❤️ Synthetic element 🐾

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❤️ Shoghi Effendi 🐾

"Shoghí Effendí Rabbání (1 March 1897 – 4 November 1957), better known as Shoghi Effendi (), was the Guardian and appointed head of the Baháʼí Faith from 1921 until his death in 1957. Shoghi Effendi spent his early life in ʻAkká (Acre). His education was directed to serving as secretary and translator to his grandfather, ʻAbdu'l-Bahá, then leader of the Baháʼí Faith and son of the religion's founder, Baháʼu'lláh. After the death of ʻAbdu'l-Bahá in 1921, the leadership of the Baháʼí community changed from that of a single individual to an administrative order with executive and legislative branches, the head of each being the Guardianship and the Universal House of Justice, respectively. Shoghi Effendi was referred to as the Guardian, and had the authority to interpret the writings of the three central figures of the religion and define the sphere of legislative authority. His writings are effectively limited to commentaries on the works of the central figures, and broad directives for the future. Future hereditary Guardians were permitted in the Baháʼí scripture by appointment from one to the next with the prerequisite that appointees be male descendants of Baháʼu'lláh. At the time of Shoghi Effendi's death, all living male descendants of Baháʼu'lláh had been declared Covenant-breakers by either ʻAbdu'l-Bahá or Shoghi Effendi, leaving no suitable living candidates. Shoghi Effendi died without appointing a successor Guardian, and the Universal House of Justice, the only institution authorized to adjudicate on situations not covered in scripture, later announced that it could not legislate to make possible the appointment of a successor to Shoghi Effendi. Shoghi Effendi was the first and last person acknowledged as Guardian of the Baháʼí Faith. Background The young Shoghi, Born in ʻAkká in the Acre Sanjak of the Ottoman Empire in March 1897, Shoghi Effendi was related to the Báb through his father, Mírzá Hádí Shírází, and to Baháʼu'lláh through his mother, Ḍíyáʼíyyih Khánum, the eldest daughter of ʻAbdu'l-Bahá. ʻAbdu'l-Bahá, who provided much of his initial training, greatly influenced Shoghi Effendi from the early years of his life. Shoghi Effendi learned prayers from his grandfather, who encouraged him to chant. ʻAbdu'l-Bahá also insisted that people address the child as "Shoghi Effendi", ("Effendi" signifies "Sir"), rather than simply as "Shoghi", as a mark of respect towards him. From his early years, Shoghi Effendi was introduced to the suffering which accompanied the Baháʼís in ʻAkká, including the attacks by Mírzá Muhammad ʻAlí against ʻAbdu'l-Bahá. As a young boy, he was aware of the desire of Sultan Abdul Hamid II (reigned 1876-1909) to banish ʻAbdu'l-Bahá to the deserts of North Africa where he was expected to perish. At one point, Shoghi Effendi was warned not to drink coffee in the homes of any of the Baháʼís in the fear that he would be poisoned. = Tablet from ʻAbdu'l-Bahá = As the eldest grandson of ʻAbdu'l-Bahá, Shoghi Effendi from his earliest childhood had a special relationship with his grandfather. According to one account, when Shoghi Effendi was only 5 years old, he pestered his grandfather to write a tablet for him, which was common practice for ʻAbdu'l-Bahá. He wrote the following for his grandson: Shoghi Effendi then set out to memorize a number of prayers, and chanted them as loud as he could. This caused family members to ask ʻAbdu'l-Bahá to quieten him down, a request which he apparently refused. = Education = Shoghi Effendi received his early education at home with the other children in the household, then attended a French Christian Brothers school in Haifa, and later boarded at another Catholic school in Beirut. Shoghi Effendi later attended the Syrian Protestant College (later known as the American University of Beirut) for his final years of high school and first years of university, where he earned an arts degree in 1918. He reports being very unhappy in school and often returned on vacations to Haifa to spend time with ʻAbdu'l-Bahá. During his studies, he dedicated himself to mastering English—adding this language to the Persian, Turkish, Arabic and French languages in which he was already fluent—so that he could translate the letters of ʻAbdu'l-Bahá and serve as his secretary. Shoghi Effendi was protected from World War I due to the neutrality of the Syrian Protestant College. Though political tensions in 1917 meant the college was closed briefly, student life continued. In the summer of 1918 ʻAbdu'l-Bahá's life was in critical danger until the entry of General Allenby's troops to Haifa. With the Armistice looming and having completed his studies Shoghi Effendi was ready to return to his grandfather. In the Autumn of 1918 Shoghi Effendi went back to Haifa to assist ʻAbdu'l-Bahá in his mounting correspondence. In a private letter to a friend from late 1918 Shoghi Effendi reflects on the untold sufferings of the War but anticipates that "this is indeed the era of service". After studying at the American University of Beirut he later went to Balliol College, Oxford, in England, where he matriculated in "Economics and Social Sciences", while still perfecting his translation skills. Shoghi Effendi was happy during his time in Balliol. Accounts from his contemporaries remember him as a cheerful and popular student. He was acquainted with future British prime minister Anthony Eden but they were not close friends. His studies were interspersed with occasional trips around the United Kingdom to meet Baháʼí communities. Shoghi Effendi was particularly touched meeting the small group of Baháʼís from Manchester. During this period Shoghi Effendi began what would be a life-long affinity to aspects of British culture such as reading The Times everyday and his love for English literature. = Death of ʻAbdu'l-Bahá and Guardianship = Shoghi Effendi before 1940 The issue of successorship to ʻAbdu'l-Bahá was in the minds of early Baháʼís, and although the Universal House of Justice was an institution mentioned by Baháʼu'lláh, the institution of the Guardianship was not clearly introduced until the Will and Testament of ʻAbdu'l-Bahá was publicly read after his death. While studying in England, on 29 November 1921, the news of ʻAbdu'l-Bahá's death reached Shoghi Effendi, which, according to Wellesley Tudor Pole, the deliverer of the cable, left him "in a state of collapse". After spending a couple of days with John Esslemont, and after some passport difficulties, he sailed from England accompanied with Sara Blomfield and his sister Ruhangiz on 16 December and arrived in Haifa on 29 December. A few days later he opened ʻAbdu'l-Bahá's Will and Testament, which was addressed to Shoghi Effendi. In the will, Shoghi Effendi found that he had been designated as "the Sign of God, the chosen branch, the Guardian of the Cause of God". He also learned that he had been designated as this when he was still a small child. As Guardian he was appointed as head of the religion, someone whom the Baháʼís had to look to for guidance. ʻAbdu'l-Bahá's Will and Testament is considered one of the three charters of the Baháʼí administrative order, and in it ʻAbdu'l-Bahá laid down the authority of the Guardian and the Universal House of Justice, the elected governing body of the Baháʼí Faith that had been written about by Baháʼu'lláh, and had not yet been established: Shoghi Effendi later expressed to his wife and others that he had no foreknowledge of the existence of the Institution of Guardianship, least of all that he was appointed as Guardian. The most he expected was perhaps, because he was the eldest grandson, ʻAbdu'l-Bahá might have left instructions as to how the Universal House of Justice was to be elected and he might have been designated as Convener of the gathering which would elect it. Accomplishments From the time of his appointment as Guardian until his death the Baháʼí Faith grew from 100,000 to 400,000 members, capitalizing on prior growth and setting the stage for more, and the countries and territories in which Baháʼís had representation went from 35 to 250. As Guardian and head of the religion, Shoghi Effendi communicated his vision to the Baháʼís of the world through his numerous letters and his meetings with pilgrims to Palestine. During the 1920s he first started to systematize and extend the Baháʼí administration throughout the world; the Baháʼí community was relatively small and undeveloped when he assumed leadership of the religion, and he strengthened and developed it over many years to support the administrative structure envisioned by ʻAbdu'l-Bahá. Under Shoghi Effendi's direction, National Spiritual Assemblies were formed, and many thousands of Local Spiritual Assemblies were created. During the 1930s he worked on projects translating the works of Baháʼu'lláh into English. Starting in 1937, he set into motion a series of systematic plans to establish Baháʼí communities in all countries. A Ten Year Crusade was carried out from 1953 to 1963 with the aim of electing the Universal House of Justice as its paramount aim. Starting in the late 1940s, after the establishment of the State of Israel, he started to develop the Baháʼí World Centre in Haifa, including the construction of the superstructure of the Shrine of the Báb and the building of the International Archives as well as beautifying the gardens at Bahji, where the Shrine of Baháʼu'lláh is located, as well as developing plans and resources to raise several of the continental Baháʼí Houses of Worship around the world; these plans continued through the 1950s. In the 1950s he also continued building the Baháʼí administration, establishing in 1951 the International Baháʼí Council to act as a precursor to the Universal House of Justice, as well as appointing 32 living Hands of the Cause — Baháʼís appointed to the highest rank of service available, whose main function was to propagate and protect the religion. He also acted as the official representative of the religion to legal authorities in Israel as well as designated other representatives to work with the UN. In a more secular cause, prior to World War II he supported the work of restoration-forester Richard St. Barbe Baker to reforest Palestine, introducing him to religious leaders from the major faiths of the region, from whom backing was secured for reforestation. Translations and writings One of Shoghi Effendi's earliest letters as Abdu'l-Bahá's amanuensis, 1919 In his lifetime, Shoghi Effendi translated into English many of the writings of the Báb, Baháʼu'lláh and ʻAbdu'l-Bahá, including the Hidden Words in 1929, the Kitáb-i-Íqán in 1931, Gleanings in 1935 and Epistle to the Son of the Wolf in 1941. He also translated such historical texts as The Dawn- breakers. His significance is not just that of a translator, but he was also the designated and authoritative interpreter of the Baháʼí writings. His translations, therefore, are a guideline for all future translations of the Baháʼí writings. The vast majority of his writings were in the style of letters with Baháʼís from all parts of the globe. These letters, of which 17,500 have been collected thus far and are believed to number a total of 30,000, ranged from routine correspondence regarding the affairs of Baháʼís around the world to lengthy letters to the Baháʼís of the world addressing specific themes. Some of his longer letters include World Order of Baháʼu'lláh, regarding the nature of Baháʼí administration, Advent of Divine Justice, regarding teaching the religion, and Promised Day is Come regarding Baháʼu'lláh's letters to world leaders. Other letters included statements on Baháʼí beliefs, history, morality, principles, administration and law. He also wrote obituaries of some distinguished Baháʼís. Many of his letters to individuals and assemblies have been compiled into several books which stand out as significant sources of literature for Baháʼís around the world. The only actual book he ever wrote was God Passes By in 1944 to commemorate the centennial anniversary of the religion. The book, which is in English, is an interpretive history of the first century of the Bábí and Baháʼí Faiths. A shorter Persian language version was also written. Leadership As a young student of twenty-four, Shoghi Effendi was initially shocked at the appointment as Guardian. He was also mourning the death of his grandfather to whom he had great attachment. The trauma of this culminated in him making retreats to the Swiss Alps. However, despite his youth, Shoghi Effendi had a clear idea of the goal he had for the religion. Oxford educated and Western in his style of dress, Shoghi Effendi was a stark contrast to his grandfather ʻAbdu'l-Bahá. He distanced himself from the local clergy and notability, and travelled little to visit Baháʼís unlike his grandfather. Correspondence and pilgrims were the way that Shoghi Effendi conveyed his messages. His talks are the subject to a great number of "pilgrim notes". He also was concerned with matters dealing with Baháʼí belief and practice — as Guardian he was empowered to interpret the writings of Baháʼu'lláh and ʻAbdu'l-Bahá, and these were authoritative and binding, as specified in ʻAbdu'l-Bahá's will. His leadership style was however, quite different from that of ʻAbdu'l-Bahá, in that he signed his letters to the Baháʼís as "your true brother", and he did not refer to his own personal role, but instead to the institution of the guardianship. He requested that he be referred in letters and verbal addresses always as Shoghi Effendi, as opposed to any other appellation. He also distanced himself as a local notable. He was critical of the Baháʼís referring to him as a holy personage, asking them not to celebrate his birthday or have his picture on display. Private life Shoghi Effendi's personal life was largely subordinate to his work as Guardian of the religion. His lack of secretarial support with the mass of correspondence had left a pattern of hard work in Haifa interspersed with occasional summer breaks to Europe—in the early years often to the Swiss Alps. In 1929 and 1940 he also travelled through Africa from south to north. In public Shoghi Effendi was variously described as aristocratic, composed and highly informed in international affairs. In private his contemporaries remembered him as warm, informal and humorous. Shoghi Effendi would sleep very little and usually ate only once a day. He was short in stature, with dark hair, an olive complexion and hazel eyes. He was noted as not resembling his grandfather ʻAbdu'l-Bahá (who was taller and had blue eyes) but his great-grandfather Baháʼu'lláh. Shoghi Effendi had a great love for the English language. He was an avid fan of English literature, and enjoyed reading the King James Bible. He was noted for speaking English in clipped received pronunciation, and Persian in an Isfahani dialect, inherited from his grandmother. Shoghi Effendi held Iranian (Persian) nationality throughout his life and travelled on an Iranian passport, although he never visited Iran. = Marriage = Mary Maxwell, known as Rúhíyyih Khánum In March 1937, Shoghi Effendi married Mary Maxwell, entitled Rúhíyyih Khánum, a Canadian. She was the only child of May Maxwell, a disciple of ʻAbdu'l-Bahá, and William Sutherland Maxwell, a Canadian architect. Shoghi Effendi had first met Mary as a girl when she came on pilgrimage with her mother in 1923. The two had begun a regular correspondence from the mid-1920s. Mary was an active Baháʼí teacher, and a letter written to Shoghi Effendi described her as "a beautiful and most refreshing girl to know".Etter-Lewis, Gwendolyn (2006). Lights of the Spirit: Historical Portraits of Black Baha'is in North America, 1898-2000. Baha'i Publishing Trust. p. 80. . Whilst on her third pilgrimage in 1937 the two began a discreet courtship. Then 26 years old, Mary was a tall, athletic woman. Mary had been living in Nazi Germany for 18 months with her cousin prior to coming to Haifa. The couple married in the room of Bahíyyih Khánum in the House of ʻAbdu'l-Bahá in Haifa. The ceremony was a short, simple and quiet one in which Rúhíyyih Khánum wore black. Very few knew the wedding was taking place apart from the witnesses and a small group of residents of Haifa. Therefore the marriage came as a great surprise to the world-wide Baháʼí community when the mother of Shoghi Effendi cabled the Baháʼís: While Shoghi Effendi and Rúhíyyih Khánum never had children, Rúhíyyih Khánum became his constant companion and helpmate; in 1941, she became Shoghi Effendi's principal secretary in English. In a rare public statement revealing his private sentiments in 1951 he described his wife as "my helpmate, my shield in warding off the darts of Covenant breakers and my tireless collaborator in the arduous tasks I shoulder". Opposition Mírzá Muhammad ʻAlí was ʻAbdu'l-Bahá's half brother and was mentioned by Baháʼu'lláh as having a station "beneath" that of ʻAbdu'l-Bahá. Muhammad ʻAli later fought ʻAbdu'l-Bahá for leadership and was ultimately excommunicated, along with several others in the Haifa/ʻAkká area who supported him. When Shoghi Effendi was appointed Guardian Muhammad ʻAli tried to revive his claim to leadership, suggesting that Baháʼu'lláh's mention of him in the Kitáb-i-'Ahd amounted to a succession of leadership. After Shoghi Effendi's death, Rúhíyyih Khánum published parts of her personal diaries to show glimpses of Shoghi Effendi's life. She recalls a great deal of pain and suffering caused by his immediate family, and Baháʼís in Haifa. Throughout Shoghi Effendi's life, nearly all remaining family members and descendants of ʻAbdu'l-Bahá were expelled by him as covenant-breakers when they didn't abide by Shoghi Effendi's request to cut contact with covenant-breakers, as specified by ʻAbdu'l-Bahá. Other branches of Baháʼu'lláh's family had already been declared Covenant-breakers in ʻAbdu'l-Bahá's Will and Testament. At the time of his death, there were no living descendants of Baháʼu'lláh that remained loyal to him. Unexpected death Shoghi Effendi's resting place in London at the New Southgate Cemetery Shoghi Effendi's death came unexpectedly in London, on 4 November 1957, as he was travelling to Britain and caught the Asian Flu, during the pandemic which killed two million worldwide, and he is buried there in New Southgate Cemetery. His wife sent the following cable: According to the framework of the Will and Testament of ʻAbdu'l-Bahá, it was not possible to appoint a successor, and the legislative body "possessing the exclusive right to legislate on matters not explicitly revealed" was not yet established in the world. Furthermore, Shoghi Effendi had left no will as attested to by the Hands of the Cause, who were required to ratify his selection. All of the 27 living Hands of the Cause unanimously signed a statement shortly after the death of Shoghi Effendi stating that he had died "without having appointed his successor..." Ministry of the Custodians, pp. 28–30 = Ministry of the Custodians = In Shoghi Effendi's final message to the Baha'i World, dated October 1957, he named the Hands of the Cause of God, "the Chief Stewards of Baháʼu'lláh's embryonic World Commonwealth."Effendi, Shoghi. Messages to the Baháʼí World: 1950–1957, p. 127 Consequently, following the death of Shoghi Effendi, the Baháʼí Faith was temporarily stewarded by the Hands of the Cause, who elected among themselves 9 "Custodians" to serve in Haifa as the head of the Faith. They reserved to the "entire body of the Hands of the Cause" the responsibility to determine the transition of the International Baháʼí Council into the Universal House of Justice, and that the Custodians reserved to themselves the authority to determine and expel Covenant-breakers. This stewardship oversaw the execution of the final years of Shoghi Effendi's ordinances of the ten year crusade (which lasted until 1963) culminating and transitioning to the election and establishment of the Universal House of Justice, at the first Baha'i World Congress in 1963. = Election of the Universal House of Justice = At the end of the Ten Year Crusade, planned by Shoghi Effendi and concluding in 1963, the Universal House of Justice was first elected. As its first order of business, the Universal House of Justice evaluated the situation caused by the fact that the Guardian had not appointed a successor. It determined that under the circumstances, given the criteria for succession described in the Will and Testament of ʻAbdu'l-Bahá, there was no legitimate way for another Guardian to be appointed. Therefore, although the Will and Testament of ʻAbdu'l-Bahá leaves provisions for a succession of Guardians, Shoghi Effendi remains the first and last occupant of this office. See also * Baháʼí administration * Baháʼí World Centre * Baháʼí divisions * Baháʼí Terraces Notes References * Further reading * McLean, Jack (2012). A Celestial Burning: A Selective Study of the Writings of Shoghi Effendi. Bahaʼi Publishing Trust, India. External links * * The work and life of Shoghi Effendi * Writings of Shoghi Effendi in English * Biography of Shoghi Effendi * Directions to the Resting Place of Shoghi Effendi * The first documentary film about his life Category:Alumni of Balliol College, Oxford Category:Iranian Bahá'ís Category:Family of Baháʼu'lláh Category:1897 births Category:1957 deaths Category:Burials at New Southgate Cemetery Category:20th-century Iranian people Category:Religious leaders in the United Kingdom Category:People of the Ottoman Empire of Iranian descent "

❤️ Slope 🐾

"Slope: m=\left( \frac{\Delta y}{\Delta x} \right)=\tan( \theta ) In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "y = mx + c". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line. The direction of a line is either increasing, decreasing, horizontal or vertical. *A line is increasing if it goes up from left to right. The slope is positive, i.e. m>0. *A line is decreasing if it goes down from left to right. The slope is negative, i.e. m<0. *If a line is horizontal the slope is zero. This is a constant function. *If a line is vertical the slope is undefined (see below). The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances, where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language, the slope m of the line is :m=\frac{y_2-y_1}{x_2-x_1}. The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of incline θ by the tangent function :m = \tan (\theta) Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1\. As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve. This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment. Definition Slope illustrated for y = (3/2)x − 1\. Click on to enlarge Slope of a line in coordinates system, from f(x)=-12x+2 to f(x)=12x+2 The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation: :m = \frac{\Delta y}{\Delta x} = \frac{\text{vertical} \, \text{change} }{\text{horizontal} \, \text{change} }= \frac{\text{rise}}{\text{run}}. (The Greek letter delta, Δ, is commonly used in mathematics to mean "difference" or "change".) Given two points (x1,y1) and (x2,y2), the change in x from one to the other is (run), while the change in y is (rise). Substituting both quantities into the above equation generates the formula: :m = \frac{y_2 - y_1}{x_2 - x_1}. The formula fails for a vertical line, parallel to the y axis (see Division by zero), where the slope can be taken as infinite, so the slope of a vertical line is considered undefined. = Examples = Suppose a line runs through two points: P = (1, 2) and Q = (13, 8). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line: :m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{13 - 1} = \frac{6}{12} = \frac{1}{2}. :Since the slope is positive, the direction of the line is increasing. Since m<1, the incline is not very steep (incline <45°). As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is :m = \frac{ 21 - 15}{3 - 4} = \frac{6}{-1} = -6. :Since the slope is negative, the direction of the line is decreasing. Since m>1, this decline is fairly steep (decline >45°). Algebra and geometry *If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form ::y = mx + b :then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, that is, the y-coordinate where the line intersects the y-axis. *If the slope m of a line and a point (x1,y1) on the line are both known, then the equation of the line can be found using the point-slope formula: ::y - y_1 = m(x - x_1). *The slope of the line defined by the linear equation ::ax + by +c = 0 :is ::-\frac {a}{b} . *Two lines are parallel if and only if they are not the same line (coincident) and either their slopes are equal or they both are vertical and therefore both have undefined slopes. Two lines are perpendicular if the product of their slopes is −1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line). *The angle θ between −90° and 90° that a line makes with the x-axis is related to the slope m as follows: ::m = \tan (\theta) :and ::\theta = \arctan (m) (this is the inverse function of tangent; see inverse trigonometric functions). =Examples= For example, consider a line running through the points (2,8) and (3,20). This line has a slope, , of :\frac {(20 - 8)}{(3 - 2)} \; = 12. One can then write the line's equation, in point-slope form: :y - 8 = 12(x - 2) = 12x - 24 or: :y = 12x - 16. The angle θ between -90° and 90° that this line makes with the -axis is :\theta=\arctan (12) \approx 85.2^{\circ} \,. Consider the two lines: and . Both lines have slope . They are not the same line. So they are parallel lines. Consider the two lines and . The slope of the first line is . The slope of the second line is . The product of these two slopes is −1. So these two lines are perpendicular. Statistics In statistical mathematics, the gradient of the least-squares regression line of best fit for a given distribution of data which is linear, numerical, and free of outliers, may be written as m = \frac{rs_y}{s_x}, where m is defined as the statistical gradient for the line of best fit (y=mx+c), r is Pearson's correlation coefficient, s_y is the standard deviation of the y-values and s_x is the standard deviation of the x-values. This may also be written as a ratio of covariances: m = \frac{\operatorname{cov}(Y,X)}{\operatorname{cov}(X,X)} Slope of a road or railway :Main articles: Grade (slope), Grade separation There are two common ways to describe the steepness of a road or railroad. One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See also steep grade railway and rack railway. The formulae for converting a slope given as a percentage into an angle in degrees and vice versa are: ::\text{angle} = \arctan \left( \frac{\text{slope}}{100\%} \right) , (this is the inverse function of tangent; see trigonometry) :and ::\mbox{slope} = 100\% \cdot \tan( \mbox{angle}), where angle is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100% or 1000‰ is an angle of 45°. A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (or "1 in 10", "1 in 20" etc.) Note that 1:10 is steeper than 1:20. For example, steepness of 20% means 1:5 or an incline with angle 11,3°. Roads and railways have both longitudinal slopes and cross slopes. File:Nederlands verkeersbord J6.svgSlope warning sign in the Netherlands File:Znak A-23.svgSlope warning sign in Poland File:Skloník-klesání.jpgA 1371-meter distance of a railroad with a 20‰ slope. Czech Republic File:Railway gradient post.jpgSteam-age railway gradient post indicating a slope in both directions at Meols railway station, United Kingdom Calculus At each point, the derivative is the slope of a line that is tangent to the curve at that point. Note: the derivative at the point A is positive where green and dash-dot, negative where red and dashed, and zero where black and solid. The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point. If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition, :m = \frac{\Delta y}{\Delta x}, is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve. For example, the slope of the secant intersecting y = x2 at (0,0) and (3,9) is 3. (The slope of the tangent at is also 3—a consequence of the mean value theorem.) By moving the two points closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus, we can determine the limit, or the value that Δy/Δx approaches as Δy and Δx get closer to zero; it follows that this limit is the exact slope of the tangent. If y is dependent on x, then it is sufficient to take the limit where only Δx approaches zero. Therefore, the slope of the tangent is the limit of Δy/Δx as Δx approaches zero, or dy/dx. We call this limit the derivative. :\frac{dy}{dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x} Its value at a point on the function gives us the slope of the tangent at that point. For example, let y=x2. A point on this function is (-2,4). The derivative of this function is dy/dx=2x. So the slope of the line tangent to y at (-2,4) is 2*(-2) = -4. The equation of this tangent line is: y-4=(-4)(x-(-2)) or y = -4x - 4. See also * Euclidean distance * Grade * Inclined plane * Linear function * Line of greatest slope * Mediant * Slope definitions * Theil–Sen estimator, a line with the median slope among a set of sample points References External links * interactive Category:Elementary mathematics Category:Analytic geometry Category:Ratios "