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1829
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Johann Doberiener in 1829 discovered the halogen triad which consisted of chlorine, bromine, and iodine. He also discovered the alkali metal triad which consisted of lithium, sodium and potassium. After this he then came up with the Law of Triads which explained that some of the middle element's properties in the triad were the average of the two outer element's properties. This only worked if the elements were in order by atomic mass. The triads are his most popular work.
http://blog.uchceu.es/eponimos-cientificos/wp-content/uploads/sites/24/2011/10/FOTO175.jpg
1864
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John Newlands in 1863, wrote a paper that organized the elements that they had at the time, which was 56, and organized them in to groups based on similar physical properties. Then, in 1864, Newlands developed his own Periodic Table and his Law of Octaves. The Law of Octaves states that any given element will exhibit analogous behavior to the eighth element following it in the table.
http://www.nndb.com/people/480/000103171/john-alexander-reina-newlands-1-sized.jpg
http://upload.wikimedia.org/wikipedia/commons/e/e5/Newlands_periodiska_system_1866.png
1870
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Demitri Mendeleev was a Russian chemist who is known for creating the Periodic Table of Elements. His table was the first to have similarities in the horizontal and vertical rows and columns of the periodic table. Other chemists were close, but Mendeleev's table was the most accurate and detailed at the time.
http://www.nndb.com/people/593/000091320/dmitri-mendeleev-1-sized.jpg
http://upload.wikimedia.org/wikipedia/commons/5/55/Mendelejevs_periodiska_system_1871.png
1870
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There is controversy of who is the "father" of the Periodic Table, and Lothar Meyer, a Russian chemist, is one of the "fathers" of the Periodic Table. Meyer had created his own, abbreviated, periodic table in one of his textbooks. Then when he had created a more detailed periodic table, Mendeleev beat him to it, by one year because Meyer's periodic table was being evaluated before publication.
http://www.chemheritage.org/Images/Main-Images-250x290/Discover/Themes/Path-to-the-Periodic-Table/meyer1.jpg
http://faculty.sdmiramar.edu/fgarces/zCourse/All_Year/Ch100_OL/aMy_FileLec/04OL_LecNotes_Ch100/03_AtomsElements/301_ElementsSymbolsPT/301_pic/MendeleevePT.gif
1895
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In 1895, Lord Rauleigh had discovered the element Argon, but it didn't fit in any of the current periodic groups.
http://images.fineartamerica.com/images-medium-large/1-3rd-baron-rayleigh-granger.jpg