Theon Of Smyrna
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Theon of
Smyrna Smyrna ( ; grc, Σμύρνη, Smýrnē, or , ) was a Greek city located at a strategic point on the Aegean coast of Anatolia. Due to its advantageous port conditions, its ease of defence, and its good inland connections, Smyrna rose to promi ...
( el, Θέων ὁ Σμυρναῖος ''Theon ho Smyrnaios'', ''gen.'' Θέωνος ''Theonos''; fl. 100 CE) was a
Greek Greek may refer to: Greece 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 ...
philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
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, structure, space, models, and change. History On ...
, whose works were strongly influenced by the
Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...
school of thought. His surviving ''On Mathematics Useful for the Understanding of Plato'' is an introductory survey of
Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
.


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
Smyrna Smyrna ( ; grc, Σμύρνη, Smýrnē, or , ) was a Greek city located at a strategic point on the Aegean coast of Anatolia. Due to its advantageous port conditions, its ease of defence, and its good inland connections, Smyrna rose to promi ...
, and art historians date it to around 135 CE.
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
refers several times in his ''
Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it canoni ...
'' to a Theon who made observations at
Alexandria Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
, 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 Lunar most commonly means "of or relating to the Moon". Lunar may also refer to: Arts and entertainment * ''Lunar'' (series), a series of video games * "Lunar" (song), by David Guetta * "Lunar", a song by Priestess from the 2009 album ''Prior t ...
impact crater An impact crater is a circular depression in the surface of a solid astronomical object formed by the hypervelocity impact of a smaller object. In contrast to volcanic craters, which result from explosion or internal collapse, impact craters ...
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 ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
. 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 Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
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 process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
. The first section, on mathematics, is most focused on what today is most commonly known as
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
:
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 a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
s,
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 a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
s,
prime number A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
s,
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
s,
abundant number In number theory, an abundant number or excessive number is a number 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 of 16. The ...
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 a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
,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 geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
.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 The ''musica universalis'' (literally universal music), also called music of the spheres or harmony of the spheres, is a philosophical concept that regards proportions in the movements of celestial bodies – the Sun, Moon, and planets – as a fo ...
" (''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
consonance In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unpl ...
s 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 ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
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 Alexander ( Gr. ) surnamed Lychnus (), was an ancient Greek rhetorician and poet. He was a native of Ephesus, from which he is sometimes called Alexander Ephesius, and must have lived shortly before the time of Strabo (i.e., the 1st century BC), ...
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 astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
. 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 ...
s, transits, conjunctions, and
eclipse An eclipse is an astronomical event that 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 ce ...
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 anti ...
to criticize him for not fully understanding the material he attempted to present.


''On Pythagorean Harmony''

Theon was a great philosopher of
harmony In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
and he discusses
semitones A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
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 half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
” with a value of 16/15 and the “
chromatic semitone In modern Western tonal music theory an augmented unison or augmented prime is the interval between two notes on the same staff position, or denoted by the same note letter, whose alterations cause them, in ordinary equal temperament, to be one ...
” 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 Greek colony of Kroton, ...
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 process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
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).


See also

*
Theon of Alexandria Theon of Alexandria (; grc, Θέων ὁ Ἀλεξανδρεύς;  335 – c. 405) was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's '' Elements'' and wrote commentaries on wor ...


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 {{Authority control 2nd-century philosophers Ancient Greek mathematicians Ancient Greek music theorists Ancient Smyrnaeans People from İzmir Neo-Pythagoreans Roman-era philosophers Philosophers of ancient Ionia Pythagoreans Middle Platonists 2nd-century mathematicians