scale (n.1) Look up scale at Dictionary.com
"skin plates on fish or snakes," c.1300, from Old French escale (12c., Modern French écale) "scale, husk," from Frankish or some other Germanic source, from Proto-Germanic *skælo "split, divide" (cf. Old High German scala "shell," Gothic skalja "tile," Old English scealu "shell, husk), from PIE root *(s)kel- "to cut, cleave, split" (cf. Latin culter "knife," scalpere "to cut, scrape;" Old Church Slavonic skolika "mussel, shell," Russian skala "rind, bark," Old English scell "shell").

In reference to humans, as a condition of certain skin diseases, it is attested from c.1400. As what falls from one's eye when blindness ends (usually figurative), it echoes Acts ix:18 (Latin tanquam squamæ, Greek hosei lepides). Verb meaning "to remove the scales from (a fish)" is attested from mid-15c.
scale (n2.) Look up scale at Dictionary.com
"pan of a balance," late 14c., earlier "drinking cup" (c.1200), from Old Norse skal "bowl, drinking cup," in plural, "weighing scale" from Proto-Germanic *skælo "split, divide" (cf. Old Norse skel "shell," Old English scalu, Old Saxon skala, Old High German scala, German Schale, Middle Dutch scale, Dutch schaal "drinking cup, bowl, shell, scale of a balance"), see scale (n.1).

The connecting sense seems to be of half of a bivalve ("split") shell used as a drinking cup or a pan for weighing. But according to Paulus Diaconus the "drinking cup" sense originated from a supposed custom of making goblets from skulls (see skull).
scale (v.) Look up scale at Dictionary.com
"to climb," late 14c., from Latin scala, from scandere "to climb" (see scan (v.)). This is also the source (perhaps via Italian scala) of the noun in the musical sense (1590s), and the meaning "proportion of a representation to the actual object" (1660s). Scale down "reduce" is attested from 1887. Scale factor is from 1948. Related: Scaled; scaling.