Pythagorean theorem

In mathematics, the Pythagorean theorem known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides; this theorem can be written as an equation relating the lengths of the sides a, b and c called the "Pythagorean equation": a 2 + b 2 = c 2, where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. The theorem, whose history is the subject of much debate, is named for the ancient Greek thinker Pythagoras; the theorem has been given numerous proofs – the most for any mathematical theorem. They are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years; the theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, indeed, to objects that are not triangles at all, but n-dimensional solids.

The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power. The two large squares shown in the figure each contain four identical triangles, the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, Q. E. D. Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, mentions the proposals of Bretschneider and Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him." Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.

If c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation: a 2 + b 2 = c 2. If the lengths of both a and b are known c can be calculated as c = a 2 + b 2. If the length of the hypotenuse c and of one side are known the length of the other side can be calculated as a = c 2 − b 2 or b = c 2 − a 2; the Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.

This theorem may have more known proofs than any other. This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles. Let ABC represent a right triangle, with the right angle located at C. Draw the altitude from point C, call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e; the new triangle ACH is similar to triangle ABC, because they both have a right angle, they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is similar to ABC; the proof of similarity of the triangles requires the triangle postulate: the sum of the angles in a triangle is two right angles, is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides: B C A B = B H B C and A C A B = A H A C.

The first result equates the cosines of the angles θ. These ratios can be written as B C 2 = A B × B H and

Henry Norris Russell

Prof Henry Norris Russell ForMemRS HFRSE FRAS was an American astronomer who, along with Ejnar Hertzsprung, developed the Hertzsprung–Russell diagram. In 1923, working with Frederick Saunders, he developed Russell–Saunders coupling, known as LS coupling. Russell was born on 25 October 1877, at Oyster Bay, New York, the son of Rev Alexander Gatherer Russell and his wife, Eliza Hoxie Norris, he studied astronomy at Princeton University, obtaining his B. A. In 1897 and his doctorate in 1899, studying under Charles Augustus Young. From 1903 to 1905, he worked at the Cambridge Observatory with Arthur Robert Hinks as a research assistant of the Carnegie Institution and came under the strong influence of George Darwin, he returned to Princeton to become an instructor in astronomy, assistant professor and research professor. He was the director of the Princeton University Observatory from 1912 to 1947 where Charlotte Moore Sitterly helped him measure and calculate the properties of stars, he died in Princeton, New Jersey on 18 February 1957 at the age of 79.

He is buried in Princeton Cemetery. In November 1908 Russell married Lucy May Cole, they had four children. Their youngest daughter, Margaret Russell, married the astronomer Frank K. Edmondson in the 1930s. Russell co-wrote an influential two-volume textbook in 1927 with Raymond Smith Dugan and John Quincy Stewart: Astronomy: A Revision of Young’s Manual of Astronomy; this became the standard astronomy textbook for about two decades. There were two volumes: the first was The Solar System and the second was Astrophysics and Stellar Astronomy; the textbook popularized the idea that a star's properties were determined by the star's mass and chemical composition, which became known as the Vogt-Russell theorem. Since a star's chemical composition changes with age, stellar evolution results. Russell dissuaded Cecilia Payne-Gaposchkin from concluding that the composition of the Sun is different from that of the Earth in her thesis, as it contradicted the accepted wisdom at the time, he realized she was correct four years after deriving the same result by different means.

In his paper Russell credited Payne with discovering that the Sun had a different chemical composition from Earth. Henry Norris Russell. "New Regularities in the Spectra of the Alkaline Earths". Astrophysical Journal. 61: 38–69. Bibcode:1925ApJ....61...38R. Doi:10.1086/142872. Henry Norris Russell. Astronomy: A Revision of Young’s Manual of Astronomy. I: The Solar System. Boston: Ginn & Co. Henry Norris Russell. "On the Composition of the Sun's Atmosphere". Astrophysical Journal. 70: 11–82. Bibcode:1929ApJ....70...11R. Doi:10.1086/143197. Henry Norris Russell. "Model Stars". Bull. Amer. Math. Soc. 43: 49–77. Doi:10.1090/S0002-9904-1937-06492-5. MR 1563489. Fellow of the American Academy of Arts and Sciences Gold Medal of the Royal Astronomical Society Lalande Prize Henry Draper Medal from the National Academy of Sciences Bruce Medal Rumford Prize Franklin Medal Janssen Medal from the French Academy of Sciences Foreign Member of the Royal Society Honorary Fellow of the Royal Society of Edinburgh Henry Norris Russell Lectureship asteroid 1762 Russell

Nicasius of Rheims

Saint Nicasius or Nicaise of Rheims was a bishop of Rheims. He is the patron saint of smallpox victims. Sources placing his death in 407 credit him with prophesying the invasion of France by the Vandals, he notified his people of this vision. When asked if the people should fight or not, Nicasius responded, "Let us abide the mercy of God and pray for our enemies. I am ready to give myself for my people." When the barbarians were at the gates of the city, he decided to attempt to slow them down so that more of his people could escape. He was killed at the altar of his church or in its doorway, he was killed with Jucundus, his lector, his deacon, Eutropia, his virgin sister. After the killing of Nicasius and his colleagues, the Vandals are said to have been frightened away from the area, according to some sources leaving the treasure they had gathered. Accounts of his martyrdom credit him with being among the cephalophores like Saint Denis. Nicasius was said to have been reciting Psalm 119: he was decapitated as he reached the verse Adhaesit pavimento anima mea and continued reciting Vivifica me Domine secundum verbum tuum after his head had fallen to the ground.

He was sometimes depicted in art walking with its miter in his hand. Sources placing his death in 451 record similar acts but concerning the Huns rather than the Vandals; these sources – but not those concerning the Vandals – further relate that Nicasius survived a bout of smallpox, suggesting this legacy of his may have been a fabrication. However, the supposed dubiousness of this claim has been made more credible by research showing a long history of smallpox in Egypt, suggestions that it spread through the Roman Empire, identification of 6th century outbreaks with the disease. From his supposed survival of smallpox, Nicasius became the patron saint of smallpox victims. One prayer ran:In the name of our Lord Jesus Christ, may the Lord protect these persons and may the work of these virgins ward off the smallpox. St. Nicaise had the smallpox and he asked the Lord whoever carried his name inscribed. O St. Nicaise! Thou illustrious bishop and martyr, pray for me, a sinner, defend me by thy intercession from this disease.

Amen. A Benedictine abbey in Rheims was named in his honor. Nicasius of Rheims The Golden Legend: The Life of Saint Nicasius San Nicasio di Reims Lives of the Saints entry on Saint Nicasius Traditional catholic page on Nicasius of Rheims