prerequisite for Harmony for the Greeks was expressed with the phrase “nothing superfluous.” This phrase contained mysterious positive qualities, which became the object of the study of the best minds. Thinkers, such as Pythagoras, sought to unravel the mystery of Harmony as something unspeakable and illuminated by mathematics.”
The Mathematics of Harmony, studied by the ancient Greeks, is still an inspiring model for modern scholars. Of decisive importance for this was the discovery of the quantitative expression Harmony, in all the amazing variety and complexity of Nature, through the golden section Φ = (1 +
It is important to emphasize one more aspect in the book [54] — the concept of Mathematics of Harmony is directly associated with the golden section, the most important mathematical discovery of the ancient science in the field of the Mathematical Harmony, which at that time was called the “division of a segment in the extreme and mean ratio.”
1.2. The Golden Section in Euclid’s Elements
Euclid’s Elements is the greatest mathematical work of the ancient Greeks. Currently, every school student knows the name of Euclid, who wrote the most significant mathematical work of the Greek epoch, Euclid’s Elements. This scientific work was created by him in the third century BC and contains the foundations of ancient mathematics: elementary geometry, number theory, algebra, the theory of proportions and relations, methods for the calculating areas and volumes, etc. In this work, Euclid summed the development of Greek mathematics and created a solid foundation for its development (Fig. 1.3).
The information about Euclid is extremely scarce. For the most reliable information about Euclid’s life, it is customary to relate the little data that is given in Proclus’ Commentaries to the first book of Euclid’s Elements. Proclus points out that Euclid lived during the time of Ptolemy I Soter, because Archimedes, who lived under Ptolemy I, mentions Euclid. In particular, Archimedes says that Ptolemy once asked Euclid if there is a shorter way of studying geometry than Elements; and Euclid replied that there was no royal path to geometry.
Fig. 1.3. Euclid.
Plato’s disciples were teachers of Euclid in Athens and in the reign of Ptolemy I (306–283 BC). Euclid taught at the newly founded school in Alexandria. Euclid’s Elements had surpassed the works of his predecessors in the field of geometry and for more than two millennia remained the main work on elementary mathematics. This unique mathematical work contained most of the knowledge on geometry and arithmetic of Euclid’s era.
Fig. 1.4. Division of a segment in the extreme and mean ratio.
1.2.1. Proposition II.11 of Euclid’s Elements
In Euclid’s Elements, we find a task that later played an important role in the development of science. It was called Dividing a segment in the extreme and mean ratio. In the Elements, this task occurs in two forms. The first form is formulated in Proposition II.11 of Book II [32].
Proposition II.11 (see Fig. 1.4). The given segment AD is divided into two unequal segments AF and FD so that the area of the square AGHF, constructed on a larger segment AF, must equal to the area of the rectangle ABCD, constructed on a larger (AF) and smaller (FD) segments.
Let’s try understanding the essence of this task by using Fig. 1.4.
If we denote the length of the larger segment AF through b (it is equal to the side of the square AGHE), and the side of the smaller segment through a (it is equal to the vertical side of the rectangle ABCD), then the condition of the Proposition II.11 can be written in the form
1.2.2. The second form of the task of the division of segment in the extreme and mean ratio
The second form follows from the first form, given by (1.1), if we will make the following transformations. Dividing both sides of the expression (1.1) first by a, and then by b, we obtain the following proportion:
Fig. 1.5. Division of a segment in extreme and mean ratio (the golden section).
The proportion (1.2) has the following geometric interpretation (Fig. 1.5). We divide the segment AB by the point C for the two inequal segments AC and CB in such a manner that the bigger segment CB so refers to the smaller segment AC, as how the whole segment AB refers to the bigger segment CB, that is,
This is the definition of the “golden section”, which is used in modern science.
We denote the proportion (1.3) by x. Then, taking into consideration that AB = AC + CB, the proportion (1.3) can be written in the following form:
from which the following algebraic equation for calculating the desired ratio x follows:
It follows from the “physical meaning” of the proportion (1.3) that the desired solution of the equation (1.4) must be a positive number, from which it follows that the solution of task of “dividing a segment in extreme and mean ratio” [32] is the positive root of equation (1.4), which we denote by Φ:
Fig. 1.6. Phidias (490–430 BC).
This is the famous irrational number that has many delightful
