Theon of Smyrna

There is almost no information about Theon's life, except that Claudius Ptolemy in the Almagest (I, 2, 275 and 296-299) mentions a number of observations of Mercury and Venus made by Theon the Mathematician under the Emperor Hadrian in 127-132. n e. In Smyrna (modern Izmir ) a statue was found, erected "by the priest Theon for his father, Theon, the Platonic philosopher"; based on style, it also dates from the reign of Emperor Hadrian.

Theon's text is based on the writings of numerous predecessors, and above all on the compilation works of the peripatetics of Adrastus of Aphrodisia and the platonist Trasilla ; in addition, the text mentions Derkillid , whose work Theon may also have used. Theon relies on the scientific results of Archimedes , Eratosthenes and Hipparchus , mentions the ancient authors of the Pythagorean tradition: Hippas , Filolaus , Architects , Aristoxen .

Theon’s tract is addressed to a wide circle of students of Platonic schools who “did not have the opportunity to practice mathematics, but still would like to study the writings of Plato” (Expos. 1.10-12 Hiller). In his essay, the genre of which he defines himself as “abbreviated exposition,” Theon sets the task to consider “the essential and necessary characteristics of the most important mathematical theorems of arithmetic, music, geometry, stereometry and astronomy, without which, as Plato said, a blissful life is impossible” (1.15 -2.1).

In the form that reached us, Theon's work consists of an introduction and three parts devoted to arithmetic, music and astronomy (parts about geometry are lost). In the introduction, Theon talks about the purpose of his essay, cites numerous quotes from Plato that speak of the benefits of studying the mathematical sciences, and also compares the learning process of Platonic philosophy with the order of transmission of the mysteries.

The first is the purification, which is acquired by studying the required mathematical sciences from childhood ... The dedication is to transfer the theorems of philosophy, logic, politics and physics. Review is the occupation of intelligible, truly existing and ideas. Crowning is considered to be the transfer of a theory from those who have acquired it to others. The fifth step is a perfect and triumphant good life, which, according to Plato himself, is likened to God as much as possible (15.8-16.2).

The arithmetic part of the treatise (17.25-46.19) is preceded by a presentation of the doctrine of one and one.

According to the Pythagorean legend, numbers are the beginning, source and root of everything. A number is a collection of units, or an ascent starting from a unit and ending in a unit, the descent of the sets. The unit is the limit quantity (the beginning and the element of the number), which, being removed from the set by weaning and isolated from it, remains lonely and unchanged: because its further dissection is impossible. If we divide the sensually perceived body into parts, in quantity it will become one from many, and if we continue to divide each part, everything will end on one; and if we further divide one into parts, these parts will produce many, and the division of the parts will end again on one (17.25-18.15) ... As a number differs from a countable one, one is one. A number is an intelligible amount, for example 5 as such and 10 as such, incorporeal and not perceived by the senses, but by the mind alone. The number is sensually perceived number - 5 horses, 5 bulls, 5 people. The unit is the intelligible idea of ​​one, and it is indivisible; and one is perceived by the senses, and they speak of him as one: one horse, one man. The beginning of the numbers is one, and the beginning of the numeral is one. And one, being perceived sensually, can be divisible to infinity, but not as a number and the beginning of numbers, but as sensually perceived. And the intelligible unit is inherently indivisible, in contrast to the sensually perceived one, divisible to infinity. Countable objects also differ from numbers, because the first are bodily, and the second are bodiless (19.13-20.5).

This distinction between the intelligible world of mathematical entities and the sensually perceived world of things is an improvement of the Pythagorean doctrine that belongs to Plato . In any case, Theon himself points out that such late Pythagoreans as Filolaus and Archit did not yet know this distinction, calling the unit - one, and one - unit.

Further, in the arithmetic section, the properties of various types of numbers are considered: even and odd, simple and compound, polygonal and solid, perfect, redundant and insufficient, third-party and diagonal. The results presented are not accompanied by any evidence.

The music section (46.20-119.21) speaks of the leading significance of numerical harmony , examines the basic elements of musical theory. Theon reports on how the Pythagoreans discovered the numerical nature of musical harmonies, discusses the famous "cosmic scale" of Plato. In relation to the theory of music, the doctrine of numerical relations, proportions and averages is also considered.

Theon’s treatise contains unique quotes from Eratosthenes ( Platonic ), Adrast , Thrasillus and other now lost ancient texts. First of all, this is the famous passage connecting the name of Plato with the task of doubling the cube (2.3-12). Further, this is a series of fragments related to the refinement of the essence of proportion, relationship and interval .

Theon also has a brief description of the Pythagorean algorithm for unfolding all, without exception, relations of inequality from the relation of equality (107.23-111.9). This algorithm is also considered by Arithmetic by Nikomah Gerassky and in the comments in it by Yamvlich . Theon's text is interesting in that it allows you to establish sources. Firstly, this is Adrast’s book, which contained some evidence. Secondly, this is the book of Eratosthenes, in which the evidence is omitted. But once it was omitted, it means it already existed before, which confirms the ancient origin of this algorithm, discovered either by mathematicians of the Platonic school, or their predecessors.

Here the ancient Pythagorean doctrine of the four and ten is transmitted, and the properties of the numbers of the first ten are discussed. Fours are the first four numbers 1 2 3 4; in total they give ten, that is, a decade. In the fourfold, the basic musical harmonies are found, from the double octave 4: 1 to the quarter 3: 4. But the Pythagoreans revered it not only for this reason, because they believed that it contained the nature of the whole, manifested primarily in geometric interpretations: one is a point , two - a straight line, three - a plane, four - a body, that is, a "whole". Calls Theon and the other fours, relating to both the world of things and the world of intelligible entities, a total of eleven.

The astronomical section (120.1-205.6) of Theon’s treatise is of an overview nature and is generally similar to similar works by Gemin and Cleomedes . This material dates back to a wide range of authors, from the Pythagoreans to Hipparchus ; part of it is also known by the Almagest of Claudius Ptolemy . It discusses the arguments in favor of the spherical shape of Heaven and Earth, sets forth the doctrine of celestial circles, examines the theory of eccentrics and epicycles and the doctrine of celestial spheres , explains the causes of solar and lunar eclipses, sets out a brief history of astronomical discoveries. In this section, Theon mentions his commentary on the State of Plato and reports that “on this clarification we have built a sphere; for Plato himself says that teaching without visual assimilation is a waste of work ”(146.3-8).

Regarding the other works of Theon, in one Arabic text it is reported that Theon wrote an essay on the correct order of Plato's dialogues, in which he accepts their distribution according to tetralogies, dating back to Thracell .

In honor of Theon of Smyrnsky named crater on the moon .

Compositions

• Theonis Smyrnaei philosophi platonici expositio rerum mathematicarum ad legendum Platonem utilium. Rec. E. Hiller. Lipsiae: Teubner, 1878 .
• Théon de Smyrne. Exposition des connaissances mathématiques utiles pour la lecture de Platon. Trad. J. Dupuis. Paris: Hachette, 1892. (Repr. Bruxelles: Culture et Civilization, 1966) (French translation)
• Theon of Smyrna. Mathematics useful for understanding Plato . Trans. R. and D. Lawlor, ed. C. Toulis ao San Diego: Wizards Bookshelf, 1979 (full English translation).
• Theon of Smyrna. Mathematics Useful for Reading Plato. Translated by A. Barker // Greek Musical Writings. Vol. 2. Cambridge, 1989, pp. 211-229 (English translation of Theon's harmonics)
• Theon of Smyrna. A statement of mathematical subjects useful in reading Plato. Per. A.I. Shchetnikova. ΣΧΟΛΗ , 3, 2009, ss. 466–558.

About him

• Shchetnikov A.I. Theon Smirn. // Ancient philosophy: an encyclopedic dictionary. Ed. M. A. Solopova. M .: Progress-Tradition, 2008. P. 724-726.
• Dillon J. Middle Platonists . St. Petersburg, 2002.S. 382-384.

• Neophthagoreanism