"Divine Proportione" by Luca Pachioli

The culture of Ancient Greece and the culture of Rome and Byzantium are two powerful flows of spiritual values, which gave sprouts of new Renaissance culture and became a cause of the origin of the Renaissance epoch "Titans". The "Titan" is the most precise word in attitude to such people, as Leonardo da Vince, Michelangelo, Nickolai Copernicus, Albert Durer and others. The honor place among them belongs to Luca Pachioli, the outstanding Italian mathematician of the Renaissance epoch.

He was born in 1445 in provincial town Borgo San-Sepolcoro, that in translation from the Italian sounds is not too joyous: "City of the Sacred Coffin".

We do not know, how much years was to the future mathematician when he began to study in the workshop of the artist Pierro della Franchesko. Franchesko's glory "rattled" in all Italy. This was the first meeting of juvenile talent with the great person. Pierro della Franchesko was the artist and mathematician, but only the second vocation of his teacher found out a respond in heart of his schoolboy. Juvenile Luca, the mathematician from the God, was loved in number world; the numbers were perceived by him as some universal key simultaneously opening an access to the true and to the beauty.

Leone Battista Alberti, architect, scientist, writer, and musician was the second great person met on the life path by Luca Pachioli. The following Alberti's words entered deeply to Luca's consciousness:

"The beauty is a certain consent and consonance of the parts in that, by which parts they are, answering to stringent number, limitation and arrangement, which are demanded by harmony, that is, by the absolute and primary beginning of the nature".

Being loved in the number world and following to Pythagor Luca Pachioli will repeat that the numbers underlay the Universe.

In 1472 Luca Pachioli becomes by the monk of the Franciscan order that gives him a possibility to be engaged in science. The events showed that he made a right decision. In 1477 he gets the Professor chair at the Perugi University.

Luca Pachioli (1454-1514)
Luca Pachioli (1454-1514)

The following portrait's description of Luca Pacioli gives a presentation about him:

"The beautiful, vigorous young man: the raised and rather broad shoulder expose inherent physical force, a potent neck and developed jaw, expressive face and eye beaming nobleness and intellect, underline force of nature. Such professor could urge to listen himself and to respect his subject".

The pedagogical work Pachioli combines with scientific activity: he starts to write the encyclopedic mathematics book. In 1494 this book was issued under the title "The Sum of arithmetic's, geometry, doctrine about proportions and relations". All material of the book is divided into two parts; the former part is dedicated to arithmetic and algebra, the latter one to geometry. One of the book sections is dedicated to problems of mathematics application to commercial business and in this part his book is a continuation of the famous Fibonacci's book "Liber abaci" (1202). In essence this mathematical work by Luca Pachioli is a sum of the mathematical knowledge of the Renaissance epoch.

Fundamental Luca Pacioli's book promoted to his glory. When in 1496 in Milan, the largest city and state of Italy, the mathematics chair was opened Luca Pachioli was invited to take this chair.

In that period Milan was the center of science and art and many outstanding scientists and artists lived and worked there. And one of them was Leonardo da Vince who became as the third great person met on the life path by Luca Pachioli. Under direct Leonardo da Vince's influence Luca Pachioli starts to write his second great book "De Divine Proportione".

This book published by Pachioli in 1509 rendered the noticeable influence on his contemporaries. Pachioli's folio was one of the first fine examples of the Italian book-printing art. The historical significance of the book consists of the fact that it was the first mathematical book dedicated to the "golden section". The book is illustrated by 60 (!) magnificent figures executed by Leonardo da Vince. The book consists of three parts: in the first part the properties of the "golden section" are given, the second part is dedicated to regular polyhedrons, the third one is dedicated to applications of the golden section in architecture.

Appealing to the "State", "Laws", and "Timey" by Plato Luca Pachioli sequentially deduces the 12 (!) different properties of the golden section. Characterizing these properties, Pachioli uses a rather strong epithets: "exclusive", "remarkable", "almost supernatural " etc. Uncovering the given proportion as the universal relation expressing a perfection of beauty in the nature and in art he calls it as "Divine Proportion" and is bent to consider it as the "instrument of thinking", as the "aesthetic canon", and as the "Main Principle of the Universe".

The Pachioli's book 'De Divine Proportione' (1509)

This book is one of the first mathematical works, in which the Christian doctrine about the God as the creator of the Universe gets a scientific substantiation. Pachioli calls the golden section as the "Divine Proportion" and selects a number of the golden proportion properties, which, in his opinion, are generic to the God himself:

"The first one consists of the fact that there is the only proportion and it is impossible to give the examples of the other kind proportions or though a little distinguished from it. This uniqueness according to the political and philosophical doctrines is the highest property of the God. The second property is property of sacred tripletness, namely, like to the God the same nature is expressed in the three persons, the Father, the Son and the Sacred Spirit, also the same proportion of this kind can take a place only for three expressions, and for the greater and smaller expressions does not exist. The third property consists of the fact that like to the God who can not be determined and clarified by any word our proportion can not be expressed neither number, accessible to us, nor any rational value and remains hidden and secret and consequently is called by mathematicians as irrational number. The fourth property consists of the fact that, just as the God never changes and presents all in all and all in each part, and our proportion for any continuous and definite large or small value can not be changed and perceived by mind. It is possible to add to the named properties the fifth property consisted of the fact that like to the God who called to the life the celestial virtue named by the fifth substance, and with its help called other four simple elements, the Earth, the Water, the Air and the Fire and with their help called to the Being any thing in the Nature, and our sacred proportion according to Plato in his "Timey" gives the formal Being to the palate, wherefore there is assigned to it the Solid called as the Dodecahedron, which cannot be constructed without our proportion".

The Dodecahedron, drawn by Leonardo da Vince for Pachioli's book 'De Divine Proportione'
The Dodecahedron, drawn by Leonardo da Vince for Pachioli's book "De Divine Proportione".

In 1510 Luca Pachioli was 65 years old. He was tired and grown old. In the library of the Bolon University the manuscript of the unpublished Pachioli book "About forces and quantities" is stored. In the foreword we found the sad phrase: "The last days of my life approach". He died in 1515 and is buried at the cemetery of his native city Borgo San-Sepolcoro.

After his death the works of the great mathematician were forgotten almost for four centuries. And when at the end of the 19th century his works become internationally known, the grateful offsets after 370-year's oblivion put the monument on his grave with the following inscription:

"To Luca Pachioli who was the friend and adviser of Leonardo da Vince and Leone Battista Alberti, who was the first scientist given to algebra the language and the frame of science, who applied his great discovery to geometry, who invented the double accounting and who gave in his mathematical works the fundamentals and invariable standards for succeeding generations".