a legend to explain how the island [Chemmis] came to float: in former times Leto, one of the eight original deities, lived in Buto, where her oracle now is, and having received Apollo, son of Osiris, as a sacred trust from Isis, she saved him from Typhon when he came there in his world-wide search, by hiding him in the island. The Egyptians say that Apollo and Artemis are the children of Isis and Dionysus, and that Leto saved them and brought them up. In Egyptian, Apollo is Horus, Demeter is Isis, Artemis is Bubastis. (Marincola, pp. 144–5/Book 2.156)
Thus Apollo is linked to the Egyptian gods Isis, Horus and Osiris.[5] The historian also writes this about Apollo and Leto:
I have often made mention of the Egyptian oracle, and I will now treat fully of it, for this it deserves. This Egyptian oracle is in a temple sacred to Leto, and is situated in a great city by the Sebennytic arm of the Nile, on the way up from the sea. The name of the city where is this oracle is Buto; I have already named it. In Buto there is a temple of Apollo and Artemis. The shrine of Leto in which is the oracle is itself very great, and its outer court is ten fathoms high. But I will now tell of what was the most marvellous among things visible there: in this precinct is the shrine of Leto, whereof the height and length of the walls is all made of a single stone slab; each wall has an equal length and height, namely, forty cubits. Another slab makes the surface of the roof, the cornice of which is four cubits broad. (Godley, p. 467/Book 2.155)
Ten fathoms is 60 Greek feet. So is 40 Greek cubits. Marincola translates “outer court is ten fathoms high” as a “gateway sixty feet high” (Marincola, page 144). The shrine is clearly a cube and is translated by Marincola as such. Curiously, Herodotus refers to the 60-foot measure in two forms: (1) as ten fathoms; and (2) as forty cubits.
Dilke transliterates the Greek for fathom as orguia, and this is sometimes used by the present writer (Dilke, p. 26).
In English, the word “fathom” not only signifies a measure of six feet, it can also mean:
1) To penetrate a mystery;
2) Outstretched arms. (Reference: Collins English Dictionary, Third Edition, Harper Collins Publishers)
Now in Greek, orguia not only signifies four cubits (6 feet), it too means “outstretched arms”. Below is an extract from A Greek-English Lexicon compiled by H. G. Liddell and R. Scott (page 1246) that shows the entry for orguia
Visually, the outstretched arms of Vitruvian Man are a prominent and intriguing feature of his layout. As frequently discussed, the arms are six feet or four cubits across. Consequently, it is not at all difficult to contemplate the possibility that the origin of the orguia measure is linked to the creation of the “well shaped man”. In forthcoming work, the way Vitruvian Man was created and for what purpose is comprehensively explained.
Herodotus says the cornice of the slab that makes the surface of the roof of the shrine is four cubits broad. Four cubits is the measure of the outstretched arms of Vitruvian Man, as well as being the measure of his height.
So—Apollo’s mother has a shrine that has the shape of a cube. Each face (a square) on the cube measures and contains:
Aspect 1. 60 x 60 Greek feet: an area of 3600 square feet
Aspect 2. 40 x 40 Greek cubits: an area of 1600 square cubits
Aspect 3. 240 by 240 Greek palms an area of 57,600 square palms.
Accordingly, each aspect is an enlarged version of the square that encloses Vitruvian Man. See the text and figure 13 in sub section 3.3.
Figure 18: Leonardo, Self-portrait,
ca. 1510: Royal Library, Windsor
A quote from LEONARDO the universal man by Alessandra Fregolent provides a meaningful end to this essay. Especially note the last sentence in the quotation.
Leonardo was also carrying on his theoretical studies, as usual directed at a variety of disciplines. From Pietro da Novellara we know that in 1504 his attention was concentrated on geometry. As in Milan, he was aided in his exploration of this field by his friend Luca Pacioli, who was in Pisa between 1500 and 1505, teaching at the university. The rules of mathematics were of overwhelming importance to Leonardo: arithmetic and geometry “embrace everything in the universe,” he wrote, and “if one of them is missing nothing can be accomplished” (Codex Madrid II). For a mind like his, capable of giving solid form to abstract intuitions, form and volumes, perceived as tangible realities, became the pretexts for the creation of increasingly new configurations of regular bodies. In the “Treatise on Geometry” (Codex Forster I), begun on July 12, 1505, he explains how to transform geometric bodies without changing their volume.
(Fregolent, pp. 102–3)
Appendix 1. Remarks on references and abbreviations
The works of classical writers often have ciphers printed in the margins of the translations to indicate a standard means of precise reference to passages. For example, in the case of Plato they indicate the pages in the edition of the philosopher’s works by Stephanus (Henri Estienne), Geneva, 1578. To accord with tradition, references to the works of classical writers are presented in this manner, often along with references to more accessible popular versions of the books. Hence, in the text of this essay, references are mostly presented in the following way:
Herodotus.
In the case of Herodotus (The Histories), the numbering system is that used by A. D. Godley and J. Marincola (from the Aubrey de Sélincourt translation), e.g.,
Godley, p. 467/Book 2.155 or
Marincola, p. 144/Book 2.155 Plato
Lee, D. (H. D. P.) The Republic, pp. 273–4/S526c – 527c
S526 indicates the page in the Stephanus edition. The letter “c” represents the position on page 526.
Plutarch. For Plutarch: Moralia V
Babbitt, pp. 209–211/S386
S386 indicates page 386 in the Books of the Moralia, the edition of Stephanus, 1572.
Vitruvius. For Vitruvius: The Ten Books on Architecture
Morgan, p. 72/Book 3.1.1
Book 3.1.1 indicates Book 3, chapter 1, passage 1.
References
Babbitt, Frank Cole. Plutarch: Moralia. Volume V. Harvard University Press, 1999 edition.
Dilke, O. A. W. (Reading the Past series) Mathematics and Measurement. British Museum Publications, London, 1991.
Fregolent, Alessandra. LEONARDO the universal man. Thunder Bay Press, California.
English translation © 2004 Mondadori, Electa S.p.A., Milan
Godley, A. D. (Transl.) Herodotus [“The Histories”]. Heinemann Ltd. London. (Harvard University Press) 1946 edition.
MacCurdy, E. The Notebooks of Leonardo da Vinci (Vol. 1) Jonathan Cape Ltd., London 1938.
Marincola, J. (Transl.) Herodotus: The Histories. Penguin Books, 1996 edition. (From the translation by Aubrey de Sélincourt)
Morgan, M. H. Vitruvius: The Ten Books on Architecture. Harvard University Press, Cambridge, Mass., 1914. (Republished by Dover Publications)
Rose, H. J. A Handbook of Greek Mythology. Published by E. P. Dutton, U.S.A., 1959.
Rowland, I. D. and Howe, T. N. Vitruvius: Ten Books on Architecture. Cambridge University Press, 1999.
Zupko, R. E. British Weights and Measures: A History from Antiquity to the Seventeenth Century. Madison University of Wisconsin Press, 1977.
Credits
(All images for educational and study purposes only.)
● Vitruvian Man by Leonardo da Vinci, Galleria dell´Accademia, Venice
● Leonardo da Vinci, Self-Portrait; Royal Library, Windsor (see Fregolent’s book, p. 124)
[1] Correct spelling is Panionian.
[2] Leonardo makes a misstatement in the quotation. The Roman pace was equal to five Roman feet, which was equal to three and one-third Roman cubits, not four. Bold face words in the quotation are the present writer’s emphasis.
[3] All of Vitruvius’s design specifications are taken from the Morris Hicky Morgan translation of The Ten Books on Architecture published by Dover. The specifications are found in Book 3, chapter 1. They are the same as in the Rowland and Howe translation.
[4] The English word “stadium” derives from the Greek stade.
[5] The Greeks regarded Leto as the birth mother of Apollo and Artemis.
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