Thousands of years ago in ancient Greece, the philosophers described the mathematical foundations of art: “No art is created without proportion, and proportion lies in number. So all art arises through number […] it is right to say: through number everything looks beautiful.”1 The mathematical speculations expressed the conviction that beauty is the highest divine factor with which man can commune in the real world, for it is present in it through measure and proportion based on numbers, which are the basis of harmony, that is, the perfect interplay of the elements of which the world as a whole is composed

This harmony is expressed in man, in the proportions of his body, and especially in the fact that, as the architect Vitruvius, who lived in the first century BC, described in his treatise On the Architecture of the Ten Books, man is the only being in the world who can simultaneously be inscribed in the square and the circle, which are the most perfect geometrical figures. Earlier, the Greek sculptor Polyclet had searched for the ideal proportions of the human body based on establishing their numerical ratios: “Beauty lies in the proportions of the parts of the body, the finger to the finger, the finger to the joint, his to the hand and all these parts one to the other” and “perfection depends on many numerical relations and small differences determine it”.2 The artistic culmination of his explorations was a sculpture depicting a young athlete holding a javelin Doryphoros.

###### Poliklet, Doryhoros, around 450 BC Sources: Wikipedia

The proportions of the young man sculpted by Polycletus are based on the ratio of the size of the head to the whole body in the ratio of 1 to 10. The model of beauty of the human body was modified by the sculptor Lysippus, who, in his sculpture *Apoksyomenos*, lengthened his proportions in a ratio of 1 to 9, making the figure more slender.

The search for beauty based on the harmony of proportions based on numerical ratios was also at the heart of the achievements of Greek architecture, particularly temples. The Parthenon, built in the 2nd half of the 5th century BC on the Acropolis in Athens, according to a design by the architect Iktinos, expressed in its most perfect form this quest for beauty based on numerical proportions derived from the module, or unit, which was the width of the column, where the principle of the so-called “golden ratio” would have been used.

The legacy of the Greek search for mathematically based objective beauty was revived in 15th-century Italy. In this way, the widely admired statues of the Apollo of Belvedere or The Laocoon Group were found.

Giovanni Dondi wrote: “Few works of the brilliant masters of antiquity have survived; those that have survived, however, are of keen interest to those who know how to do so […] And when one compares with those works what is done today, it becomes clear that the creators of those things were superior to the artists of today in their innate talent, and that they knew how to apply their skills better”.3

Living in the 2nd half of the 15th century, the eminent mathematician and humanist Luca Pacioli, in his work *De divina proportione* published in 1495 in Venice, wrote: “We shall speak first of the proportions of man, for from the human body derive all measurements and in it can be found all the relations and proportions through which God reveals the greatest mysteries of nature. Having considered the proper arrangement of the human body, the ancients gave all their works, above all their temples, proportions in accordance with it. For in the human body they found two main shapes without which it is impossible to accomplish anything, namely the circle, which is the most perfect and measurable […] and the equilateral square .”4 The Venetian edition was decorated with engravings designed by Leonardo da Vinci himself.

Among the illustrations in the volume there are Renaissance letter designs based on the Roman capital, a typeface model used in antiquity. They have been inscribed in mathematical diagrams that determine their proportions.

There are also engravings showing the head of a man written in the figure of an equilateral triangle.

Leonardo referred directly to the Vitruvian model of a man inscribed simultaneously in a square and a circle, and he produced, around 1490, his famous drawing depicting a standing naked man framed in two ways. He wrote: “Vitruvius, the architect, put the message in his work on architecture that the dimensions of man in nature are composed so that […] the hand is equal to four fingers [and] man is equal to 24 hands and these dimensions are in his buildings.”5

Centuries later, Le Corbusier, a prominent modern architect, presented the Modulor in 1951, a model of the system of proportions on which modern architecture was to be based. Le Corbusier based on a model of the man of the future, measuring 183 cm, and with his arm raised, 226. This value was to determine the minimum height of a residential storey, which the architect used in the Unité ïHabitation (Apartment Unit), a multi-storey residential building in Marseille that he designed in 1947.

The ancient Greek ideal of beauty based on number, and embodied in works of art, became one of the most important artistic models, restored after centuries in the Renaissance and, as it turns out, still inspiring artists.

##### Bibliography

- Alicja Kuczyńska, Philosophy and theory of beautification Marsilia Ficina, Warszawa, 1970
- Alicja Kuczyńska, Beautification. Myth and reality, Warszawa, 1972
- Alicja Kuczyńska, Art as philosophy in the culture of the Italian Renaissance, Warszawa, 1988
- Michael Levey, The early Renaissance, Warszawa, 1970
- Michael Levey, The High Renaissance, Warszawa, 1980
- Maria Rzepińska, Cinquecento, Warszawa, 1988
- Władysław Tatarkiewicz, A history of six concepts,Warszawa, 2011
- Władysław Tatarkiewicz, History of aesthetics, volumes 1-3, Warszawa, 2009

[1] Mirosław Bogdan, Beauty and forms of relevance in architecture, Scientific Journals of the Silesian University of Technology, series: Architecture, j. 19, 1992, p.10

[2] Kazimierz Michałowski, How Greeks created the arts, Warsaw, 1970, p. 128

[3] Jan Białostocki, Thinkers, chroniclers and artists on art. From antiquity to 1500, Warsaw, 1978, p.340

[4] Jan Białostocki, Theorists, writers and artists on art 1500 – 1600, Warsaw, 1985, p.23

[5] Luba Ristujczina, Leonardo da Vinci, Dragon, 2022, p. 245