Dobereiner (1829) found that some groups of three elements have similar properties, for example, lithium, sodium and potassium; the atomic mass (mass number) for the second element was the average of the first and third element.
For example, lithium has an atomic number of 7 and potassium has an atomic number of 39. The average is (7 + 39)/2 = 23. The atomic number of sodium is 23.
The same was found for some other groups of three elements, for example, carbon (12), nitrogen (14) and oxygen (16).
These groups became known as Dobereiner's Triads.
John Newlands (1863) noticed that by arranging the elements in order of increasing atomic mass, every eighth element seemed to have similar properties. He proposed a similarity with music, where the eighth note is an octave above the first one.
This idea became known as Newlands' Octaves.
It did not work for the fourth period with the transition metals. It works for the lighter elements (atomic number less than 19) because eight electrons completes the other shell.
Dmitri Mendeleev is credited as being the Father of the Modern Periodic Table. In 1869, he arranged the 50 (or so) known elements in order of atomic mass, putting the elements with similar properties in the same vertical group, and leaving gaps for unknown elements, yet to be discovered.
When the unknown elements were later discovered, they were found to have the properties predicted by Mendeleev's table.
Knowing nothing of protons, nuclei or atomic number, Dmitri Mendeleev's periodic table was broadly correct.
The modern periodic table is very useful for giving a summary of the atomic structure and the electron structure of all the elements.