THE PERIODIC TABLE
The Periodic Table is a representation of all known chemical elements, arranged by increasing weight (left to right) and grouped by chemical similarity (top to bottom). The "periodic" refers to chemical properties.
At the time of Dalton (~1808) some 36 chemical elements where know and by the 1860 Karlsruhe Conference sbout 60 had been identified . The conference convinced most chemists of the existence of atoms and ushered in a decade of classification efforts . Mendeleev's table of 1869  succeeded where many others had failed, in large part because his rules had forced gaps in his table, thereby predicting the existence of scandium (element 21), gallium (element 31) and germanium (element 32). All 3 duly discovered within the next 20 years.
The case for the existence of atoms of the element was now compelling, but the table was still a classification of elements, not atoms. The distinction is subtle and is the following: the fact that an element exists does not automatically imply that it is finitely divisible, it could be divisible forever, in which case atoms would not exist.
 Elementymology & Elements Multidict, Development of the chemical symbols and the Periodic Table, Peter van der Krogt.
The Laws of Chemistry
The modern understanding of atoms began with Bohr's 1913 theory of the Hydrogen atom. By 1925 the new Quantum Mechanics could successfully tackle atoms more complex than Hydrogen and that same year Pauli formulated the exclusion principle. The periodic table now made sense. The laws of chemistry had been found and physicists moved on to study the nucleus.
Elements 115 and 113 are the latest entries in the table.