Eutocius of Ascalon ( // ; Greek : Εὐτόκιος ὁ Ἀσκαλωνίτης; c. 480s – c. 520s) was a Palestinian-Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian Conics.
Little is known about the life of Eutocius. He was born in Ascalon, then in Palestina Prima. He lived during the reign of Justinian. Eutocius became head the school of philosophy in Athens following Ammonius and he was succeeded in this position by Olympiodorus, possibly as early as 525.He traveled to the greatest scientific centers of his time, including Alexandria, to conduct research on Archimedes' manuscripts.
He wrote commentaries on Apollonius and on Archimedes. The surviving works of Eutocius are:
Historians owe much of their knowledge of Archimedes' solution of a cubic by means of intersecting conics, alluded to in The Sphere and Cylinder, to Eutocius and his commentaries. Eutocius dedicated his commentary on Apollonius' Conics to Anthemius of Tralles, also a mathematician, and architect of the Hagia Sophia patriarchal basilica in Constantinople.
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered to be the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including: the area of a circle; the surface area and volume of a sphere; area of an ellipse; the area under a parabola; the volume of a segment of a paraboloid of revolution; the volume of a segment of a hyperboloid of revolution; and the area of a spiral.
Anthemius of Tralles was a Greek from Tralles who worked as a geometer and architect in Constantinople, the capital of the Byzantine Empire. With Isidore of Miletus, he designed the Hagia Sophia for Justinian I.
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.
Isidore of Miletus was one of the two main Byzantine Greek architects that Emperor Justinian I commissioned to design the cathedral Hagia Sophia in Constantinople from 532 to 537. The creation of an important compilation of Archimedes' works has been attributed to him. The spurious Book XV from Euclid's Elements has been partly attributed to Isidore of Miletus.
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Apollonius of Perga was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Gottfried Wilhelm Leibniz stated “He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.”
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In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.
Greek mathematics refers to mathematics texts written during 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 mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by Greek culture and the Greek language. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: máthēmaAttic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θi.ma], meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is an important difference between Greek mathematics and those of preceding civilizations.
Diocles was a Greek mathematician and geometer.
Abu-Abdullah Muhammad ibn Īsa Māhānī was a Persian mathematician and astronomer born in Mahan, and active in Baghdad, Abbasid Caliphate. His known mathematical works included his commentaries on Euclid's Elements, Archimedes' On the Sphere and Cylinder and Menelaus' Sphaerica, as well as two independent treatises. He unsuccessfully tried to solve a problem posed by Archimedes of cutting a sphere into two volumes of a given ratio, which was later solved by 10th century mathematician Abū Ja'far al-Khāzin. His only known surviving work on astronomy was on the calculation of azimuths. He was also known to make astronomical observations, and claimed his estimates of the start times of three consecutive lunar eclipses were accurate to within half an hour.
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In geometry, tangent circles are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration and maximizing the use of materials.