Theon of Smyrna (fl. 100 CE) was a Greek

Greece

Greece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....

 philosopher and mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, whose works were strongly influenced by the Pythagorean

Pythagoras

Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...

 school of thought. His surviving On Mathematics Useful for the Understanding of Plato is an introductory survey of Greek mathematics

Greek mathematics

Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

.

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

Smyrna was an ancient city located at a central and strategic point on the Aegean coast of Anatolia. Thanks to its advantageous port conditions, its ease of defence and its good inland connections, Smyrna rose to prominence. The ancient city is located at two sites within modern İzmir, Turkey...

, and art historians date it to around 135 CE. Ptolemy

Ptolemy

Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

 refers several times in his Almagest

Almagest

The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

to a Theon who made observations at Alexandria

Alexandria

Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...

, but it is uncertain whether he is referring to Theon of Smyrna. The lunar

Moon

The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

 impact crater

Impact crater

In the broadest sense, the term impact crater can be applied to any depression, natural or manmade, resulting from the high velocity impact of a projectile with a larger body...

 Theon Senior

Theon Senior (crater)

Theon Senior is a lunar impact crater that is located to the northwest of the crater Delambre, and south of D'Arrest. It forms a matching pair with Theon Junior, about two crater diameters to the south-southeast. The satellite crater Theon Senior A can be found to the north...

 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

Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

. 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 language

Arabic is a name applied to the descendants of the Classical Arabic language of the 6th century AD, used most prominently in the Quran, the Islamic Holy Book...

 translation.

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

Harmony

In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...

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

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

: odd numbers, even numbers, prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s, perfect number

Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

s, abundant numbers, and other such properties.

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

Musica universalis

Musica universalis is an ancient philosophical concept that regards proportions in the movements of celestial bodies—the Sun, Moon, and planets—as a form of musica . This 'music' is not usually thought to be literally audible, but a harmonic and/or mathematical and/or religious concept...

" (hē en kosmō harmonia kai hē en toutō harmonia). The "music of numbers" is a treatment of temperament and harmony using ratio

Ratio

In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

s, proportions, and means; the sections on instrumental music concerns itself not with melody but rather with interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

s and consonance

Consonance and dissonance

In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...

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

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

 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

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

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

Occultation

An occultation is an event that occurs when one object is hidden by another object that passes between it and the observer. The word is used in astronomy . It can also refer to any situation wherein an object in the foreground blocks from view an object in the background...

s, transits

Astronomical transit

The term transit or astronomical transit has three meanings in astronomy:* A transit is the astronomical event that occurs when one celestial body appears to move across the face of another celestial body, hiding a small part of it, as seen by an observer at some particular vantage point...

, conjunctions, and eclipse

Eclipse

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

s. However, the quality of the work led Otto Neugebauer to criticize him for not fully understanding the material he attempted to present.