The whole family grows out of one concrete noun: עַיִן, the eye. From the eye comes the spring (a spring is, beautifully, the eye of the earth where water looks out), and from the eye comes looking closely, the verb לְעַיֵּן is to study, to examine a text the way you'd peer at something. A darker branch is hostility: to fix someone with a hard eye. One organ, and around it cluster water, scholarship, and the evil eye.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
Two living verbs, two branches of the eye. The pa'al carries the hard, hostile gaze; the pi'el carries the careful, scholarly gaze. The other binyanim are not lexicalized for this root.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.