Menelaus’ major contribution to the rising science of trigonometry was contained in his Sphaerica, in three books. It is this work which entitles him to be regarded as the founder of spherical trigonometry and the first to have disengaged trigonometry from spherics and astronomy and to have made it a separate science.
Who is Menelaus trigonometry?
Menelaus of Alexandria was a Greek astronomer, scientist, and mathematician who lived around 100 CE. Menelaus made a significant and lasting contribution to the fields of astronomy, geometry, and trigonometry. His major work, the Spherics survives and presents what is today called Menelaus’ Theorem.
What is Sphaerica?
contribution to trigonometry
…in Book 1 of the Sphaerica, a three-book treatise by Menelaus of Alexandria (c. 100 ce) in which Menelaus developed the spherical equivalents of Euclid’s propositions for planar triangles.
Who discovered Menelaus Theorem?
Russell, John Wellesley (1905). “Ch. 1 §6 “Menelaus’ Theorem””. Pure Geometry.
Why did Helen choose Menelaus?
Before her marriage to Menelaus, Helen lived with Leda and Leda’s husband, King Tyndareus of Sparta. … To prevent any violence against her future husband, the Greek warrior Odysseus made his countrymen swear to protect the man she agreed to wed. Helen chose Menelaus, who later became king of Sparta.
Who is the wife of Menelaus?
Menelaus, in Greek mythology, king of Sparta and younger son of Atreus, king of Mycenae; the abduction of his wife, Helen, led to the Trojan War. During the war Menelaus served under his elder brother Agamemnon, the commander in chief of the Greek forces.
Is Menelaus real?
Menelaus (also Menelaos) is a figure from ancient Greek mythology and literature who was the king of Sparta and the husband of beautiful Helen, whose abduction by the Trojan prince Paris sparked off the legendary Trojan War.
Did Paris and Menelaus fight?
In Book 3, Menelaus challenges Paris to a duel for Helen’s return. Menelaus soundly beats Paris, but before he can kill him and claim victory, Aphrodite spirits Paris away inside the walls of Troy.
Why does Clytemnestra get angry with Agamemnon?
Clytemnestra starts an affair
While he was away, Clytemnestra started an affair with Aegisthus, with whom she plotted against her husband. Clytemnestra was angry with her husband, both because of her daughter’s sacrifice, as well as because Agamemnon had killed her first husband and taken her by force.
Who introduced spherical triangle?
Menelaus of Alexandria, (flourished 1st century ad, Alexandria and Rome), Greek mathematician and astronomer who first conceived and defined a spherical triangle (a triangle formed by three arcs of great circles on the surface of a sphere).
Who contributed to trigonometry?
The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as “the father of trigonometry.” Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.
What did Ptolemy contribute to trigonometry?
He compiled a star catalog and the earliest surviving table of a trigonometric function and established mathematically that an object and its mirror image must make equal angles to a mirror.
What is nine point circle in triangle?
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. … The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).
What is the converse of Menelaus Theorem?
Menelaus’ theorem relates ratios obtained by a line cutting the sides of a triangle. The converse of the theorem (i.e. three points on a triangle are collinear if and only if they satisfy certain criteria) is also true and is extremely powerful in proving that three points are collinear.
How do you prove Ptolemy’s Theorem?
In order to prove his sum and difference forumlas, Ptolemy first proved what we now call Ptolemy’s theorem. Ptolemy’s theorem: For a cyclic quadrilateral (that is, a quadrilateral inscribed in a circle), the product of the diagonals equals the sum of the products of the opposite sides. AC BD = AB CD + AD BC.
Is Odysseus a Grecian?
Famed for his courage, intelligence, and leadership, Odysseus (Roman name: Ulysses) was one of the great pan-Hellenic heroes of Greek mythology. … In Greek mythology, Odysseus was the son of Laertes and Antikleia (or Anticlea) and the King of Ithaca, leader of the Kephallenians.
How was Menelaus chosen as Helen’s husband?
Tyndareus readily agreed, and Odysseus proposed that, before the decision was made, all the suitors should swear a most solemn oath to defend the chosen husband against whoever should quarrel with him. After the suitors had sworn not to retaliate, Menelaus was chosen to be Helen’s husband.
Who killed Agamemnon?
Clytemnestra, in Greek legend, a daughter of Leda and Tyndareus and wife of Agamemnon, commander of the Greek forces in the Trojan War. She took Aegisthus as her lover while Agamemnon was away at war. Upon his return, Clytemnestra and Aegisthus murdered Agamemnon.
Did Helen cheat on Menelaus?
Helen of Troy, Greek Helene, in Greek legend, the most beautiful woman of Greece and the indirect cause of the Trojan War. … When Paris was slain, Helen married his brother Deiphobus, whom she betrayed to Menelaus once Troy was captured.
How did Zeus seduce Europa?
There, Zeus cast off the shape of the white bull, and back into his human form, made Europa his lover beneath a simple cypress tree. … At last, Zeus reproduced the shape of the white bull, used by Zeus to seduce Europa, in the stars. Even today we can recognize its shape in the constellation Taurus.
Who did Agamemnon kidnap?
During the Trojan War (prior to the actions described in Homer’s Iliad), Agamemnon took Chryses’ daughter Chryseis (Astynome) from Moesia as a war prize. When Chryses attempted to ransom her, Agamemnon refused to return her.
Is Odysseus a real person?
Archaeologists believe they have found the palace of Odysseus, the legendary Greek king of Ithaca and hero of Homer’s epic poem. They believe that the 8th BC century palace which they have discovered in Ithaca, in the Ionian Seas west of mainland Greece, proves that he was a real historical figure.
Was Achilles A Spartan or Trojan?
In Greek mythology, Achilles was the strongest warrior and hero in the Greek army during the Trojan War. He was the son of Peleus, king of the Myrmidons, and Thetis, a sea nymph. The story of Achilles appears in Homer’s Iliad and elsewhere.
Why did Menelaus and Agamemnon split ways?
During and after the war, Agamemnon and Menelaus were in charge. They disagreed about when they should go back home and what route they should take. They separated, and some people went with Agamemnon and some with Menelaus.
Who did Aphrodite save?
In his epic of the Trojan War, Homer tells how Aphrodite intervened in battle to save her son Aeneas, a Trojan ally. The Greek hero Diomedes, who had been on the verge of killing Aeneas, attacked the goddess herself, wounding her on the wrist with his spear and causing the ichor to flow.
Who is Priam in the Odyssey?
Priam, in Greek mythology, the last king of Troy. He succeeded his father, Laomedon, as king and extended Trojan control over the Hellespont. He married first Arisbe (a daughter of Merops the seer) and then Hecuba, and he had other wives and concubines.
Who married hippodamia?
Hippodamia married Pelops, son of King Tantalus of Lydia, and their daughters were Astydameia, Nicippe, Lysidice, Mytilene, and Eurydice, and their sons were Atreus, Thyestes, Pittheus, Alcathous, Troezen, Hippalcimus, Copreus, Dias, and Hippasus.
Was Clytemnestra jealous of Cassandra?
Aeschylus claimed Clytemnestra—like any queen unwilling to tolerate such concubines in her husband, did it out of pure jealousy. Cassandra: His two-footed lioness, who mates with a wolf in the absence of the noble lion, will slay me, miserable as I am.
Are Helen and Clytemnestra twins?
Helen of Troy, the most beautiful woman on the planet, is the daughter of Leda and Zeus. Helen, with “the face that launched a thousand ships.” Her lesser known twin is Clytemnestra. The two sisters go on to marry brothers, King Menelaus and King Agamemnon.
What are napiers rules?
Definition of Napier’s rule
: either of two rules in spherical trigonometry: the sine of any part is equal to the product of the tangents of the adjacent parts and the sine of any part is equal to the product of the cosines of the opposite parts.
Is plane a trigonometry?
Plane trigonometry deals with the relations between the angles and sides of triangles of which the three vertices are located on the surface of a plane and the sides are straight lines (a straight line being the shortest connection between two points on the plane).
What is Girard’s Theorem?
Girard’s theorem states that the area of a spherical triangle is given by the spherical excess: , where the interior angles of the triangle are , , , and the radius of the sphere is 1. … A spherical lunar-shaped gap is formed with angle , whose area is double the area of the original spherical triangle.
How was sine derived?
The word sine derives from Latin sinus (“bend”, “bay”, “the hanging fold of the upper part of a toga “, “the bosom of a garment”). The use of sinus originates in twelfth-century European translations of the Arabic word jaib (“pocket” or “fold”).
Where did Sin Cos Tan come from?
The functions sine and cosine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin.
Why is Hipparchus the father of trigonometry?
Hipparchus | |
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Died | c. 120 BC (around age 70) Rhodes, Roman Republic |
Occupation | Astronomer Mathematician Geographer |
Where did Nicolaus Copernicus live?
On February 19, 1473, Nicolaus Copernicus is born in Torun, a city in north-central Poland on the Vistula River. The father of modern astronomy, he was the first modern European scientist to propose that Earth and other planets revolve around the sun.
Who discovered diameter of Earth?
The first person to determine the size of Earth was Eratosthenes of Cyrene, who produced a surprisingly good measurement using a simple scheme that combined geometrical calculations with physical observations. Eratosthenes was born around 276 B.C., which is now Shahhat, Libya. He studied in Athens at the Lyceum.
How long did Ptolemy live?
Ptolemy | |
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Κλαύδιος Πτολεμαῖος | |
Ptolemy ‘the Alexandrian’, as depicted in a 16th-century engraving | |
Born | c. 100 AD Egypt, Roman Empire |
Died | c. 170 (aged 69–70) AD Alexandria, Egypt, Roman Empire |
Who discovered the 9 point circle?
In 1765, the eminent Swiss-born mathematician Leonhard Euler showed that nine points constructed from a triangle lie on a circle. This circle would come to be called the Feuerbach circle after Karl Wilhelm Feuerbach, a professor at the gymnasium in Erlangen, Germany.
Who invented the nine-point circle?
The nine-point circle, discovered by Feuerbach in about 1820, contains the midpoints of the three sides of the triangle and the feet of the three altitudes. Its center is the midpoint of the segment joining the orthocenter and the circumcenter of the triangle, and consequently lies on the Euler line.