text: Manson International and Showmen, Inc. produced a 95-minute English-dubbed adaptation of the film, titled "Warriors of the Wind", which was released theatrically in the United States by New World Pictures on 13 June 1985, followed by a VHS release in December 1985. In the late 1980s, Vestron Video would re-release the film in the UK and First Independent Video would re-release it again in 1993, with another minute cut from the film. The voice actors and actresses were not credited and were not informed of the film's plotline, and the film was heavily edited to market it as a children's action-adventure film, although the film received a PG rating just like Disney's later English dub. Consequently, part of the film's narrative meaning was lost: some of the environmentalist themes were diluted as was the main subplot of the Ohmu, altered to turn them into aggressive enemies. Most of the characters' names were changed, including the titular character who became Princess Zandra. The United States poster and VHS cover featured a cadre of male characters who are not in the film, riding the resurrected God Warrior—including a still-living Warrior shown briefly in a flashback. Approximately 22 minutes of scenes were cut for the film's North American release.


title: Nausicaä of the Valley of the Wind (film)

text: Dissatisfied with "Warriors of the Wind", Miyazaki eventually adopted a strict "no-edits" clause for further foreign releases of the company's films. On hearing that Miramax co-chairman Harvey Weinstein would attempt to edit "Princess Mononoke" to make it more marketable, Toshio Suzuki sent an authentic katana with a simple message: "No cuts". "Warriors of the Wind" also prompted Miyazaki to allow translator Toren Smith of Studio Proteus to create an official, faithful translation of the "Nausicaä" manga for Viz Media.


text: On 18 October 2003, Cindy and Donald Hewitt, the scriptwriters of Walt Disney Pictures' English dubs of "Spirited Away" and "Porco Rosso", announced that a more faithful English version of "Nausicaä" was in pre-production at Disney, and that Patrick Stewart and Uma Thurman had been cast. Natalie Portman was originally intended to voice Nausicaä, but Alison Lohman was eventually cast in the role. The dub was directed by Disney executive Rick Dempsey.


text: "Nausicaä" was released on DVD by Buena Vista Home Entertainment on February 22, 2005, for Region 1. This DVD includes both Disney's English dub and the Japanese audio track with English subtitles. Optimum Home Entertainment released the film in Region 2 and the Region 4 DVD is distributed by Madman Entertainment. A remastered Blu-ray sourced from a 6K filmscan was released on 14 July 2010 in Japan. It includes an uncompressed Japanese LPCM stereo track, the Disney-produced English dub and English subtitles. On 18 October 2010, a Blu-ray version was released in Region B by Optimum Home Entertainment. The film was released on Blu-ray in the United States and Canada on March 8, 2011, by Walt Disney Studios Home Entertainment. The Blu-ray earned $334,473 in retail sales during its first week of release in the United States. GKIDS and Shout! Factory re-issued the film on Blu-ray and DVD on October 31, 2017, along with "Castle in the Sky". A Limited Edition steelbook release of the film's DVD and Blu-ray was released in the United States on 25 August 2020.


text: Spain first released two versions of the cut film, both called "Guerreros del Viento" ("Warriors of the Wind") with the first in 1987 and again in 1991, and then a version of the original uncut film under the "Nausicaä del Valle del Viento" title in 2010. France has had both versions of the movie appear with two cut versions named "La Princesse des Étoiles" ("The Princess of the Stars") and "Le vaisseau fantôme" ("The Ghost Ship"): the uncut film had a regular and collector's DVD set released on 18 April 2007. In Germany UFA released the 86-minute long cut version on VHS as "Sternenkrieger" (literally "Star Warriors") in 1986 and Universum Anime released the uncut DVD release on 5 September 2005. The 2007 Hungarian release, titled "Nauszika - A szél harcosai" ("Nausicaä - The Warriors of the Wind") is uncut despite the title's reference. The Korean DVD release of the uncut film was on 3 March 2004. China has had three releases of "Nausicaä": the first on Video CD and two DVD releases. In Italy the film, titled "Nausicaä della Valle del vento", was first aired uncut on Rai 1 on 6 January 1987 with a first dub, but this version was reaired only a few times and then never officially published; a planned DVD release around 2003 by Buena Vista Italia was eventually cancelled. "Nausicaä" had a theatrical distribution and a DVD release with a new dub by Lucky Red in 2015.


text: In Japan, the film grossed about 1.48billion () at the box office, earning 742million in distributor rental income. Its 2020 re-release grossed a further $6,393,174 in Japan, adding up to grossed in Japan.


text: Overseas, the film grossed $1,720,214 from theatrical releases in seven countries between 2006 and 2017, including $1,521,343 in France alone. Combined, the film grossed worldwide.


text: In terms of box office admissions, the film sold 914,767 tickets in Japan up until 2006 and 342,235 tickets in Europe, adding up to at least tickets sold in Japan (not including the 2020 re-release) and Europe.


text: "Nausicaä of the Valley of the Wind" received critical acclaim. The film is frequently ranked among the best animated films in Japan, and is seen by critics as a seminal influence on the development of anime, as the film's success led to the foundation of Miyazaki's Studio Ghibli and several other anime studios. Theron Martin of Anime News Network praised the film for its character designs, as well as Hayao Miyazaki's direction and Joe Hisaishi's score. He also said that the film "deserves a place on any short list of all-time classic anime movies." Common Sense Media, which serves to inform parents about media for children, rated the film positively and cited its good role models and positive messages, but also cautioned parents about its dramatic setting and violent scenes. At Metacritic, the film has a weighted average score of 86 out of 100 based on 7 critics, indicating "universal acclaim". Helen McCarthy in "500 Essential Anime Movies" praised the animation techniques of Miyazaki, stating that "the real strength of this film is the script, packed with incident, excitement and passion, and the soundtrack" of Joe Hisaishi.


text: "Final Fantasy" creator Hironobu Sakaguchi has cited the manga and film as an influence on his video game series; the horseclaws in the film were used as an inspiration for the Chocobos in the games. Sega's Yukio Futatsugi has cited the film an inspiration for his 1995 rail shooter "Panzer Dragoon", as he was an avid fan of Miyazaki's work. Numerous games have used Ohmu-like creatures, assumed to be reference to the film including "Metal Slug 3", "Cyber Core", and "Viewpoint". The game "Crystalis", known in Japan as "God Slayer: Haruka Tenkū no Sonata" (ゴッド・スレイヤー はるか天空のソナタ), shares common elements with the film, including an insect that resembles an Ohmu. The film "" (2015) also shares common elements with the film, including similarities between the protagonists Nausicaa and Rey (such as their personalities and headwear), and a number of strikingly similar scenes.


text: Manga author Katsura Hoshino regarded it as her favorite anime film to the point of having watched it multiple times when she was young. In 2001, the Japanese magazine "Animage" elected "Nausicaä of the Valley of the Wind" the 43rd best anime production of all time. It was the highest-ranking film in a 2006 poll of the greatest animations conducted at the Japan Media Arts Festival, voted by 80,000 attendees. The film was named 2nd "Greatest Japanese Animated Film of All Time" by Japanese film magazine "Kinema Junpo" in 2009.


text: Various gliders are seen in the film and the protagonist, Nausicaä, uses a jet-assisted one-person glider-shaped machine with folding wings. According to the accompanying film book released in Japan, the glider is called , the German word meaning "gull". An official scale model lists it as having an approximate wingspan of 5.8 meters (1/20 model measured to be 29 cm), while the design notes indicate it has a mass of only 12 kg. In 2004, the Japanese-led OpenSky Aircraft Project began attempts to build a real-life, working personal jet glider based on the glider from the film. Two full-size gliders with no power source carrying the code name M01 and M02, with a half-sized jet-powered remote-controlled mock-up called "moewe 1/2", were built. The designer and tester of the project refused the official endorsement of the project by Studio Ghibli and Hayao Miyazaki, noting that he did not want to cause trouble for them if an accident occurred. A jet powered version (registration number JX0122) was finally able to take off under its own power for the first time on 3 September 2013.


text: The film's score was composed by Joe Hisaishi, while the titular theme song "Kaze no Tani no Naushika" was written by Takashi Matsumoto, composed by Haruomi Hosono and sung by Narumi Yasuda. Additionally, the song "Nausicaä's Requiem" was performed by the then-four-year-old Mai Fujisawa, Joe Hisaishi's daughter, who has also collaborated with Studio Ghibli on the vocal album for "Kiki's Delivery Service" and the image album for "Ponyo on the Cliff by the Sea". Numerous soundtracks and albums relating to the film have been released.


text: Miyazaki's manga version of "Nausicaä" was written over a period of 12 years, with breaks taken to work on Studio Ghibli films. Serialized in Tokuma Shoten's Animage magazine, the first chapter was published in February 1982 and the last chapter in March 1994. Miyazaki adapted and altered the work for the film because only sixteen chapters of the manga were written at the time of the film's production. The manga would continue to be produced until the seventh and final book was released on 15 January 1995. The English localization was initially done by Toren Smith and Dana Lewis of Studio Proteus. After Miyazaki resumed production of the manga, Viz Media chose a new team and continued to release the rest of manga.


text: Three video games were released based on the manga and the film, all of which were developed by Technopolis Soft and published by Technopolis Soft and Tokuma Shoten and are released in 1984 on popular Japanese computer systems. The first game, "Nausicaä's Close Call", also known as "Nausicaä in the Nick of Time", (ナウシカ危機一髪, Naushika Kiki Ippatsu) is a Japanese shoot 'em up video game developed and published by Technopolis Soft for the NEC PC-6001. The second game, "Nausicaä of the Valley of the Wind" and known by its title screen as "Nausicaä Adventure Game" (風の谷のナウシカ, Kaze no Tani no Naushika), is an adventure game developed by Technopolis Soft for the NEC PC-8801. The third game, "Never Forget to Nausicaä Game Forever" (忘れじのナウシカ・ゲーム, Wasure ji no Naushika Gemu) for the MSX is the most well-known of the releases and has been frequently and erroneously referred to as a game where the player kills the Ohmu. These games signaled the end of video game adaptations for Hayao Miyazaki's films. The only other games based on Miyazaki films were the LaserDisc arcade game "Cliff Hanger" and the MSX2 platform-adventure game "Lupin III: The Castle of Cagliostro", both of which were based on "The Castle of Cagliostro". Luke Plunkett describes these "two awful adaptations" as the reason Miyazaki does not allow further video game adaptations of his films.


text: An art book, , was released by Tokuma Shoten on 20 June 1984. It contains artwork during the early stages of production of the film and commentary of assistant director Kazuyoshi Katayama. was released by Tokuma Shoten on 5 September 1995. The art book contains artwork of the manga in watercolor, examples of storyboards for the film, autographed pictures by Hayao Miyazaki and interviews on the birth of "Nausicaä". The book has been translated in English and French. Two bunkobon volumes containing the story boards were released, on 31 March 1984. In 2001, the "Nausicaä" storyboards were re-released, bundled into a single, larger, volume as part 1 of the Studio Ghibli Story boards collection. A selection of layout designs for the film was also incorporated in the "Studio Ghibli Layout Designs" exhibition tour, which started in the Museum of Contemporary Art Tokyo (28 July 2008 to 28 September 2008) and subsequently travelled to different museums around Japan and Asia, concluding in the Fukuoka Asian Art Museum (12 October 2013 to 26 January 2014). The exhibition catalogues contain annotated reproductions of the displayed artwork. Tokuma Shoten released a film comic, in four volumes, one each week from 20 November 1990 to 20 December 1990. A two-volume children's version was released on 31 March 1998. A kabuki play adaptation, covering the events of the movie, was performed in December 2019.


title: Quadratic equation

text: where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quadratic.) The numbers , , and are the "coefficients" of the equation and may be distinguished by respectively calling them, the "quadratic coefficient", the "linear coefficient" and the "constant" or "free term".


title: Quadratic equation

text: The values of that satisfy the equation are called "solutions" of the equation, and "roots" or "zeros" of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included; and a double root is counted for two. A quadratic equation can be factored into an equivalent equation


title: Quadratic equation

text: expresses the solutions in terms of , , and . Completing the square is one of several ways for getting it.


title: Quadratic equation

text: Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.


title: Quadratic equation

text: Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.


title: Quadratic equation

text: A quadratic equation with real or complex coefficients has two solutions, called "roots". These two solutions may or may not be distinct, and they may or may not be real.


title: Quadratic equation

text: It may be possible to express a quadratic equation as a product . In some cases, it is possible, by simple inspection, to determine values of "p", "q", "r," and "s" that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if or . Solving these two linear equations provides the roots of the quadratic.


title: Quadratic equation

text: For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. If one is given a quadratic equation in the form , the sought factorization has the form , and one has to find two numbers and that add up to and whose product is (this is sometimes called "Vieta's rule" and is related to Vieta's formulas). As an example, factors as . The more general case where does not equal can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.


title: Quadratic equation

text: Except for special cases such as where or , factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.


title: Quadratic equation

text: which represents a well-defined algorithm that can be used to solve any quadratic equation. Starting with a quadratic equation in standard form,


title: Quadratic equation

text: Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula. The mathematical proof will now be briefly summarized. It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:


title: Quadratic equation

text: Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as or  , where has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.


title: Quadratic equation

text: A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.


title: Quadratic equation

text: A lesser known quadratic formula, as used in Muller's method provides the same roots via the equation


title: Quadratic equation

text: This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is .


title: Quadratic equation

text: One property of this form is that it yields one valid root when , while the other root contains division by zero, because when , the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form for the other root. On the other hand, when , the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form .


title: Quadratic equation

text: It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by , which is always possible since is non-zero. This produces the "reduced quadratic equation":


title: Quadratic equation

text: The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is:


title: Quadratic equation

text: In the quadratic formula, the expression underneath the square root sign is called the "discriminant" of the quadratic equation, and is often represented using an upper case or an upper case Greek delta:


title: Quadratic equation

text: A quadratic equation with "real" coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:


title: Quadratic equation

text: Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.


title: Quadratic equation

text: The function is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of , , and . As shown in Figure 1, if , the parabola has a minimum point and opens upward. If , the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The "-coordinate" of the vertex will be located at formula_23, and the "-coordinate" of the vertex may be found by substituting this "-value" into the function. The "-intercept" is located at the point .


title: Quadratic equation

text: The solutions of the quadratic equation correspond to the roots of the function , since they are the values of for which . As shown in Figure 2, if , , and are real numbers and the domain of is the set of real numbers, then the roots of are exactly the -coordinates of the points where the graph touches the -axis. As shown in Figure 3, if the discriminant is positive, the graph touches the -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the -axis.


title: Quadratic equation

text: In the special case where the quadratic has only one distinct root ("i.e." the discriminant is zero), the quadratic polynomial can be factored as


title: Quadratic equation

text: If the parabola intersects the -axis in two points, there are two real roots, which are the -coordinates of these two points (also called -intercept).


title: Quadratic equation

text: If the parabola is tangent to the -axis, there is a double root, which is the -coordinate of the contact point between the graph and parabola.


title: Quadratic equation

text: If the parabola does not intersect the -axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be.


title: Quadratic equation

text: Let and be respectively the -coordinate and the -coordinate of the vertex of the parabola (that is the point with maximal or minimal -coordinate. The quadratic function may be rewritten


title: Quadratic equation

text: Let be the distance between the point of -coordinate on the axis of the parabola, and a point on the parabola with the same -coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is , and their imaginary part are . That is, the roots are


title: Quadratic equation

text: Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages). In this context, the quadratic formula is not completely stable.


title: Quadratic equation

text: This occurs when the roots have different order of magnitude, or, equivalently, when and are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root. To avoid this, the root that is smaller in magnitude, , can be computed as formula_34 where is the root that is bigger in magnitude.


title: Quadratic equation

text: A second form of cancellation can occur between the terms and of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.


title: Quadratic equation

text: The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.


title: Quadratic equation

text: Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation.


title: Quadratic equation

text: The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation.


title: Quadratic equation

text: Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.


title: Quadratic equation

text: The equation given by Fuss' theorem, giving the relation among the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an ex-tangential quadrilateral.


title: Quadratic equation

text: Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation.


title: Quadratic equation

text: Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the Third Dynasty of Ur. In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:


title: Quadratic equation

text: The steps given by Babylonian scribes for solving the above rectangle problem, in terms of and , were as follows:


title: Quadratic equation

text: In modern notation this means calculating formula_38, which is equivalent to the modern day quadratic formula for the larger real root (if any) formula_39 with , , and .


title: Quadratic equation

text: Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots. Rules for quadratic equations were given in "The Nine Chapters on the Mathematical Art", a Chinese treatise on mathematics. These early geometric methods do not appear to have had a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work "Arithmetica", the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.


title: Quadratic equation

text: In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." ("Brahmasphutasiddhanta", Colebrook translation, 1817, page 346) This is equivalent to


title: Quadratic equation

text: The "Bakhshali Manuscript" written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ). Muhammad ibn Musa al-Khwarizmi (9th century), possibly inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process. He also described the method of completing the square and recognized that the discriminant must be positive, which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions, as well as irrational numbers as solutions. Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.


title: Quadratic equation

text: The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation. His solution was largely based on Al-Khwarizmi's work. The writing of the Chinese mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published "La Géométrie" containing the quadratic formula in the form we know today.


title: Quadratic equation

text: between the roots of a quadratic polynomial and its coefficients. They result from comparing term by the relation


title: Quadratic equation

text: The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, the vertex's -coordinate is located at the average of the roots (or intercepts). Thus the -coordinate of the vertex is


title: Quadratic equation

text: The -coordinate can be obtained by substituting the above result into the given quadratic equation, giving


title: Quadratic equation

text: For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If , then , and we have the estimate:


title: Quadratic equation

text: These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large ), which causes round-off error in a numerical evaluation. The figure shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse.


title: Quadratic equation

text: This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see Step response).


title: Quadratic equation

text: In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots. Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations.


title: Quadratic equation

text: It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution. Consider the following alternate form of the quadratic equation,


title: Quadratic equation

text: where the subscripts and correspond, respectively, to the use of a negative or positive sign in equation [1]. Substituting the two values of or found from equations [4] or [5] into [2] gives the required roots of [1]. Complex roots occur in the solution based on equation [5] if the absolute value of exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone. Calculating complex roots would require using a different trigonometric form.


title: Quadratic equation

text: If the quadratic equation formula_29 with real coefficients has two complex roots—the case where formula_63 requiring "a" and "c" to have the same sign as each other—then the solutions for the roots can be expressed in polar form as


title: Quadratic equation

text: The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients , , are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient or SA. If is the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA.


title: Quadratic equation

text: The Carlyle circle, named after Thomas Carlyle, has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.


title: Quadratic equation

text: The formula and its derivation remain correct if the coefficients , and are complex numbers, or more generally members of any field whose characteristic is not . (In a field of characteristic 2, the element is zero and it is impossible to divide by it.)


title: Quadratic equation

text: in the formula should be understood as "either of the two elements whose square is , if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic . Even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.


title: Quadratic equation

text: In a field of characteristic , the quadratic formula, which relies on being a unit, does not hold. Consider the monic quadratic polynomial


title: Quadratic equation

text: over a field of characteristic . If , then the solution reduces to extracting a square root, so the solution is


title: Quadratic equation

text: In the case that , there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root of to be a root of the polynomial , an element of the splitting field of that polynomial. One verifies that is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic are


title: Quadratic equation

text: For example, let denote a multiplicative generator of the group of units of , the Galois field of order four (thus and are roots of over . Because , is the unique solution of the quadratic equation . On the other hand, the polynomial is irreducible over , but it splits over , where it has the two roots and , where is a root of in .


title: Achelousaurus

text: Achelousaurus () is a genus of centrosaurine ceratopsid dinosaur that lived during the Late Cretaceous Period of what is now North America, about 74.2 million years ago. The first fossils of "Achelousaurus" were collected in Montana in 1987, by a team led by Jack Horner, with more finds made in 1989. In 1995, Achelousaurus horneri was described and named by Scott D. Sampson; the generic name means "Achelous lizard", in reference to the Greek deity Achelous, and the specific name refers to Horner. The genus is known from a few specimens consisting mainly of skull material from individuals, ranging from juveniles to adults.


title: Achelousaurus

text: A large centrosaurine, "Achelousaurus" supposedly was about long, with a weight of about . As a ceratopsian, it walked on all fours, had a short tail and a large head with a hooked beak. It had a bony neck-frill at the rear of the skull, which sported a pair of long spikes, which curved towards the outside. Adult "Achelousaurus" had rough bosses (roundish protuberances) above the eyes and on the snout where other centrosaurines often had horns in the same positions. These bosses were covered by a thick layer of keratin, but their exact shape in life is uncertain. Some researchers hypothesize that the bosses were used in fights, with the animals butting each other's heads, as well as for display.


title: Achelousaurus

text: Within the Ceratopsia, "Achelousaurus" lies within the clade Pachyrostra (or "thick-snouts"). It has been suggested that it was the direct descendant of the similar genus "Einiosaurus" (which had spikes but no bosses) and the direct ancestor of "Pachyrhinosaurus" (which had larger bosses). The first two genera would be transitional forms, evolving through anagenesis from "Styracosaurus". There has been debate about this theory, with later discoveries showing that "Achelousaurus" is closely related to "Pachyrhinosaurus" in the group Pachyrhinosaurini. "Achelousaurus" is known from the Two Medicine Formation and lived in the island continent of Laramidia. As a ceratopsian, "Achelousaurus" would have been a herbivore and it appears to have had a high metabolic rate, though lower than that of modern mammals and birds.


title: Achelousaurus

text: All known "Achelousaurus" specimens were recovered from the Two Medicine Formation in Glacier County, Montana during excavations conducted by the Museum of the Rockies, which still houses the specimens. The discoveries came about by an accidental chain of events. In the spring of 1985, paleontologist John "Jack" R. Horner was informed that he would no longer be allowed to exploit the Willow Creek site, where he had studied the "Maiasaura" Egg Mountain nesting colony for six years. Having already made extensive arrangements for a new field season, he was suddenly forced to seek an alternative site. Horner had always been intrigued by the field diaries of Charles Whitney Gilmore who had reported the discovery of dinosaur eggs at Landslide Butte in 1928, but never published on them. In this locality, Gilmore had employed George Fryer Sternberg to excavate skeletons of the horned dinosaurs "Brachyceratops" and "Styracosaurus ovatus". That summer, Horner obtained the permission of the Blackfeet Indian Tribal Council to prospect for fossils on Landslide Butte, which is part of the Blackfeet Indian Reservation; it was the first paleontological investigation there since the 1920s. In August 1985, Horner's associate Bob Makela discovered a rich fossil site on the land of the farmer Ricky Reagan, which was called the Dinosaur Ridge Quarry and contained fossils of horned dinosaurs. On 20 June 1986, Horner and Makela returned to the Blackfeet Indian Reservation and resumed work on the Dinosaur Ridge Quarry, which proved to contain, apart from eggs, more than a dozen skeletons of a horned dinosaur later named "Einiosaurus". In August 1986, at a nearby site – the Canyon Bone Bed on the land of Gloria Sundquist, east of the Milk River – Horner's team discovered another "Einiosaurus" bone bed. Part of the discoveries made on this occasion was an additional horned dinosaur skull, specimen MOR 492, that later would be referred to (i.e., formally assigned to) "Rubeosaurus", the genus name in 2010 given to "Styracosaurus ovatus".


title: Achelousaurus

text: During the field season of 1987 (early July), volunteer Sidney M. Hostetter located another horned dinosaur skull near the Canyon Bone Bed, specimen MOR 485. By the end of August, it had been secured and was driven on a grain truck to the Museum of the Rockies in Bozeman. On 23 June 1988, another site was discovered in the vicinity – the Blacktail Creek North. In the summer of 1989, graduate student Scott D. Sampson joined the team, wanting to study the function of the frill display structures in horned dinosaurs. At the end of June 1989, Horner, his son Jason and his head preparator Carrie Ancell discovered horned dinosaur specimen MOR 591, a subadult skull and partial postcranial skeleton, near the Blacktail Creek.


title: Achelousaurus

text: It was initially assumed that all the horned dinosaur material recovered by the expeditions could be assigned to a single "styracosaur" species distinct from "Styracosaurus albertensis", as the fossils represented a limited geological time period, then estimated at half a million years. Raymond Robert Rogers, who was studying the stratigraphy of the bone beds, referred to it as a "Styracosaurus" sp. (of undetermined species) in 1989. "Styracosaurus ovatus" – though sometimes considered an invalid "nomen dubium" – had already been found in the area by G. F. Sternberg and was an obvious candidate. But also the possibility was taken into account that the finds were of a species new to science. This species was informally named "Styracosaurus makeli" in honor of Bob Makela, who had died in a traffic accident just days before the discovery of specimen MOR 485. In 1990, this name, as an invalid "nomen nudum", appeared in a photo caption in a book by Stephen Czerkas.


title: Achelousaurus

text: Horner, an expert on the Hadrosauridae family, had less affinity for other kinds of dinosaurs. In 1987 and 1989, horned dinosaur specialist Peter Dodson was invited to investigate the new ceratopsian finds. In 1990, the fossil material was seen by Dodson as strengthening the case for the validity of a separate "Styracosaurus ovatus", to be distinguished from "Styracosaurus albertensis".


title: Achelousaurus

text: Meanwhile, Horner had come to a more complex view of the situation. He still thought that the fossil material had been part of a single population but concluded that this had developed over time as a chronospecies evolving into a series of subsequent taxa. In 1992, Horner, David Varricchio, and Mark Goodwin published an article in "Nature" based on the six-year field study of sediments and dinosaurs from Montana. They proposed that the expeditions had uncovered three "transitional taxa" spanning the gap between the already known "Styracosaurus" and "Pachyrhinosaurus". For the moment, they declined to name these taxa. The oldest form was indicated as "Transitional Taxon A," mainly represented by skull MOR 492. Then came "Taxon B" – the many skeletons of the Dinosaur Ridge Quarry and the Canyon Bone Bed. The youngest was "Taxon C," represented by skull MOR 485 and the horned dinosaur fossils of the Blacktail Creek.


title: Achelousaurus

text: Sampson had continued his studies of the material since 1989. In 1994, in a talk during the annual meeting of the Society of Vertebrate Paleontology, he named "Taxon C" as a new genus and species, "Achelousaurus horneri"; although his abstract was published, it did not identify a holotype, a name-bearing specimen, and genera published in conference abstracts are not considered valid by the ICZN. In 1995, in a subsequent article, Sampson formally named "Achelousaurus horneri", indicating specimen MOR 485 as the holotype. The generic name consists of the words "Achelous", the name of a Greek mythological figure, and "saurus", which is Latinized Greek for lizard. Achelous (Ἀχελῷος) is a Greek river deity and a shapeshifter who was able to transform himself into anything. During a fight with Hercules, the mythical hero, Achelous took the form of a bull, but lost the battle when one of his horns was removed. This allusion is a reference to the supposedly transitional traits of the dinosaur and the characteristic loss of horns through ontogenetic and phylogenetic development, and thus through individual change and evolution. Dodson, in 1996, praised the generic name for being original and intelligent. The specific name honors Jack Horner, for his research on the dinosaurs of the Two Medicine Formation in Montana. Sampson also named "Taxon B" as the genus "Einiosaurus" in the same article wherein "Achelousaurus" was described. He said paleontologists needed to be cautious when naming new ceratopsian genera because their intraspecific variation (i.e., variation within a species) might be mistaken for interspecific differences (between species). Until 1995, only one new genus of centrosaurine dinosaur had been named since "Pachyrhinosaurus" in 1950, namely "Avaceratops" in 1986. "Achelousaurus" thus holds particular importance for being one of the few ceratopsid genera named in the late twentieth century.


title: Achelousaurus

text: The holotype specimen MOR 485 was collected by Hostetter and Ray Rogers from the Landslide Butte Field Area about northwest of Cut Bank. In 1995 Sampson described it as the partial skull of an adult animal including the nasal and supraorbital (region above the eye socket) bosses (roundish protuberances instead of horns), and the parietal bones. Additionally, MOR 485 preserves some bones of the skull rear and sides, which in 2009 were listed by Tracy L. Ford as a right squamosal bone, the left squamosal, both maxillae, both lacrimal bones, both quadrate bones, both palatine bones, the braincase and the basioccipital bone. In 2015, Leonardo Maiorino reported that as part of the same specimen a fragmentary lower jaw has been catalogued as MOR 485-7-12-87-4. A right squamosal bone from another adult individual was recovered from the same Canyon Bone Bed site as MOR 485 (and catalogued under the same number), but only reported in 2010. Two other specimens were collected on the Blacktail Creek, to the south of Cut Bank and referred to "Achelousaurus" by Sampson in 1995. Specimen MOR 591 is a partial skull and an about 60% complete skeleton of a sub-adult specimen that includes the vertebral column, pelvis, sacrum and a femur. It also includes lower jaws, catalogued as MOR 591-7-15-89-1. Both skull and lower jaws are nearly complete, lacking only the braincase and occipital region. MOR 591 is smaller than the holotype with a skull base length of about . Specimen MOR 571 includes a partial skull and lower jaws with associated ribs and vertebrae of an adult. The skull consists of only the parietals, and the lower jaws are limited to their upper rear bones, the surangulars and articulars. A fifth specimen is MOR 456.1, a subadult. None of the specimens were of an advanced individual age. According to Andrew McDonald and colleagues, the "Achelousaurus" finds represented single individuals, not bone beds.


title: Achelousaurus

text: In addition to fossils that have been unequivocally assigned to "Achelousaurus", some other material has been found of which the identity is uncertain. A centrosaurine ceratopsid specimen with bosses from the Dinosaur Park Formation (specimen TMP 2002.76.1) found in 1996 was suggested to belong to a new taxon in 2006, but may instead belong to "Achelousaurus" or "Pachyrhinosaurus". Since it is missing the parietal bones, which are used to diagnose centrosaurines, it is not possible to assign it to any genus with confidence. In 2006, it was also proposed that "Monoclonius lowei", a dubious species based on a skull (specimen CMN 8790) from the Dinosaur Park Formation, could be a sub-adult specimen of "Styracosaurus", "Achelousaurus" or "Einiosaurus", with which it is roughly contemporaneous. In addition, some indeterminate specimens from the Two Medicine Formation – such as fragmentary skull MOR 464 or snout MOR 449 – may belong to "Achelousaurus" or the two other roughly contemporary ceratopsids "Einiosaurus" and "Rubeosaurus".


title: Achelousaurus

text: "Achelousaurus" is estimated to have been long with a weight of . The skull of an adult individual (holotype specimen MOR 485) was estimated to have been long. This puts it in the same size-range as other members of the Centrosaurinae subgroup of ceratopsians that lived during the Campanian age. It was about as large as its close relative "Einiosaurus", but with a much heavier build. "Achelousaurus" approached the robustness of one of the largest and most heavily built horned dinosaurs known: "Triceratops".


title: Achelousaurus

text: As a ceratopsid, "Achelousaurus" would have been a quadrupedal animal with hoofed digits and a shortened, downwards swept tail. Its very large head, which would have rested on a straight neck, had a hooked upper beak, very large nasal openings, and long tooth rows developed into dental batteries that contained hundreds of appressed and stacked individual teeth. In the tooth sockets, new teeth grew under the old ones, each position housing a column of teeth posed on top of each other. "Achelousaurus" had 25 to 28 such tooth positions in each maxilla (upper jaw bone).


title: Achelousaurus

text: In 1995, when describing the species, Sampson gave a formal list of four traits that distinguish "Achelousaurus" from its centrosaurine relatives. Firstly, adult individuals have nasal bones with a boss on top that is relatively small and thin, and heavily covered with pits; secondly, adult individuals do not have true horns above the eye sockets but relatively large bosses with high ridges; thirdly, not yet fully grown individuals, or subadults, have true horncores (the bony part of the horns) above the eye sockets with the inward facing surface being concave; and fourthly, the parietal bones of the neck shield have a single pair of curved spikes sticking out from the rear margin to behind and to the outside.


title: Achelousaurus

text: Besides these unique characteristics, Sampson pointed out additional differences with two very closely related forms. The frill spikes of "Achelousaurus" are more outwards oriented than the spikes of "Einiosaurus", which are medially curved; the spikes of "Achelousaurus" are nevertheless less directed to the outside than the comparable spikes of "Pachyrhinosaurus". "Achelousaurus" also differs from "Pachyrhinosaurus" in its smaller nasal boss that does not reach the frontal bones at its rear. Apart from the skull, no features of the skeleton are known that distinguish "Achelousaurus" from other members of the Centrosaurinae.


title: Achelousaurus

text: Horned dinosaurs mainly differ from each other in their horns, which are located on the snout and above the eyes, and in the large skull frill, which covers the neck like a shield. "Achelousaurus" exhibited the build of derived ("advanced") centrosaurines, which are typified by short brow horns or bosses, combined with elaborate frill spikes. The general frill proportions are typically centrosaurine, with a wide rounded squamosal bone at the side, which expanded towards the rear. It also shares the typical frill curvature with a top surface that is convex from side to side and concave from front to rear.


title: Achelousaurus

text: Adult "Achelousaurus" skulls had a rugose, heavily pitted boss on the snout or nasal region, where many other ceratopsids had a horn. Such a boss is often called "pachyostotic", i.e. consisting of thickened bone. But describing it as a thick "boss" can be misleading: in fact, it forms a wide depression with a thin bone floor and irregular excavations, though it is less depressed than with "Pachyrhinosaurus". The nasal boss covered about two-thirds of the top surface of the nasal bones. The boss was similar to that seen in the related genus "Pachyrhinosaurus", though narrower, shorter, and less high. It covered 27% of the total skull length, was 30% longer than the nostril-eye socket distance and was about twice as long as the eye socket. Its rear edge did not reach the level of the eye socket. The nasal boss extended forward, where it fused onto the nasal and premaxilla bones (of the upper jaw) at the front of the snout, though the nasal bone itself did not fuse with the premaxilla. The boss of specimen MOR 485 furthermore had an excavation (or cavity) at its front end. The horn core that formed the boss may have developed by either becoming procurved (i.e. bent forward) during growth, like the horn of the related "Einiosaurus", until it fused onto the nasal bone; or from a simple, erect horn, which later extended its base forward over the snout region, as in "Pachyrhinosaurus". The nasal bone formed the top of a large bony nostril. From the rear edge of that nostril a sharp process stuck out to the front. The snout was – compared to that of "Einiosaurus" – relatively wide at the level of the rear nostrils. The lacrimal bone, in front of the eye socket, was thickened, mainly on the inner surface while the outer surface was featureless apart from a crater-like excavation.


title: Achelousaurus

text: Adult skulls also possess large, rugose, and oval bosses on the supraorbital region above the eyes, instead of the horns of other ceratopsids. The supraorbital bosses extended from the postorbital bone forward to incorporate the triangular palpebral and prefrontal bones, and had high transverse ridges around the middle, which were thick at their base and thin towards their top. The palpebral bones strongly stood out, forming an "antorbital buttress". The fused prefrontals did not reach the nasal boss, forming a distinctive transverse saddle-shaped groove separating the nasal boss from the supraorbital bosses. This groove extended backwards, separating the supraorbital bosses from each other and forming a T-shape in top view. These bosses were similar to those of "Pachyrhinosaurus", but with taller ridges and more pronounced rugosities. The long and low supraorbital horncores of the sub-adult specimen MOR 591 were similar to those of sub-adult "Einiosaurus" and "Pachyrhinosaurus". They had a concave surface on the inner side as with "Pachyrhinosaurus"; ridges on the postorbital bones were present that may indicate a beginning transition to bosses.


title: Achelousaurus

text: The skull roof of "Achelousaurus" had a midline cavity, with an opening at the top called the frontal fontanelle, a feature found in all ceratopsids, which have a "double" skull roof formed by the frontal bones folding towards each other between the brow horn bases. This cavity formed sinuses that extended below the supraorbital bosses, which were therefore relatively thin internally, being thick from the outside to the cavity roof. This cavity appears to have partially closed over as an animal aged, with only the rear part of the fontanelle being open in the adult specimen MOR 485.


title: Achelousaurus

text: Like that of all other ceratopsids, the skull of "Achelousaurus" had a parietosquamosal frill or "neck shield", which was formed by the parietal bones at the rear and the squamosal bones at the sides. The parietal is one of the main bones used to distinguish centrosaurine taxa from each other and resolve relationships between them, whereas the squamosal is very similar across taxa. In "Achelousaurus", the squamosal bone was much shorter than the parietal. Of its inner margin the rear portion formed a step in relation to the front part, with the suture between the squamosal and the parietal showing a kink to behind at the level of the rear supratemporal fenestra, a typical centrosaurine trait. The squamosal and the jugal bone, by touching each other, excluded the quadratojugal from the edge of the lateral temporal fenestra, i.e. the opening at the rear of the skull side.


title: Achelousaurus

text: The frill of "Achelousaurus" had two conspicuous large spikes that were directed backwards and curved to the sides away from each other. During the 1990s, it was increasingly understood that such spikes on the parietals were not random growths but specific traits that could be used to determine the evolution of horned dinosaurs, if only it could be analyzed how they corresponded among species. Sampson, in the paper describing "Achelousaurus" in 1995, therefore introduced a generalized numbering system for such parietal processes, counting them from the midline to the side of the frill. This was applied to the Centrosaurinae as a whole in 1997. The large spikes of "Achelousaurus" correspond to "Process 3" spikes of other centrosaurines and were similar to those of "Einiosaurus", though curved more to the sides, similar to "Pachyrhinosaurus". They were shorter and thinner than the corresponding spikes of "Styracosaurus". Between these spikes, on both sides of the central frill notch, were two small tab-like processes ("Process 2") that were directed towards the midline. Innermost "Process 1" spikes, as present in "Centrosaurus", are lacking with "Achelousaurus". The frill had two large paired openings, the parietal fenestrae, with a midline parietal bar between them. A linear row of rounded swellings ran along the top of the parietal bar, which may be homologous to the spikes and horns in the same area of some "Pachyrhinosaurus" specimens. A row of relatively small processes ran along the parietal shield margin from the "Process 3" spikes outwards, for a total per side of seven. They were largely equal in size, causing the P4 process to be reduced in comparison to the P3. These lower processes appear to have been capped by epoccipitals, bones that lined the frills of ceratopsids. In "Achelousaurus" these epoccipitals, which start as separate skin ossifications or osteoderms, fuse with the underlying frill bone to form spikes, at least in the third position. In 2020, it was denied that these processes were separate ossifications. In the most mature individuals, the front-most P6 and P7 processes would be less imbricated relative to each other, rotated around their longitudinal axes.


title: Achelousaurus

text: The bosses on the skull of "Achelousaurus" may have been covered in a keratinous sheath in life, but their shape in a living animal is uncertain. In 2009, the paleontologist Tobin L. Hieronymus and colleagues examined correlations between skull morphology, horn, and skin features of modern horned animals, and examined the skull of centrosaurine dinosaurs for the same correlates. They proposed that the rugose bosses of "Achelousaurus" and "Pachyrhinosaurus" were covered by thick pads of cornified (or keratinized) skin, similar to the boss of modern muskoxen ("Ovibos moschatus"). The nasal horncore of adult "Achelousaurus" had an upward slant and its upper surface had correlates for a thick epidermal (outer layer of skin) pad that graded into correlates for a cornified sheath on the sides. A thick pad of epidermis may have grown from the V-shaped pitted notch at the tip of the nasal horncore. The growth direction of the nasal pad would have been towards the front. The supraorbital bosses may have had a thick pad of epidermis, which grew at a sideways angle similar to the curved horncores of "Coronosaurus", as indicated by the orientation of the "fins" or ridges on the bosses. That the supraorbital bosses lacked a sulcus (or furrow) at their bases indicates that their horn pads stopped at the wrinkled edges of the bosses. The pitting might indicate a softer growing layer connecting the hard inner bone with the hard horn sheath. In addition, correlates for a rostral scale in front of the nasal boss and scale rows along the parietal midline and supraorbital-squamosal region were identified.


title: Achelousaurus

text: In 1992, the study by Horner "" tried to account for the fact that within a limited geological period of time (about half a million years) there had been a quick succession of animal communities in the upper Two Medicine Formation. Normally, this would be interpreted as a series of invasions, with the new animal types replacing the old ones. But Horner noted that the newer forms often had a strong similarity to the previous types. This suggested to him that he had discovered a rare proof of evolution in action: the later fauna was basically the old one but at a more evolved stage. The various types found were not distinct species but transitional forms developed within a process of anagenesis. This conformed to the assumption, prevalent at the time, that a species should last about two to three million years. A further indication, according to Horner, was the failure to identify true autapomorphies – unique traits that prove a taxon is a separate species. The fossils instead showed a gradual change from basal (or ancestral) into more derived characters.


title: Achelousaurus

text: The horned dinosaurs discovered by Horner exemplified this phenomenon. In the lowest layers of the Two Medicine Formation, below the overlaying Bearpaw Formation, "Transitional Taxon A" was present. It seemed to be identical to "Styracosaurus albertensis", differing from it only in the possession of just a single pair of parietal spikes. The middle layers, below the Bearpaw, contained "Transitional Taxon B" that also had a single spike pair but differed in the form of its nasal horn that curved to the front over the anterior branches of the nasal bones. In the upper strata, below the Bearpaw, "Transitional Taxon C" had been excavated. It too had a spike pair but now the nasal horn was fused with the front branches. The upper surface of the horn was elevated and very rough. The orbital horns showed coarse ridges. Subsequently, "Taxon A" was named "Stellasaurus", "Taxon B" became "Einiosaurus", while "Taxon C" became "Achelousaurus". In 1992, Horner "et al." did not name these as species for the explicit reason that the entire evolutionary sequence was seen as representing a grade of transitional ceratopsians between "Styracosaurus albertensis", known from the Judith River Formation, and the derived, hornless "Pachyrhinosaurus" from the Horseshoe Canyon Formation, which had the spike pair and bosses on the nose and above the eyes, as well as additional frill ornamentation. In 1997, Horner referred to the three taxa as "centrosaurine 1.", "centrosaurine 2." and "centrosaurine 3.".


title: Achelousaurus

text: Horner thought he had found the mechanism driving this evolution, elaborating on ideas he had developed even before he had investigated Landslide Butte. The animals were living on a narrow strip on the east-coast of Laramidia, bordering the Western Interior Seaway and constrained in the west by the high proto-Rocky Mountains. During the Bearpaw Transgression sea levels were rising, steadily reducing the width of their coastal habitat from about to . This led to stronger selection pressures, the severest for "Achelousaurus" which lived during the phase that the coastal strip was at its narrowest. The lower number of individuals that the smaller habitat could have sustained constituted a population bottleneck, making rapid evolution possible. Increased sexual selection would have induced changes in the sexual ornamentation such as spikes, horns and bosses. A reduced environmental stress by lower sea levels on the other hand, would be typified by adaptive radiation. That sexual selection had indeed been the main mechanism would be proven by the fact that young individuals of all three populations were very similar: they all had two frill spikes, a small nasal horn pointing to the front, and orbital horns in the form of slightly elevated knobs. Only in the adult phase did they begin to differ. According to Horner, this also showed that the populations were very closely related.


title: Achelousaurus

text: Horner did not perform an exact cladistic analysis determining the relationship between the three populations. Such an analysis calculates which evolutionary tree implies the lowest number of evolutionary changes and therefore is the most likely. He assumed that this would result in a tree in which the types were successive branches. Such a tree would, as a consequence of the method used, never show a direct ancestor-descendant relationship. Many scientists believed such a relation could never be proven anyway. Horner disagreed: he saw the gradual morphological changes as clear proof that, in this case, the evolution of one taxon into another, without a splitting of the populations, could be directly observed. Evolutionists in general would be too hesitant to recognize this. Such a transition is called anagenesis; he posited that, if the opposite, cladogenesis, could not be proven, a scientist was free to assume an anagenetic process.


title: Achelousaurus

text: Basing himself on revised data, Sampson in 1995 estimated that the layers investigated represented a longer period of time than the initially assumed 500,000 years: after the deposition of Gilmore's "Brachyceratops" quarry, 860,000 years would have passed, and after the "Einiosaurus" beds 640,000 years, until the maximal extent of the Bearpaw transgression. He did not adopt Horner's hypothesis of anagenesis but assumed speciation took place, with the populations splitting. These time intervals were still short enough to indicate that the rate of speciation must have been high, which might have been true of all centrosaurines of the late Campanian.


title: Achelousaurus

text: In 1996, Dodson raised two objections to Horner's hypothesis. Firstly, the possession of just one pair of main spikes seemed more basal than the presence of three pairs, as with "Styracosaurus albertensis". This suggested to him that the "Einiosaurus"–"Achelousaurus" lineage was a separate branch within the Centrosaurinae. Secondly, he was concerned that "Einiosaurus" and "Achelousaurus" were a case of sexual dimorphism, one type being the males, the other the females. This would be suggested by the short geological time interval between the layers their fossils had been found in, which was estimated by him at about 250,000 years. But if the hypothesis were true, it would be perhaps the best example of fast evolution in the Dinosauria.


title: Achelousaurus

text: In 2010, Horner admitted that specimen TMP 2002.76.1 seemed to indicate that "Achelousaurus" was not descended from "Einiosaurus", as it preceded both in age, and yet had a nasal boss. But he stressed that even if the lineages split off, its ancestor might have resembled "Einiosaurus". Furthermore, it might still be possible that "Einiosaurus" was a direct descendant of "Rubeosaurus". Also, the process of rapid displacements and extinctions of species could in his opinion still be elegantly explained by a westward expansion of the Bearpaw Sea.


title: Achelousaurus

text: The process of anagenesis was affirmed by John Wilson and Jack Scannella in 2016, who studied the ontogenetic changes in horned dinosaurs. They compared a small "Einiosaurus" specimen, MOR 456 8-8-87-1, with "Achelousaurus" specimen MOR 591. Both proved to be quite similar, with the main differences being a longer face in MOR 456 8-8-87-1, and a sharper supraorbital horncore in MOR 591. They concluded that "Achelousaurus" was likely the direct descendant of "Einiosaurus". The more adult "Einiosaurus" individuals approached the "Achelousaurus" morphology. The differences between the two taxa would have been caused by heterochrony – differential changes in the speed the various traits developed during the lifetime of an individual. Since Wilson and colleagues found in 2020 that "Stellasaurus" (Horner's "Taxon A") was intermediate between "Styracosaurus" and "Einiosaurus" in morphology and stratigraphy, they could not discount that it was a transitional taxon within an anagenetic lineage.


title: Achelousaurus

text: In 1995, Sampson formally placed "Achelousaurus" in the Ceratopsidae, more precisely the Centrosaurinae. In all analyses, "Einiosaurus" and "Achelousaurus" are part of the clade Pachyrhinosaurini. By definition, "Achelousaurus" is a member of the clade Pachyrostra (or "thick-snouts"), in which it is united with "Pachyrhinosaurus". In 2010, Gregory S. Paul assigned "A. horneri" to the genus "Centrosaurus", as "C. horneri". This has found no acceptance among other researchers, with subsequent taxonomic assessments invariably keeping the generic name "Achelousaurus".


title: Achelousaurus

text: Sampson felt, in 1995, that there was not enough evidence to conclude that "Achelousaurus" was a direct descendant of "Einiosaurus". Unlike Horner, he decided to perform a cladistic analysis to establish a phylogeny. This showed an evolutionary tree wherein "Achelousaurus" split off between "Einiosaurus" and "Pachyrhinosaurus", as Horner had predicted. Contrary to Horner's claim, "Styracosaurus albertensis" could not have been a direct ancestor, as it was a sister species of "Centrosaurus" in Sampson's analysis.


title: Achelousaurus

text: Subsequent studies have sought to determine the precise relationships within this part of the evolutionary tree, with conflicting results regarding the question whether "Styracosaurus albertensis" or "Einiosaurus" might have been in the direct line of ascent to "Achelousaurus". In 2005, an analysis by Michael Ryan and Anthony Russell found "Styracosaurus" more closely related to "Achelousaurus" than to "Centrosaurus". This was confirmed by analyses by Ryan in 2007, Nicholas Longrich in 2010, and Xu et al. in 2010. The same year Horner and Andrew T. McDonald moved "Styracosaurus ovatus" to its own genus, "Rubeosaurus", finding it a sister species of "Einiosaurus", while "Styracosaurus albertensis" was again located on the "Centrosaurus" branch. They also assigned specimen MOR 492, the basis of "Taxon A", to "Rubeosaurus". In 2011, a subsequent study by Andrew T. McDonald in this respect replicated the outcome of his previous one, as did a publication by Andre Farke et al. In 2017, J.P. Wilson and Ryan further complicated the issue, concluding that MOR 492 ("Taxon A") was not referable to "Rubeosaurus" and announcing that yet another genus would be named for it. Wilson and colleagues moved MOR 492 to the new genus "Stellasaurus" in 2020, which therefore corresponds to "Taxon A". Their study found "Rubeosaurus ovatus" to be the sister species of "Styracosaurus albertensis", and concluded "Rubeosaurus" to be synonymous with "Styracosaurus".


title: Achelousaurus

text: Before "Achelousaurus" was described, "Pachyrhinosaurus canadensis" had been considered a solitary form among centrosaurines, set apart from them by its unusual bosses. "Achelousaurus" gave evolutionary context to the Canadian species, while expanding the temporal and geographical range for what came to be seen as "pachyrhinosaurs." In all analyses, "Achelousaurus" and "Pachyrhinosaurus" were sister groups. In 2008, another closely related species was named, "Pachyrhinosaurus lakustai". In that study, the term "Pachyrhinosaurs" was used for the clade consisting of "Achelousaurus" and "Pachyrhinosaurus". When "Pachyrhinosaurus perotorum" was described in 2012, the clade name Pachyrostra was coined, uniting the two genera; "Achelousaurus" is the basalmost pachyrostran. Shared derived traits (or synapomorphies) of the group are an enlarged nasal ornamentation and a change of the nasal and brow horns into bosses. At the end of the Campanian, there seems to have been a trend of pachyrostrans replacing other centrosaurines. Also in 2012, the clade Pachyrhinosaurini was named, consisting of species more closely related to "Pachyrhinosaurus" or "Achelousaurus" than to "Centrosaurus". Apart from "Einiosaurus" and "Rubeosaurus", this included "Sinoceratops" and "Xenoceratops", according to a 2013 study.


title: Achelousaurus

text: Cladistic analyses develop gradually, reflecting new discoveries and insights. Their results can be shown in a cladogram, with the relationships found ordered in an evolutionary tree. The cladogram below shows the phylogenetic position of "Achelousaurus" in a cladogram from Wilson and colleagues, 2020.


title: Achelousaurus

text: In 1995, Sampson noted that earlier studies had found that the horns and frills of ceratopsians most likely had a function in intraspecific display and combat, and that these features would therefore have resulted from sexual selection for successful mating. Likewise, in 1997 Horner concluded that such ornamentation was used by males to establish dominance and that females would have preferred well-equipped males as their offspring would then inherit these traits, conferring a reproduction benefit. Dodson thought that in the Centrosaurinae in general the display value of the frill had been reduced compared to the nasal and supraorbital ornamentation. Sampson in 1995 rejected the possibility that the difference in skull ornamentation between "Einiosaurus" and "Achelousaurus" represented sexual dimorphism, for three reasons. Firstly, the extensive "Einiosaurus" bone beds did not contain any specimens with bosses, as would have been expected if one of the sexes had them. Secondly, "Einiosaurus" and "Achelousaurus" are found in strata of a different age. Thirdly, in a situation of sexual dimorphism usually only one of the sexes shows exaggerated secondary sexual characters. "Einiosaurus" and "Achelousaurus" however, each have developed a distinct set of such traits.


title: Achelousaurus

text: Hieronymus, in 2009, concluded that the nasal and supraorbital bosses were used for butting or ramming the head or the flank of a rival. The bone structure indicates that the bosses were covered by cornified pads as in modern muskoxen, suggesting dominance fights similar to those of members of the Caprinae subfamily. In the latter group, an evolutionary transition can be observed, where the originally straight horns become more robust, padded, and increasingly curved downwards. The evolution from horncores into bosses in Centrosaurinae would likewise have reflected a change in fighting technique, from clashing to high-energy head-butting. Head-butting would have been an expensive and risky behavior. Opponents would have engaged this way only after assessing each other's strengths visually. For this reason, Hieronymus considered it unlikely that the bosses served for species recognition as this was already guaranteed by the innate species-specific display rituals preceding a real – instead of a ritual – fight. The bosses would have evolved for actual combat, part of a social selection in which individuals competed for scarce resources such as mates, food and breeding grounds.


title: Achelousaurus

text: Previously it had been suggested that the fusion of the first three neck vertebrae, such as seen in the mature specimen MOR 571, might have been a paleopathology, an instance of the disease spondyloarthropathy, but in 1997 it was concluded that it was more likely a normal ontogenetic trait, the vertebrae growing together to form a so-called "syncervical" to support the heavy head. All three main known specimens have syncervicals consisting of three fused neck vertebrae; the trait could have been inherited from a smaller ancestor using a stiffer neck for burrowing or food acquisition.


title: Achelousaurus

text: It has been claimed that ceratopsian dinosaurs were herding animals, due to the large number of known bone beds containing multiple members of the same ceratopsian species. In 2010, Hunt and Farke pointed out that this was mainly true for centrosaurine ceratopsians. Horner assumed that the horned dinosaurs at Landslide Butte lived in herds which had been killed by drought or disease. Dodson concluded that the fact that the "Achelousaurus" bone beds were monospecific (containing only one species) confirmed the existence of herds.


title: Achelousaurus

text: There has long been debate about the thermoregulation of dinosaurs, centered around whether they were ectotherms ("cold-blooded") or endotherms ("warm-blooded"). Mammals and birds are homeothermic endotherms, which generate their own body heat and have a high metabolism, whereas reptiles are heterothermic ectotherms, which receive most of their body heat from their surroundings. A 1996 study examined the oxygen isotopes from bone phosphates of animals from the Two Medicine Formation, including the juvenile "Achelousaurus" specimen MOR 591. δO values of phosphate in vertebrate bones depend on the δO values in their body water and the temperature when the bones were deposited, making it possible to measure fluctuations in temperature for each bone of an individual when they were deposited. The study analyzed seasonal variations in the body temperature and differences in temperature between skeletal regions, to determine whether the dinosaurs maintained their temperature seasonally. A varanid lizard fossil sampled for the study showed isotopic variation consistent with it being an heterothermic ectotherm. The variation of the dinosaurs, including "Achelousaurus", was consistent with them being homeothermic endotherms. The metabolic rate of these dinosaurs was likely not as high as that of modern mammals and birds, and they may have been intermediate endotherms.


title: Achelousaurus

text: "Achelousaurus" is known from the Two Medicine Formation, which preserves coastal sediments dating from the Campanian stage of the Late Cretaceous Period, between 83 and 74 million years ago. "Achelousaurus" specimens are found in the highest levels of the formation, probably closer to the end of that timeframe, 74 million years ago.


title: Achelousaurus

text: The Two Medicine Formation is typified by a warm semiarid climate. Its layers were deposed on the east coast of the Laramidia island continent (which consisted of western North America). The high cordillera in the west, combined with predominantly western winds, would have caused a rain shadow, limiting annual rainfall. Rain would mainly have fallen during the summer, when convection storms flooded the landscape. The climate would thus also have been very seasonal, with a long dry season and a short wet season. Vegetation would have been sparse and a little varied. In such conditions, horned dinosaurs would have been dependent on oxbow lakes for a continuous supply of water and food – the main river channels tending to run dry earlier – and perished in them during severe droughts when the animals concentrated around the last watering holes, causing bone beds to form. The brown paleosol in which the horned dinosaurs were found – a mixture of clay and coalified wood fragments – resembles that of modern seasonally dry swamps. The surrounding vegetation might have consisted of about high conifer trees. "Achelousaurus" ate much smaller plants, though: a 2013 study determined that ceratopsid herbivores on Laramidia were restricted to feeding on vegetation with a height of or lower.


title: Achelousaurus

text: More or less contemporary dinosaur genera of the area included "Prosaurolophus", "Scolosaurus", "Hypacrosaurus", "Einiosaurus" and tyrannosaurids of uncertain classification. As proven by tooth marks, horned dinosaur fossils in the Landslide Butte Field Area had been scavenged by a large theropod predator, which Rogers suggested were "Albertosaurus".


title: Achelousaurus

text: The exact composition of the fauna "Achelousaurus" was part of is uncertain, as its fossils have not been discovered in direct association with other taxa. Its intermediate anagenetic position suggests that "Achelousaurus" shared its habitat with forms roughly found in the middle or at the end of the time range of its formation. As with horned dinosaurs, Horner assumed he had found transitional taxa in other dinosaur groups of the Two Medicine Formation. One of these was a form in between "Lambeosaurus" and "Hypacrosaurus"; in 1994 he would name it "Hypacrosaurus stebingeri". Today, "Hypacrosaurus stebingeri" is no longer seen as having evolved through anagenesis because autapomorphies of the species have been identified. Horner saw some pachycephalosaur skulls as indicative for a taxon in between "Stegoceras" and "Pachycephalosaurus"; these have not been consistently referred to a new genus. Finally, Horner thought there was a taxon present that was transitional between "Daspletosaurus" and "Tyrannosaurus". In 2017, tyrannosaurid remains from the Two Medicine Formation were named as a new species of "Daspletosaurus": "Daspletosaurus horneri". The 2017 study considered it plausible that "D. horneri" was a direct descendant of "D. torosus" in a process of anagenesis, but rejected the possibility that "D. horneri" was the ancestor of "Tyrannosaurus".


title: Achelousaurus

text: Other ceratopsians from the Two Medicine Formation include "Einiosaurus" and "Stellasaurus". In addition, remains of other indeterminate and dubious centrosaurines, including "Brachyceratops", are known from the formation and though they may represent younger stages of the three valid genera, this is not possible to demonstrate. Whereas Horner assumed that "Einiosaurus" and "Achelousaurus" were separate in time, in 2010 Donald M. Henderson considered it possible that at least their descendants or ancestors were overlapping or sympatric and thus would have competed for food sources unless there had been niche partitioning. The skull of "Achelousaurus" was more than twice as strong than that of "Einiosaurus" in its bending strength and torsion resistance. This might have indicated a difference in diet to avoid competition. The bite strength of "Achelousaurus", measured as an ultimate tensile strength, was 30.5 newtons per square millimeter (N/mm) at the maxillary tooth row and 18 N/mm at the beak. Wilson and colleagues found that since the Two Medicine centrosaurines were separated stratigraphically, they were therefore possibly not contemporaneous. However, in 2021 a study by Wilson and Scannella pointed out that specimen MOR 591 was of a younger individual age than the "Einiosaurus" skull MOR 456 8-8-87-1, but of the same size. If MOR 591 could indeed be referred to "Achelousaurus", this might indicate this genus reached its adult size more quickly.


title: Achelousaurus

text: The indeterminate specimen TMP 2002.76.1 is from the Dinosaur Park Formation and, if it belongs to "Achelousaurus", the genus would be the stratigraphically oldest known pachyrhinosaurine taxon. "Achelousaurus" would then also be the only Campanian ceratopsid known from more than one formation. Both animals occur right below the marine shales of the Bearpaw Formation, but due to longitudinal differences, TMP 2002.76.1 is about 500,000 years older than the "Achelousaurus" fossils from the Two Medicine Formation.


title: David Reimer

text: David Reimer (born Bruce Peter Reimer; 22 August 1965 – 4 May 2004) was a Canadian man born male but raised as a girl following medical advice and intervention after his penis was severely injured during a botched circumcision in infancy.


title: David Reimer

text: The psychologist John Money oversaw the case and reported the reassignment as successful and as evidence that gender identity is primarily learned. The academic sexologist Milton Diamond later reported that Reimer's realization that he was not a girl crystallized between the ages of 9 and 11 years and that he was living as a male by age 15. Well known in medical circles for years anonymously as the "John/Joan" case, Reimer later went public with his story to help discourage similar medical practices. At age 38, he committed suicide after suffering severe depression.


title: David Reimer

text: David Reimer was born in Winnipeg, Manitoba, on 22 August 1965, the elder of identical twin boys. He was originally named Bruce, and his identical twin was named Brian. Their parents were Janet and Ron Reimer, a couple of Mennonite descent who had married the previous December. At the age of six months, after concern was raised about how both of them urinated, the boys were diagnosed with phimosis. They were referred for circumcision at the age of seven months. General practitioner Dr. Jean-Marie Huot performed the operation using the unconventional method of electrocauterization, but the procedure did not go as doctors had planned, and David's penis was burned beyond surgical repair. The doctors chose not to operate on Brian, whose phimosis soon cleared without surgical intervention.


title: David Reimer

text: The parents, concerned about their son's prospects for future happiness and sexual function without a penis, took him to Johns Hopkins Hospital in Baltimore in early 1967 to see John Money, a psychologist who was developing a reputation as a pioneer in the field of sexual development and gender identity, based on his work with intersex patients. Money was a prominent proponent of the "theory of gender neutrality"—that gender identity developed primarily as a result of social learning from early childhood and that it could be changed with the appropriate behavioural interventions. The Reimers had seen Money being interviewed in February 1967 on the Canadian news program "This Hour Has Seven Days", during which he discussed his theories about gender.


title: David Reimer

text: At the time, surgical construction of the vagina was more advanced than construction of the penis, and Money believed that Reimer would be happiest in adulthood living as a woman with functioning genitalia. Additionally, for Money, a case where identical twin boys were involved where one could be raised as a girl provided a perfect test of his theories.


title: David Reimer

text: Money and the Hopkins family team persuaded the baby's parents that sex reassignment surgery would be in Reimer's best interest. At the age of 22 months, David underwent a bilateral orchidectomy, in which his testes were surgically removed and a rudimentary vulva was fashioned. David was reassigned to be raised as female and given the name Brenda (similar to his birth name, "Bruce"). Psychological support for the reassignment and surgery was provided by John Money, who continued to see Reimer annually for consultations and to assess the outcome. This reassignment was considered an especially important test case of the social learning concept of gender identity for two reasons: first, Reimer's identical twin brother, Brian, made an ideal control because the brothers shared genes, family environments, and the intrauterine environment; second, this was reputed to be the first reassignment and reconstruction performed on a male infant who had no abnormality of prenatal or early postnatal sexual differentiation.


title: David Reimer

text: Money continued to supervise and report on the twins' gender development as the "John/Joan case" until the twins were 13 years old.


title: David Reimer

text: According to John Colapinto, who published a biography of Reimer in 2001, the sessions with Money included childhood sexual rehearsal play: Money theorized that reproductive behavior formed the foundation of gender, and that "play at thrusting movements and copulation" was a key aspect of gender development in all primates. Starting at age six, according to Brian, the twins were forced to act out sexual acts, with David playing the female role—Money made Reimer get down on all fours, and Brian was forced to "come up behind [him] and place his crotch against [his] buttocks". Money also forced Reimer, in another sexual position, to have his "legs spread" with Brian on top. On "at least one occasion" Money took a photograph of the two children doing these activities.


title: David Reimer

text: When either child resisted these activities, Money would get angry. Both Reimer and Brian recall that Money was mild-mannered around their parents, but ill-tempered when alone with them. When they resisted inspecting each other's genitals, Money got very aggressive. Reimer says, "He told me to take my clothes off, and I just did not do it. I just stood there. And he screamed, 'Now!' Louder than that. I thought he was going to give me a whupping. So I took my clothes off and stood there shaking."


title: David Reimer

text: Money's rationale for these various treatments was his belief that "childhood 'sexual rehearsal play was important for a "healthy adult gender identity".


title: David Reimer

text: Both Reimer and Brian were traumatized by the therapy, with Brian speaking about it "only with the greatest emotional turmoil," and Reimer unwilling to speak about the details publicly, although his wife, Jane Fontane, stated that Reimer had privately told her the same story. Brian was found dead of a drug overdose at 36, and Reimer died by suicide at age 38. Reimer's parents state that Money's methodology was responsible for both deaths.


title: David Reimer

text: Money never commented publicly on Colapinto's book or on Reimer's suicide before his death, although colleagues said he was "mortified" by the case.


title: David Reimer

text: For several years, Money reported on Reimer's progress as the "John/Joan case". Money wrote, "The child's behavior is so clearly that of an active little girl and so different from the boyish ways of her twin brother."


title: David Reimer

text: The twins attended Glenwood School in Winnipeg, with David then attending R.B. Russell Vocational High School, from the age of 14. He eventually ceased attending the school and was tutored privately.


title: David Reimer

text: By the age of 13 years, Reimer was experiencing suicidal depression and he told his parents he would take his own life if they made him see Money again. Finally, on 14 March 1980, Reimer's parents told him the truth about his gender reassignment, following advice from Reimer's endocrinologist and psychiatrist. At 14, having been informed of his past by his father, Reimer decided to assume a male gender identity, calling himself David. He underwent treatment to reverse the reassignment, including testosterone injections, a double mastectomy, and phalloplasty operations.


title: David Reimer

text: Reimer worked in a slaughterhouse and then worked doing odd jobs. On 22 September 1990, he married Jane Fontane and would adopt her three children. His hobbies included camping, fishing, antiques and collecting old coins.


title: David Reimer

text: His case came to international attention in 1997 when he told his story to Milton Diamond, an academic sexologist who persuaded Reimer to allow him to report the outcome in order to dissuade physicians from treating other infants similarly. Soon after, Reimer went public with his story and John Colapinto published a widely disseminated and influential account in "Rolling Stone" magazine in December 1997. The article won the National Magazine Award for Reporting.


title: David Reimer

text: This was later expanded into "The New York Times" best-selling biography "As Nature Made Him: The Boy Who Was Raised as a Girl" (2000), in which Colapinto described how—contrary to Money's reports—when living as Brenda, Reimer did not identify as a girl. He was ostracized and bullied by peers (who dubbed him "cavewoman"), and neither frilly dresses nor female hormones made him feel female.


title: David Reimer

text: In addition to his difficult lifelong relationship with his parents, Reimer experienced unemployment and the death of his brother Brian from an overdose of antidepressants on 1 July 2002. On 2 May 2004, his wife Jane told him she wanted to separate. On the morning of 4 May 2004, Reimer drove to a grocery store's parking lot in his hometown of Winnipeg and shot himself in the head with a sawed-off shotgun. He was 38 years old. He was buried in St. Vital Cemetery in Winnipeg.


title: David Reimer

text: For the first 30 years after Money's initial report that the reassignment had been a success, Money's view of the malleability of gender became the dominant viewpoint among physicians and doctors, reassuring them that sexual reassignment was the correct decision in certain instances, resulting in thousands of sexual reassignments. Researcher Mary Anne Case argues that Money's view on gender also fueled the rise of the anti-gender movement.


title: David Reimer

text: Diamond's report and Colapinto's subsequent book about Reimer influenced several medical practices, reputations, and even current understanding of the biology of gender. The case accelerated the decline of sex reassignment and surgery for unambiguous XY infants with micropenis, various other rare congenital malformations, or penile loss in infancy.


title: David Reimer

text: Colapinto's book described unpleasant childhood therapy sessions, implying that Money had ignored or concealed the developing evidence that Reimer's reassignment to female was not going well.


title: David Reimer

text: The case has also been treated by Judith Butler in their 2004 book "Undoing Gender", which examines gender, sex, psychoanalysis, and the medical treatment of intersex people.


title: David Reimer

text: The BBC science series "Horizon" based two episodes on his life. "The Boy Who Was Turned into a Girl" aired in 2000 and "Dr Money and the Boy with No Penis" in 2004.


title: David Reimer

text: A 2001 episode of the PBS documentary series "Nova" entitled "Sex: Unknown" investigated David's life and the theory behind the decision to raise him as female.


title: David Reimer

text: An episode of BBC Radio 4 "Mind Changers", "Case Study: John/Joan - The Boy Who Was Raised as a Girl", discusses the impact on two competing psychological theories of nature vs. nurture.


title: Roberto Durán

text: Roberto Durán Samaniego (born June 16, 1951) is a Panamanian former professional boxer who competed from 1968 to 2001. He held world championships in four weight classes: lightweight, welterweight, light middleweight and middleweight, as well as reigns as the undisputed and lineal lightweight champion, and the lineal welterweight champion. He is also the second boxer to have competed over a span of five decades, the first being Jack Johnson. Durán was known as a versatile, technical brawler and pressure fighter, which earned him the nickname of ""Mano de Piedra"" ("Hands of Stone") for his formidable punching power and excellent defense.


title: Roberto Durán

text: In 2002, Durán was voted by "The Ring" magazine as the fifth greatest fighter of the last 80 years, while boxing historian Bert Sugar rated him as the eighth greatest fighter of all time. The Associated Press voted him as the best lightweight of the 20th century, with many considering him the greatest lightweight of all time. Durán retired for good in January 2002 at age 50, following a car crash in Argentina in October 2001, after which he had required life saving surgery. He had previously retired in November 1980, June 1984 and August 1998, only to change his mind. Durán ended his career with a professional record of 119 fights, 103 wins, and 70 knockouts. From May 1971 up until his second fight against Sugar Ray Leonard in November 1980, as well as in his fight against Wilfred Benítez in January 1982, Durán was trained by legendary boxing trainer Ray Arcel.


title: Roberto Durán

text: Roberto Durán was born on June 16, 1951, in Guararé, Panama. His mother, Clara Samaniego, was a native of Guararé, and his father, Margarito Durán Sánchez, was from Arizona, United States, and of Mexican descent. He was raised in the slums of El Chorrillo in the district "La Casa de Piedra" (The House of Stone), in Panama City. He began sparring with experienced boxers at the Neco de La Guardia gymnasium when he was only eight years old.


title: Roberto Durán

text: Durán competed as an amateur, compiling a record of 29–3 (other sources say 18–3 or 13–3), with all 3 losses coming in Durán's first 3 amateur fights. Following his amateur career, Durán made his professional debut in February 1968 at the age of 16.


title: Roberto Durán

text: Durán won his first 31 consecutive professional fights, and scored knockout victories over future Featherweight Champion Ernesto Marcel and former Super Featherweight Champion Hiroshi Kobayashi, culminating in his first title bout in June 1972, where he defeated Ken Buchanan in Madison Square Garden for the WBA Lightweight Championship. Durán, as a 2-to-1 underdog, scored a knockdown against the defending champion just fifteen seconds into the opening round and battered him throughout the bout. He was well ahead on all three cards as the bell rang to end the 13th round, at which time Durán (apparently not hearing the bell) continued to throw a couple of extra punches as Buchanan lay on the ropes. As Durán continued punching, the referee, Johnny LoBianco, grabbed him to pull him away. He pulled down on Durán's arms, which led to a seemingly accidental low blow. Buchanan dropped to the canvas in pain. His trainer Gil Clancy later said he had believed the blow to have been caused by a knee. Durán was not disqualified from the bout; instead, he was deemed as winner by thirteenth-round technical knockout. Columnist Red Smith of "The New York Times" wrote that LoBianco had to award the victory to Durán, even if the punch was a low blow, as "anything short of pulling a knife is regarded indulgently" in American boxing. Buchanan said he left the fight "with sore balls".


title: Roberto Durán

text: Durán followed up on his title winning performance with several non-title matches. Later that year, in another non-title bout, he lost a ten-round decision to Esteban De Jesús. Durán got back on track with successful title defenses against Jimmy Robertson, Hector Thompson and future Lightweight Champion Guts Ishimatsu. In 1974, Durán avenged his loss to De Jesus with a brutal eleventh round knock out. In 1976, he defeated future Light Welterweight Champion Saoul Mamby. Overall, Durán made twelve successful defenses of his title (eleven coming by knockout) and amassed a record of 62–1, his last defense coming in 1978 when Durán fought a third bout with De Jesus in a unification match wherein Durán once again knocked out De Jesus and captured his WBC Lightweight Championship. Durán gave up the Undisputed Lightweight Championship in February 1979.


title: Roberto Durán

text: Vacating the Lightweight title was a buildup for an attempt at the Welterweight title. Durán earned wins against former WBC Welterweight Champion Carlos Palomino and Zeferino Gonzales, among others, setting the stage for a title bout against then-undefeated WBC Welterweight Champion Sugar Ray Leonard. The venue chosen was the Olympic Stadium in Montreal (the same location where Leonard won an Olympic gold medal during the 1976 Summer Olympics). Durán resented the fact that he was getting only one-fifth of the money that Leonard was getting, despite the fact that Durán was entering the bout with an incredible 71–1 record and seen by many as the best boxer of the decade of the 1970s. To the surprise of Leonard and his camp, who had expected a warm homecoming from the place where Leonard had won Olympic gold, Leonard only got a mixed reception in Montreal, while Durán was incredibly popular with the crowd, with Leonard later admitting that Durán's popularity in Canada "threw me for a loop". On June 20, 1980, Durán captured the WBC Welterweight title by defeating Leonard via a 15-round unanimous decision (145–144, 148–147, 146–144), although it was incorrectly announced as a majority decision in the ring with the 148–147 scorecard being incorrectly announced as 147–147. The fight became known as "The Brawl in Montreal".