To be exact the phrase of Plato says: “Everything that is good is beauty, and beauty is not without relations or regular proportions.” Also in Philebus 64c he says: “The measure (metron) and proportion (Symmetria) performed everywhere beauty and perfection “
The objective of classical ancient art is imitation of nature. But nature is a κόσμος, kosmos, orderly, harmonious and beautiful whole, and beauty is only possible if there is proportion.
So the issue of the ratio is a key concern in the reflection of philosophers and artists. They call συμμετρἰα, symmetria, to proportion, with a meaning therefore different from that currently has the word.
Pythagoras and Theano of Croton, one philosopher and mathematician who some believe was the wife of Pythagoras, theorized about ratio or "reason for the ends and means." It was certainly very important oriental influence of Mesopotamia and Egypt in this and many other issues that the Greeks developed.
Euclides defined so the cited reason or proportion
A right line is said to be cut at a point in extreme and mean ratio when the whole line is to the greater segment as the greater segment is to the less. ( Books of the Elements of Euclid, VI, Definition 3; translated by John Casey)
This is what in the Renaissance was called "golden ratio". It is also called golden number, extreme and mean ratio, golden section, divine proportion). It is represented by the Greek letter φ(fi) (lowercase) or Φ (fi) (uppercase), in honor of the Greek sculptor Phidias. It is an irrational number whose value is set in the Renaissance in 1.61803398874989 …
This ratio is in geometric figures and various beings of nature. This proportion was applied for example in the construction of temples.
These proportions are set or measured with a stick or cane κανών called "canon". From this function, the word came to mean "rule, law, model, canon .... ).
So it is soon established that the canon of the human body must have certain proportions to be beautiful.
This whole matter of proportions, beauty and the canon of beauty is very influenced or it is related to the philosophical problem of knowledge of things. Men do not looking both realistic imitation of concrete things, that is, the shadows of the things that Plato says, but the gist of the thing itself . So the "canon" is not found in a particular thing or being, but that, observed several, the ideal model is established. In the case of man, the “canon” shall be set by the observation of multiple individuals and in a process of abstraction at which the essential is sought.
According to the physician Galen, the famous sculptor Polykleitos theorized about it in a treaty called Canon and precisely he reflected it in a statue of a naked man who is identified with the Doryphoros; in the canon of Polykleitos man's height equivalent to 7 heads.
The Greco-Roman physician Galen (130-200) tells us in the Latin version of his "The nine books on the doctrines of Hippocrates and Plato, De placitis Hippocratis et Platonis libri novem, V 449 (vol I, recensuit et explanavit I, Müller, Lipsiae, BG Teubneri, 1874), p. 426:
"He thought that beauty does not reside in the right proportion of the elements but of the parties; ie finger to finger and all fingers to the palm of the hand and the first part of the palm and these to the elbow, the elbow to the arm and finally all these to everything else, as it is stated in writing in the Canon of Polykleitos. In this book Polykleitos, after teaching all body proportions, confirmed his presentation with a work making a statue as shown in his book, which he called with the same name as the book, ie "canon". Thus, the beauty of the body, according to the opinion of all physicians and philosophers, is the appropriate proportion of the members, health in the proportion of each of the elements, whatever "
Claudii Galeni, “ pulchritudinem autem non in elementorum sed in partium apta congruentia, digiti scilicet ad digitum digitorumque omnium ad palmam palmaeque primam partem et harum ad cubitum, cubiti ad brachium, omnium denique ad omne positam esse censuit, quemadmodum in Polycliti canone litteris mandatum est. In eo libro Polyclitus cum omnes corporis proportiones docuisset, opere demum orationem confirmavit ex libri praeceptis statuam fabricatus, quam eodem nomine,quo librum, canona appellavit. Pulchritudo igitur corporis ex omnium medicorum philosopoorumque sententia in membrorum apta convenientia consistit, sanitas in elementorum, quaecumque sunt, inter se congruentia.
Note: in ancient times, the works of Galen were not translated into Latin; so at the end of the Empire they fell into oblivion, because few people could read Greek, except in the East; then some works were translated into Arabic and from there to Latin; in the Renaissance they are translated into Latin and even in some cases it is tried to translate some only in Arabic existing work into Greek original, which was lost.
For the Greeks symmetry and harmonic proportions of the body are synonymous with health.
Then Euphranor of Itsmia, painter and sculptor, wrote another treaty about Symmetria,
altering the canon of Polykleitos; in the canon of Lysippos the body has seven heads and a half, instead seven only.
Precisely measurement units are taken from parts of the human body and its proportions: they are elbows, arms, hands and feet, which may be different at each artistic school.
In Roman times Vitruvius collects these theories and measures and their relationships in his De Architectura, III, 1:
The design of Temples depends on symmetry, the rules of which Architects should be most careful to observe. Symmetry arises from proportion, which the Greeks call ἀναλογία. Proportion is a due adjustment of the size of the different parts to each other and to the whole; on this proper adjustment symmetry depends. Hence no building can be said to be well designed which wants symmetry and proportion. In truth they are as necessary to the beauty of a building as to that of a well formed human figure, which nature has so fashioned, that in the face, from the chin to the top of the forehead, or to the roots of the hair, is a tenth part of the height of the whole body. From the chin to the crown of the head is an eighth part of the whole height, and from the nape of the neck to the crown of the head the same. From the upper part of the breast to the roots of the hair a sixth; to the crown of the head a fourth. A third part of the height of the face is equal to that from the chin to under side of the nostrils, and thence to the middle of the eyebrows the same; from the last to the roots of the hair, where the forehead ends, the remaining third part. The length of the foot is a sixth part of the height of the body. The fore-arm a fourth part. The width of the breast a fourth part.
Similarly have other members their due proportions, by attention to which the ancient Painters and Sculptors obtained so much reputation. Just so the parts of Temples should correspond with each other, and with the whole. The navel is naturally placed in the centre of the human body, and, if in a man lying with his face upward, and his hands and feet extended, from his navel as the centre, a circle be described, it will touch his fingers and toes. It is not alone by a circle, that the human body is thus circumscribed, as may be seen by placing it within a square. For measuring from the feet to the crown of the head, and then across the arms fully extended, we find the latter measure equal to the former; so that lines at right angles to each other, enclosing the figure, will form a square.º
If Nature, therefore, has made the human body so that the different members of it are measures of the whole, so the ancients have, with great propriety, determined that in all perfect works, each part should be some aliquot part of the whole; and since they direct, that this be observed in all works, it must be most strictly attended to in temples of the gods, wherein the faults as well as the beauties remain to the end of time. It is worthy of remark, that the measures necessarily used in all buildings and other works, are derived from the members of the human body, as the digit, the palm, the foot, the cubit, and that these form a perfect number, called by the Greeks τέλειος.a The ancients considered ten a perfect number, because the fingers are ten in number, and the palm is derived from them, and from the palm is derived the foot. Plato, therefore, called ten a perfect number, Nature having formed the hands with ten fingers, and also because it is composed of units called μονάδες in Greek, which also advancing beyond ten, as to eleven, twelve, &c. cannot be perfect until another ten are included, units being the parts whereof such numbers are composed. The mathematicians, on the other hand, contend for the perfection of the number six, because, according to their reasoning, its divisors equal its number: for a sixth part is one, a third two, a half three, two-thirds four, which they call δίμοιρος; the fifth in order, which they call πεντάμοιρος, five, and then the perfect number six. When it advances beyond that, a sixth being added, which is called ἔφεκτος, we have the number seven. Eight are formed by adding a third, called triens, and by the Greeks, ἐπίτριτος. Nine are formed by the addition of a half, and thence called sesquilateral; by the Greeks ἡμιόλιος; if we add the two aliquot parts of it, which form ten, it is called bes alterus, or in Greek ἐπιδίμοιρος. The number eleven, being compounded of the original number, and the fifth in order is called ἐπιπεντάμοιρος. The number twelve, being the sum of the two simple numbers, is called διπλασίων. Moreover, as the foot is the sixth part of a man's height, they contend, that this number, namely six, the number of feet in height, is perfect: the cubit, also, being six palms, consequently consists of twenty-four digits.
(Translated by Joseph Gwilt, London: Priestley and Weale, 1826. From http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Vitruvius/home.html)
Aedium compositio constat ex symmetria, cuius rationem diligentissime architecti tenere debent. ea autem paritur a proportione, quae graece ἀναλογία dicitur. proportio est ratae partis membrorum in omni opere totiusque commodulatio, ex qua ratio efficitur symmetriarum. namque non potest aedis ulla sine symmetria atque proportione rationem habere compositionis, nisi uti ad hominis bene figurati <speciem> membrorum habuerit exactam rationem.
Corpus enim hominis ita natura composuit uti os capitis a mento ad frontem summam et radices imas capilli esset decimae partis, item manus palma ab articulo ad extremum medium digitum tantundem, caput a ad súmmum verticen octavae; cum cervicibus imis ad imas radices capillorum sextae, <a medio pectore>ad summum verticem quartae. ipsius autem oris altitudinis tertia est pars ab imo mento ad imas nares, nasus ab imis naribus ad finem medio superciliorum tantundem, ab ea fine ad imas radices capilli frons efficitur item tertiae partis. pes vero altitudinis corporis sextae, cubitus quartae, pectus item quartae. reliqua quoque membra suas habent commensus proportiones, quibus etiam antiqui pictores et statuarii nobiles usi magnas et infinitas laudes sunt adsecuti.
Similiter vero sacrarum aedium membra ad universam totius magnitudinis summam ex partibus singulis convenientissimum debent habere commensus responsum. item corporis centrum medium naturaliter est umbilicus. inamque si homo conlocatus fuerit supinus manibus et pedibus pansis circinique conlocatum centrum in umbilico eius, circumagendo rotundationem utrarumque manuum et pedum digiti linea tangentur. non minus quemadmodum schema rotundationis in corpore efficitur, item quadrata designatio in eo invenietur. nam si a pedibus imis ad summum caput mensum erit eaque mensura relata fuerit ad manus pansas, invenietur eadem latitudo uti altitudo, quemadmodum areae quae ad normam sunt quadratae.
Ergo si ita natura composuit corpus hominis uti proportionibus membra ad summam figurationem eius respondeant, cum causa constituisse videntur antiqui ut etiam in operum perfectionibus singulorum membrorum ad universae figurae speciem habeant commensus exactionem. igitur cum in omnibus operibus ordines traderent, <tum> maxime in aedibus deorum, <quorum> operum et laudes et culpae aeternae solent permanere.
Nec minus mensurarum rationes, quae in omnibus operibus videntur necessariae esse, ex corporis membris collegerunt, uti digitum palmum pedem cubitum, et eas distrubuerunt in perfectum numerum, quam Graeci τελειον dicunt, perfectum autem antiqui instituerunt numerum qui decem dicitur. namque ex manibus digitorum numero ab palmo pes est inventus. si autem in utrisque palmis ex articulis ab natura decem sunt perfecti, etiam Platoni placuit esse eum numerum ea re perfectum qui ex singularibus rebus, quae μοναδες apud Graecos dicuntur, perficitur decussis. simul autem undecim aut duodecim sunt facti, quod superaverint, non possunt esse perfecti, donec ad alterum decussim perveniant. singulares enim res particulae sunt eius numeri.
Mathematici vero contra disputantes ea re perfectum dixerunt esse numerum qui sex dicitur, quod is numerus habet partitiones eorum rationibus numero convenientes sic, sextantem unum, trientem duo, semissem tria [bessem quam διμοιρον dicunt quattuor, quintarium quam πενταμοιρον dicunt quinque, perectum sex. cum ad superlationem crescat, supra sex adiecto asse ἐφεκτον, cum facta sunt octo quod est tertia adiecta tertiarum alterum qui ἐπιτριτος dicitur, dimidia adiecta cum facta sunt novem sesquialterum qui ἡμιολιος appellatur, duabus partibus additis et decussi facto bessem alterum quam ἐπιδιμοιρον vocitant, in undecim numero quod adiecti sunt quinque quintarium quam ἐπιπεμπτον dicunt, duodecim autem quod ex duobus numeris simplicibus est effectus διπλαχιον].
Non minus etiam quod pes hominis altitudinis sextam habet partem, id est ex eo quod perficitur pedum numero corporis sexis altitudinis terminatio, eum perfectum constituerunt, cubitumque animadverterunt ex sex palmis constare digitisque XXIIII.
This text has been essential in Western culture: Leonardo da Vinci picked it up in his Treatise on Painting and according to it he made his famous drawing "Vitruvian Man" made around 1490 that is conserved in the Galleria dell'Accademia in Venice.
Vitruvius had an enormous importance for the development of the entire Renaissance architecture. Who has not felt impressed by this work of Leonardo so often reproduced in a thousand different contexts?
Well, Leonardo reproduces it and he comments it at the top and at the bottom of the drawing. Besides a tribute to Classical Antiquity, Leonardo tries to emphasize the scientific and mathematical nature of artistic representation.
What he wrote by abbreviations at the top says:
"Vetruvio, architect, puts in his work on architecture that the measurements of man are in nature distributed in this manner, that is: a palm is four fingers, a foot is four palms, a cubit is six palms, four cubits make a man, a pace is four cubits, a man is 24 palms; and these measurements are in his buildings.
If you open your legs enough that your head is lowered by one-fourteenth of your height and raise your hands enough that your extended fingers touch the line of the top of your head, know that the centre of the extended limbs will be the navel, and the space between the legs will be an equilateral triangle".
At the bottom it is written:
The length of the outspread arms is equal to the height of a man. From the hairline to the bottom of the chin is one-tenth of the height of a man; from below the chin to the top of the head is one-eighth of the height of a man; from above the chest to the top of the head is one-sixth of the height of a man; from above the chest to the hairline is one-seventh of the height of a man; from below the knee to the root of the penis is a quarter of the height of a man; the distances from below the chin to the nose and the eyebrows and the hairline are equal to the ears and to one-third of the face.
The maximum width of the shoulders is a quarter of the height of a man; from the breasts to the top of the head is a quarter of the height of a man; the distance from the elbow to the tip of the hand is a quarter of the height of a man; the distance from the elbow to the armpit is one-eighth of the height of a man; the length of the hand is one-tenth of the height of a man; the root of the penis is at half the height of a man; the foot is one-seventh of the height of a man; from below the foot to below the knee is a quarter of the height of a man.
The texts are not absolutely coincident because Leonardo gives few different measurements and adds some new, but they speak for themselves.
Well, Vitruvius tells us that man's height is equal to the width with his hands extended , therefore height and width form a square. He also tells us that by extending his arms and legs forms a circle. Many artists tried to integrate the human figure, forming the square and the circle, in one drawing. Leonardo found the solution: the center of the square is in the genitals; the center of the circle is in the navel. And oh great ! The ratio between the side of the square and the circle side is golden.