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94-465: Between the highway and Wiehl lies the biggest Wiehl industrial area, covering about 81 ha: Wiehl-Bomig. The river of the same name, Wiehl, flows by the town in an east–west direction. Wiehl is first recorded in 1131 under the name of Wila . On the 1575 Mercator map it is shown as Wiell . Wiehl was eventually allocated in the 1604 Treaty of Siegburg to the Barony of Homburg and was subordinated with it to

188-435: A compass rose or protractor, and the corresponding directions are easily transferred from point to point, on the map, e.g. with the help of a parallel ruler . Because the linear scale of a Mercator map in normal aspect increases with latitude, it distorts the size of geographical objects far from the equator and conveys a distorted perception of the overall geometry of the planet. At latitudes greater than 70° north or south,

282-400: A specialist —is used to describe a person with a general approach to knowledge. The term universal genius or versatile genius is also used, with Leonardo da Vinci as the prime example again. The term is used especially for people who made lasting contributions in at least one of the fields in which they were actively involved and when they took a universality of approach. When a person

376-442: A Mercator map printed in a book might have an equatorial width of 13.4 cm corresponding to a globe radius of 2.13 cm and an RF of approximately ⁠ 1 / 300M ⁠ (M is used as an abbreviation for 1,000,000 in writing an RF) whereas Mercator's original 1569 map has a width of 198 cm corresponding to a globe radius of 31.5 cm and an RF of about ⁠ 1 / 20M ⁠ . A cylindrical map projection

470-474: A comprehensive historical overview of the ascension and decline of the polymath as, what he calls, an "intellectual species". He observes that in ancient and medieval times, scholars did not have to specialize. However, from the 17th century on, the rapid rise of new knowledge in the Western world—both from the systematic investigation of the natural world and from the flow of information coming from other parts of

564-491: A different relationship that does not diverge at  φ  = ±90°. A transverse Mercator projection tilts the cylinder axis so that it is perpendicular to Earth's axis. The tangent standard line then coincides with a meridian and its opposite meridian, giving a constant scale factor along those meridians and making the projection useful for mapping regions that are predominately north–south in extent. In its more complex ellipsoidal form, most national grid systems around

658-459: A map in Mercator projection that correctly showed those two coordinates. Many major online street mapping services ( Bing Maps , Google Maps , Mapbox , MapQuest , OpenStreetMap , Yahoo! Maps , and others) use a variant of the Mercator projection for their map images called Web Mercator or Google Web Mercator. Despite its obvious scale variation at the world level (small scales), the projection

752-401: A median latitude, hk = 11.7. For Australia, taking 25° as a median latitude, hk = 1.2. For Great Britain, taking 55° as a median latitude, hk = 3.04. The variation with latitude is sometimes indicated by multiple bar scales as shown below. The classic way of showing the distortion inherent in a projection is to use Tissot's indicatrix . Nicolas Tissot noted that the scale factors at

846-580: A mix of occupations or of intellectual interests, Ahmed urges a breaking of the "thinker"/"doer" dichotomy and the art/science dichotomy. He argues that an orientation towards action and towards thinking support each other, and that human beings flourish by pursuing a diversity of experiences as well as a diversity of knowledge. He observes that successful people in many fields have cited hobbies and other "peripheral" activities as supplying skills or insights that helped them succeed. Ahmed examines evidence suggesting that developing multiple talents and perspectives

940-436: A new model of education that better promotes creativity and innovation: "we must focus education on principles, methods, and skills that will serve them [students] in learning and creating across many disciplines, multiple careers, and succeeding life stages". Peter Burke , Professor Emeritus of Cultural History and Fellow of Emmanuel College at Cambridge, discussed the theme of polymathy in some of his works. He has presented

1034-510: A point on a map projection, specified by the numbers h and k , define an ellipse at that point. For cylindrical projections, the axes of the ellipse are aligned to the meridians and parallels. For the Mercator projection, h  =  k , so the ellipses degenerate into circles with radius proportional to the value of the scale factor for that latitude. These circles are rendered on the projected map with extreme variation in size, indicative of Mercator's scale variations. As discussed above,

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1128-466: A ship's bearing in sailing between locations on the chart; the region of the Earth covered by such charts was small enough that a course of constant bearing would be approximately straight on the chart. The charts have startling accuracy not found in the maps constructed by contemporary European or Arab scholars, and their construction remains enigmatic; based on cartometric analysis which seems to contradict

1222-423: A single issue. Ahmed cites biologist E. O. Wilson 's view that reality is approached not by a single academic discipline but via a consilience between them. One argument for studying multiple approaches is that it leads to open-mindedness . Within any one perspective, a question may seem to have a straightforward, settled answer. Someone aware of different, contrasting answers will be more open-minded and aware of

1316-418: A small portion of the spherical surface without otherwise distorting it, preserving angles between intersecting curves. Afterward, this cylinder is unrolled onto a flat plane to make a map. In this interpretation, the scale of the surface is preserved exactly along the circle where the cylinder touches the sphere, but increases nonlinearly for points further from the contact circle. However, by uniformly shrinking

1410-408: A society, company, community, guild, corporation , etc". At this time, universities did not specialize in specific areas, but rather trained students in a broad array of science, philosophy, and theology. This universal education gave them a grounding from which they could continue into apprenticeship toward becoming a master of a specific field. When someone is called a "Renaissance man" today, it

1504-448: A straight segment. Such a course, known as a rhumb (alternately called a rhumb line or loxodrome) is preferred in marine navigation because ships can sail in a constant compass direction. This reduces the difficult, error-prone course corrections that otherwise would be necessary when sailing a different course. For small distances (compared to the radius of the Earth), the difference between

1598-463: A typology of polymathy, ranging from the ubiquitous mini-c polymathy to the eminent but rare Big-C polymathy, as well as a model with some requirements for a person (polymath or not) to be able to reach the highest levels of creative accomplishment. They account for three general requirements—intelligence, motivation to be creative, and an environment that allows creative expression—that are needed for any attempt at creativity to succeed. Then, depending on

1692-412: Is R  cos  φ , the corresponding parallel on the map must have been stretched by a factor of ⁠ 1 / cos φ ⁠ = sec φ . This scale factor on the parallel is conventionally denoted by k and the corresponding scale factor on the meridian is denoted by  h . The Mercator projection is conformal . One implication of that is the "isotropy of scale factors", which means that

1786-474: Is a specific parameterization of the cylindrical equal-area projection . In response, a 1989 resolution by seven North American geographical groups disparaged using cylindrical projections for general-purpose world maps, which would include both the Mercator and the Gall–Peters. Practically every marine chart in print is based on the Mercator projection due to its uniquely favorable properties for navigation. It

1880-463: Is also commonly used by street map services hosted on the Internet, due to its uniquely favorable properties for local-area maps computed on demand. Mercator projections were also important in the mathematical development of plate tectonics in the 1960s. The Mercator projection was designed for use in marine navigation because of its unique property of representing any course of constant bearing as

1974-437: Is an individual whose knowledge spans many different subjects, known to draw on complex bodies of knowledge to solve specific problems. Embodying a basic tenet of Renaissance humanism that humans are limitless in their capacity for development, the concept led to the notion that people should embrace all knowledge and develop their capacities as fully as possible. This is expressed in the term Renaissance man , often applied to

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2068-418: Is considered the principal responsible for rekindling interest in polymathy in the scientific community. His works emphasize the contrast between the polymath and two other types: the specialist and the dilettante. The specialist demonstrates depth but lacks breadth of knowledge. The dilettante demonstrates superficial breadth but tends to acquire skills merely "for their own sake without regard to understanding

2162-411: Is contrasted with the idea of narrowness, specialization, and the restriction of one's expertise to a limited domain. The possession of comprehensive knowledge at very disparate areas is a hallmark of the greatest polymaths. Depth refers to the vertical accumulation of knowledge and the degree of elaboration or sophistication of one's sets of one's conceptual network. Like Robert Root-Bernstein, Araki uses

2256-472: Is helpful for success in a highly specialised field. He cites a study of Nobel Prize-winning scientists which found them 25 times more likely to sing, dance, or act than average scientists. Another study found that children scored higher in IQ tests after having drum lessons, and he uses such research to argue that diversity of domains can enhance a person's general intelligence. Ahmed cites many historical claims for

2350-513: Is meant that rather than simply having broad interests or superficial knowledge in several fields, the individual possesses a more profound knowledge and a proficiency, or even an expertise, in at least some of those fields. Some dictionaries use the term "Renaissance man" to describe someone with many interests or talents, while others give a meaning restricted to the Renaissance and more closely related to Renaissance ideals. Robert Root-Bernstein

2444-516: Is presented in a 2018 article with two main objectives: The model, which was designed to reflect a structural model, has five major components: Regarding the definition of the term polymathy, the researcher, through an analysis of the extant literature, concluded that although there are a multitude of perspectives on polymathy, most of them ascertain that polymathy entails three core elements: breadth, depth and integration. Breadth refers to comprehensiveness, extension and diversity of knowledge. It

2538-481: Is specified by formulae linking the geographic coordinates of latitude  φ and longitude  λ to Cartesian coordinates on the map with origin on the equator and x -axis along the equator. By construction, all points on the same meridian lie on the same generator of the cylinder at a constant value of x , but the distance y along the generator (measured from the equator) is an arbitrary function of latitude, y ( φ ). In general this function does not describe

2632-494: Is well-suited as an interactive world map that can be zoomed seamlessly to local (large-scale) maps, where there is relatively little distortion due to the variant projection's near- conformality . The major online street mapping services' tiling systems display most of the world at the lowest zoom level as a single square image, excluding the polar regions by truncation at latitudes of φ max  = ±85.05113°. (See below .) Latitude values outside this range are mapped using

2726-565: The Diatribae upon the first part of the late History of Tithes of Richard Montagu in 1621. Use in English of the similar term polyhistor dates from the late 16th century. The term "Renaissance man" was first recorded in written English in the early 20th century. It is used to refer to great thinkers living before, during, or after the Renaissance . Leonardo da Vinci has often been described as

2820-505: The House of Sayn-Wittgenstein . In 1815, the Congress of Vienna assigned the little Homburg territory, which practically only consisted of the municipalities of Wiehl and Nümbrecht , to Prussia . At that time, Wiehl was still an agriculturally oriented settlement with a village character. Those in the population who could not earn a living from the land had to serve as migratory labour . Not until

2914-566: The equator ; the closer to the poles of the Earth, the greater the distortion. Because of great land area distortions, critics like George Kellaway and Irving Fisher consider the projection unsuitable for general world maps. It has been conjectured to have influenced people's views of the world: because it shows countries near the Equator as too small when compared to those of Europe and North America, it has been supposed to cause people to consider those countries as less important. Mercator himself used

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3008-504: The gifted people of that age who sought to develop their abilities in all areas of accomplishment: intellectual, artistic, social, physical, and spiritual. In Western Europe, the first work to use the term polymathy in its title ( De Polymathia tractatio: integri operis de studiis veterum ) was published in 1603 by Johann von Wowern , a Hamburg philosopher. Von Wowern defined polymathy as "knowledge of various matters, drawn from all kinds of studies ... ranging freely through all

3102-424: The globe in this section. The globe determines the scale of the map. The various cylindrical projections specify how the geographic detail is transferred from the globe to a cylinder tangential to it at the equator. The cylinder is then unrolled to give the planar map. The fraction ⁠ R / a ⁠ is called the representative fraction (RF) or the principal scale of the projection. For example,

3196-407: The integral of the secant function , The function y ( φ ) is plotted alongside φ for the case R  = 1: it tends to infinity at the poles. The linear y -axis values are not usually shown on printed maps; instead some maps show the non-linear scale of latitude values on the right. More often than not the maps show only a graticule of selected meridians and parallels. The expression on

3290-470: The "Old Town Hall"). After the Second World War Wiehl integrated the influx of refugees who had been bombed out or expelled. In the municipal reorganisation in 1969, it was combined with Bielstein and Drabenderhöhe and, in 1971, 840 years after its first record, it was granted town rights. Since that time it has grown steadily, partly through the arrival of immigrants and through policies encouraging

3384-638: The Chinese Song dynasty may have been drafted on the Mercator projection; however, this claim was presented without evidence, and astronomical historian Kazuhiko Miyajima concluded using cartometric analysis that these charts used an equirectangular projection instead. In the 13th century, the earliest extant portolan charts of the Mediterranean sea, which are generally not believed to be based on any deliberate map projection, included windrose networks of criss-crossing lines which could be used to help set

3478-473: The Mercator projection can be found in many world maps in the centuries following Mercator's first publication. However, it did not begin to dominate world maps until the 19th century, when the problem of position determination had been largely solved. Once the Mercator became the usual projection for commercial and educational maps, it came under persistent criticism from cartographers for its unbalanced representation of landmasses and its inability to usefully show

3572-442: The Mercator projection inflates the size of lands the further they are from the equator . Therefore, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator. Nowadays the Mercator projection is widely used because, aside from marine navigation, it is well suited for internet web maps . Joseph Needham , a historian of China, speculated that some star charts of

3666-420: The Mercator projection is practically unusable, because the linear scale becomes infinitely large at the poles. A Mercator map can therefore never fully show the polar areas (but see Uses below for applications of the oblique and transverse Mercator projections). The Mercator projection is often compared to and confused with the central cylindrical projection , which is the result of projecting points from

3760-481: The Mercator projection is the unique projection which balances this East–West stretching by a precisely corresponding North–South stretching, so that at every location the scale is locally uniform and angles are preserved. The Mercator projection in normal aspect maps trajectories of constant bearing (called rhumb lines or loxodromes ) on a sphere to straight lines on the map, and is thus uniquely suited to marine navigation : courses and bearings are measured using

3854-535: The North and South poles, and the contact circle is the Earth's equator . As for all cylindrical projections in normal aspect, circles of latitude and meridians of longitude are straight and perpendicular to each other on the map, forming a grid of rectangles. While circles of latitude on the Earth are smaller the closer they are to the poles, they are stretched in an East–West direction to have uniform length on any cylindrical map projection. Among cylindrical projections,

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3948-458: The Web Mercator. The Mercator projection can be visualized as the result of wrapping a cylinder tightly around a sphere, with the two surfaces tangent to (touching) each other along a circle halfway between the poles of their common axis, and then conformally unfolding the surface of the sphere outward onto the cylinder, meaning that at each point the projection uniformly scales the image of

4042-431: The advantages of polymathy. Some of these are about general intellectual abilities that polymaths apply across multiple domains. For example, Aristotle wrote that full understanding of a topic requires, in addition to subject knowledge, a general critical thinking ability that can assess how that knowledge was arrived at. Another advantage of a polymathic mindset is in the application of multiple approaches to understanding

4136-549: The aesthetic and structural/scientific connections between mathematics, arts and the sciences. In 2009, Sriraman published a paper reporting a 3-year study with 120 pre-service mathematics teachers and derived several implications for mathematics pre-service education as well as interdisciplinary education. He utilized a hermeneutic-phenomenological approach to recreate the emotions, voices and struggles of students as they tried to unravel Russell's paradox presented in its linguistic form. They found that those more engaged in solving

4230-510: The archetype of the Renaissance man, a man of "unquenchable curiosity" and "feverishly inventive imagination". Many notable polymaths lived during the Renaissance period, a cultural movement that spanned roughly the 14th through to the 17th century that began in Italy in the Late Middle Ages and later spread to the rest of Europe. These polymaths had a rounded approach to education that reflected

4324-563: The association of the municipalities of Wiehl and Bielstein from the Bielsteiner coat of arms. Wiehl is twinned with: Mercator map The Mercator projection ( / m ər ˈ k eɪ t ər / ) is a conformal cylindrical map projection first presented by Flemish geographer and mapmaker Gerardus Mercator in 1569. In the 18th century, it became the standard map projection for navigation due to its property of representing rhumb lines as straight lines. When applied to world maps,

4418-744: The basis of creative giftedness ask not 'who is creative?' but 'what is the basis of creative thinking?' From the polymathy perspective, giftedness is the ability to combine disparate (or even apparently contradictory) ideas, sets of problems, skills, talents, and knowledge in novel and useful ways. Polymathy is therefore the main source of any individual's creative potential". In "Life Stages of Creativity", Robert and Michèle Root-Bernstein suggest six typologies of creative life stages. These typologies are based on real creative production records first published by Root-Bernstein, Bernstein, and Garnier (1993). Finally, his studies suggest that understanding polymathy and learning from polymathic exemplars can help structure

4512-544: The big picture—and for analysis. He says: "It takes a polymath to 'mind the gap' and draw attention to the knowledges that may otherwise disappear into the spaces between disciplines, as they are currently defined and organized". Bharath Sriraman , of the University of Montana, also investigated the role of polymathy in education. He poses that an ideal education should nurture talent in the classroom and enable individuals to pursue multiple fields of research and appreciate both

4606-404: The broader applications or implications and without integrating it". Conversely, the polymath is a person with a level of expertise that is able to "put a significant amount of time and effort into their avocations and find ways to use their multiple interests to inform their vocations". A key point in the work of Root-Bernstein and colleagues is the argument in favor of the universality of

4700-447: The concept of dilettancy as a contrast to the idea of profound learning that polymathy entails. Integration, although not explicit in most definitions of polymathy, is also a core component of polymathy according to the author. Integration involves the capacity of connecting, articulating, concatenating or synthesizing different conceptual networks, which in non-polymathic persons might be segregated. In addition, integration can happen at

4794-499: The creative process. That is, although creative products, such as a painting, a mathematical model or a poem, can be domain-specific, at the level of the creative process, the mental tools that lead to the generation of creative ideas are the same, be it in the arts or science. These mental tools are sometimes called intuitive tools of thinking. It is therefore not surprising that many of the most innovative scientists have serious hobbies or interests in artistic activities, and that some of

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4888-555: The domain of choice, more specific abilities will be required. The more that one's abilities and interests match the requirements of a domain, the better. While some will develop their specific skills and motivations for specific domains, polymathic people will display intrinsic motivation (and the ability) to pursue a variety of subject matters across different domains. Regarding the interplay of polymathy and education, they suggest that rather than asking whether every student has multicreative potential, educators might more actively nurture

4982-515: The equal-area sinusoidal projection to show relative areas. However, despite such criticisms, the Mercator projection was, especially in the late 19th and early 20th centuries, perhaps the most common projection used in world maps. Atlases largely stopped using the Mercator projection for world maps or for areas distant from the equator in the 1940s, preferring other cylindrical projections , or forms of equal-area projection . The Mercator projection is, however, still commonly used for areas near

5076-458: The equator where distortion is minimal. It is also frequently found in maps of time zones. Arno Peters stirred controversy beginning in 1972 when he proposed what is now usually called the Gall–Peters projection to remedy the problems of the Mercator, claiming it to be his own original work without referencing prior work by cartographers such as Gall's work from 1855. The projection he promoted

5170-471: The fields of the disciplines, as far as the human mind, with unwearied industry, is able to pursue them". Von Wowern lists erudition, literature, philology , philomathy , and polyhistory as synonyms. The earliest recorded use of the term in the English language is from 1624, in the second edition of The Anatomy of Melancholy by Robert Burton ; the form polymathist is slightly older, first appearing in

5264-561: The form of the Web Mercator projection . Today, the Mercator can be found in marine charts, occasional world maps, and Web mapping services, but commercial atlases have largely abandoned it, and wall maps of the world can be found in many alternative projections. Google Maps , which relied on it since 2005, still uses it for local-area maps but dropped the projection from desktop platforms in 2017 for maps that are zoomed out of local areas. Many other online mapping services still exclusively use

5358-418: The geometrical projection (as of light rays onto a screen) from the centre of the globe to the cylinder, which is only one of an unlimited number of ways to conceptually project a cylindrical map. Since the cylinder is tangential to the globe at the equator, the scale factor between globe and cylinder is unity on the equator but nowhere else. In particular since the radius of a parallel, or circle of latitude,

5452-431: The geometry of corresponding small elements on the globe and map. The figure below shows a point P at latitude  φ and longitude  λ on the globe and a nearby point Q at latitude φ  +  δφ and longitude λ  +  δλ . The vertical lines PK and MQ are arcs of meridians of length Rδφ . The horizontal lines PM and KQ are arcs of parallels of length R (cos  φ ) δλ . The corresponding points on

5546-405: The globe radius R . It is often convenient to work directly with the map width W  = 2 π R . For example, the basic transformation equations become The ordinate y of the Mercator projection becomes infinite at the poles and the map must be truncated at some latitude less than ninety degrees. This need not be done symmetrically. Mercator's original map is truncated at 80°N and 66°S with

5640-459: The growth of industry. German traditional electric wiring company Merten has it productions facility in Wiehl. Merten is part of Schneider Electric since 2006. The coat of arms served the power Homburg as a basis of today's Wiehler coat of arms. It consists of a two tower castle with open gate and portcullis. The unresolved Knight of St John of Jerusalem cross over the right lower tower was taken at

5734-490: The ideals of the humanists of the time. A gentleman or courtier of that era was expected to speak several languages, play a musical instrument , write poetry , and so on; thus fulfilling the Renaissance ideal . The idea of a universal education was essential to achieving polymath ability, hence the word university was used to describe a seat of learning. However, the original Latin word universitas refers in general to "a number of persons associated into one body,

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5828-436: The impossibility of determining the longitude at sea with adequate accuracy and the fact that magnetic directions, instead of geographical directions , were used in navigation. Only in the middle of the 18th century, after the marine chronometer was invented and the spatial distribution of magnetic declination was known, could the Mercator projection be fully adopted by navigators. Despite those position-finding limitations,

5922-702: The individual and wider society. It suggests that the complex problems of the 21st century need the versatility, creativity, and broad perspectives characteristic of polymaths. For individuals, Ahmed says, specialisation is dehumanising and stifles their full range of expression whereas polymathy "is a powerful means to social and intellectual emancipation" which enables a more fulfilling life. In terms of social progress, he argues that answers to specific problems often come from combining knowledge and skills from multiple areas, and that many important problems are multi-dimensional in nature and cannot be fully understood through one specialism. Rather than interpreting polymathy as

6016-445: The intellectual climate, it has since then been more common to find "passive polymaths", who consume knowledge in various domains but make their reputation in one single discipline, than "proper polymaths", who—through a feat of "intellectual heroism"—manage to make serious contributions to several disciplines. However, Burke warns that in the age of specialization, polymathic people are more necessary than ever, both for synthesis—to paint

6110-451: The isotropy condition implies that h = k = sec φ . Consider a point on the globe of radius R with longitude λ and latitude φ . If φ is increased by an infinitesimal amount, dφ , the point moves R dφ along a meridian of the globe of radius R , so the corresponding change in y , dy , must be hR dφ = R  sec  φ dφ . Therefore y′ ( φ ) =  R  sec  φ . Similarly, increasing λ by dλ moves

6204-728: The limitations of their own knowledge. The importance of recognising these limitations is a theme that Ahmed finds in many thinkers, including Confucius , Ali ibn Abi Talib , and Nicolas of Cusa . He calls it "the essential mark of the polymath." A further argument for multiple approaches is that a polymath does not see diverse approaches as diverse, because they see connections where other people see differences. For example da Vinci advanced multiple fields by applying mathematical principles to each. Aside from Renaissance man , similar terms in use are homo universalis ( Latin ) and uomo universale ( Italian ), which translate to 'universal man'. The related term generalist —contrasted with

6298-416: The map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata : "A new and augmented description of Earth corrected for the use of sailors". This title, along with an elaborate explanation for using the projection that appears as a section of text on the map, shows that Mercator understood exactly what he had achieved and that he intended the projection to aid navigation. Mercator never explained

6392-410: The mathematical principle of the rhumb line or loxodrome, a path with constant bearing as measured relative to true north, which can be used in marine navigation to pick which compass bearing to follow. In 1537, he proposed constructing a nautical atlas composed of several large-scale sheets in the equirectangular projection as a way to minimize distortion of directions. If these sheets were brought to

6486-404: The maximum latitude attained must correspond to y  = ± ⁠ W / 2 ⁠ , or equivalently ⁠ y / R ⁠  =  π . Any of the inverse transformation formulae may be used to calculate the corresponding latitudes: The relations between y ( φ ) and properties of the projection, such as the transformation of angles and the variation in scale, follow from

6580-531: The method of construction or how he arrived at it. Various hypotheses have been tendered over the years, but in any case Mercator's friendship with Pedro Nunes and his access to the loxodromic tables Nunes created likely aided his efforts. English mathematician Edward Wright published the first accurate tables for constructing the projection in 1599 and, in more detail, in 1610, calling his treatise "Certaine Errors in Navigation". The first mathematical formulation

6674-514: The most innovative artists have an interest or hobbies in the sciences. Root-Bernstein and colleagues' research is an important counterpoint to the claim by some psychologists that creativity is a domain-specific phenomenon. Through their research, Root-Bernstein and colleagues conclude that there are certain comprehensive thinking skills and tools that cross the barrier of different domains and can foster creative thinking: "[creativity researchers] who discuss integrating ideas from diverse fields as

6768-547: The multicreative potential of their students. As an example, the authors cite that teachers should encourage students to make connections across disciplines, use different forms of media to express their reasoning/understanding (e.g., drawings, movies, and other forms of visual media). In his 2018 book The Polymath , British author Waqas Ahmed defines polymaths as those who have made significant contributions to at least three different fields. Rather than seeing polymaths as exceptionally gifted, he argues that every human being has

6862-418: The oblique Mercator in order to keep scale variation low along the surface projection of the cylinder's axis. Although the surface of Earth is best modelled by an oblate ellipsoid of revolution , for small scale maps the ellipsoid is approximated by a sphere of radius a , where a is approximately 6,371 km. This spherical approximation of Earth can be modelled by a smaller sphere of radius R , called

6956-538: The paradox also displayed more polymathic thinking traits. He concludes by suggesting that fostering polymathy in the classroom may help students change beliefs, discover structures and open new avenues for interdisciplinary pedagogy. Michael Araki is a professor at the UNSW Business School at the University of New South Wales, Australia. He sought to formalize in a general model how the development of polymathy takes place. His Developmental Model of Polymathy (DMP)

7050-430: The personality level, when the person is able to integrate their diverse activities in a synergic whole, which can also mean a psychic (motivational, emotional and cognitive) integration. Finally, the author also suggests that, via a psychoeconomic approach, polymathy can be seen as a "life project". That is, depending on a person's temperament, endowments, personality, social situation and opportunities (or lack thereof),

7144-435: The point R cos φ dλ along a parallel of the globe, so dx = kR cos φ dλ = R dλ . That is, x′ ( λ ) =  R . Integrating the equations with x ( λ 0 ) = 0 and y (0) = 0, gives x(λ) and y(φ) . The value λ 0 is the longitude of an arbitrary central meridian that is usually, but not always, that of Greenwich (i.e., zero). The angles λ and φ are expressed in radians. By

7238-464: The point scale factor is independent of direction, so that small shapes are preserved by the projection. This implies that the vertical scale factor, h , equals the horizontal scale factor, k . Since k = sec φ , so must h . The graph shows the variation of this scale factor with latitude. Some numerical values are listed below. The area scale factor is the product of the parallel and meridian scales hk = sec φ . For Greenland, taking 73° as

7332-428: The polar regions. The criticisms leveled against inappropriate use of the Mercator projection resulted in a flurry of new inventions in the late 19th and early 20th century, often directly touted as alternatives to the Mercator. Due to these pressures, publishers gradually reduced their use of the projection over the course of the 20th century. However, the advent of Web mapping gave the projection an abrupt resurgence in

7426-477: The potential to become one: that people naturally have multiple interests and talents. He contrasts this polymathic nature against what he calls "the cult of specialisation". For example, education systems stifle this nature by forcing learners to specialise in narrow topics. The book argues that specialisation encouraged by the production lines of the Industrial Revolution is counter-productive both to

7520-588: The project of a polymathic self-formation may present itself to the person as more or less alluring and more or less feasible to be pursued. James C. Kaufman , from the Neag School of Education at the University of Connecticut, and Ronald A. Beghetto, from the same university, investigated the possibility that everyone could have the potential for polymathy as well as the issue of the domain-generality or domain-specificity of creativity. Based on their earlier four-c model of creativity, Beghetto and Kaufman proposed

7614-778: The projection define a rectangle of width  δx and height  δy . For small elements, the angle PKQ is approximately a right angle and therefore The previously mentioned scaling factors from globe to cylinder are given by Since the meridians are mapped to lines of constant x , we must have x = R ( λ − λ 0 ) and δx  =  Rδλ , ( λ in radians). Therefore, in the limit of infinitesimally small elements Polymath A polymath ( Greek : πολυμαθής , romanized :  polymathēs , lit.   'having learned much'; Latin : homo universalis , lit.   'universal human') or polyhistor ( Greek : πολυΐστωρ , romanized :  polyīstor , lit.   'well-learned')

7708-409: The result that European countries were moved toward the centre of the map. The aspect ratio of his map is ⁠ 198 / 120 ⁠ = 1.65. Even more extreme truncations have been used: a Finnish school atlas was truncated at approximately 76°N and 56°S, an aspect ratio of 1.97. Much Web-based mapping uses a zoomable version of the Mercator projection with an aspect ratio of one. In this case

7802-410: The resulting flat map, as a final step, any pair of circles parallel to and equidistant from the contact circle can be chosen to have their scale preserved, called the standard parallels ; then the region between chosen circles will have its scale smaller than on the sphere, reaching a minimum at the contact circle. This is sometimes visualized as a projection onto a cylinder which is secant to (cuts)

7896-455: The rhumb and the great circle course is negligible. Even for longer distances, the simplicity of the constant bearing makes it attractive. As observed by Mercator, on such a course, the ship would not arrive by the shortest route, but it will surely arrive. Sailing a rhumb meant that all that the sailors had to do was keep a constant course as long as they knew where they were when they started, where they intended to be when they finished, and had

7990-469: The right of the second equation defines the Gudermannian function ; i.e., φ  = gd( ⁠ y / R ⁠ ): the direct equation may therefore be written as y  =  R ·gd ( φ ). There are many alternative expressions for y ( φ ), all derived by elementary manipulations. Corresponding inverses are: For angles expressed in degrees: The above formulae are written in terms of

8084-416: The same scale and assembled, they would approximate the Mercator projection. In 1541, Flemish geographer and mapmaker Gerardus Mercator included a network of rhumb lines on a terrestrial globe he made for Nicolas Perrenot . In 1569, Mercator announced a new projection by publishing a large world map measuring 202 by 124 cm (80 by 49 in) and printed in eighteen separate sheets. Mercator titled

8178-463: The scholarly consensus, they have been speculated to have originated in some unknown pre-medieval cartographic tradition, possibly evidence of some ancient understanding of the Mercator projection. German polymath Erhard Etzlaub engraved miniature "compass maps" (about 10×8 cm) of Europe and parts of Africa that spanned latitudes 0°–67° to allow adjustment of his portable pocket-size sundials . The projection found on these maps, dating to 1511,

8272-595: The second half of the 19th century were the conditions created for a significant increase in population. In 1860, the water power of the River Wiehl was first utilised by the Ohler Hammer Mill; in 1895, the river was used to generate electricity; the place was connected to the railway network in 1897 and the BPW Bergische Achsen factory was founded in 1898. The Nazi era left Wiehl a new town hall (today called

8366-431: The sphere onto a tangent cylinder along straight radial lines, as if from a light source placed at the Earth's center. Both have extreme distortion far from the equator and cannot show the poles. However, they are different projections and have different properties. As with all map projections , the shapes or sizes are distortions of the true layout of the Earth's surface. The Mercator projection exaggerates areas far from

8460-405: The sphere, though this picture is misleading insofar as the standard parallels are not spaced the same distance apart on the map as the shortest distance between them through the interior of the sphere. The original and most common aspect of the Mercator projection for maps of the Earth is the normal aspect, for which the axis of the cylinder is the Earth's axis of rotation which passes through

8554-522: The world use the transverse Mercator, as does the Universal Transverse Mercator coordinate system . An oblique Mercator projection tilts the cylinder axis away from the Earth's axis to an angle of one's choosing, so that its tangent or secant lines of contact are circles that are also tilted relative to the Earth's parallels of latitude. Practical uses for the oblique projection, such as national grid systems, use ellipsoidal developments of

8648-399: The world—was making it increasingly difficult for individual scholars to master as many disciplines as before. Thus, an intellectual retreat of the polymath species occurred: "from knowledge in every [academic] field to knowledge in several fields, and from making original contributions in many fields to a more passive consumption of what has been contributed by others". Given this change in

8742-538: Was publicized around 1645 by a mathematician named Henry Bond ( c.  1600 –1678). However, the mathematics involved were developed but never published by mathematician Thomas Harriot starting around 1589. The development of the Mercator projection represented a major breakthrough in the nautical cartography of the 16th century. However, it was much ahead of its time, since the old navigational and surveying techniques were not compatible with its use in navigation. Two main problems prevented its immediate application:

8836-424: Was stated by John Snyder in 1987 to be the same projection as Mercator's. However, given the geometry of a sundial, these maps may well have been based on the similar central cylindrical projection , a limiting case of the gnomonic projection , which is the basis for a sundial. Snyder amended his assessment to "a similar projection" in 1993. Portuguese mathematician and cosmographer Pedro Nunes first described

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