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= Theodosius of Bithynia = Theodosius of Bithynia (Greek: Θεοδόσιος; c. 160 BC – c. 90 BC) was a Greek astronomer and mathematician who wrote the Sphaerics, a three-volume text on the spherical geometry called Sphaerics.

Life
Born in Tripolis, in Bithynia. Bithynia is a place located in Turkey along the northwestern coast of the Aegean. Theodosius is cited by Vitruvius as having invented a sundial suitable for any place on Earth.[1] His Sphaerics provided the mathematics for spherical astronomy, and may have been based on a work by Eudoxus of Cnidus. Francesco Maurolico translated his works in the 16th century. It contains about 60 propositions in total and many definitions. In addition to the Sphaerics, two other works by Theodosius have survived: On Habitations (De Habitationibus), describing the appearances of the heavens at different climes, and On Days and Nights (De Diebus et noctibus), a study of the apparent motion of the Sun. All of his works can be tied back to the works of others such as Euclid, Autolycus, Hipparchus, and Menelaus. Ultimately most of his works were incorporated into textbooks which is why they still exist today.

Theodosius of Bithynia lived most likely during the first or second century BC. He is most famously known for the Sphaerics, a textbook on the geometry of the sphere and minor astronomical and astrological aspects [1]. It is Menelaus, a Greek mathematician and astronomer who is famous for conceiving and defining a spherical triangle[2], who is quoted saying that Theodosius is the author of the Sphaerics[3]. His works were tranated by Naṣīr al-Dīn al-Ṭūsī, Qusṭā ibn Lūqā in Bagdad [4]. The three books translated were the Sphaerics, On days and nights, and On hanitations [5] . These works become known as The intermediates and the purpose was to teach students in schools[6]. The collection was at one point edited by al-Ṭūsī [7] . All of these works would become a part of The little astronomy, an intermediate after Euclid’s Elements and before the Almagest [8].

Works
Theodosius most recognized work was a compilation of three-volume text on spherical geometry, Sphaerica (translated Sphaerics), which satisfied the need for astronomers of the time to tackle the subject of sphericity. There have been many debates over the success and credibility of this book because while it does provide valuable information on everything spherical, many would argue that it contains almost no original philosophy from Theodosius. The Mathematician, T. Heath called him a “laborious compiler” and Otto Neugebauer pointed out that his theories seldom treat more than what is obvious and his proofs do little more than reword the conjecture.12 Another notable fact is that Theodosius left out the great-circle triangle, which was an important structure at the time and still remains as so today. Throughout the compilation, he rarely admits to the assumptions used and the composition of the ideas. Nonetheless, despite criticism, this text did exactly what Theodosius aimed for it to and that was explain the phenomenon of spherical geometry. The thoughts he compiles are well-organized and serve the purpose of responding to the need of one text that contains all information about this science.

He is also noted as the inventor of a sundial suitable for any specific region, as noted by Vitruvius, a Roman architect, engineer and author most famous for treatise On Architecture [1]. While there are no details of Theodosius every creating a sundial, there is little information discrediting what Vitruvius stated. Theodosius’ contributions consist of three works that make up the Sphaerics. The book contains to trigonometry and it was written to aid Euclid’s Elements in making up for the lack of results on the geometry if the sphere [2]. The works are named Sphaerica, De habitationibus and De diebus [3]. It was Sphaerica and De habitationibus that were translated from Arabic into Latin by Gerard of Cremona in the twelfth century [4]. The Greek manuscripts were translated into Arabic around the tenth century [5]. In 1518 a Latin version was printed[6]. It was not long after that in 1529 Johannes Vögelin improved the translation of the Spherics [7]. Again in 1586, Christoph Clavius produced his own translation and commentary of the works [8]. It was not until 1721 that a English version was produced [9]. In his writings Theodosius defines a sphere to be a solid figure with the property that an point on its surface is a constant distance from the center of the sphere[10]. In his works he is also cited as proving that for a spherical triangle with angles A, B, C, and sides with a, b, c, side a is opposite angle A[11]. This is equivalent to the tangent of a equals sin of b times tangent of A.

The work of Theodosius is similar to the two works of Autylocus and both of their findings resemble those discussed in Euclid’s Elements. Books 1 and 2.1-10 of the Spherics are a strict translation of book 3 of the elements from the circle to the sphere.13


 * Sphaerics:
 * Many would say that this work is closely tied to Euclid’s Elements because of the first three definitions he uses in this text. The definitions define a center of a sphere, diameter of a sphere, and the poles of a sphere and these three definitions alone are in Euclid’s book. He not only has many similarities with Euclid, but also has his own original thoughts within his text as well. For example, he thought of spheres as a bunch of circles intersecting each other at angles.
 * De Habitationibus:
 * A written text that predated Ptolemaic spherical astronomy. It was composed of over 30 propositions and was translated from Greek to Arabic in the late ninth century.  It literally describes the lengths of the day and night times which were observed through the sun with respect to the tropics.
 * De Diebus et noctibus:
 * This book also predates Ptolemaic astronomy and was also translated from Greek to Arabic in the late ninth century. It also talked about the lengths of day and night with respect to the tropics  Other claims Theodosius makes in his works, particularly On days and nights, are that the day last for seven months at the north pole and night last five months [1] . His work On days and nights aims to explain how the rotation of the Earth affects the universe. The work is divided into two books with the first being composed of thirteen propositions and the second having nineteen propositions[2] . In On days and nights, Theodosius explains his beliefs how the views of the stars and lengths of night and day are all affected by the location of the observer[3].  Theodosius believed that it was day if the sun was less than 15 degrees below the horizon [4] . Theodosius came to the conclusion in his book that if the year equals an irrational number of days than stellar phases show not annual pattern [5] . Theodosius is also credited with a work on astronomy in which he gives a commentary on Archimedes’ Mechanics [6] . These works have been deemed lost but little fragments have survived as seen in Description of Houses, a piece of work that deals with problems in architecture.

Spherical Geometry
Spherical geometry was used mainly in the Middle Ages and Renaissance. In comparison to Euclid’s Elements of Geometry, Theodosius work satisfied the need for spherical geometry. With his three volume Spherics. Spherics is not given much praise by modern writers. Mathematician T. Heath describes Theodosius as, “simply a laborious compiler.” [1] He backs up this claim be explaining that there was hardly any original information in his work. Otto Negebauer, an Austrian American mathematician known for research on the history of astronomy as well as other sciences, explains that Theodosius never recognizes the significance of the great circle triangle, his theorems only explain the obvious and he seldom admits his own assumptions [2]. A misconception of Theodosius is that he wrote s commentary on the chapter of Theudas and Skeptical Cahpters [3]. This was not the Theodosius of Bithynia but rather a sceptic philosopher of the second century AD with the same name  who wrote both works [4].

= Menelaus of Alexandria = Menelaus of Alexandria (/ˌmɛnɪˈleɪəs/; Greek: Μενέλαος ὁ Ἀλεξανδρεύς, Menelaos ho Alexandreus; c. 70 – 140 CE) was a Greek[1] mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines.

Life and works
Sphaericorum libri tres

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. [15].

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.

Most of Menelaus's work did not survive. The majority of Menelaus's work regarding the science of trigonometry was stored and can be found in Sphaerica. Out of all of Menelaus's works, 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.[16] With Sphaerica, Menelaus is recognized as the founder of spherical trigonometry and as the first to remove the use of spherics and astronomy from trigonometry.


 * In Book 1, Menelaus introduces and establishes the foundation of mathematical treatment of spherical triangles in comparison to the mathematical treatment of plane triangles as accepted by Euclid. Menelaus also founded the use of arcs of great circles on the sphere rather than the use of arcs of parallel circles. This in turn changed and was a crucial milestone in the development of spherical trigonometry.
 * In Book 2, there was a focus on establishing the theorems used in spherical astronomy and their functions.
 * In Book 3, Menelaus's theorem and spherical trigonometry are discussed. Book 3 contains the first known variant of Menelaus’s theorem which over time became imperative to spherical trigonometry and astronomy. Menelaus’s theorem states that when there is a line that crosses the three sides of a triangle, the product of three non-adjacent is equal to the product of the remaining three segments. Menelaus’s theorem is said to have possibly been known by others, however, Menelaus was able to share the theorem with others, making the theorem acknowledged with his name.

The lunar crater Menelaus is named after him.

Life
Theon was referred to different things by many famous Greek philosophers, astronomers, and great thinkers. Ptolemy call him, “Theon the mathematician”, “the old Theon” by Theon of Alexandria, and “Theon the platonist” by Proclus. Little is known about the life of Theon of Smyrna. However, one of the things that is known about Theon is that he wrote his works in an effort to understand “Platonic doctrine”.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.[1] 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. Many of Theon's works have been lost. He has been mentioned in Ibn al-Nadīm’s Fihrist [1]. In the work Ibn al-Nadīm, mentions a treatise Theon wrote about Plato's writings and the exact order they should be read in. 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.[2]  In all of his works, his main goal was to teach basic math principles


 * Exposition
 * This text can be broken down into three separate books, namely: Arithmetic,  Astronomy, and Music. This work was meant to bridge the gap between math and nature and open the door for understanding of Plato’s philosophy.
 * Theon is most famous for the Expositio. It provides citations from earlier works. The book helps explain the connection between arithmetic, geometry, stereometry, music, and astronomy for philosophy students [1]. Theon explains in his work how music is divided into three distinct parts: instrumental, musical intervals expressed numerically, and the harmony of the universe [1]. Theon claims in his work to not have invented any specific musical property but he only means to expand on his predecessors such as: Thrasyllus, Adrastus, Aristoxenus, Hippasus, Eudoxus, and Plato [1]. Theon discusses the following topics within his text: Eratosthenes’ Platonikos, Adrastus ideas are talked about in the astronomical portion of the writing, Hipparchos considered the onventor of the epicyclic hypothesis by Theon, and fragments from Eudemus on pre-Socratic astronomy [1].

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,[3] 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.[4]

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. 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 and 4/3. His contributions greatly contributed to the fields of music and physics.