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For instance, consider the elements on the left side of the table. If you ignore Hydrogen (H) at the top (which tends to be a law unto itself), they are Lithium (Li), Sodium (Na), Potassium (K), Rubidium (Rb), Caesium (Cs) and Francium (Fr). These elements are metals, and they all show the same chemical behaviour to a greater or lesser extent. For instance, they all react with water (less so in the case of Lithium, more so as you go down the group, until you get to Caesium which reacts explosively with water). They all form alkaline oxides, ions with a single positive charge etc. This is why they have been arranged above one another in the table.
These vertical arrangements are called groups, and they have different numbers of elements in them. That is why they are arranged the way they are.
The elements should really be slotted in where that gap is just after the second group. The Lanthanides should go in just after Barium (Ba) and the Actinides should go in just after Radium (Ra). The connecting lines show where the two periods should slot in:

The reason that we don't write the table like this is because it would make it too wide to be easily readable. Instead, we "chop the legs off the corpse so it will fit in the coffin" and write the two periods beneath:

Eventually he hit upon the solution. He claimed that the answer came to him in a dream, though how much credence we should give to that is debatable. Anyway, he arranged the elements in the pattern that we now call the periodic table, even though it had a great many gaps in it!
However, the fact that there were large numbers of gaps in Mendeleyev's table suggests that there were as yet unknown elements that should go in those gaps. He was overjoyed when the elements Scandium, Germanium and Gallium were discovered in the 1870s and 1880s as they fitted perfectly where he predicted in gaps in the table, and they had the properties that he predicted they should have.
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