mathematics

Theon of Smyrna

Theon of Smyrna is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Theon of Smyrna rather than just read about it. In short: Theon of Smyrna (Greek: Θέων ὁ Σμυρναῖος Theon ho Smyrnaios, gen. Θέωνος Theonos; fl. 100 CE) was a Greek philosopher and mathematician, 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.

Key takeaways

Theon of Smyrna belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Theon of Smyrna to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Theon of Smyrna from memory before moving on to harder problems.

Reference excerpt

Theon of Smyrna (Greek: Θέων ὁ Σμυρναῖος Theon ho Smyrnaios, gen. Θέωνος Theonos; fl. 100 CE) was a Greek philosopher and mathematician, 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.

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, and art historians date it to around 135 CE. Ptolemy refers several times in his Almagest to a Theon who made observations at Alexandria, but it is uncertain whether he is referring to Theon of Smyrna. The lunar impact crater 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. 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 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. The first section, on mathematics, is most focused on what today is most commonly known as number theory: odd numbers, even numbers, prime numbers, perfect numbers, abundant numbers, 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 denominators of which are Pell numbers. It is also one of the sources of our knowledge of the origins of the classical problem of Doubling the cube. 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 ratios, 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 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. Here Theon affirms the spherical shape and large size of the Earth; he also describes the occultations, transits, conjunctions, and eclipses. However, the quality of the work led Otto Neugebauer to criticize him for not fully understanding the material he attempted to present.

On Pythagorean Harmony Theon was a great philosopher of harmony and he discusses semitones in his treatise. There are several semitones used in Greek music, but of this variety, there are two that are very common. The “diatonic semitone” 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 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 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).

Illustrations

Worked examples

Example 1 — a first encounter with Theon of Smyrna

Start with the simplest possible case. Write down what Theon of Smyrna claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Theon of Smyrna before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Theon of Smyrna ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Theon of Smyrna

In research: Theon of Smyrna appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Theon of Smyrna in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Theon of Smyrna is common in secondary-school and first-year university syllabi. It links to neighbouring topics 2nd-century Greek philosophers, 2nd-century mathematicians, Ancient Greek mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Theon of Smyrna outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Theon of Smyrna in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Theon of Smyrna means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Theon of Smyrna out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Theon of Smyrna in simple terms?

Why does Theon of Smyrna matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Theon of Smyrna?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Theon of Smyrna.