ḱley-
“to lean, to incline”lean, incline, recline
Root for leaning/inclining, yielding Latin clinare, Greek klinein, English lean.
Discussion
The PIE root *ḱley- (to lean, to incline, to rest against) produced a family of words spanning physical posture, emotional tendency, and geographical terminology — all united by the image of tilting toward something.
Latin clīnāre (to lean, to bend, to incline) continues the root and generated one of the richest English word families: incline (to lean toward), decline (to lean away from, hence to refuse or to diminish), recline (to lean back), and the medical term clinical (klīnikḗ, originally "pertaining to the bed" — klinē being a bed or couch, the thing you lean on. Clinical medicine was bedside medicine). Climate (from Greek klīma, "an inclination" — originally the angle of the sun's rays at a given latitude, hence a band of the earth at a particular tilt, hence the weather conditions of a region) embeds the leaning metaphor in geography.
Greek klī́nein (κλίνειν, "to lean, to incline, to make lie down") gave: clinic (as above), climax (klîmax, "a ladder" — a series of ascending inclinations, later the highest point of an ascent), and the combining form -clinal/-cline in geological and botanical vocabulary.
The English word lean itself (OE hleonian/hlinian, from PGmc *hlainijaną) descends from the same PIE root through the Germanic branch, with Grimm's Law converting *ḱ to *h (later lost in most positions). The word ladder (OE hlǣdder, from PGmc *hlaidrō) may also be connected — the ladder as "the thing you lean against."
Latin collis (hill — that which inclines upward) may derive from a related form, though some scholars connect it to *ḱelh₁- (to rise) instead.
The root's conceptual contribution is the image of inclination as both physical and psychological: to be inclined toward something is both to lean your body and to tend your mind. English uses incline, inclination, and decline in both registers seamlessly — a metaphorical duality that traces back to the PIE root.
Notes
Pokorny 600-601. English clinic, decline, incline.