21. Mercator's Projection nicolaus mercator (no relation to Gerardus) actually offered a prize for thesolution. The problem was eventually solved in 1668 by James Gregory, http://www.math.ubc.ca/~israel/m103/mercator/mercator.html

Mercator's Projection The Mercator projection was invented by Gerardus Mercator, a Flemish mapmaker. His name is a latinized version of Gerhard Kramer This is his famous world map of 1569. A modern Mercator projection map The property of the Mercator projection map that made it useful to navigators is that it preserves angles. Lines of constant compass heading (called rhumb lines by sailors) are straight lines on this map. Say you're in Vancouver and want to get to Honolulu, Hawaii. You don't have GPS, just a compass. Just draw a straight line on the map between where you are and where you want to go, and measure the angle (almost exactly 45 degrees in this case). So (after a small detour around Vancouver Island) you head southwest, and off you go. If you keep heading in this direction, you ought to get there. Earlier maps were drawn on a simple grid: each degree of latitude or longitude is the same size. This method wouldn't work there: rhumb lines would be curves. Mercator knew that to give his map this desirable property, he had to make the lines of latitude farther apart as you go away from the equator. But he didn't say exactly how he did it. He almost certainly didn't have a formula, but might have done it using approximate measurements on a globe. He just said It is for these reasons that we have progressively increased the degrees of latitude towards each pole in proportion to the lengthening of the parallels with reference to the equator.

22. The Galileo Project JE Hofmann, nicolaus mercator (Kauffman), sein Leben und Wirken, vorzugsweiseals Mathematiker (in Academie der Wissenschaften und der Literatur in Mainz http://galileo.rice.edu/Catalog/NewFiles/mercator_nic.html

Mercator, Nicolaus [Niklaus Kauffman] 1. Dates Born: Eutin (?), Schleswig-Holstein, ca. 1619 Died: Paris, 14 Jan 1687 Dateinfo: Birth Uncertain Lifespan: 2. Father Occupation: Schoolmaster Martin Kauffman was probably a shoolmaster at Oldenburg, Holstein. No information on financial status. 3. Nationality Birth: Eutin [?], Schleswig-Holstein, then Denmark Career: England Death: Paris, France 4. Education Schooling: Rostock, M.Phil. (list as M.A.), Leiden 1632, matriculated at Univ. of Rostock. 1641, received M.Phil. I assume a B.A. Studied for a short time at Leiden. 5. Religion Affiliation: Lutheran : Lutheran (assumed) 6. Scientific Disciplines Primary: Mathematics, Astronomy Subordinate: Navigation, Astrology While at Copenhagen, Mercator produced several textbooks on elementary astronomy, geography, and spherical trigonometry. Later he did a Latin translation of Kinckhuysen's Dutch Algebra. His Logarithmotechnia, 1668, with a series for calculating logarithms, spurred Newton to set his methods down in De analysi. Mercator's Hypothesis astronomia nova, 1664, combined Kepler's ellipses with his vicarious hypothesis. His Institutiones astronomicae, 1676, gave a good exposition of contemporary astronomical theory. He exchanged letters with Newton on lunar theory, and he demolished Cassini's method of determining the line of apsides of a planetary orbit from three solar sightings.

23. List Of Scientists By Field Translate this page mercator, Gerardus. mercator, nicolaus. mercator, nicolaus. Merica, Paul Dyer.Merica, Paul Dyer. Merrett, Christopher. Merrett, Christopher http://www.indiana.edu/~newdsb/m.html

Maanen, Adriaan van Macallum, Archibald Byron Macallum, Archibald Byron MacArthur, Robert Helmer Macaulay, Francis Sowerby Macbride, David Macbride, David MacBride, Ernest William MacCullagh, James MacCulloch, John MacCulloch, John Mach, Ernst Mach, Ernst Mach, Ernst Machatschki, Felix Karl Ludwig Machatschki, Felix Karl Ludwig Machatschki, Felix Karl Ludwig Macheboeuf, Michel Macheboeuf, Michel Maclaurin, Colin Maclean, John Maclear, Thomas MacLeod, Colin Munro MacLeod, Colin Munro Macleod, John James Rickard Maclure, William MacMahon, Percy Alexander Macmillan, William Duncan Macmillan, William Duncan Macquer, Pierre Joseph Macrobius, Ambrosius Theodosius Magalotti, Lorenzo Magati, Cesare Magellan, Jean-Hyacinthe de Magellan, Jean-Hyacinthe de Magellan, Jean-Hyacinthe de Maggi, Bartolomeo Magini, Giovanni Antonio Magini, Giovanni Antonio Magini, Giovanni Antonio Magiotti, Raffaello Magiotti, Raffaello Magiotti, Raffaello Magnenus, Johann Chrysostom Magnenus, Johann Chrysostom Magni, Valeriano Magnitsky, Leonty Filippovich Magnol, Pierre Magnus, Heinrich Gustav

24. New Dictionary Of Scientific Biography mercator, nicolaus Merrill, George Perkins Messier, Charles Meton Michell, JohnMiller, William Allen Milne, Edward Arthur Mineur, Henri http://www.indiana.edu/~newdsb/astor.html

Make Suggestions Astronomical Sciences Abbe, Cleveland Abetti, Antonio Abney, William de Wiveleslie Abraham Bar ?iyya Ha-Nasi Ab?'l-Waf al-B?zj?n?, Mu?ammad Ibn Mu?ammad Ibn Ya?y? Ibn Ism?l Ibn al- ?Abb?s Acyuta Pira?i Adams, John Couch Adelard of Bath Ailly, Pierre d' Airy, George Biddell Aitken, Robert Grant Albrecht, Carl Theodor Alfonso El Sabio Alzate y Ramírez, José Antonio Anaximander Andoyer, Henri André, Charles Louis François Angelus, Johannes Ångstrüm, Anders Jonas Anthelme, Voituret Antiphon Antoniadi, Eugène M. Apian, Peter Arago, Dominique François Jean Aratus of Soli Argelander, Friedrich Wilhelm August Argoli, Andrea Aristarchus of Samos Aristotle Aristyllus Arrest, Heinrich Louis d' Asada G?ry? Autolycus of Pitane Auwers, Arthur Julius Georg Friedrich von Auzout, Adrien Baade, Wilhelm Heinrich Walter Babinet, Jacques Backlund, Jüns Oskar Bailey, Solon Irving Bailly, Jean-Sylvain Baily, Francis Banachiewicz, Thaddeus Ban? M?s? Baranzano, Giovanni Antonio

25. Rappersvill Map Colllection Cardinal nicolaus of Kuza with some Pole probably Jan Dlugosz the historian, mercator maps are easily identifiable due to precise line and attractive http://www.muzeum-polskie.org/ptolemeo-eng.htm

Ptolemy's "Geographia" Alexandrian scientist Claudius Ptolemaeus (90 - 168 AD.) was the first geographer to attempt a description of the world by way of cartographic presentation. His Geographia , including 26 maps or numbered tables, was a considerable achievement at the time. The work provided mathematical foundations for creating maps, and listed known rivers, peoples and settlements, giving their terrestrial coordinates. In order to collect so many data, Ptolemy co-operated with other scientists and made use of the works by his predecessors, such as his teacher Marios of Tyre. The presentation of Polish territories on the maps appended to the Geographia was far from perfect. Ptolemy called that part of the word SARMATIA, which referred to lands inhabited by Sanna-tians, or tribes of Pers sian origin. Later, in 15th - 18th centuries, many writers considered Sarmatians to be forefathers of Polish nobility. Regrettably, Ptolemy found no followers and his work fell into oblivion until the beginnings of the 15th century. However, it attracted interest of Arab scientists, who even translated it. Their works in this area failed to influence the development of cartography, in contrast to Arab achievements in other fields, astronomy in particular. In the Middle Ages, cartography was a "forgotten science," although it must have been known in some monasteries, and perhaps it had even been a subject of study there. Possibly, the data from

26. Rappersvill Map Colllection The Northern map was probably used by mercator for his version of the map of While working on his million map , Wapowski was supported by nicolaus http://www.muzeum-polskie.org/kart-pol-eng.htm

Bernard Wapowski and Waclaw Grodecki. Origins of Polish cartography. It seems that Wapowski's map was well known to the elite of European cartographers of that time and it was used as a source of information about the south-eastern and north-estern parts of the Continent, including the area of Poland and Hungary. These followers include Gerard Mercator, Henry Zell, Caspar Vopelius, Jacob Gaspaldi and Olaf Magnus, who authored an interesting map of Scandinavia. It is also highly probable that the map Polonia et Ungaria, included by Sebastian Muenster into his Base l edi-tion of Ptolemy's Geographia and Muenster's own work Cosmographia, was a reduced and simplified version of the "Southern" map by Wapowski. Afterwards, it's various versions were copied into many works. Giacomo Gastaldi, one of the leading Italian cartographers, placed his version of that map, very carefially woodcut, although small in size, in his edition of Geographia of 1546. The "Northern" map was probably used by Mercator for his version of the map of Lithuania and was published in various editions of his Atlas in the years 1595 - 1630. The twin-chart map by Wapowski is surprisingly precise while specifying the coor-dinates for the mouths of the Dnieper and the Dniester, as it corrects the error by Ptole-my. Ptolemy's map moves the Black Sea by 4 degrees westwards and extends it unnaturally, which makes the whole continent rather oblong. Regrettably, the next generations of cartographers did not follow Wapowski, modelling their works after Mercator, who for unknown reasons continued to place the mouths of the Dnieper and Dniester respectably by 3 and 1/2 degree and 2 and 1/2 degree farther eastwards, even after Wapowski's map was published. Wapowski's map, however, failed to show an accurate reflection ofthe regions of Polesie, Wolyn' and Grodno, where the author placed large, non-existent bogs of Amadotia, Sarmatia and the Cronos Lake. and also many rivers flowing therefrom. It seems that Wapowski did not support the theory that rivers have to spring from mountain tops.

27. Mercaptopurine - Definition Of Mercaptopurine By The Free Online Dictionary, The mercator projection mercator projection mercator s chart mercator s projection mercator, Gerardus mercator, Gerardus mercator, nicolaus http://www.thefreedictionary.com/mercaptopurine

Domain='thefreedictionary.com' word='mercaptopurine' Your help is needed: American Red Cross The Salvation Army join mailing list webmaster tools Word (phrase): Word Starts with Ends with Definition subscription: Dictionary/ thesaurus Computing dictionary Medical dictionary Legal dictionary Financial dictionary Acronyms Columbia encyclopedia Wikipedia encyclopedia Hutchinson encyclopedia mercaptopurine Also found in: Medical Columbia Wikipedia 0.03 sec. Page tools Printer friendly Cite / link Email Feedback mer·cap·to·pu·rine (m r-k p t -py r n) n. A purine analogue, C H N S, that acts as an antimetabolite by interfering with purine synthesis, used primarily in the treatment of acute leukemia. Thesaurus Legend: Synonyms Related Words Antonyms Noun mercaptopurine - a drug (trade name Purinethol) that interferes with the metabolism of purine and is used to treat acute lymphocytic leukemia Purinethol antimetabolite - an antineoplastic drug that inhibits the utilization of a metabolite immune suppressant drug immunosuppressant immunosuppressive drug - a drug that lowers the body's normal immune response brand brand name marque trade name - a name given to a product or service Mentioned in References in classic literature No references found No references found Dictionary/thesaurus browser Full browser Mercantile paper mercantile system mercantilism Mercaptal ... mercapto- mercaptopurine Mercat Mercatante Mercator Mercator projection ... Mercaptobenzothiazole mercaptopurine mercapturic acid Mercara MERCASREP MERCASSUM ... Mercator's projection Word (phrase):

28. User:Gerritholl/mathematicians - Wikipedia, The Free Encyclopedia Menger Pietro Mengoli - Dmitrii Menshov - Charles Méray - Gerardus mercator -nicolaus mercator - James Mercer - Charles Merrifield - Winifred Merrill http://en.wikipedia.org/wiki/User:Gerritholl/mathematicians

Wikimedia needs your help in its 21-day fund drive. See our fundraising page Over US$220,000 has been donated since the drive began on 19 August. Thank you for your generosity! User:Gerritholl/mathematicians From Wikipedia, the free encyclopedia. User:Gerritholl edit Mathematicians Ernst Abbe Niels Henrik Abel Abraham bar Hiyya Max Abraham ... Antoni Zygmund Retrieved from " http://en.wikipedia.org/wiki/User:Gerritholl/mathematicians Views User page Discussion Edit this page History Personal tools Create account / log in Navigation Main Page Community portal Current events Recent changes ... Donations Search Toolbox What links here Related changes Upload file Special pages ... Permanent link This page was last modified 01:33, 19 August 2005. All text is available under the terms of the GNU Free Documentation License (see for details). About Wikipedia

29. Wikipedia:List Of Encyclopedia Topics/Biographies M2 - Wikipedia, The Free Encyc nicolaus mercator, mercator, nicolaus. nicolaus mercator (1619? 1687), Danish mathematician; discoverer eponym of mercator s series http://en.wikipedia.org/wiki/Wikipedia:List_of_encyclopedia_topics/Biographies_M

Wikimedia needs your help in the final days of its fund drive. See our fundraising page The Red Cross and other charities also need your help. Wikipedia:List of encyclopedia topics/Biographies M2 From Wikipedia, the free encyclopedia. Wikipedia:List of encyclopedia topics Contents Section 11 Section 12 Section 13 Section 14 ... edit Section 11 Maximilian '''Maximilian''' ([[1832]] - [[1867]]), [[Mexican]] emperor 1864-1867 Maximilian I '''Maximilian I''' ([[1459]] - [[1519]]), Holy [[Roman]] emperor 1493-1519 Maximilian II, Holy Roman Emperor '''Maximilian II''' ([[1527]] - [[1576]]), Holy [[Roman]] emperor 1564-1576 James Maxton Maxton, James James Clerk Maxwell Maxwell, James Clerk '''James Clerk Maxwell''' ([[1831]] - [[1879]]), [[Scottish]] [[physicist]] Robert Maxwell Maxwell, Robert (Ian, Jan Ludwig Hoch Francis Henry, Sir May May, Francis Henry, Sir '''Francis Henry, Sir May''' ([[1860]] - [[1922]]), [[British]] [[administrator]]; [[governor]] of Hong Kong 1912-1919 Rollo May May, Rollo Thomas Erskine, Sir May May, Thomas Erskine, Sir '''Thomas Erskine, Sir May''' ([[1815]] - [[1886]]), [[English]] jurist Vladimir Vladimirovich Mayakovsky Mayakovsky, Vladimir Vladimirovich

30. Interactive Mathematics Miscellany And Puzzles Newton s Exponential Series; nicolaus mercator s Logarithmic Series; Newton s Sineand Cosine Series; Andre s Derivation of the Secant and Tangent Series http://www.cut-the-knot.com/books/dorrie/content.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Contents ARITHMETICAL PROBLEMS Archimedes' Problema Bovinum The Weight Problem of Bachet de Meziriac Newton's Problem of the Fields and Cows Berwick's Problem of the Seven Sevens Kirkman's Schoolgirl Problem The Bernoulli-Euler Problem of the Misaddressed letters Euler's Problem of Polygon Division Lucas' Problem of the Married Couples Omar Khayyam's Binomial Expansion Cauchy's Mean Theorem Bernoulli's Power Sum Problem The Euler Number. Newton's Exponential Series Nicolaus Mercator's Logarithmic Series Newton's Sine and Cosine Series Andre's Derivation of the Secant and Tangent Series Gregory's Arc Tangent Series Buffon's Needle Problem The Fermat-Euler Prime Number Theorem The Fermat Equation The Fermat-Gauss Impossibility Theorem The Quadratic Reciprocity Law Gauss' Fundamental Theorem of Algebra Sturm's Problem of the Number of Roots Abel's Impossibility Theorem The Hermite-Lindemann Transcedence Theorem PLANIMETRIC PROBLEMS Euler's Straight Line The Feuerbach Circle Castillon's Problem Malfatti's Problem Monge's Problem The Tangency Problem of Apollonius Macheroni's Compass Problem Steiner's Straight-edge Problem The Deliaii Cube-doubling Problem Trisection of an Angle The Regular Heptadecagon Archimedes' Determination of the Number Pi Fuss' Problem of the Chord-Tangent Quadrilateral Annex to a Survey Alhazen's Billiard Problem PROBLEMS CONCERNING CONIC SECTIONS AND CYCLOIDS An Ellipse from Conjugate Radii An Ellipse in a Parallelogram

31. 1668: Information From Answers.com Logarithmotechnia by nicolaus mercator b. Niklaus Kaufmann, Holstein, Denmark, c.1619, d. Paris, January 14, 1687 includes various calculations of http://www.answers.com/topic/1668

showHide_TellMeAbout2('false'); Arts Business Entertainment Games ... More... On this page: US Literature Wikipedia Mentioned In Or search: - The Web - Images - News - Blogs - Shopping In the year Astronomy Giovanni Cassini's Ephemerides bononienses Mediceorum siderum ("timetable of the Medicean stars") contains his computation of the movements and eclipses of the four satellites of Jupiter. See also 1664 Astronomy 1677 Astronomy Newton invents the reflecting telescope, building the first telescope based on a mirror (reflector) instead of a lens (refractor). See also 1663 Astronomy Biology Francesco Redi [b. Aresso (Italy), February 18, 1626, d. Pisa (Italy), March 1, 1697] disproves the idea that maggots arise spontaneously in rotten meat in one of the first controlled experiments in science. By showing that flies are not produced spontaneously from rotting meat, he partially disproves the theory of spontaneous generation. The theory will be definitively shown to be false by Louis Pasteur in 1859. (See biography John Mayow [b. Bray, England, December 1641, d. London, September 1679] recognizes the similarities between respiration and combustion.

32. 1666: Information From Answers.com et minimis ( geometrical exercises concerning maximums and minimums ), laterreprinted as an appendix to nicolaus mercator s Logarithmotechnia (1668). http://www.answers.com/topic/1666

showHide_TellMeAbout2('false'); Arts Business Entertainment Games ... More... On this page: US Literature Wikipedia Mentioned In Or search: - The Web - Images - News - Blogs - Shopping In the year Astronomy Giovanni Cassini observes the polar ice caps of Mars. See also 1664 Astronomy Communication The Acad©mie Royale des Sciences is founded in Paris. Christiaan Huygens, along with 19 other scientists, is elected as a founding member. After the French Revolution, the Royale is dropped and the character of the academy changes. It later becomes the Institut de France. See also 1660 Communication 1697 Communication Computers Disserto de arte combinatoria ("discussion of the combinatorial art") by Gottfried Wilhelm Leibniz contains his suggestion, based on the work of Raymond Lully, that a mathematical language of reasoning be developed. It will not be until the 19th century that George Boole and others develop this idea further. Some of the outgrowths of this idea find application in computer science and artificial intelligence research. See also 1854 Computers Cheddar cheese is invented in the English village of Cheddar.

33. Antique, Old, Rare And Historic Maps & Prints Of France Published in the first Cloppenburgh edition of Gerhard mercator s Atlas Minor Artois and Boulenois by the famous french cartographer nicolaus Sanson. http://www.bergbook.com/htdocs/Cache523.htm

Antique Maps of France St. Omer St. Omer A very good example. Verso French text. Vol. IV / #8. [Stock No.:11376] Full description click here to image to enlarge or use our new feature for ZOOMING click on image to UK £ Select this Antique Map to order Artois Artois Ortelius, Abraham Artesia. Iacobo Surhonio Montano auctore. Antwerpen 1588 [38 x 49 cm] Copper-engraving, published in the first Spanish edition of the 'Theatrum Orbis Terrarum' by Abraham Ortelius. Finely handcolored in wash and outline, when published. Detailled map of Artois, with many cities and place names. The engraved cities are shown as miniature cityviews, a few place names heightend in gold. Among the place names we find St. Omer, Arras, Renty, Monstreul etc. Ornated with a large decorative title cartouch, as well two further cartouches with a dedication and a milage scale. This map is in very fine original handcolor, typical for a Spanish edition in that period. Abraham Ortelius engraved this map after the cartographic sketch by Iacobo Surbonio Montano. A very good copy in fine condtion, a minor discoloration in the upper centerfold. Koe. Ort 23 / #37 (115). Broe. 73.

34. Antique, Old, Rare And Historic Maps & Prints Of Austria mercator, G. (163040). UK £. Select this Antique Map to order Fer, de nicolaus.Veue Septentrionale De Vienne en Autiche.- Veue Meridionale De Vienne http://www.bergbook.com/htdocs/Cache512.htm

Antique Maps of Austria Carintia Ortelius, Abraham Carinthiae Ducatus, et Goritiae Palatinatus, Wolf. Lazio auctore. Antwerpen, Diesth 1574 [33,4 x 24,3 cm] Copper-engraving, handcolored in wash when published. Decorative and early map of Carinthia published in a 1574 Latin text edition of the 'Theatrum Orbis Terrarum' by A. Ortelius. A highly decorative early map, towns are still shown as miniature engraved city views, mountains, woods, rivers and lakes are decorative engraved. The map was engraved after the cartographic source by Wolfgang Lazius. In the center of the map we find the city of Villach, with the river Trau, as well the Woehrd lake, the Ossiacher lake and the Milstretter lake. The map reaches in the north from Salzburg towards the neighbouring Friaul in the south. In excellent condition. Koe. Ort.12; Broe. 143 a. [Stock No.:16634] Full description click here to image to enlarge or use our new feature for ZOOMING click on image to UK £ Select this Antique Map to order Austria - Oesterreich Ortelius, Abraham.

35. History Of Astronomy: Persons (M) mercatorGymnasium Duisburg (in German). mercator, nicolaus Kauffman,Niklaus (c.1619-1687). Biographical data and references Short biography and http://www.astro.uni-bonn.de/~pbrosche/persons/pers_m.html

History of Astronomy Persons History of Astronomy: Persons (M) Deutsche Fassung Mac Laurin, Colin: see Maclaurin, Colin MacDonnell, William John (1842-1910) Short biographical data and archival sources Mach, Ernst (1838-1916) Very short biography and references (Or see German version Very short biography (Eric Weisstein's Treasure Trove) Maclaurin [Mac Laurin], Colin (1698-1746) Short biography and references (MacTutor Hist. Math.) Short biography (Eric Weisstein's Treasure Trove) Short biography (in French) Very short biography (infoplease.com) Family history Family history (under construction) Find more about Maclaurin with Alta Vista Biographical data (in Estonian) Madsen, Hans Frandsen (1843-1937) Short biographical data and reference Very short biography (under construction) Biography of his son John Percival Vissing Madsen (1879-1969) and briefly about himself Maestlin, Michael (1550-1631) Biographical data and references Short biography Maffei, Francesco Scipione (1675-1755) Short biography and references From the Catholic Encyclopedia, 1913

36. Astronomiegeschichte: Personen (M) Translate this page mercator, nicolaus Niklaus Kauffman (c.1619-1687). mercator-Gymnasium Duisburg.Mitchell, Maria (1818-1889). Kurzbiographie. Mukai Genshô (1609-1677) http://www.astro.uni-bonn.de/~pbrosche/persons/pers_m-d.html

Geschichte der Astronomie Personen Astronomiegeschichte: Personen (M) English Version Mach, Ernst (1838-1916) Sehr kurze Biographie und Literaturhinweise Marinoni, Johann Jakob [von] (1676-1755) Sehr kurze Biographie und Literaturhinweise Mauderli, Sigmund (1876-1962) Verzeichnis von Archivalien Meissner, Karl Wilhelm (1891-1959) Biographie (in German) Aus Physiker und Astronomen in Frankfurt Melanchthon, Phillip (1497-1560) Kurzbiographie Melanchthon-Jahr 1997 Mit einer Biographie Kurzbiographie und Melanchthonhaus Bretten Melanchthonhaus Bretten Lutherstadt Wittenberg, Melanchthon-Denkmal Mercator, Gerardus [Gerhard Kremer] (1512-1594) Mercator-Feier 1962 im Mercator-Gymnasium Mit einer Biographie. Mercator-Gymnasium Duisburg Mercator, Nicolaus [Niklaus Kauffman] (c.1619-1687) Mercator-Gymnasium Duisburg Mitchell, Maria (1818-1889) Kurzbiographie , von Wolfgang Michel Wolfgang R. Dick

37. Mercator_N Translate this page mercator (latinisiert aus Kauffmann), nicolaus, * ca. Eine Verwechslung mitGerhard mercator liegt nahe, da nicolaus in einer 1666 erschienenen Arbeit http://www.geophys.tu-bs.de/geschichte/mercatorn.html

Mercator (latinisiert aus Kauffmann) , Nicolaus, Literatur Hofmann, J. E.: Nicolaus Mercator - sein Leben und Wirken, Akad. d. Wiss. u. d.Literatur in Mainz, Abh. d. Math.-Natwiss. Kl., Jahrgang 3, S. 43-103, 1950. Hutton, Charles: Mathematical and Philosophical Dictionary, Vol. 2, S. 96-98, 153. London, 1795; Nachdruck Hildesheim, 1971. .

38. Benjamin Wardhaugh: History nicolaus mercator, while retaining a strong sense of the special identity ofmusical intervals as ratios, elaborated a specific way of measuring and http://www.benjaminwardhaugh.co.uk/academic/texts/proportionsabstract.html

B enjamin W ardhaugh On proportions of proportions: one aspect of the application of mathematics to music in the seventeenth century Abstract (Full text to appear online shortly at www.bshs.org.uk under '2005 Annual Conference'.) During the seventeenth century a major change took place in the use of mathematics to describe musical pitch. Since antiquity musical intervals had been identified with numerical ratios, and ratios had been considered significantly distinct from simple numbers. One consequence of this was that since different ratios were mutually incommensurable, so were different musical intervals. A 'common measure' for intervals was desirable for musical purposes, but was generally held to be strictly impossible, even meaningless, on mathematical grounds. This problem of measuring ratios was occasionally attempted, for instance by Nicole Oresme in the fourteenth century, but it was not until the mid-seventeenth century, prompted both by the invention of logarithms and the increasing urgency of issues arising from musical practice, that it was solved. Nicolaus Mercator, while retaining a strong sense of the special identity of musical intervals as ratios, elaborated a specific way of measuring and comparing those ratios, as well as identifying a common measure for them. Isaac Newton, apparently independently, devised a very similar scheme. Mercator's innovation had consequences for the meaning of numerical precision as applied to musical pitch, in relation both to realisation in practice and the limits of the ear's judgement. The possibility of quantifying the aural sense of intervals' relative size raised new questions about the role of the hearing in making music-theoretical knowledge. Finally, this incident sheds some light on the nature and assumptions of the discipline called 'harmonics' in the seventeenth century.

39. Proof That Log 2 = 0 to nicolaus mercator and gives dozens of other formulas for log(2). 1/4 was first calculated by nicolaus mercator Logarithmotechnia, 1667. http://mcraefamily.com/MathHelp/JokeProofLog2Zero.htm

Using this proof, you can double any number without changing its value! Proof that log 2 = * * note: "log", used here, means natural log, which is sometimes written "ln". This proof is extremely important, because it can be used to show that 2=1. It works this way: log 2 = 0, from the proof, above. log 1 = 0, because e log 2 = log 1, because both logs are equal to e log 2 = e log 1 , taking the exponent of both sides 2 = 1, because e log a = a Internet References I got an email questioning the statement that log 2 = 1-1/2+1/3-1/4+..., so I found these references to support this fact, which, by the way, has nothing to do with the error in the proof, above. http://numbers.computation.free.fr/Constants/Log2/log2Formulas.html attributes log(2)=1-1/2+1/3-1/4 to Nicolaus Mercator and gives dozens of other formulas for log(2). http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/harmonic-series gives a sketch of the proof that this series is the integral of dy/y from 1 to 2, which is log(2). H(n) = 1 + 1/2 + 1/3 + ... + 1/n.

40. Untitled tex2html_wrap_inline262 In 1668, nicolaus mercator (16201687) published hisLogarithmotechnica in which appeared the power series for the logarithm. http://www.math.tamu.edu/~don.allen/history/precalc2/precalc2.html

Next: About this document April 3, 1997 Early Calculus II Frans van Schooten , (1615-1660), Netherlands, had succeeded his father as professor of mathematics at Leyden. Because the original by Descartes was difficult to read, Van Schooten made a careful and clear translation of Descartes' La Geometrie into Latin, the preferred language of scholars. Partly the reason for this was so that his students could understand it. In 1659-1661, an expanded version was published. Geometria a Renato Des Cartes Two additional additions appeared in 1683 and 1695. It is reasonable to say that although analytic geometry was introduced by Descartes, it was established by Schooten. Jan de Witt , (1625-1672), the Grand Pensionary of Holland, was a colleague of Schooten. He wrote in his earlier years Elementa curvarum , a work in two parts. The first part (Part I) was on the kinematic and planimetric definitions of the conic sections. Among his ideas are the focus-directrix ratio definitions. The term `directrix' is original with De Witt. Part II, on the other hand, makes such a systematic use of coordinates that it has justifiably been called the first textbook on analytic geometry. (Descartes'

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