Menelaus of Alexandria ( // ; Greek : Μενέλαος ὁ Ἀλεξανδρεύς, Menelaos ho Alexandreus; c. 70 – 140 CE) was a Greek mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines.
Although very little is known about Menelaus's life, it is supposed that he lived in Rome, where he probably moved after having spent his youth in Alexandria. He was called Menelaus of Alexandria by both Pappus of Alexandria and Proclus, and a conversation of his with Lucius, held in Rome, is recorded by Plutarch.
Ptolemy (2nd century CE) also mentions, in his work Almagest (VII.3), two astronomical observations made by Menelaus in Rome in January of the year 98. These were occultations of the stars Spica and Beta Scorpii by the moon, a few nights apart. Ptolemy used these observations to confirm precession of the equinoxes, a phenomenon that had been discovered by Hipparchus in the 2nd century BCE.
Sphaerica is the only book that has survived, in an Arabic translation. Composed of three books, it deals with the geometry of the sphere and its application in astronomical measurements and calculations. The book introduces the concept of spherical triangle (figures formed of three great circle arcs, which he named "trilaterals") and proves Menelaus' theorem on collinearity of points on the edges of a triangle (which may have been previously known) and its analog for spherical triangles. It was later translated by the sixteenth century astronomer and mathematician Francesco Maurolico.
The lunar crater Menelaus is named after him.
The titles of a few books by Menelaus have been preserved:
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).
Hipparchus of Nicaea was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of precession of the equinoxes. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. He is known to have been a working astronomer between 162 to 127 BC.
Claudius Ptolemy was a mathematician, astronomer, natural philosopher, geographer and astrologer who wrote several scientific treatises, three of which were of importance to later Byzantine, Islamic and Western European science. The first is the astronomical treatise now known as the Almagest, although it was originally entitled the Mathematical Treatise and then known as The Great Treatise. The second is the Geography, which is a thorough discussion of the geographic knowledge of the Greco-Roman world. The third is the astrological treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatiká (Ἀποτελεσματικά) but more commonly known as the Tetrábiblos from the Koine Greek (Τετράβιβλος) meaning "Four Books" or by the Latin Quadripartitum.
An astrolabe is an ancient astronomical instrument that was a handheld model of the universe. Its various functions also make it an elaborate inclinometer and an analogue calculation device capable of working out several kinds of problems in astronomy. Historically used by astronomers, it is able to measure the altitude above the horizon of a celestial body, day or night; it can be used to identify stars or planets, to determine local latitude given local time, to survey, or to triangulate. It was used in classical antiquity, the Islamic Golden Age, the European Middle Ages and the Age of Discovery for all these purposes.
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 of all time, it canonized a geocentric model of the Universe that was accepted for more than 1200 years from its origin in Hellenistic Alexandria, in the medieval Byzantine and Islamic worlds, and in Western Europe through the Middle Ages and early Renaissance until Copernicus. It is also a key source of information about ancient Greek astronomy.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sphere" are used for the surface together with its 3-dimensional interior.
Al-Ṣābiʾ Thābit ibn Qurrah al-Ḥarrānī was a Syrian Arab mathematician, physician, astronomer, and translator who lived in Baghdad in the second half of the ninth century during the time of the Abbasid Caliphate.
Abu Nasri Mansur ibn Ali ibn Iraq was a Persian Muslim mathematician and astronomer. He is well known for his work with the spherical sine law.
Pappus of Alexandria was one of the last great Greek mathematicians of antiquity, known for his Synagoge (Συναγωγή) or Collection, and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than what can be found in his own writings: that he had a son named Hermodorus, and was a teacher in Alexandria.
Jacob ben Machir ibn Tibbon, of the Ibn Tibbon family, also known as Prophatius.
Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ABC, and a transversal line that crosses BC, AC, and AB at points D, E, and F respectively, with D, E, and F distinct from A, B, and C. Using signed lengths of segments, the theorem states
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.
Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the language of scholarship throughout the Hellenistic world following the conquests of Alexander. This phase of Greek astronomy is also known as Hellenistic astronomy, while the pre-Hellenistic phase is known as Classical Greek astronomy. During the Hellenistic and Roman periods, much of the Greek and non-Greek astronomers working in the Greek tradition studied at the Musaeum and the Library of Alexandria in Ptolemaic Egypt.
The Banū Mūsā brothers, namely Abū Jaʿfar, Muḥammad ibn Mūsā ibn Shākir ; Abū al‐Qāsim, Aḥmad ibn Mūsā ibn Shākir ; and Al-Ḥasan ibn Mūsā ibn Shākir, were three ninth-century Persian scholars who lived and worked in Baghdad. They are known for their Book of Ingenious Devices on automata and mechanical devices. Another important work of theirs is the Book on the Measurement of Plane and Spherical Figures, a foundational work on geometry that was frequently quoted by both Islamic and European mathematicians.
Abū ʿAbd Allāh Muḥammad ibn Muʿādh al-Jayyānī was a mathematician, Islamic scholar, and Qadi from Al-Andalus. Al-Jayyānī wrote important commentaries on Euclid's Elements and he wrote the first known treatise on spherical trigonometry.
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).
Tantrasamgraha, or Tantrasangraha, is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics. The treatise was completed in 1501 CE. It consists of 432 verses in Sanskrit divided into eight chapters. Tantrasamgraha had spawned a few commentaries: Tantrasamgraha-vyakhya of anonymous authorship and Yuktibhāṣā authored by Jyeshtadeva in about 1550 CE. Tantrasangraha, together with its commentaries, bring forth the depths of the mathematical accomplishments the Kerala school of astronomy and mathematics, in particular the achievements of the remarkable mathematician of the school Sangamagrama Madhava. In his Tantrasangraha, Nilakantha revised Aryabhata's model for the planets Mercury and Venus. His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century.
Muḥyī al‐Milla wa al‐Dīn Yaḥyā Abū ʿAbdallāh ibn Muḥammad ibn Abī al‐Shukr al‐Maghribī al‐Andalusī was an Andalusī astronomer, astrologer and mathematician of the Islamic Golden Age. He belonged to the group of astronomers associated with the Maragheh observatory, most notably Nasir al-Din al-Tusi. In astronomy, Ibn Abi al-Shukr carried out a large‐scale project of systematic planetary observations, which led to the development of several new astronomical parameters.
Abū Muḥammad Jābir ibn Aflaḥ was an Arab Muslim astronomer and mathematician from Seville, who was active in 12th century al-Andalus. His work Iṣlāḥ al-Majisṭi influenced Islamic, Jewish, and Christian astronomers.
Hypsicles was an ancient Greek mathematician and astronomer known for authoring On Ascensions (Ἀναφορικός) and the Book XIV of Euclid's Elements. Hypsicles lived in Alexandria.