1.1.4. The musical harmony of Pythagoras and the music of the spheres
Pythagoreans made wonderful discoveries in music. Pythagoras found that the most pleasant to ear consonances are obtained only when the lengths of the strings, that produce these consonances, have ratios as the first natural numbers 1, 2, 3, 4, 5, 6, that is, 1:1, 1:2 (unison and octave), 2:3, 3:4 (quint and quart), 4:5, 5:6 (thirds), etc. The discovery he made (the law of consonances) shocked Pythagoras. It was this discovery that first pointed out the existence of numerical patterns in Nature, and it was this that served as a starting point in the development of Pythagorean philosophy and in the formation of their basic thesis: “Everything is a Number.” Therefore, the day when Pythagoras discovered the law of consonances, was declared by the German physicist A. Sommerfeld the birthday of theoretical physics.
The discovery of mathematical regularities in musical consonances was the first “experimental” confirmation of the Pythagorean doctrine of Number. From this moment, the music and the related doctrine of Harmony began to occupy a central place in the Pythagorean system of knowledge. The idea of musical relations soon acquired the “cosmic scales” among the Pythagoreans and grew into the idea of Universal Harmony.
The Pythagoreans began to assert that the entire Universe is organized on the basis of simple numerical relationships and that the moving planets demonstrate “the music of the heavenly spheres” and ordinary music is merely a reflection of Universal Harmony prevailing everywhere. Thus, music and astronomy had been reduced by the Pythagoreans to the analysis of numerical relations, that is, to arithmetics and geometry. All four MATEMs (arithmetics, geometry, harmonics and spherics) began to be considered mathematical and called by one word — “mathematics.”
1.1.5. Once again about the term of the Mathematics of Harmony
As mentioned in the preface, for the first time, the term Mathematics of Harmony was introduced in a short article “Harmony of spheres”, published in The Oxford Dictionary of Philosophy [103]. This doctrine, often attributed to Pythagoras, leads to the unification of mathematics, music, and astronomy. Its essence is the fact that the celestial solids, being huge objects, must produce music during their movement. The perfection of the heavenly world requires that this music must be harmonious; it is hidden from our ears only because it is always present. The Mathematics of Harmony was a Central Discovery of Immense Importance for the Pythagoreans.
Thus, the concept of the Mathematics of Harmony in [103] is associated with the “Harmony of spheres”, which was also called the Harmony of the World or world music. The “Harmony of spheres” is an ancient and medieval doctrine about the musical and mathematical structure of the Cosmos, which goes back to the Pythagoras and Plato philosophical tradition.
Another mention about the Mathematics of Harmony, applied to ancient Greek mathematics, is found in the book by Vladimir Dimitrov A New Kind of Social Science. The Study of Self-Organization of Human Dynamics, published in 2005 [54]. Here is a quote from this book:
“The 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