Why is there only two elements in the 1st period of Mendeleev’s periodic system, and only eight in the 2nd?

the period in the periodic table of chemical elements of DI Mendeleev shows how many electron layers are in the atom. Since the first electron layer can contain only 2 electrons (1 s ^ 1 and 1s ^ 2 configurations), then there are only 2 atoms in the first period. And the second electron layer has a maximum of 8 electrons (2s ^ 2 2p ^ 6), accordingly, the atoms have the following configurations of the outer layer:
Li: 2s ^ 1.
Be: 2s ^ 2.
B: 2s ^ 2 2p ^ 1.
C: 2s ^ 2 2p ^ 2.
N: 2s ^ 2 2p ^ 3.
O: 2s ^ 2 2p ^ 4.
F: 2s ^ 2 2p ^ 5.
Ne: 2s ^ 2 2p ^ 6.