Theon of

Smyrna Smyrna ( ; , or ) was an Ancient Greece, Ancient Greek city located at a strategic point on the Aegean Sea, Aegean coast of Anatolia, Turkey. Due to its advantageous port conditions, its ease of defence, and its good inland connections, Smyrna ...

( ''Theon ho Smyrnaios'', ''gen.'' Θέωνος ''Theonos''; fl. 100 CE) was a

Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...

philosopher Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...

and

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, whose works were strongly influenced by the Pythagorean school of thought. His surviving ''On Mathematics Useful for the Understanding of Plato'' is an introductory survey of

Greek mathematics Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical antiquity, classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities ...

Life

Little is known about the life of Theon of Smyrna. A bust created at his death, and dedicated by his son, was discovered at

, and art historians date it to around 135 CE.

Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...

refers several times in his ''

Almagest The ''Almagest'' ( ) is a 2nd-century Greek mathematics, mathematical and Greek astronomy, astronomical treatise on the apparent motions of the stars and planetary paths, written by Ptolemy, Claudius Ptolemy ( ) in Koine Greek. One of the most i ...

'' to a Theon who made observations at

Alexandria Alexandria ( ; ) is the List of cities and towns in Egypt#Largest cities, second largest city in Egypt and the List of coastal settlements of the Mediterranean Sea, largest city on the Mediterranean coast. It lies at the western edge of the Nile ...

, but it is uncertain whether he is referring to Theon of Smyrna.James Evans, (1998), ''The History and Practice of Ancient Astronomy'', New York, Oxford University Press, 1998, p. 49 The lunar

impact crater An impact crater is a depression (geology), depression in the surface of a solid astronomical body formed by the hypervelocity impact event, impact of a smaller object. In contrast to volcanic craters, which result from explosion or internal c ...

Theon Senior is named for him.

Works

Theon wrote several commentaries on the works of mathematicians and philosophers of the time, including works on the philosophy of

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

. Most of these works are lost. The one major survivor is his ''On Mathematics Useful for the Understanding of Plato''. A second work concerning the order in which to study Plato's works has recently been discovered in an

Arabic Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...

translation."Theon of Smyrna" entry in John Hazel, 2002, ''Who's who in the Greek world'', page 37. Routledge

''On Mathematics Useful for the Understanding of Plato''

His ''On Mathematics Useful for the Understanding of Plato'' is not a commentary on Plato's writings but rather a general handbook for a student of mathematics. It is not so much a groundbreaking work as a reference work of ideas already known at the time. Its status as a compilation of already-established knowledge and its thorough citation of earlier sources is part of what makes it valuable. The first part of this work is divided into two parts, the first covering the subjects of numbers and the second dealing with music and

harmony In music, harmony is the concept of combining different sounds in order to create new, distinct musical ideas. Theories of harmony seek to describe or explain the effects created by distinct pitches or tones coinciding with one another; harm ...

. The first section, on mathematics, is most focused on what today is most commonly known as

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

odd number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...

abundant number In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total ...

s, and other such properties. It contains an account of 'side and diameter numbers', the Pythagorean method for a sequence of best rational approximations to the

square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...

,T. Heath "A History of Greek Mathematics", p.91
the denominators of which are

Pell number In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , an ...

s. It is also one of the sources of our knowledge of the origins of the classical problem of

Doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first ...

.L. Zhmud ''The origin of the history of science in classical antiquity'', p.84
The second section, on music, is split into three parts: music of numbers (''hē en arithmois mousikē''), instrumental music (''hē en organois mousikē''), and " music of the spheres" (''hē en kosmō harmonia kai hē en toutō harmonia''). The "music of numbers" is a treatment of temperament and harmony using

ratio In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

s, proportions, and means; the sections on instrumental music concerns itself not with melody but rather with intervals and consonances in the manner of Pythagoras' work. Theon considers intervals by their degree of consonance: that is, by how simple their ratios are. (For example, the

octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...

is first, with the simple 2:1 ratio of the octave to the fundamental.) He also considers them by their distance from one another. The third section, on the music of the cosmos, he considered most important, and ordered it so as to come after the necessary background given in the earlier parts. Theon quotes a poem by Alexander of Ephesus assigning specific pitches in the chromatic scale to each planet, an idea that would retain its popularity for a millennium thereafter. The second book is on

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

. Here Theon affirms the spherical shape and large size of the Earth; he also describes the

occultation An occultation is an event that occurs when one object is hidden from the observer by another object that passes between them. The term is often used in astronomy, but can also refer to any situation in which an object in the foreground blocks f ...

s, transits, conjunctions, and

eclipse An eclipse is an astronomical event which occurs when an astronomical object or spacecraft is temporarily obscured, by passing into the shadow of another body or by having another body pass between it and the viewer. This alignment of three ...

s. However, the quality of the work led

Otto Neugebauer Otto Eduard Neugebauer (May 26, 1899 – February 19, 1990) was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences as they were practiced in an ...

to criticize him for not fully understanding the material he attempted to present.

''On Pythagorean Harmony''

Theon was a great philosopher of

and he discusses semitones in his treatise. There are several semitones used in

Greek music The music of Greece is as diverse and celebrated as its History of Greece, history. Greek music separates into two parts: Greek folk music, Greek traditional music and Byzantine music. These compositions have existed for millennia: they originat ...

, but of this variety, there are two that are very common. The “

diatonic semitone A semitone, also called a minor second, half step, or a half tone, is the smallest interval (music), musical interval commonly used in Western tonal music, and it is considered the most Consonance and dissonance#Dissonance, dissonant when sounde ...

” with a value of 16/15 and the “ chromatic semitone” with a value of 25/24 are the two more commonly used semitones (Papadopoulos, 2002). In these times,

Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...

did not rely on irrational numbers for understanding of harmonies and the logarithm for these semitones did not match with their philosophy. Their logarithms did not lead to irrational numbers, however Theon tackled this discussion head on. He acknowledged that “one can prove that” the tone of value 9/8 cannot be divided into equal parts and so it is a number in itself. Many Pythagoreans believed in the existence of irrational numbers, but did not believe in using them because they were unnatural and not positive integers. Theon also does an amazing job of relating quotients of integers and musical intervals. He illustrates this idea in his writings and through experiments. He discusses the Pythagoreans method of looking at

harmonies In music, harmony is the concept of combining different sounds in order to create new, distinct musical ideas. Theories of harmony seek to describe or explain the effects created by distinct pitches or tones coinciding with one another; harm ...

and consonances through half-filling vases and explains these experiments on a deeper level focusing on the fact that the octaves, fifths, and fourths correspond respectively with the fractions 2/1, 3/2, and 4/3. His contributions greatly contributed to the fields of music and physics (Papadopoulos, 2002).

Notes

Bibliography

*Theon of Smyrna: ''Mathematics useful for understanding Plato; translated from the 1892 Greek/French edition of J. Dupuis by Robert and Deborah Lawlor and edited and annotated by Christos Toulis and others; with an appendix of notes by Dupuis, a copious glossary, index of works, etc.'' Series: ''Secret doctrine reference series'', San Diego : Wizards Bookshelf, 1979. . 174pp. *E.Hiller
Theonis Smyrnaei: expositio rerum mathematicarum ad legendum Platonem utilium
Leipzig:Teubner, 1878, repr. 1966. *J. Dupuis
Exposition des connaissances mathematiques utiles pour la lecture de Platon
1892. French translation. *Lukas Richter:"Theon of Smyrna". Grove Music Online, ed. L. Macy. Accessed 29 Jun 05
(subscription access)
* * Papadopoulos, Athanase (2002). Mathematics and music theory: From Pythagoras to Rameau. ''The Mathematical Intelligencer'', 24(1), 65–73. doi:10.1007/bf03025314

External links

at wilbourhall.org {{Authority control 2nd-century Greek philosophers Ancient Greek mathematicians Ancient Greek music theorists Ancient Smyrnaeans Neo-Pythagoreans Philosophers of ancient Ionia Middle Platonists 2nd-century mathematicians