Diocles (; c. 240 BC – c. 180 BC) 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 ...

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 ...

and geometer.

Life and work

Although little is known about the life of Diocles, it is known that he was a contemporary of

Apollonius Apollonius () is a masculine given name which may refer to: People Ancient world Artists * Apollonius of Athens (sculptor) (fl. 1st century BC) * Apollonius of Tralles (fl. 2nd century BC), sculptor * Apollonius (satyr sculptor) * Apo ...

and that he flourished sometime around the end of the 3rd century BC and the beginning of the 2nd century BC. Diocles is thought to be the first person to prove the focal property of the

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

. His name is associated with the geometric

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

called the

Cissoid of Diocles In geometry, the cissoid of Diocles (; named for Diocles (mathematician), Diocles) is a cubic plane curve notable for the property that it can be used to construct two Geometric mean, mean proportionals to a given ratio. In particular, it can b ...

, which was used by Diocles to solve the 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 ...

. The curve was alluded to by

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...

in his commentary on

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

and attributed to Diocles by

Geminus Geminus of Rhodes (), was a Greek astronomer and mathematician, who flourished in the 1st century BC. An astronomy work of his, the ''Introduction to the Phenomena'', still survives; it was intended as an introductory astronomy book for students ...

as early as the beginning of the 1st century. Fragments of a work by Diocles entitled ''On burning mirrors'' were preserved by

Eutocius Eutocius of Ascalon (; ; 480s – 520s) was a Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian ''Conics''. Life and work Little is known about the life of Eutocius. He was born in Ascalon, t ...

in his commentary of

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

' ''On the Sphere and the Cylinder'' and also survived 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 of the lost Greek original titled ''Kitāb Dhiyūqlīs fī l-marāyā l-muḥriqa'' (lit. “The book of Diocles on burning mirrors”). Historically, ''On burning mirrors'' had a large influence on Arabic mathematicians, particularly on

al-Haytham Ḥasan Ibn al-Haytham ( Latinized as Alhazen; ; full name ; ) was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the princ ...

, the 11th-century polymath of Cairo whom Europeans knew as "Alhazen". The treatise contains sixteen propositions that are proved by

conic sections A conic section, conic or a quadratic curve is a curve obtained from a Conical surface, cone's surface intersecting a plane (mathematics), plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is ...

. One of the fragments contains propositions seven and eight, which is a solution to the problem of dividing a sphere by a plane so that the resulting two volumes are in a given ratio. Proposition ten gives a solution to the problem of doubling the cube. This is equivalent to solving a certain

cubic equation In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d=0 in which is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...

. Another fragment contains propositions eleven and twelve, which use the cissoid to solve the problem of finding two mean proportionals in between two magnitudes. Since this treatise covers more topics than just burning mirrors, it may be the case that ''On burning mirrors'' is the aggregate of three shorter works by Diocles. In the same work, Diocles, just after demonstrating that the parabolic mirror could focus the rays in a single point, he mentioned that It is possible to obtain a lens with the same property.Toomer.

Notes

References

*Heath, Sir Thomas, ''A History of Greek Mathematics'' (2 vols.) Dover Publications, Inc. (1980), Oxford (1921) . * G. J. Toomer, "Diocles On Burning Mirrors", ''Sources in the History of Mathematics and the Physical Sciences'' 1 (New York, 1976). * *Malik, Saira (2021-01-01).
Diocles
. ''Encyclopaedia of Islam, THREE'' {{DEFAULTSORT:Diocles Ancient Greek geometers Year of birth unknown Year of death unknown 3rd-century BC Greek mathematicians 2nd-century BC Greek mathematicians