Dobereiner (1780-1849) in 1817 saw trends and similarities among severals groups of elements in threes, and observed that the atomic weight of one was apparently a near average of the atomic weights of the other two.
Examples of Dobereiner's triads:
Chlorine = 35.5, Bromine = 80, Iodine = 127 (average of Cl and I = 81)
Calcium = 40, Strontium = 88, Barium = 137 (average of Ca and Ba = 88.5)
Lithium = 7, Sodium = 23, Potassium = 39 (average of Li and K = 23)
Dobereiner first expressed his ideas as early as in 1817, but the first published explanation of his system of triads appeared in the paper, "An Attempt to Group Elementary Substances according to Their Analogies," in Annalen der Physik und Chemie 15: 301-307 (1829), an extract from which can be read at this linked site.
Dumas (1800-1884) in 1859 extended Dobereiner's triads into families of elements in fours.
Although the atomic weights within a family did not increase uniformly, a similar incremental pattern could be seen when comparing families.
Examples of Dumas' families:
Fluorine = 19, Chlorine = 35.5, Bromine = 80, Iodine = 127; 19+ 16.5 = 35.5 + 44.5= 80 + 47 = 127
Magnesium = 24, Calcium = 40, Strontium = 88, Barium = 137; 24 + 16 = 40 + 48 = 88 + 49 = 137