Mendeleev's predicted elements
Dmitri Mendeleev published a periodic table of the chemical elements in 1869 based on properties that appeared with some regularity as he laid out the elements from lightest to heaviest. When Mendeleev proposed his periodic table, he noted gaps in the table and predicted that as-then-unknown elements existed with properties appropriate to fill those gaps. He named them eka-boron, eka-aluminium and eka-silicon with atomic masses 44, 68 and 72 respectively.
Numerology
The discovery of atomic triads, an early attempt to sort the elements into some logical order by their physical properties, was once considered a form of numerology, and yet ultimately led to the construction of the periodic table. Here the atomic weight of the lightest element and the heaviest are summed, and averaged, and the average is found to be very close to that of the intermediate weight element. This didn't work with every triplet in the same group, but worked often enough to allow later workers to create generalizations.